Origin of an unusual systematic variation in the heteroepitaxy of Ag on Ni-the roles of twinning and step alignment

A systematic variation in the orientation relationship (OR) of Ag films grown on Ni substrates previously discovered by a combinatorial approach is analyzed using concepts of grain boundaries, surface science and phase transformations. On roughly half of all Ni substrate orientations, Ag adopts a “special” OR that varies systematically from a twin OR, which develops on substrates lying along the (111)-(210) line of the standard stereographic triangle (SST), to the so-called oct-cube OR, which arises exclusively on (100) substrates. On the other half of Ni substrate orientations, Ag adopts the standard cube-on-cube OR. The special ORs are modeled by a) linear interpolation, b) 1D edge-to-edge matching and c) 2D transformation strains, using the twin relationship as a reference. The 1D model shows that the systematic variation in the special ORs can be understood in terms of a growth mechanism by attachment at steps and a lattice rotation due to the difference between the step heights in the substrate and the film. The 2D model explains why this mechanism does not apply to substrates on which Ag displays the cube-on-cube OR.


INTRODUCTION
The orientation relationship (OR) that develops when a thin film is equilibrated on a single crystal substrate is essential to a fundamental understanding of the factors that control thin film growth and texture.
A great deal of research has been devoted to understanding the structure and OR at the interface between film and substrate [1][2][3]. Most of the studies have been dedicated to a rationalization of the interfaces that develop between new phases/precipitates and their surrounding matrices during phase transformations [2,4].
These interfaces between different phases often develop as a result of phase separation such as martensitic transformations or precipitation reactions. Here, the major question is how the orientation and structure of the interface (or habit) plane, is determined by the need to accommodate the structural mismatch across the interface by lattice defects [1,[5][6][7][8][9][10]. In phase transformations, the new phases are subjected to 3-dimensional (3D) constraints by being fully embedded in the matrix, and generally display a single OR with the matrix that must adapt to the orientations of all of its interfacial planes. This is different from the case of heteroepitaxy, of interest here, which addresses the OR of an A-film on a B-substrate of a given surface orientation (2D constraint). This may be thought of as an inverse problem: rather than finding the optimum interface for a given OR, the challenge is to find the optimum OR for a given interface. Different ORs may develop when a film is equilibrated on substrates of different orientations. Such behavior was recently reported for Ag films grown on Ni substrates [11,12], but was not previously fully explained. The theories that have been developed to describe ORs in epitaxy have identified a set of variables (lattice parameters, elastic constants, interfacial energies, etc.), which drive the OR between two phases abutting at a hetero-interface. Most of the discussion has concentrated on the role of terraces, steps and defects (such as dislocations) to accommodate structural differences or lattice mismatch across the interface [5][6][7][8][9][13][14][15].
On the experimental side, most studies of epitaxy have been limited to the behavior displayed by films on single crystal substrates of low-index or vicinal orientations. However, the combinatorial approach [16] of using electron backscatter diffraction (EBSD) to make statistical measurements of A films on polycrystalline B substrates has greatly expanded the range of accessible substrate orientations (e.g. [11,17,18]).
In two recent studies, we have compared experimental determinations by EBSD of the ORs of Ag particles equilibrated on some two hundred different Ni substrate orientations [11] with ORs obtained by computer simulation on twelve distinct Ni substrate orientations [12]. The simulations were performed by molecular dynamics (MD) in conjunction with embedded atom method (EAM) potentials [19] and showed excellent agreement with the experimental results [11]. More recently, MD simulations for the same system have also explored the different ORs that develop under 2D vs. 3D constraints [20].
In this paper, we aim at providing an original physical explanation of the ORs adopted by Ag on Ni.
We first summarize how the ORs of Ag films observed on an uncommonly large set of Ni orientations spanning the whole SST can be sorted primarily into two kinds, namely "special" and cube-on-cube. We then present an analysis of the data by merging approaches used by the grain boundary and interface science communities, together with those used in phase transformations. Our analysis does not aim at giving a detailed description of the structure of each interface as found in [21] but rather at understanding the general trends. The physical outcome is surprisingly simple and powerful for a more general approach to heteroepitaxy.

