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An O-lattice approach to the migration of crystalline interfaces
Scfipta Materialia, Vol. 37, No. 5, pp. 543-548, 1997 Elsevier Science Ltd Copyright © 1997 Aeta Metallurgica Inc. Printed in the USA. All rights reserved 1359-6462/97 $17.00 + .00
Pergamon PII S1359-6462(97)00144-9
AN O-LATTICE APPROACH TO THE M I G R A T I O N OF C R Y S T A L L I N E I N T E R F A C E S W.-Z. Zhang÷ and G.R. Purdy* *C-CORE, Memorial University of Newfoundland St. John's, Canada, A1B 3X5 "Department of Materials Science and Engineering, McMaster University, Hamilton, Canada, L8S 4L7 (Received January 20, 1997) (Accepted March 6, 1997) Introduction In this contribution, we investigate certain aspects of the geometry of migration of an interface linking two crystals of differing orientation and structure. The crystals are related by a three-dimensional misfit. A number of simplifying assumptions will be made: The preferred habit plane is taken to be parallel to a primary O-lattice plane for a given orientation relationship [1]. Assuming in the first instance that the equilibrium interface is a rigid plane, the virtual variation of its misfit with normal boundary displacement is defined and discussed in terms of Bollmann's O-lattice theory [2]. This virtual misfit is first used as a first estimate of the variation of interfacial free energy with displacement, and then, following Cahn [3] as the basis of an estimate of a critical driving force, below which the interface can advance only by the lateral motion of ledges. (This assumption amounts to the neglect of the variation of the chemical component of interfacial energy with the interface displacement.) In a previous paper [4], we advanced arguments to suggest that a residual component of misfit at growth ledges will be the rule, rather than the exception. Here we consider first the formal consequences of the rigid O-lattice assumption, and then discuss the limitations of this assumption when applied to the process of lateral growth. Rigid O-Lattice We begin by making the usual assumption that the O-lattice is fixed referred to an O-point as the origin. Consider the simple O-lattice of Figure 1. If there is no misfit in the direction normal to the plane of the figure, the O-lattice becomes an O-line lattice. The interfacial misfit of any point in a boundary lying along the primary O-lattice vector x° (q-q in Fig. lb) is in the direction of the Burgers vector b L. The misfit between each pair of adjacent O-points can be completely accommodated by the misfit dislocation located at the edge of the cell wall between the O-points. The resulting boundary is then free from long range strain. This boundary can possess this equilibrium structure only when it lies along x °. Thus on displacing the boundary, for instance, from q-q to q'-q', we find an additional component of the misfit, defined as follows: Any point in q'-q" boundary within the centre cell, i.e., at the a'-a" segment in Fig. lc, can be defined by: v = px°/Ix° I + ct,
543
(1)
544
AN O-LATTICE APPROACH TO THE MIGRATION
Vol. 37, No. 5
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Figure 1. Misfit displacement in a boundary with respect to a rigid O-lattice: (a) an O-lattice formed by overlapping rotated lattices; (b) misfit displacement on q-q and q'-q" boundaries; (c) positions of boundary during migration.
where t is a unit vector parallel to the cell wall, and p and c are scalars defined in Fig. lc. Suppose the t w o lattices in an orthonormal coordinate are related by ~ xp = A x . ,
(2)
where x. and xp are vectors in the lattices ~ and p, respectively, and A is a matrix defining the transformation. The misfit displacement associated with v is defined by d = "Iv, = pbUlxOl
(3a)
+ cdt,
•We use • and p to define vectors from different lattices, and numbers to denote vectors within one lattice. Lower-case letters in bold face are used to denote column vectors for vectors in direct space. Upper-case letters in bold face denote matrixes. Row vectors marked by a prime are used to define reciprocal vectors.
Vol. 37, No. 5
AN O-LATTICE APPROACH TO THE MIGRATION
545
T = I - A ~,
(3b)
b L = Tx °,
(3c)
d t = Tt.
