UNDERSTANDING THE STRUCTURE OF GRAIN BOUNDARIES. A HIGHER DIMENSIONAL APPROACH.

David Romeu.

Departamento de Materia Condensada, Instituto de Física UNAM. P.O. Box 20-364
México 01000 D.F.
romeu@fisica.unam.mx

Summary

A formalism to describe the crystallography of interfaces based on the representation of the dichromatic pattern in a higher dimensional space is presented. The interfacial structure is obtained when the resulting single crystal hyper-lattice is projected into physical space. It is shown that a new lattice called P-lattice associated with displacements in perpendicular space is required for the proper understanding of interfaces. The formalism allows the characterization of interfaces from the symmetry of the hyper lattice. It is also shown that interfaces and quasicrystals are equivalent systems and can both be regarded as a region in space where two (or more) lattices compete to contribute their points to the system, the strip-projection method solves the conflict and determines the final structure.

Keywords: Grain Boundaries, quasicrystals, crystallography, dislocations.

Resumen

Palabras clave: Fronteras de grano, cuasicristales, cristalografía, dislocaciones.

1. Introduction

Since many properties of materials depend on phenomena occurring at grain boundaries (GBs), a large amount of work has been carried out in order to determine their structure. Unfortunately there is at present no general interfacial theory or model capable of explaining the structure, and hence the properties of general GBs.
The purpose of this paper is to show that using the tools developed for the study of quasicrystals in higher dimensional spaces, a step forward in the creation of a general crystallographic theory of interfaces can be given. Using the example of twist GBs in the cubic system, it will be shown, that representing interfaces as the projection of an hyper-lattice in 6 dimensions (6D) allows a complete crystallographic description of interfaces which permits their characterization and the study of their symmetry related properties as it does for quasicrystals. Moreover, it will be shown that interfaces and quasicrystals are equivalent systems since they are described by the same set of equations.
Although higher dimensional spaces have been used before in the context of interfaces, the work has been limited to either non-periodic interfaces (i.e., interfaces in quasicrystals) or in crystalline GBs with irrational rotation angles [1].
At first glance it may seem that introducing extra dimensions into the problem may complicate the issue unnecessarily, yet it actually makes it simpler. For example, it is easier to envisage (both conceptually and mathematically) structural properties, such as symmetry, of a single crystal in 6D than those of a dichromatic pattern (2 interpenetrated lattices) in 3D, which is difficult to visualize and requires the introduction of color groups to describe its symmetry. Moreover, since the end result of the method is a pair of simple equations in 3D (see Eq. 3) from which all measurable quantities such as dislocation spacings, domain symmetries, etc., are derived, actual calculations can be performed without any reference to higher dimensions.

2. Lattices involved in the Description of Interfaces.

3. The Modified Strip-Projection Method.

The method presented here consists of a slightly modified version of the Strip method of Katz and Duneau [5]. In the present version, the two crystal lattices are embedded in 6D space where they combine into a single crystal hyper-lattice. The main difference with the original method resides in that there, a high symmetry (cubic) hyper lattice in n dimensions is sought with unit vectors that project onto a star vector (a set of n 3D vectors) in the physical space (denoted E//).
One limitation of the original approach is that although the star vector has the symmetry of the quasicrystal, the vectors in it are not directly related to the atomic structure. This means that the end result is a set of tiles unrelated to the underlying atomic structure. In the present version, the star vector is the set of base vectors of two interpenetrating lattices, so that they are directly related to the physical structure. This produces a lower symmetry hyper-lattice that projects into stable atomic structures [3].

3-1. Embedding Crystal Lattices in 6D.

The 6D hyper-lattice L is built as follows: first, the base vectors of two arbitrary crystal lattices L₁, L₂ with base vectors (a₁, a₂, a₃) and (a₄, a₅, a₆) are embedded in 6 dimensional space, in such a way that each lattice L1 and L2 live in its own (3D) subspace V₁, V₂ as shown in Fig. 1. This generates a single crystal lattice in 6D whose points have coordinates x = (x₁, x₂, x₃, x₄, x₅, x₆), where (x₁, x₂, x₃) is a point in L₁ and (x₄, x₅, x₆) is in L₂. In other words, two lattices L₁, L₂ competing for space in E// cooperate in 6D to form a single crystal hyper lattice L. The hyper lattice is then projected back into E// and E^. The projection mechanism acts like a referee who determines the structure in the region of conflict, i.e. the interface. To avoid an excessive number of points in E//, only those hyper points falling within a small 6D region around E// called a strip are projected (see Fig. 1). The exact size and shape of the strip depend upon the structure of L₁ and L₂ and their atomic decoration.
If S₁, S₂ are the structure matrices of L₁, L₂, then the 6D hyper-lattice L is defined by the 6D structure matrix S given in terms of the standard basis of R⁶ as:

    (1)

The first 3 column vectors of S have the form (x₁, x₂, x₃,0, 0, 0), span a 3D subspace V₁ and generate the sublattice L₁; while the last 3 columns (0,0,0, x₄, x₅, x₆) generate the sublattice L₂ in the 3D subspace V2. Clearly V1|¯ V2 = V = R⁶. Note that L₁, L₂ now live in 3D subspaces different from the physical space (E^//) and are expanded by the square root of two as required for proper projection [6]. This factor however, is irrelevant and shall be ignored in what follows.

Figure 1 Schematic representation of the projection of an hyper point x in 2D space into the 1D orthogonal subspaces E^// and E^{^}, x results of the embedding in 2D of the lattice points x⁽¹⁾, x⁽²⁾ in L₁, L₂. All spaces are to be envisaged as 3D spaces.

The decisive step consists in realizing that V = V₁ V₂ = E^// E^{^}, so that x can also be written x = (x^//, x^{^}). Hence (x⁽¹⁾, x⁽²⁾ ) and (x^//, x^{^}) refer to the same hyper-lattice point expressed in different bases of V: one being the crystals coordinate system and the other, the coordinate system of the real space E^// and its orthogonal complement E^{^}.
The x^// component of x is given by P(x) where P is an orthogonal projector [6] given in terms of the standard basis of R⁶ by the block matrix

      (2)

with I being the identity matrix in 3D. The perpendicular space component x^ is in turn given by P^(x) with P^ = I - P. Using Eqs.1 and 2 we obtain the fundamental result:

     (3)

These equations are the heart of the process. They make it possible, given two (or more) interpenetrating lattices in physical space, to find the best fit lattice [3,4] as the set of points {x^//}. Eq. 3 makes clear that while E^// contains physical lattice points, E^{^} contains displacements and must therefore be associated with the displacement space called b-space by Bollmann [7]. The projection of x onto real space, x^//, given by the average position of x⁽¹⁾ and x⁽²⁾ is clearly a "compromise" position. It has been pointed out that this lowers the energy of the GB in metals by acting as a buffer between two lattices meeting at the interface [3]. The x^{^} component on the other hand is a measure of the local strain or "frustration" between two points that would have liked (energy wise) to occupy the same position. Clearly the interfacial energy should be lower in regions of small local strain. In particular, x^{^} = 0, x⁽¹⁾ = x⁽²⁾, corresponds to a coincidence point with no associated strain. Note that every point x^// in the interface has an associated strain given by x^{^} so that the configuration of points in E^{^} is a reduced representation of the strain in E^//. When the two lattices are incommensurate (non periodic) the distances |x⁽¹⁾ - x⁽²⁾ | can be arbitrarily small (but not zero except at the origin) and E^{^} becomes densely filled. For rational, periodic orientations, a discrete lattice is formed in E^{^} which can be used to characterize the interface.

Not all hyper lattice points are projected into E^// since this would overcrowd the physical space. Only those points contained in a bounded region around E^// called "strip" are projected (see Fig. 1). The strip selects hyper points with small x^{^} component, i.e., hyper points with a small associated strain called quasi coincidences [3]. If the strip is chosen so that | x⁽¹⁾ - x⁽²⁾ | is less that half the interatomic distance, the structures obtained in E^// are equal to those obtained from the previously introduced GCSN model [3, 4].
The intersection of the strip and E^{^} defines a bounded region of E^{^} called window. Only the hyper points whose x^{^} component lies within this region get projected into physical space. For example, in the case of rotation related lattices there are no displacements along the rotation axis r. Hence, the component of x^{^} parallel to r is zero and the window is a 2D plane segment. For cubic twist boundaries rotated through the angle q, the window is defined by the intersections of the plane normal to r and two cubes rotated by ±q/2 as shown in Fig. 2.

Figure 2. Projection window of cubic <001> twist GBs defined by the intersection of the basal planes of two cubes rotated by ±q/2. The edge of the cubes is ½ the minimum interatomic distance. The rotation axis is normal to the page.

The shape of the window depends on the symmetries of L₁ and L₂, the transformation relating them and to some extent the nature of the atomic interactions. Using different window shapes and sizes, it is possible to (crudely) model diverse forms of interatomic interactions. 