PREVIOUS RESULTS
We begin with a summary of previous results presented in refs [11,12]. The Ni surfaces on which the Ag ORs have been measured by EBSD are presented in Fig. 1, in the form of a standard stereographic triangle (SST) with corners at (100), (111) and (110), and (hkl) plane indices defined such that h≥k, k≥l, l≥0.  [11] indicated by the points, and the measured ORs of the Ag film indicated by the colors of the points: pink = cube-on-cube, black = twin, orange = oct-cube, and green = "special" (see text). Notice the even distribution of Ni poles but the clear separation between the two major families of ORs (pink and green).
From the uniform distribution of Ni poles in Fig. 1 it is apparent that Ni substrate surface orientations were sampled fairly evenly across the entire orientation space. In contrast, the associated ORs of Ag on Ni, denoted by the colors of the symbols, are clearly divided into separate regions of the SST. The pink poles in the right segment indicate a cube-on-cube OR (OR C) in which the three cube axes of the Ag film are aligned with those of the Ni substrate. The green poles in the left segment represent a family of "special" ORs (OR S) which undergo a gradual but systematic transition between two delimiting ORs indicated by the orange square at the Ni(100) orientation (bottom left) and the central (111)  to provide a connection between the two interfacial planes, Ag(HKL) and Ni(hkl), that form the observed 4 hetero-interface in the case of OR S. As far as the two parallel directions in the above description of OR S are concerned (i.e., Ag [11 0] and Ni[01 1]) the EBSD data analysis [11] has shown that these two <110> directions are those closest to lying within the interfacial plane.
One important point about the distribution of Ag interfacial planes that needs to be emphasized is that their orientations on all possible Ni substrate orientations are confined to just the (111)-(110)-(210) subtriangle of the SST. While it is obvious that for Ni substrates in the (111)-(110)-(210) subtriangle, where Ag adopts OR C, the orientation of Ag must also fall in that subtriangle, it turns out that on Ni substrates in the (100)-(111)-(210) subtriangle, where OR S prevails, Ag orientations are also confined to the same (111)-(110)-(210) region of orientation space. This rather striking and unexpected result for the ORs displayed by a relatively simple FCC-on-FCC heteroepitaxial system is in need of an interpretation. This explains our focus on relating the interfacial orientation of Ag(HKL) to that of Ni(hkl) in the region where OR S is observed, and to seek a rationale for the factors that may stabilize OR C in other regions of orientation space.
and (d) superimposed stereograms of Ag (red points) and Ni (blue points) with the coincident Ag(11 0 and Ni (01 1) poles circled in green on the edge of the stereograms. Triangle, square and diamond symbols represent poles corresponding to {111}, {100} and {110} planes, respectively. The round symbols in (c) represent the Ni(210) (blue) and Ag(120) (red). In (b) the stereograms are centered on the Ni(111) and Ag(111) planes in a twin OR; in (c) the stereograms are still in twin OR but centered on Ni(100) by rotating both the Ag and Ni stereograms of (b) together by 54.7° about their common <110> axes (circled in green); and in (d) the stereograms are centered on Ni(100) and Ag(111 ) in an oct-cube OR (i.e. a relative rotation of 15.8° of the Ag stereogram in (c) about the common <110> axis). The blue dashed triangles identify the SST of Ni, and the red dashed triangle in (c) is the Ag (111 )-(111)-(120) triangle which contains all Ag poles that result from the observed ORs. In addition, the Ni(111) and Ag(111) poles have been circled in brown in b, c, and d, for easy identification.

5
In the particular case of the gradual transition of OR S observed for Ag on Ni substrates lying along the left edge of the SST, from Ni(100) to Ni(111), an explanation of the OR was provided in our previous study [12], and is summarized here. Consider the sketch in Fig. 2a showing an example of OR S, with a Ag film (red) on top of a Ni substrate (blue). Both crystals are viewed along the common <110> directions (Ag 11 0 and Ni 01 1 ). The range of orientations along the left edge of the SST of Fig. 1 (100) and Ni(111), the angular gap between Ni(hkl) and Ag(HKL) is given by where  is the angle between Ni(hkl) and Ni(111) and  is the angle between Ag(HKL) and Ag(111) planes (all three angles taken to be positive). As shown previously [11,12], both the observed experimental and simulated ORs are well described by a linear model: The stereograms in Figs