(3d)
where
b L is a Burgers vector of the reference lattice a, and c d t is the misfit that cannot be accommodated by the
original misfit dislocations. For different positions along the boundary with the O-cell, only p varies, but the residual misfit ct is invariable. It can also be shown that the residual misfit is identical in different Ocells, as can be seen from Fig. lb. In this example, the two lattices are related by a rotation, and so the Ocell is the Wigner-Seitz cell of the O-lattice [1]. Consequently, x ° is parallel to the normal of the cell wall e°" and perpendicular to t, and so c defines directly the displacement of the boundary. In a general case, x° and t may not be perpendicular to each other. If the angle between them is v and the displacement o f the interface is s, c in (3a) can be replaced by s/sin(v) so that: d = pbL/Ix° I + sO/sin(v).
(4)
For a given O-line structure V is fixed; the unbalanced misfit will always increase for small displacements of the boundary from the primary O-lattice plane. The interfacial energy of the rigid interface is postulated to increase with the magnitude of lattice misfit. The magnitude of misfit associated with v can be determined from Id 12= (plbLI/Ix°l)2 + (sldtl/sin(v)) 2.
(5)
This relation follows because b L and d t are perpendicular to each other, since c °' --- bL'T/IbLI2 [1] and e°'t = bL'dt/lbLI2 = 0. At any position in the boundary specified by p, Idl increases with s. Therefore, the average misfit interfacial energy is expected to increase also with s. We assume further that the misfit and misfit energy continue to increase to their maxima at position m-m, equidistant from adjacent O-lattice planes. Consequently, this energy is a periodic function of interfacial position, defined by s, with periodicity equal to the spacing of the O-lattice planes containing x °. The spacing of a set of O-lattice plane is h = 1/IP°'l . Here pO, is a reciprocal vector representing the O-lattice planes defined by [ 1] pO, = g.'T
(6a)
and g.', representing planes in lattice a, satisfies the following two conditions. First, g~'biL = 0,
(6b)
where bi L is a Burgers vector associated with a dislocation in the boundary. Depending on the number of the sets of dislocations in the boundary, i can take a different value from 1 up to 3. Condition (6b) must hold for all Burgers vectors. The second condition of g~' is determined from pO'Xn° = g,,'b.L = 1, where x, ° defines an O-point in the next plane. It is related to the Burgers v e c t o r bn L by Tx, ° = bn L.
(6c)
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AN O-LATTICE APPROACH TO THE MIGRATION
Vol. 37, No. 5
The above analysis can be extended to a boundary parallel to a general primary O-lattice plane containing a two-dimensional distribution of the O-points [1]. Any point in the boundary can be expressed as a sum of two vector components: xp in the primary O-lattice plane, and xt in an O-cell wall. In a fashion similar to the derivation in (1)-(6), it can be shown that only the misfit associated with Xp is contained in the plane g," and that this can be taken up by the misfit dislocations; the misfit corresponding to xt does not lie in the g," plane and remains unbalanced by misfit dislocations. As demonstrated in the twodimensional example, it is this second part that contributes to the increase of the misfit due to the deviation of the boundary from the primary O-lattice plane, and by inference to the interfacial energy increase. In a special case, when there is no misfit in the O-lattice plane, the same analysis can be applied simply by letting the Burgers vector be a zero vector, g," is not unique for O-line and O-plane types of intersections. However, it can be proven that pC, is uniquely defined when g," meets condition (6b) and (6c) because of special properties of the invariant line [5] and the invariant plane [6]. The main point is that, in general, the normal displacement of a rigid interface would, if it occurred, be accompanied by periodic variations in interracial energy, with period equal to the spacing of O-lattice planes. Next, following Cahn's analysis [3], we take the following fn'st approximation for the variation of specific interfacial free energy with position, s: o,{s} -- (l+g{s})Omin,
(7)
where the dimensionless periodic function, g{s} = (Omax -Omin)(l-cos(2~s/h))/2Omin,
(8)
and Omin are the maximum and minimum values of the average interfacial free energies respectively. The variation of the total energy per unit area due to a uniform displacement of an interface, 6s, is given approximately by [3] a n d Omax
8F = (AF, + doJds)Ss,
(9)
where AFv is the driving force, and daJcls can be regarded as the lattice resistance to the migration of the interface. The critical driving force, AFt, permitting a uniform motion of an interface is given by setting 6F = 0. It corresponds to the maximum value of da,/ds, o r amindg/ds, and is determined from - A F e r = g(O'ma x - Omin)/h.