Figure 3. Projection of a complete bicrystal. Crystals 1 and 2 are obtained by projecting the points (x⁽¹⁾, 0 ) and (0, x⁽²⁾ ). The interfacial region, results from the projection of the hyper-points (x⁽¹⁾, x⁽²⁾ ) inside the strip.
This method can be used to model not only the interface (the region where the two lattices coexist), but a complete bicrystal as shown in Fig. 3.

It must be stressed that the procedure above is completely general and describes equally well quasicrystals and interfaces (see figs. 4-6). The only difference with the conventional strip-projection method lies in the symmetry of the hyper-lattice. As has been shown, by giving up the cubic symmetry used in the quasicrystal field, an hyper-lattice can be chosen to be a combination or "marriage" of points belonging to two or more lattices in physical space.
An interface can then be understood as a region where these lattices, each representing the ground state of a crystalline system are forced to coexist. In this view, the strip-projection method acts like a judge that determines the final structure in the region of conflict. But since the formalism is equally applicable to quasicrystals, it is also possible to envisage a quasicrystal as interpenetrated lattices in 3D. If such lattices were found, then the strip-projection method could be useful in inferring the atomic structure of quasicrystals.
Clearly, the average position given by (Eq. 3) is expected to give a low energy structure for metallic close packed structures where the interatomic potential is isotropic [3]. This is probably why quasicrystals have only been found in metal alloys. For covalent systems, the same interpretation should in principle be valid, although the formalism would have to include a way of accounting for bonding anisotropy.

5. Application to Rotation Related Interfaces.

In the following, the above formalism will be applied to <001> twist interfaces in the cubic system as an example of how it can be used to obtain a crystallographic description of interfaces. It will be seen that in order to properly understand the structure of interfaces, it is necessary to introduce a new lattice closely related to the O-lattice, called "phason" lattice. The phason or P-lattice is to intervening GBs what the O lattice is to delimiting. But first, it is important to notice that, with no loss of generality, all interfaces can be assumed to be periodic.
It is seldom realized that the term irrational orientation is a mathematical concept, with no more physical significance than that of "point". This is because the set of rational numbers is dense, which means that any irrational number can be approximated as accurately as desired by a fraction.
For a given irrational orientation (rotation angle) corresponding to a non-periodic interface, one can find an infinite number of rational orientations corresponding to periodic (coincidence) boundaries within any experimental accuracy. The same reasoning applies to the problem of epitaxial interfaces between lattices with incommensurate unit cells. Therefore, for all practical purposes, all GBs can be considered to be periodic, although the period can be arbitrarily large.

5.1. Rational Rotations.

If all GBs are periodic, we can use Ranganathan's equation [9] to describe any boundary. According to Ranganathan [9], a CSL is obtained between two cubic lattices related by a rotation through q around <hkl> if q can be expressed as

     (4)

where p and q are arbitrary of co-prime positive integers and N = h² + k² + l². Coincidence boundaries are characterized by their index number S, the inverse of the density of coincident points, which is an odd integer given by

    (5)

divided by 2 until odd. One can alternatively write Eq. 4 as

    (6)

where x = Round[cot(q/2)], Round is the closest integer function, d = cot(q/2) - x is the fractional part of cot(q/2) contained in the interval (0.5, 0.5] and x = x + d. Equation 6 generalizes Eq. 4 since it can be used to describe non CSL interfaces by letting d acquire irrational values (see fig. 6). Its most significant advantage however, is that unlike p and q, x and d are given in terms of the measurable angle q and are directly related to structural properties [3]. In terms of x and d the index number S becomes

    (7)

When d is rational, we have a periodic (coincidence) GB and Eq. 6 reduces to Eq. 4. When d = 0 (p=1), x = x, and Eq. 6 becomes:

(8)

qx marks the position of special singular orientations called delimiting. Delimiting boundaries have "special" properties such as minimum dislocation content [3]. In this case x = x and

     (9)

d is a measure of the angular distance of a GB with rotation angle q to the singular orientation qx attained when d=0 and it is therefore called the deviation parameter.
Note that in Eq.7, the integer p takes part in the definition of S which makes it ill defined since p is not accessible experimentally. This means that the index number is only well defined for delimiting boundaries as given by Eq. 9 where it is unambiguously given in terms of x which depends on the measurable angle qx.
Clearly, when the rotation angle is small, x + d must be large so that the fractional part d can be neglected in comparison with x. Therefore, small angle boundaries, which are usually regarded as belonging to a class of their own, belong to the class of delimiting boundaries.
Delimiting orientations are thus defined as those for which d = 0 in Eq. 6. Alternatively, delimiting boundaries could be defined as those possessing primary dislocations, i.e., dislocations with crystalline Burgers vectors. In contrast, orientations for which d | 0, called intervening [2] contain dislocations with non-crystalline Burgers vectors. Accordingly, low angle boundaries should have dislocation networks with crystalline Burgers vectors in agreement with experimental evidence.