ANALYSIS of OR S
The clear separation between OR C and OR S is apparent from the green and pink points in the SST shown in Fig. 1. OR S denotes a continuous range of ORs from OR T, displayed by Ag on Ni orientations that lie along the (111)-(210) line (see Fig. 1), to OR O, which occurs on Ni(100). We hypothesize that the same gradual orientation change that occurs for Ni substrates lying along the left edge of the SST, and described in section 2, also occurs along all possible "orientation trajectories" that run from OR T at any point along the An example of the use of the above procedure to obtain the computed orientation of the Ag interfacial plane, Ag(HKL), for the case of OR S, is given in Appendix A. The experimental value of Ag(HKL) may be obtained directly from the experimental EBSD data [11] as the a 13 , a 23 and a 33 coefficients of the orientation matrix [22]. The mean absolute deviation between the calculated and measured values of Ag(HKL) obtained from over 90 measurements averages 2.3°. This is rather good agreement, given that the experimental error in measuring the Ni(hkl) can be as large as about 3°. ‡ R() was obtained from the standard expression for a rotation  around a [01 1] direction, followed by a cubic symmetry operation (a fourfold rotation) in order to match the specific indices to those chosen in the figures.

DISCUSSION
The linear model of the previous section is independent of any physical or chemical attributes of Ni and Ag. At first glance, this appears to indicate that strains or defects due to lattice mismatch are unimportant, and that the phenomenon is controlled entirely by geometric or crystallographic parameters. This conclusion is surprising because lattice mismatch is known to play a key role in the structure and properties of interfaces, and the mismatch between the Ag and Ni lattices is about 16%, which is significant.
To solve this apparent paradox, we explore the possible role of lattice mismatch for the interface in some detail. First, we investigate the effect of size misfit at steps in a 1-dimensional (1D) approach, followed by a more complete 2D approach. It will be shown that these models provide additional physical insight into the origin of the systematically variable OR S while remaining fully consistent with the experimentally observed data.

The role of steps and lattice mismatch in OR S
As indicated above, our previous work [11] has noted that the steps on both sides of the Ni/Ag interfaces are parallel to each other and that this is consistent with the observation that the Ni[01 1] and Ag [11 0] directions that lie closest to the interfacial planes in both the Ni substrate and the Ag film are aligned (to within 0.9°±0.8°). This suggests that film growth occurs by atomic attachment at step edges, leading to a natural alignment of terraces and steps. However, the difference in lattice parameters between the substrate and the film implies that at any step, there is a mismatch in the spacing of terrace planes, which can lead to a macroscopic rotation of the lattices around the step direction [21,23]. The macroscopic lattice rotation is related to the mismatch and the step density. This allows the prediction of the Ag interface orientation (HKL) for any Ni surface orientation (hkl), where Ag(HKL) and Ni(hkl) are described by their angles  and  to the (111) planes in Ag or Ni, respectively, as defined earlier (see Fig.   2a). Consider Fig. 4; if the distance between step edges is L and the step height is d, then the angle of the stepped interface plane to the nearby terrace plane is sin = d Ag /L in Ag and sin = d Ni /L in Ni. This yields the following relationships between  and : and the rotation is obtained from Eq. 1. As written here in Eq. 6, and described in Fig. 4, the interfacial geometry has been simplified for the case where the terraces on both sides of the interface are taken to be (111) planes. The general case is somewhat more complex, as described in more detail below in section 4.2.1.3. Thus, if the width L is identical in the two crystals § , the difference in step height d leads to different inclinations  and , with an angular gap of . Bringing the two crystals into contact along the interface requires a rotation  to close the angular gap. This rotation ensures that the terrace planes meet edge-toedge in the interface [23].
The relationship of this geometrical description to moiré patterns and the reciprocal space ∆g description is summarized in Appendix B.