(lo)
In the present analysis, the minimum surface energy, Omin is associated with a boundary lying in the primary O-lattice plane. It has been shown that the planar density of the O-points serves as a good approximation for the density of the misfit dislocations [1]. For a given O-lattice structure the density of the O-points in the plane is proportional to interplanar spacing h (=l/[p °" I). 0a is equivalent to the Burgers vector content in the boundary plane [1].) Applying the (approximate) relation between interfacial energy and density of the O-lattice points,
Om~.=rh, with r constant. Equation (10) becomes
(11)
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or
AN O-LATTICE APPROACH TO THE MIGRATION
547
-AFcr = n(omJh - r),
(12a)
-AFcr = nr(Om~x/Omi, 1).
(12b)
-
A boundary of maximum average interfacial free energy am~xwould then lie equidistant between adjacent primary O-lattice planes. At this position planes g=" are in the worst matching position with the correlated planes gp, as can be seen from a moir6 pattern formed by overlapping these planes [1]. This will hold no matter which set of primary O-lattice planes is under consideration. Since the extra misfit energy is mainly caused by the misfit which is out of the g=" plane, we are tempted to suggest that o.~, is independent of the selection of primary O-lattice plane and therefore independent ofh. Then, Eqn. (12a) would relate the critical driving force directly to the periodicity of the O-lattice. Accordingly, an interface lying parallel to those primary O-lattice planes with the smallest periodicity would require the largest critical driving force for its uniform advancement. Based on (12b), an interface with the lowest energy would then be the more likely to require a lateral mode for its migration, which seems intuitively reasonable. In the rigid O-lattice, in order that the boundary maintain its low energy configuration during lateral migration, the step heights must be exact multiples of the interplanar spacing of the primary O-lattice planes. We term these steps equilibrium steps, because they connect boundary planes in equilibrium positions. Ideally, there is no unbalanced misfit, and no long-range strain associated with these steps. If the steps are rather high, the riser of the step may also become a facet parallel to a (second) primary O-lattice plane. Deformable O-Lattice
According to Eqn. (4), when the misfit is accommodated by dislocations located at the cell walls, a boundary at s = 0 does not possess a long range strain. This is not generally true when s,0; then there is a net residual misfit, e.g., cTt in every O-cell in the entire facet (Fig. lb and lc). However, the postulated rigid crystal system, while it is of value in the discussion of the critical driving force for ledge growth, and in defining equilibrium growth steps is not of general application since crystals are not ideally rigid. If the crystals are permitted to translate locally, the equilibrium structure of a facet can be realized at positions other than the equilibrium ones. This effect can be regarded as a local translation of the O-lattice plane to a new position of the boundary plane. In this sense, we consider the O-lattice to be deformable: As long as the O-lattice plane coincides with the boundary plane, the residual misfit in this major area of the boundary is removed. However, where the crystal translation terminates or changes direction or magnitude at a non-equilibrium step or the edge of the boundary, there will be a local crystal distortion, or singularity. The singularity, which is very similar to a partial dislocation bounding a stacking fault, is equivalent to a transformation dislocation [7]; it will in general possess a long range strain field. Depending on whether the strain derived from a succession of ledges is cumulative or not, the system may exhibit differing types of behaviour. The translation of a crystal lattice can be determined according to its relationship with the displacement of the O-lattice, as discussed in detail elsewhere [8, 9]. In the case of cumulative behaviour, it can be shown for example [9] that the accumulated translation is equivalent to the shape deformation in the phenomenological theory of martensite crystallography (PTMC) [ 10, 11]. When migration of a boundary is accompanied by collective translations, boundaries in other orientations cannot have equilibrium structures, because they bear the long range strain. Therefore, only one boundary, i.e., the habit plane, is capable of an equilibrium structure. Non-cumulative translations can also occur when a boundary plane departs from the ideal primary Olattice plane. Such translations are probably not uncommon in real systems, likely occurring whenever the boundary is displaced via semicoherent steps whose heights do not agree with the multiples of the spacing of the corresponding primary O-lattice planes.