6. The Crystallography of Interfaces.

6.1. The O-lattice.

The usefulness of the O-lattice resides in that it provides the dislocation content of interfaces. This however, is only true for delimiting boundaries. It has been known for some time that intervening GBs near "special" orientations posses a network of so called secondary dislocations [10] with Burgers vectors belonging to the DSC lattice [7]. For such boundaries, the O-lattice does not describe the dislocation content, instead, the secondary O-lattice [7] (see below) is used to account for the secondary dislocation network. The problem is that until now no definition of "special" GB (d = 0) had been given, so that the understanding of intervening GBs has been incomplete. In fact, all intervening GBs (d | 0) have a network of secondary dislocations with non crystalline Burgers vectors regardless of their angular distance to the d = 0 "special" orientations.
Burgers vectors of secondary dislocations are of the form (x⁽¹⁾ - x⁽²⁾ )/2 and are thus contained in E^{^} (see Eq. 3). Displacements in E^{^} are known in the quasicrystal jargon as phasons, We shall see that by considering the Phason lattice (which contains the secondary O-lattice), it is possible to fully describe the structure of intervening boundaries.
The O-lattice transformation O for rotation related interfaces is given by the inverse of the displacement field I - (R_<hkl>q)^-1 with I being the identity matrix and R_<hkl>q a rotation through the angle q around the axis <hkl>. O can be expressed in several ways depending on the variables in terms of which it is written, the most useful representations is given in the in median lattice in terms of x:

     (10)

6.2. CSL and DSC Lattices.

The CSL for <001> twist boundaries can have 2 orientations [3]. One is the "parallel" orientation in which the CSL is given by the O-lattice rotated by 90° and scaled by 2p. The other is the inclined orientation in which the CSL is the O-lattice rotated by 45° and scaled by p.
Since S is only well defined for delimiting GBs we shall consider the CSL of delimiting boundaries only. In this case, the CSL and O-lattices are parallel when x in Eq. 8 is even and inclined otherwise. Hence, if we define

     (11)

and

     (12)

where mod is the modulus function we have:

    (13)

Where Ox has been obtained from Eq. 10 after making d = 0, x = x. Note n=1 for x odd and 2 otherwise.

     (14)

6-3. P and O Lattices.

According Bollmann, the secondary O-lattice provides the secondary dislocation network and therefore the structure of boundaries near "special" orientations at qx. In order to derive the Secondary O-lattice, Bollmann rightly assumed that displacements by a vector of the DSC lattice (D_x) of the nearby "special" orientation which we now know is at q_x, should preserve the structure of the interface and concluded that secondary dislocations should have Burgers vectors belonging to D_x. Following Bollmann, the secondary O-lattice of an intervening GB at q = q_x + Dq is given by S_x = O_yD_x (see Eq. 10, 14), with y being the value of x resulting from substituting Dq = q - q_x in Eq. 6. S_x is thus given in the median lattice by:

  (15)

Eq.15 can be expressed in terms of the structure matrix of the intervening O-lattice Ox at q as:

  (16)

In other words, the secondary O-lattice can be obtained applying F2-n to the conventional O-lattice scaled by the inverse of the deviation parameter, a far simpler operation than calculating Oy and Dx. We now see the convenience of the notation introduced in section 5. The Secondary O-lattice vectors, marked with thin arrows in Fig. 5 join intervening domains of the same type while the P-lattice joins domains of different type. The P-lattice is given, in terms of the O-lattice, by

(17)

Finally, using Eqs 16 and 17 we have:

Clearly, d and thus Sx and Px diverge for delimiting boundaries. Note all lattices can be expressed in terms of the intervening O-lattice and the deviation parameter.

7. The Crystallography of Twist Grain Boundaries.

In this section a brief description of the result of applying the strip-projection formalism to <001> twist boundaries in the cubic system will be given. For reasons of space, only the main geometrical features will be described with no detailed crystallographic analysis such as the symmetry of the domains. However, it should be clear to the reader that such analysis follows straightforwardly from the data given.

7-1. Delimiting Boundaries.