Advantages of the twin over the cube-on-cube OR
From the simple argument presented in Fig. 4, one might expect a lattice rotation  for all vicinal surfaces in heteroepitaxial systems where the step height depends on the lattice parameters. However, this effect is observed for OR S only, but not for OR C. To understand this critical difference it is necessary to consider the mismatch of the other lattice planes across the interface. This requires an extension to a 2D description of the 1D model based on edge-to-edge matching of a single set of planes (the terraces).  (100), the Ag lattice must be rotated by an angle =2° to allow the (111) terrace planes of Ag and Ni to meet edge-to-edge along the interface, as suggested by the 1D model (Fig. 4). Figure 5 illustrates two possible 2D views of the near-interface region. Figure 5a is an OR C representation of the interface, where the near-horizontal {111} terrace planes meet edge to edge, but the mismatch of the planes inclined to the interface is apparent. In contrast, Fig. 5b illustrates the structure expected for OR S, where the stacking of {111} terrace planes across the interface is reversed in § Due to the discrete nature of a crystal, , ,  and L are not strictly continuous variables.

The role of lattice mismatch in
twin-like fashion. Here the small rotation of =2° that allows terrace planes to meet edge-to-edge also improves the match of other planes across the interface, as is evident from Fig. 5b.

∆g representation of edge-to-edge matching in reciprocal space
Appendix B shows that the difference vector ∆g between two reciprocal lattice vectors defines the unique interface (perpendicular to this ∆g vector) where the two corresponding lattice planes meet edge-toedge. This is a 1D matching criterion. To compare the behavior of ∆g vectors for different ORs in 2D, Fig. 6 shows superimposed diffraction patterns of Ni (blue) and Ag (red) seen along the common <110> close-

Asymmetry of <110> edges in the TLK model
The asymmetry of rotations can also be seen in the crystallography of ledges on both sides of the Ni/Ag interface. Using the terrace-ledge-kink (TLK) model of surfaces [27], it is possible to decompose any surface orientation (hkl) into terraces and steps, and to further decompose any steps into ledges and kinks.
According to van Hove and Somorjai [28], any (hkl) surface may be decomposed into three linearly independent microfacet orientations, with the microfacet of largest area being identified as the terrace, the one with intermediate area as the ledge, and the one with smallest area as the kink. We choose the (100),    7a shows a cuboctahedral crystal shape (calculated using Wulffmaker [29]) which illustrates that, for this choice of facets, there can only be two kinds of junctions between microfacets depending on whether they connect two {111}s or a {111} to a {100} facet, but all of these junctions must run along <110> type directions. In Fig. 7a, the two types of junctions are marked in pink and green, respectively.  To understand the observed ORs in the framework of this TLK model, consider the lattice rotations described in Figure 6. The lattice rotation from Fig. 6b to 6a corresponds to the change from OR T to OR O, which allows edge-to-edge matching of all or most lattice planes crossing the interface (∆g vectors aligned).
This describes the behavior at steps between mixed facets (i.e. green <110> edges), which leads to a Ag lattice relative rotation of between 0 and 15.8° around the <110> common direction. By comparison, a lattice rotation in the opposite sense (Fig. 6c), would lead to increased mismatch between most Ag and Ni planes crossing the interface (diverging ∆g vectors), and is therefore unfavorable. This describes the behavior at steps between {111} facets, as found in the pink region along the right edge of the SST.
Along the entire line separating the green and pink halves of the SST, OR T is maintained and hence no lattice rotation is observed. This may seem surprising since along this line, going from (111) to (210), the relative area of (100) microfacets connected by a green <110> edge increases. However, the area of (111 ) microfacets connected by a pink <110> edge increases in equal proportion, and it can be argued that any gain from a lattice rotation around the green edge would be offset by the cost of accommodating this rotation