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AN O-LATTICE APPROACH TO THE MIGRATION
Vol. 37, No. 5
Discussion We have explored the application of classical (rigid) O-lattice theory to the problem of migration of crystal interfaces. The approach has proven of value in defining a periodic variation of virtual misfit, which accompanies the continuous normal displacement of a planar interface through a rigid O-lattice. The period of the three-dimensional misfit in the direction normal to the boundary is equal to the spacing of the primary O-planes. We assume a conceptual misfit interfacial energy to vary with the positions of the interface in the same period. The approach is of value in helping to understand a virtual resistant force against motion of the interface. Thus, uniform migration of the interface requires a critical driving force, below which interface motion must occur by lateral migration of ledges. In addition, the rigid O-lattice concept permits the definition of an equilibrium growth ledge, which joins terraces of equilibrium structure and possesses no long-range strain field. However, we believe that although equilibrium facets will be the rule, equilibrium ledges will be the exception in real cases of crystal growth. Growth ledges will then generally correspond to singularities, which can be described by the local translation of a deformable O-lattice. They will possess long-range strain fields, whose nature and cumulative behaviour will determine whether or not the interface migration results in an accumulated strain. Unfortunately, the introduction of a degree of freedom in the analysis also introduces a degree of ambiguity, due to the different magnitudes and directions of strain accompanying the choice of different ledge displacements in the deformable O-lattice. It is this long-range strain field that is the primary cause of elastic interactions among growth ledges, and, when ledges of like sign accumulate, it is the cause of surface relief accompanying precipitate growth. We have only discussed the occurrence of growth ledges, not their means of migration. It is to be expected that the change of the lateral position of a step is often accompanied by an energy variation [3, 12, 13]. This will correspond to another critical driving force. Whether a step is movable will then depend on the balance between this second critical driving force and the available driving force [13].
Acknowledgments One of the authors (WZZ) wishes to express her thanks to Dr. J.I. Clark (of C-CORE) for his indispensable support, to Dr. W. Bollmann (Geneva) for continuous encouragement and stimulating discussions, and to Dr. V. Perovic for kindly providing useful reference materials. The support of the Natural Science and Engineering Research Council of Canada and of Ontario Hydro Research to a project from which the present work was initiated is gratefully acknowledged.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. I0. 11. 12. 13.
W.-Z. Zhang and G.R. Purdy, Phil. Mag. 68, 279 (1993). W. Bollmann, Crystal Defects and Crystalline Interfaces, Springer, Berlin, (1970). J.W. Cahn, Acta Metall. 8, 554 (1960). G.R. Purdy, and W.-Z. Zhang, Metall. Mater. Trans. 25A, 1875 (1994). W.-Z. Zhang and G.R. Purdy, Phil. Mag. 68A, 291 (1993). A.G. Khachaturyan, Theory of Structural Transformations in Solids, Wiley, New York, p. 164 (! 983). J.W. Christian, Dislocations and Properties of Real Materials, The Institute of Metals, London (1985). W.-Z. Zhang to be submitted to Phil. Mag. W.-Z. Zhang and G.C. Weatherly to be submitted to Acta Mater. J.S. Bowles and J.K. Mackenzie, Acta Metall. 2, 129 (1954). J.K. Mackenzie and J.S. Bowles, Acta Metall. 2, 138 (1954). J.M. Howe and N. Prabhu, ActaMetall. Mater. 38, 881 (1990). Y.J.M. Brechet, V. Perovic and G.R. Purdy, Scripta Mater., 35, 72 (1996).