The interface of <001> delimiting GBs consists of a single plane (in contrast to the <011> case where it can be stepped [4]) of atomic domains that have the structure of a 001 crystal plane (a square pattern corresponding to S = 1; q = 0). Delimiting domains are separated by a network of primary dislocations. An example of delimiting boundaries for the two possible orientations is shown in Fig 4. All delimiting GBs have the same structure, and differ only in the size of the domains, which decreases with qx. Although all domains have the same structure, they differ in the relative displacement of their centers with respect to the O-point at their center. Domains labeled A, B and C, with O-points colored black, gray and white are shifted by the vectors with coordinates {0,0}, ½{1,1} and ½{0,1} respectively, given in terms of the unit vectors of the median lattice.
The inclined orientation has O-points of types A and B only, while the parallel orientation also has domains type C. Thick vectors indicate the O-lattice and thin long arrows represent the CSL unit vectors. Note the CSL coincides with the sublattice defined by O-points type A.

7-2. Intervening Boundaries.

Figure 4. Wire Figure (lines joining nearest neighbors) of delimiting simple cubic 001 twist boundaries for parallel (top) and inclined (bottom) relative orientations of the O and CSL lattices. O-Lattice: short thick arrows. CSL: Long narrow arrows. Note the domain periodicity is the same as the CSL.

Figure 5. Wire figure of intervening interfacial structure in the FCC cubic system. Top: parallel orientation corresponding to S = 4369 showing domains A, B,C. Bottom: inclined state S = 21389 showing domains A and B. The Secondary O-lattice vectors, marked with long thin arrows join domains of same type (A-A) while the P-lattice (short thick arrows) joins domains of different types. The CSL unit cell of S = 4369 is drawn, but that of S = 21389 is too large to be shown. The properties of intervening GBs are determined by the domain periodicity and not by the CSL.

7-3. General Considerations.

Considering the domain arrangement of delimiting and intervening boundaries, it is clear that they are isomorphic. Delimiting GBs are fully characterized by the Ox and Cx lattices, while intervening boundaries are characterized by the lattices Px and Sx. Note Cx and Sx, join domains of the same type, while Ox and Px join domains of different types. Comparing Eqs. 13, 16 and 17 it is clear that the secondary O-lattice is to intervening Gbs what the CSL is to delimiting boundaries and that the same relation holds for Ox and Px.

It is well known that the size of the smallest O-lattice vectors gives the spacing between primary dislocations, as confirmed by Fig. 4. Fig. 5 shows that the secondary dislocation spacing s should be given by the magnitude of the smallest P-lattice vectors, which can be expressed in terms of the angles as

(19)

and letting Dq = q - qx we obtain:

(20)

which are in agreement with experimental observations [3,10] and support the conclusion of section 7-3.

8. Structure vs. Properties.

The properties of both intervening and delimiting GBs depend on the period and structure of domains, and not on the period of the CSL, except for delimiting boundaries where the domain and CSL periods coincide.
This is a straightforward conclusion since as we have seen, even an infinitesimal variation in the misorientation angle q, can greatly change the periodicity of the GB. The period of the domain structure as evidenced by measurements of dislocation spacings on the other hand, varies smoothly with the angle. We therefore see that the coincidence lattice concept is only useful for delimiting boundaries.
Burgers vectors can be determined by applying the Frank-Bilby equation to Eqs. 10 and 16. Primary dislocations have a crystalline Burgers vector of the form x(i), entirely contained in the physical space E//, and their spacing is given by the size of the smallest O-lattice vectors (Eq. 10). The displacement associated to secondary dislocations on the other hand is of the form ½ (x(1) - x(2)) which is contained in the perpendicular space E^ (see Eq. 3) and their spacing is given by the size of the smallest P-lattice vectors (Eq. 16). ]]> This implies that dislocations between intervening domains do not belong to the DSC lattice whose vectors are contained in E//. Such dislocations are in fact partial dislocations, and the difference in the structure of adjacent domains corresponds to the stacking fault associated with this type of dislocations. The displacements associated with the secondary O-lattice vectors on the other hand, do belong to the DSC lattice and must therefore preserve the symmetry. That is why they connect domains with equal symmetry as shown in Fig. 5.

Figure 6 Grain boundary with quasicrystalline octagonal symmetry obtained with q=45°, x=3, d = . Grey circles indicate atomic positions.

9- Conclusions.

A formalism capable of describing the crystallography of interfaces has been presented. The formalism is based on the representation of the dichromatic pattern in a higher dimensional space where it becomes a single crystal lattice and then projecting it to physical space. It has been shown that a new lattice called P-lattice associated with displacements in perpendicular space is required to understand the structure of general boundaries. The formalism permits the characterization of interfaces from the symmetry of the hyper lattice, which determines the symmetry of the interface. It has also been shown that interfaces and quasicrystals are equivalent systems and can both be regarded as a region in space where two (or more) lattices compete to contribute their points to the final structure.

Acknowledgements

Support from CONACYT through grant 25125-A is acknowledged.

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