Twin correspondence and 2D transformation matrix
In this section, we employ the phase transformations approach to minimize the interfacial mismatch for heteroepitaxial films using the strain ellipse, e.g. [9,30,31], which provides a graphical representation of the transformation strain for a given lattice correspondence.
Lattice correspondence relationships are essential to our understanding of phase transformations, e.g. [8,32]. For cubic crystals, the most common is the cube correspondence, used as a reference to describe the misorientation between grains across grain boundaries, and the Bain correspondence, used to describe the relation between FCC austenite and BCT martensite, or more generally, between FCC and BCC crystals. Any The twin correspondence for Ag-Ni is apparent from Fig. 6, where ∆g vectors have been drawn to connect corresponding g-vectors. One may think of the ∆g vectors as a graphical shorthand notation for "g 1 corresponds to g 2 ". Once two sets of corresponding planes have been chosen, the lattice correspondence is fully determined (the fact that the common <110> directions define the axis of rotation and remain parallel throughout reduces the correspondence to 2-dimensions) ** . Fig. 6b illustrates why this is referred to as the   The strain ellipse shown in the center (Fig. 8b) depicts the transformation in OR T. A rotation of the Ag ellipses by 15.8° relative to the stationary Ni circle results in OR O (Fig. 8a). Here the eigenvectors enclose an angle of ~60° each in real space (hollow) and reciprocal space (solid). It can be seen that eigenvectors in real space are pairwise orthogonal to eigenvectors in reciprocal space [31].
Note that OR O is special because at =15.8° a direction in the Ni(100)/Ag(111 ) interface is strainfree (an eigenvector with ∆g=0, i.e. the small hollow arrow vanishes) † † . It can be seen that all strain (or ∆g) vectors are aligned in OR O and nearly aligned in OR T. By comparison, they diverge for the hypothetical OR at =-15.8° shown in Fig. 8c.
As the OR changes from Fig. 8a to b, eigenvectors move closer to each other in both, real and reciprocal space until they coincide (i.e. become degenerate) at the critical value of =0° in Fig. 8b. Any † † To be exactly strain-free (an invariant line), the corresponding eigenvalue  i must equal unity, which requires =15.1° (see Eq. 9). An explicit solution for the case of an invariant line has been given by Xiao and Howe [30] (their Eq. 8).
further rotation eliminates all eigenvectors, as shown for an arbitrarily chosen rotation of -15.8° in Fig. 8c. An animated graphic to illustrate the behavior of eigenvectors during lattice rotation is provided in the supplementary materials.
In phase transformations, the question typically asked is which interface direction (and associated lattice rotation ) has the lowest energy [1,4,8]. A precipitate will tend to maximize its dimension along this direction [31]. This criterion generally predicts an invariant line strain with a unique interface and lattice orientation. However, in the current problem of thin film growth, the substrate orientation (hkl) is fixed, and the question becomes which lattice rotation  has the lowest energy for a fixed substrate orientation (hkl).
We postulate that for a given substrate orientation (hkl) along the left edge of the SST, the interface must contain an eigenvector of the transformation because any strain with a component normal to the interface would accumulate long-range elastic stresses or require dislocation arrays to accommodate the strain. Naturally, the interface with the smaller of the two eigenstrains is preferable. It can be shown that the interface containing the smaller of the two eigenstrains is normal to the larger of the two eigenstrains in reciprocal space (i.e., the larger ∆g vector also visible in Fig. 8).
The eigenvector in the interfaces is the 2D equivalent of the 1D criterion of edge-to-edge matching.
The two criteria predict very similar ORs, but only the 2D analysis can explain why no lattice rotations are observed for OR C, even if the terrace planes in the two phases have different spacings. Consider the transformation matrix describing OR C, which is a pure expansion of magnitude =1. 16. For a pure expansion, the strain ellipse becomes a circle, and all displacements are radial. Hence every vector is an eigenvector, and any rotation  will rotate all the strain vectors (or ∆g vectors), thus eliminating all eigenvectors (which by definition are unrotated), similar to the case shown in Fig. 8c. This implies that for any interface, the 2D strain will have components normal to the interface, even if one set of planes meets edge-to-edge (see Fig. 5a). This would lead to prohibitive long-range stresses. Thus, the orientation relationship for OR C is effectively locked by the fact that any lattice rotation eliminates all eigenvectors. This is clearly different for the twin correspondence, where a suitable lattice rotation can be chosen to align the smaller eigenvector with the substrate surface, using Eq. 10, which also determines the appropriate rotation .
It is interesting to note that the directions of the eigenvectors (Eq. 10) are independent of the lattice parameter ratio  whereas the eigenvalues (Eq. 9) are directly proportional to  (see also animated graphics V1 and V2 in supplementary materials). This implies that the limiting rotation that brings eigenvectors into coincidence does not depend on the lattice parameters, and hence the range of allowable orientations is independent of the combination of materials, for a given lattice correspondence. The range of rotations from OR T to OR O is given by the twin lattice correspondence alone and is thus valid for all heteroepitaxial systems, independent of lattice parameter ratio. This explains why the same ORs are found in a number of heteroepitaxial systems [34]. However, the eigenvalues depend directly on the lattice parameters, and since only small strains (eigenvalues close to =1) can be accommodated in the interface, it is not unreasonable that only a limited range of systems exhibits OR O. This also offers an interesting solution to the apparent paradox between the conclusions of the linear approximation (rotation independent of ) and 2D model (rotation dependent on ).