Pergamon PII S1359-6462(97)00144-9
AN O-LATTICE APPROACH TO THE M I G R A T I O N OF C R Y S T A L L I N E I N T E R F A C E S W.-Z. Zhang÷ and G.R. Purdy* *C-CORE, Memorial University of Newfoundland St. John's, Canada, A1B 3X5 "Department of Materials Science and Engineering, McMaster University, Hamilton, Canada, L8S 4L7 (Received January 20, 1997) (Accepted March 6, 1997) Introduction In this contribution, we investigate certain aspects of the geometry of migration of an interface linking two crystals of differing orientation and structure. The crystals are related by a three-dimensional misfit. A number of simplifying assumptions will be made: The preferred habit plane is taken to be parallel to a primary O-lattice plane for a given orientation relationship [1]. Assuming in the first instance that the equilibrium interface is a rigid plane, the virtual variation of its misfit with normal boundary displacement is defined and discussed in terms of Bollmann's O-lattice theory [2]. This virtual misfit is first used as a first estimate of the variation of interfacial free energy with displacement, and then, following Cahn [3] as the basis of an estimate of a critical driving force, below which the interface can advance only by the lateral motion of ledges. (This assumption amounts to the neglect of the variation of the chemical component of interfacial energy with the interface displacement.) In a previous paper [4], we advanced arguments to suggest that a residual component of misfit at growth ledges will be the rule, rather than the exception. Here we consider first the formal consequences of the rigid O-lattice assumption, and then discuss the limitations of this assumption when applied to the process of lateral growth. Rigid O-Lattice We begin by making the usual assumption that the O-lattice is fixed referred to an O-point as the origin. Consider the simple O-lattice of Figure 1. If there is no misfit in the direction normal to the plane of the figure, the O-lattice becomes an O-line lattice. The interfacial misfit of any point in a boundary lying along the primary O-lattice vector x° (q-q in Fig. lb) is in the direction of the Burgers vector b L. The misfit between each pair of adjacent O-points can be completely accommodated by the misfit dislocation located at the edge of the cell wall between the O-points. The resulting boundary is then free from long range strain. This boundary can possess this equilibrium structure only when it lies along x °. Thus on displacing the boundary, for instance, from q-q to q'-q', we find an additional component of the misfit, defined as follows: Any point in q'-q" boundary within the centre cell, i.e., at the a'-a" segment in Fig. lc, can be defined by: v = px°/Ix° I + ct,
543
(1)
544
AN O-LATTICE APPROACH TO THE MIGRATION
Vol. 37, No. 5
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Figure 1. Misfit displacement in a boundary with respect to a rigid O-lattice: (a) an O-lattice formed by overlapping rotated lattices; (b) misfit displacement on q-q and q'-q" boundaries; (c) positions of boundary during migration.
where t is a unit vector parallel to the cell wall, and p and c are scalars defined in Fig. lc. Suppose the t w o lattices in an orthonormal coordinate are related by ~ xp = A x . ,
(2)
where x. and xp are vectors in the lattices ~ and p, respectively, and A is a matrix defining the transformation. The misfit displacement associated with v is defined by d = "Iv, = pbUlxOl
(3a)
+ cdt,
•We use • and p to define vectors from different lattices, and numbers to denote vectors within one lattice. Lower-case letters in bold face are used to denote column vectors for vectors in direct space. Upper-case letters in bold face denote matrixes. Row vectors marked by a prime are used to define reciprocal vectors.
Vol. 37, No. 5
AN O-LATTICE APPROACH TO THE MIGRATION
545
T = I - A ~,
(3b)
b L = Tx °,
(3c)
d t = Tt.
(3d)
where
b L is a Burgers vector of the reference lattice a, and c d t is the misfit that cannot be accommodated by the
original misfit dislocations. For different positions along the boundary with the O-cell, only p varies, but the residual misfit ct is invariable. It can also be shown that the residual misfit is identical in different Ocells, as can be seen from Fig. lb. In this example, the two lattices are related by a rotation, and so the Ocell is the Wigner-Seitz cell of the O-lattice [1]. Consequently, x ° is parallel to the normal of the cell wall e°" and perpendicular to t, and so c defines directly the displacement of the boundary. In a general case, x° and t may not be perpendicular to each other. If the angle between them is v and the displacement o f the interface is s, c in (3a) can be replaced by s/sin(v) so that: d = pbL/Ix° I + sO/sin(v).
(4)
For a given O-line structure V is fixed; the unbalanced misfit will always increase for small displacements of the boundary from the primary O-lattice plane. The interfacial energy of the rigid interface is postulated to increase with the magnitude of lattice misfit. The magnitude of misfit associated with v can be determined from Id 12= (plbLI/Ix°l)2 + (sldtl/sin(v)) 2.
(5)
This relation follows because b L and d t are perpendicular to each other, since c °' --- bL'T/IbLI2 [1] and e°'t = bL'dt/lbLI2 = 0. At any position in the boundary specified by p, Idl increases with s. Therefore, the average misfit interfacial energy is expected to increase also with s. We assume further that the misfit and misfit energy continue to increase to their maxima at position m-m, equidistant from adjacent O-lattice planes. Consequently, this energy is a periodic function of interfacial position, defined by s, with periodicity equal to the spacing of the O-lattice planes containing x °. The spacing of a set of O-lattice plane is h = 1/IP°'l . Here pO, is a reciprocal vector representing the O-lattice planes defined by [ 1] pO, = g.'T
(6a)
and g.', representing planes in lattice a, satisfies the following two conditions. First, g~'biL = 0,
(6b)
where bi L is a Burgers vector associated with a dislocation in the boundary. Depending on the number of the sets of dislocations in the boundary, i can take a different value from 1 up to 3. Condition (6b) must hold for all Burgers vectors. The second condition of g~' is determined from pO'Xn° = g,,'b.L = 1, where x, ° defines an O-point in the next plane. It is related to the Burgers v e c t o r bn L by Tx, ° = bn L.
(6c)
546
AN O-LATTICE APPROACH TO THE MIGRATION
Vol. 37, No. 5
The above analysis can be extended to a boundary parallel to a general primary O-lattice plane containing a two-dimensional distribution of the O-points [1]. Any point in the boundary can be expressed as a sum of two vector components: xp in the primary O-lattice plane, and xt in an O-cell wall. In a fashion similar to the derivation in (1)-(6), it can be shown that only the misfit associated with Xp is contained in the plane g," and that this can be taken up by the misfit dislocations; the misfit corresponding to xt does not lie in the g," plane and remains unbalanced by misfit dislocations. As demonstrated in the twodimensional example, it is this second part that contributes to the increase of the misfit due to the deviation of the boundary from the primary O-lattice plane, and by inference to the interfacial energy increase. In a special case, when there is no misfit in the O-lattice plane, the same analysis can be applied simply by letting the Burgers vector be a zero vector, g," is not unique for O-line and O-plane types of intersections. However, it can be proven that pC, is uniquely defined when g," meets condition (6b) and (6c) because of special properties of the invariant line [5] and the invariant plane [6]. The main point is that, in general, the normal displacement of a rigid interface would, if it occurred, be accompanied by periodic variations in interracial energy, with period equal to the spacing of O-lattice planes. Next, following Cahn's analysis [3], we take the following fn'st approximation for the variation of specific interfacial free energy with position, s: o,{s} -- (l+g{s})Omin,
(7)
where the dimensionless periodic function, g{s} = (Omax -Omin)(l-cos(2~s/h))/2Omin,
(8)
and Omin are the maximum and minimum values of the average interfacial free energies respectively. The variation of the total energy per unit area due to a uniform displacement of an interface, 6s, is given approximately by [3] a n d Omax
8F = (AF, + doJds)Ss,
(9)
where AFv is the driving force, and daJcls can be regarded as the lattice resistance to the migration of the interface. The critical driving force, AFt, permitting a uniform motion of an interface is given by setting 6F = 0. It corresponds to the maximum value of da,/ds, o r amindg/ds, and is determined from - A F e r = g(O'ma x - Omin)/h.