Quantitative comparison of different models and experimental data
In addition to the three models presented above -the linear, the 1D and the 2D model -this interface can also be described by the topological model of Pond et al. [35], which assumes that the terraces are strained to be coherent and the steps are spaced to compensate for the coherency strain. In this model a geometrically invariant interface is relaxed into coherent patches separated by disconnections, which have both step and dislocation character. The relaxed interface contains alternating regions of compression and expansion, which cancel each other at a small distance from the interface. A comparison between the phenomenological theory of martensitic transformations (PTMC) [36] and the topological model initially indicated that the predicted ORs were essentially identical, but the habit planes were significantly different [35]. However, this difference in habit planes was almost entirely due to a sign error, and when corrected, the predictions of the topological model agreed with those of the PTMC [8]. For the case of Ag on Ni, the large 16% strain in the terrace plane is unlikely to be accommodated elastically, but the formalism of the topological model remains valid.
In this model the rotation angle  is mainly due to the difference in step height, which determines the z-component of the Burgers vector associated with a disconnection. This is also the concept underlying the edge-to-edge matching, and it is not therefore surprising to find that the different approaches predict essentially the same ORs, even for the present case, where the interface is not generally an invariant plane.
For a more quantitative comparison of the different models, we use a plot of the angle between the interface and the Ni(111) plane versus the rotation angle . Each of the models assumes that interfaces in the OR S regime are described by these two angles. It is the relationship between  and  that distinguishes the models from each other. We can therefore use a plot of  versus  to compare the models with each other and with the experimental data.
The linear model assumes that = (70.5/54.7)  (Eq. 2), and since =-(Eq. 1), we can write  as: The edge-to-edge matching in the 1D model requires that sin =1.16 sin  (Eq. 6). Using =-, we find: 19 The topological model (Eq B.8 in [35]  ) proposes that: where b z is the difference in step heights and h is the smaller of the two step heights.
Finally, the 2D model finds the eigenvectors  1,2 = (m,1), as defined in Eq. 10, where: , B=(2 cos  -s sin ), and can be found from the dot product of  with the x-axis: These four models are plotted in Fig. 9, along with the experimentally measured data points from OR S. Although the data points are scattered, there is a systematic trend that shows a clear correlation between  and . This trend is nearly linear but presents an apparent curvature, as predicted by the three non-linear models. The error bars in  represent the experimental uncertainty of positioning the sample in the microscope. The error bars in  are much smaller because they represent relative orientations of Ni and Ag for the same position of the sample in the microscope. From this graph, we can conclude that the data support the models based on edge-to-edge matching, whether it is in the 1D sense, as in the topological approach, or in the 2D eigenvector approach. However, the scatter in the data is too large to favor one model over the others.
From the superposition of the light and dark green lines in Fig. 9 it is apparent that the 1D edge-to-  Note that the model discussed by Pond et al. [35] describes an invariant line strain, which is achieved by allowing the ratio of step heights to vary. For a constant step height ratio (1.16 for Ag(111) on Ni(111)), Eq. B.8 in [35] describes the strain compensation normal to the interface (edge-to-edge matching), even if it does not compensate for the strain parallel to the interface. It can be shown that without the invariant line constraint, the topological model predicts a relationship between  and  that is essentially identical to that of the 2D model. Figure 9. Plot of  versus  comparing four models with the experimental data. The linear model is shown in blue (Eq. 11), the 2D model is shown in red (Eq. 14), the 1D edge-to-edge matching model is shown as a dark green dashed line (Eqs. 2 and 12) and the topological model is shown in light green (Eq. 13). The discontinuity around 25° is due to the change of reference frames -from Ag(111) on Ni(111) (left) to Ag(111) on Ni(100) (right). The two thin dashed vertical lines indicate the positions of the invariant lines. The experimental data points were obtained by orientation imaging, using Euler angles to determine Ni(hkl) and Ag(HKL), finding  from the angle between Ni(hkl) and the nearest Ni(111), finding  from the angle between Ag(HKL) and the nearest Ag(111), and  from =. Error bars are indicated at top left.
The largest difference between the various models occurs in the low-angle regime, for <10°. This is where the 2D model predicts = 0.7°, the 1D model predicts = 1°, and the linear model predicts = 3°.
Even within experimental error limits, these differences should be measurable. It will therefore be instructive to assess interfaces in this regime by high-resolution electron microscopy to ascertain their precise structure and ORs. A detailed examination of interfaces in this neighborhood is expected to reveal the distinction between the structures predicted by the different models.
The largest discrepancy between the data and the models appears in the middle of the range for  between 30 and 40°. In this range, the data points lie consistently above all the predictions. Experimental observations of interfaces in this range could shed light on a mechanism of accommodation that has not been considered in any of the models. This could include, for instance, periodic stacking faults, which in Ag bicrystals have been shown to lead to dissociated grain boundaries stabilized by the 9R phase [37].