(lo)
In the present analysis, the minimum surface energy, Omin is associated with a boundary lying in the primary O-lattice plane. It has been shown that the planar density of the O-points serves as a good approximation for the density of the misfit dislocations [1]. For a given O-lattice structure the density of the O-points in the plane is proportional to interplanar spacing h (=l/[p °" I). 0a is equivalent to the Burgers vector content in the boundary plane [1].) Applying the (approximate) relation between interfacial energy and density of the O-lattice points,
Om~.=rh, with r constant. Equation (10) becomes
(11)
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or
AN O-LATTICE APPROACH TO THE MIGRATION
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-AFcr = n(omJh - r),
(12a)
-AFcr = nr(Om~x/Omi, 1).
(12b)
-
A boundary of maximum average interfacial free energy am~xwould then lie equidistant between adjacent primary O-lattice planes. At this position planes g=" are in the worst matching position with the correlated planes gp, as can be seen from a moir6 pattern formed by overlapping these planes [1]. This will hold no matter which set of primary O-lattice planes is under consideration. Since the extra misfit energy is mainly caused by the misfit which is out of the g=" plane, we are tempted to suggest that o.~, is independent of the selection of primary O-lattice plane and therefore independent ofh. Then, Eqn. (12a) would relate the critical driving force directly to the periodicity of the O-lattice. Accordingly, an interface lying parallel to those primary O-lattice planes with the smallest periodicity would require the largest critical driving force for its uniform advancement. Based on (12b), an interface with the lowest energy would then be the more likely to require a lateral mode for its migration, which seems intuitively reasonable. In the rigid O-lattice, in order that the boundary maintain its low energy configuration during lateral migration, the step heights must be exact multiples of the interplanar spacing of the primary O-lattice planes. We term these steps equilibrium steps, because they connect boundary planes in equilibrium positions. Ideally, there is no unbalanced misfit, and no long-range strain associated with these steps. If the steps are rather high, the riser of the step may also become a facet parallel to a (second) primary O-lattice plane. Deformable O-Lattice
According to Eqn. (4), when the misfit is accommodated by dislocations located at the cell walls, a boundary at s = 0 does not possess a long range strain. This is not generally true when s,0; then there is a net residual misfit, e.g., cTt in every O-cell in the entire facet (Fig. lb and lc). However, the postulated rigid crystal system, while it is of value in the discussion of the critical driving force for ledge growth, and in defining equilibrium growth steps is not of general application since crystals are not ideally rigid. If the crystals are permitted to translate locally, the equilibrium structure of a facet can be realized at positions other than the equilibrium ones. This effect can be regarded as a local translation of the O-lattice plane to a new position of the boundary plane. In this sense, we consider the O-lattice to be deformable: As long as the O-lattice plane coincides with the boundary plane, the residual misfit in this major area of the boundary is removed. However, where the crystal translation terminates or changes direction or magnitude at a non-equilibrium step or the edge of the boundary, there will be a local crystal distortion, or singularity. The singularity, which is very similar to a partial dislocation bounding a stacking fault, is equivalent to a transformation dislocation [7]; it will in general possess a long range strain field. Depending on whether the strain derived from a succession of ledges is cumulative or not, the system may exhibit differing types of behaviour. The translation of a crystal lattice can be determined according to its relationship with the displacement of the O-lattice, as discussed in detail elsewhere [8, 9]. In the case of cumulative behaviour, it can be shown for example [9] that the accumulated translation is equivalent to the shape deformation in the phenomenological theory of martensite crystallography (PTMC) [ 10, 11]. When migration of a boundary is accompanied by collective translations, boundaries in other orientations cannot have equilibrium structures, because they bear the long range strain. Therefore, only one boundary, i.e., the habit plane, is capable of an equilibrium structure. Non-cumulative translations can also occur when a boundary plane departs from the ideal primary Olattice plane. Such translations are probably not uncommon in real systems, likely occurring whenever the boundary is displaced via semicoherent steps whose heights do not agree with the multiples of the spacing of the corresponding primary O-lattice planes.
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AN O-LATTICE APPROACH TO THE MIGRATION
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Discussion We have explored the application of classical (rigid) O-lattice theory to the problem of migration of crystal interfaces. The approach has proven of value in defining a periodic variation of virtual misfit, which accompanies the continuous normal displacement of a planar interface through a rigid O-lattice. The period of the three-dimensional misfit in the direction normal to the boundary is equal to the spacing of the primary O-planes. We assume a conceptual misfit interfacial energy to vary with the positions of the interface in the same period. The approach is of value in helping to understand a virtual resistant force against motion of the interface. Thus, uniform migration of the interface requires a critical driving force, below which interface motion must occur by lateral migration of ledges. In addition, the rigid O-lattice concept permits the definition of an equilibrium growth ledge, which joins terraces of equilibrium structure and possesses no long-range strain field. However, we believe that although equilibrium facets will be the rule, equilibrium ledges will be the exception in real cases of crystal growth. Growth ledges will then generally correspond to singularities, which can be described by the local translation of a deformable O-lattice. They will possess long-range strain fields, whose nature and cumulative behaviour will determine whether or not the interface migration results in an accumulated strain. Unfortunately, the introduction of a degree of freedom in the analysis also introduces a degree of ambiguity, due to the different magnitudes and directions of strain accompanying the choice of different ledge displacements in the deformable O-lattice. It is this long-range strain field that is the primary cause of elastic interactions among growth ledges, and, when ledges of like sign accumulate, it is the cause of surface relief accompanying precipitate growth. We have only discussed the occurrence of growth ledges, not their means of migration. It is to be expected that the change of the lateral position of a step is often accompanied by an energy variation [3, 12, 13]. This will correspond to another critical driving force. Whether a step is movable will then depend on the balance between this second critical driving force and the available driving force [13].
Acknowledgments One of the authors (WZZ) wishes to express her thanks to Dr. J.I. Clark (of C-CORE) for his indispensable support, to Dr. W. Bollmann (Geneva) for continuous encouragement and stimulating discussions, and to Dr. V. Perovic for kindly providing useful reference materials. The support of the Natural Science and Engineering Research Council of Canada and of Ontario Hydro Research to a project from which the present work was initiated is gratefully acknowledged.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. I0. 11. 12. 13.
W.-Z. Zhang and G.R. Purdy, Phil. Mag. 68, 279 (1993). W. Bollmann, Crystal Defects and Crystalline Interfaces, Springer, Berlin, (1970). J.W. Cahn, Acta Metall. 8, 554 (1960). G.R. Purdy, and W.-Z. Zhang, Metall. Mater. Trans. 25A, 1875 (1994). W.-Z. Zhang and G.R. Purdy, Phil. Mag. 68A, 291 (1993). A.G. Khachaturyan, Theory of Structural Transformations in Solids, Wiley, New York, p. 164 (! 983). J.W. Christian, Dislocations and Properties of Real Materials, The Institute of Metals, London (1985). W.-Z. Zhang to be submitted to Phil. Mag. W.-Z. Zhang and G.C. Weatherly to be submitted to Acta Mater. J.S. Bowles and J.K. Mackenzie, Acta Metall. 2, 129 (1954). J.K. Mackenzie and J.S. Bowles, Acta Metall. 2, 138 (1954). J.M. Howe and N. Prabhu, ActaMetall. Mater. 38, 881 (1990). Y.J.M. Brechet, V. Perovic and G.R. Purdy, Scripta Mater., 35, 72 (1996).