Comments on heteroepitaxy
The conventional view on the heteroepitaxy of FCC films is that they either display {111} planes parallel to the substrate, or if the substrate is also FCC they tend to adopt OR C. Our studies of Ag ORs on more than two hundred Ni-substrate orientations [11] have shown a much more complex heteroepitaxial system, even in this presumably simple FCC-on-FCC system. While it has been known for some time that Ag equilibrated on Ni(100) adopts a (111) interfacial orientation, i.e. the so-called oct-cube OR (OR O) [38], it has been shown more recently that this OR prevails in other FCC-on-FCC systems where the deposit possesses a lattice constant that is larger than that of the substrate by more than about 10% [34]. What our previous work [11,12,20] has uncovered is that there also exists an infinite set of "special" ORs (OR S) that develop as a transition from OR T, which prevails on Ni substrates oriented along the (111) to (210) line of the SST, to OR O at Ni(100).
The present paper has focused on the development of a model for relating the orientations of Ag to those of the Ni substrates in the regime of the "special" ORs. However, a classical axis-angle analysis of OR S, starting from a cube-on-cube reference and rotating about a common axis, as is usually performed for grain boundaries (e.g., [39]), does not lend itself to a simple interpretation. Except in the region of Ni(hkl) close to (100), such analyses yield a twin OR, within some apparent inaccuracy which can easily be attributed to experimental error. Only by going outside the description of ORs as disorientations and by using the twin correspondence as a reference does the simplicity of OR S as a systematic rotation around a common <110> direction reveals itself. In this model we use linear interpolation to relate plane normals Ag(HKL) to Ni(hkl) and to obtain the corresponding OR via the rotation angle . The predictions of this linear model agree to within ~2° with the experimental observations.
The second approach builds on the TLK model typically used in surface science. Ag films grow by atomic attachment at surface steps on terraces of the Ni substrate. The key role of <110> step alignment in this system was emphasized in previous work [11,12]. Because terrace planes in Ag and Ni have different lattice spacings, the mismatch at each step can cause a small macroscopic rotation of the two lattices. This rotation increases as the step density increases with increasing inclination of the surface to the terrace planes.
The same geometry may also be viewed as edge-to-edge matching of the terrace planes and can be described by either a moiré pattern in direct space or a ∆g vector in reciprocal space. However, this 1D model cannot explain why edge-to-edge matching does not lead to rotations in the OR C domain of the SST.
The third approach employs a 2D model generally used in phase transformations. It can be considered an extension of the edge-to-edge matching. Referred to the twin correspondence, the 2D transformation matrix can be written explicitly. OR T and OR O are both described by the same lattice correspondence and may be considered end-members of a continuous range of orientations (OR S). The range of possible orientations is given by the disappearance of eigenvectors at a critical rotation of 20° from OR T. The same criterion prohibits any rotation at all for the cube correspondence. This explains the difference between the two domains in the SST, characterized by OR S and OR C.
It is interesting to compare the interpretation of our results with some recent detailed X-ray diffraction measurements by Bellec et al. [15] of the OR adopted by Ag on a Ni(11 9 9) substrate. They find that Ag adopts a (9 9 7) orientation on the substrate, which corresponds to our angles  and  of 5.6° and 6.5°, respectively. For a Ni(11 9 9) substrate orientation, and with our Eq. 2, we compute an angle  of 7.2°, and 22 with our Eq. 6 we obtain  = 6.5°. Thus, the agreement with those results is quite good. Bellec et al. also report that the FCC stacking of the Ag layer is reversed compared to that of the Ni substrate, as would be expected for OR T.
It is also interesting to note that in the case of the heteroepitaxy of oxide films on oxide substrates, it has been found, for SnO 2 films on columbite, that there exist two distinct domains of columbite substrate orientation in the columbite SST, which accommodate two different polymorphs of SnO 2 [18]. This observation is similar to ours, even though we observe no change in crystallographic structure here between Ag in OR S and Ag in OR C. In the case of the oxides, it could be that a better interfacial accommodation of the two polymorphs determines their stabilization.
Finally, with the advent of EBSD, and its capacity for simplifying the ease of acquiring data over the complete ranges of substrate orientation, one may speculate that many more examples of such unexpected orientation relationships remain to be discovered.

SUMMARY
This work builds on previous combinatorial observations of Ag films on Ni substrates of many different orientations, which identified two distinct ORs: the traditional cube-on-cube OR is seen on roughly half of all surface orientations, and a special OR on the other half. Remarkably, the special OR varies continuously and systematically, ranging from a twin relationship to the so-called oct-cube relationship.
The systematic nature of this variation becomes apparent only by going outside the traditional framework of disorientations and the standard stereographic triangle. Using the twin correspondence as a reference frame, the variable orientation can be described as a gradual rotation from =0 to 15.8° around a common <110> direction, which lies close to, but not always in, the interface.
We present explicit expressions for the lattice rotation and corresponding interface plane normals for the entire orientation space covered by the special OR. This expression is based on an extension of a linear interpolation approach previously used for surface orientations along a line in orientation space. In the framework of the terrace-ledge-kink model of surface science the role of steps for atomic attachment helps understand the alignment of a set of <110> directions. This mechanism also implies a macroscopic lattice rotation due to the mismatch in the spacing of terrace planes between Ag and Ni. The rotation due to this mismatch leads to edge-to-edge matching of terrace planes across a vicinal interface. An extension of this 1D approach to 2D employs the transformation matrix typically used in phase transformations. This approach also explains why the OR varies systematically only for the twin correspondence but remains locked for the cubeon-cube correspondence.

Appendix B. Relationship between moiré patterns, edge-to-edge plane matching and ∆g vectors
The superposition of two sets of lattice planes of spacing d 1 and d 2 , rotated by an angle  gives rise to moiré fringes, as shown in Fig. B1a. The corresponding diffraction vectors g 1 , g 2 and ∆g are shown in Fig. B1b. The moiré fringes are normal to ∆g and their spacing is inversely proportional to |∆g|. In Fig. B1c, an interface has been placed between the two lattices in a plane parallel to the moiré fringes, i.e. normal to ∆g. It is apparent that in such an interface the corresponding planes meet edge-to-edge. This simple geometrical tool greatly simplifies the analysis of small rotations or changes in lattice parameter. In particular, it allows representation of the lattice rotation necessary for vicinal surfaces, where a small difference in lattice spacing of the terrace planes must be accommodated at steps. As the substrate vicinal surface rotates by an angle , the two lattices must rotate by a relative angle , if the terrace planes are to meet edge-to-edge in the interface (see Eq. 6).
The reciprocal space analysis shown in Fig. B1 for the 1D case of a single set of corresponding planes can be extended to the 2D case of a transformation between two lattices. This has been discussed in some detail in the context of optimal interfaces between two dissimilar lattices [5][6][7][8][9][22][23][24][25]33,40]. Figure B1. Representation in direct and reciprocal space of the relationship between lattice rotation, interface plane and edge-to-edge plane matching; a) moiré pattern generated by superposition of two lattices with spacings d 1 and d 2 rotated through an angle ; b) reciprocal space representation of the direct-space geometry in a), with g-vectors perpendicular to the corresponding planes; c) the two lattices are drawn meeting in an interface parallel to the moiré fringes to show that for this interface, the two planes meet edgeto-edge. Note that the difference vector ∆g = g 2 -g 1 lies perpendicular to the interface plane, at an angle  to g 1 [5][6][7][8][9][22][23][24][25]33,40].