Nanoscale waviness of low-angle grain boundaries

Abstract

Low-angle grain boundaries (LAGBs) are ubiquitous in natural and man-made materials and profoundly affect many of their mechanical, chemical, and electrical properties. The properties of LAGBs are understood in terms of their constituent dislocations that accommodate the small misorientations between grains. Discrete dislocations result in a heterogeneous local structure along the boundary. In this article, we report the lattice rotation across a LAGB in olivine (Mg_1.8Fe_0.2SiO₄) measured at the nanometer scale by using quantitative high-resolution transmission electron microscopy. The analysis reveals a grain boundary that is corrugated. Elastic calculations show that this waviness is independent of the host material and thus a general feature of LAGBs. Based on our observations and analysis, we provide equations for the boundary position, local curvature, and the lattice rotation field for any LAGB. These results provide the basis for a reexamination of grain-boundary properties in materials such as high-temperature superconductors, nanocrystalline materials, and naturally deformed minerals.

The properties of materials are influenced by the presence of grain boundaries. Low-angle grain boundaries (LAGBs) consist of arrays of discrete dislocations that separate two crystals of slightly different orientation (1). The dislocation model for LAGBs was first proposed by Taylor (2) 70 years ago and further developed by Read and Shockley (3) to predict dislocation density from macroscopic boundary geometry. Read and Shockley used their model to calculate boundary energy based on the summed elastic energy of the individual dislocations. The dislocation model remains the basis for understanding LAGB structures and interpreting their properties.

In this article, we reexamine the structure of LAGBs by using a combination of high-resolution transmission electron microscopy (HRTEM) and geometric phase analysis, a recently developed technique that is sensitive to small displacements of lattice fringes in HRTEM images (4). Previous analyses of LAGBs by HRTEM relied on visual inspection of images (5). Phase analysis has been used to accurately measure displacements around a [100] dislocation in olivine to <0.09 Å (6) and a [110] dislocation in Si to 0.03 Å (7). The technique also has been used to examine the distribution of dislocations along heterophase interfaces and LAGBs in semiconducting materials (8, 9). Here we employ the phase technique to analyze LAGBs in naturally deformed olivine, a widespread rock-forming mineral that is an important constituent of Earth's upper mantle and crust.

Materials and Methods

The olivine sample (Mg_1.8Fe_0.2SiO₄; space group Pbnm) is from a mantle xenolith collected in San Carlos, AZ. Grain boundaries in ion-milled samples were imaged with a JEOL 4000EX transmission electron microscope. We observed arrays parallel to (100) consisting primarily of [100] dislocations and mixed arrays nearly parallel to () consisting of [100], , and dislocations. The dislocations with b = and commonly occur dissociated into partial dislocations with b_p = ½<101>. In such cases the dissociation widths are small (<3 nm). The dislocations probably result from a combination of [100] and dislocations in the grain boundary, as was suggested by Christie and Ardell (10). Such grain boundaries are common in natural and experimentally deformed olivine samples (10–12). We present a detailed analysis of a mixed-type sessile boundary (Fig. 1a).

Fig. 1.

HRTEM image of a LAGB and corresponding rotation map. (a) The [010] zone-axis HRTEM image shows faint strain contrast around dislocations that are oriented parallel to the viewing direction. The dislocations comprise a LAGB nearly parallel to () that accommodates a 1.5° rotation between the two adjacent crystals. ⊤ indicates the orientation of the Burgers vector. The brace indicates a pair of partial dislocations. (Inset) The digital diffractogram of the image indicates the average orientation of the sample. (b) The magnified view of the region marked “b” in a shows a [100] dislocation (arrowhead is pointing in the direction parallel to the Burgers vector). (c) The magnified view of the region marked “c” in a shows closely spaced [100] and dislocations (vertical arrowhead). The dislocation is weakly dissociated (horizontal arrowheads). Both b and c show 12 × 12 nm fields of view. (d) The rotation map gives the local orientation of the lattice across the LAGB. The field of view is identical to the HRTEM image. Green corresponds to a rotation of –0.75°, and red corresponds to +0.75° relative to the median lattice. The waviness of the boundary is evident from the rotation contours. Bright areas are the rotation maxima and give the positions of dislocation cores (maximum rotation = 12°; minimum rotation =–12°). Owing to the sensitivity of the phase technique, even dislocations that do not exhibit strain contrast and are invisible in a become visible in the rotation map (compare upper left corners of d and a). Contours are every ±0.3° on either side of the 0° contour to ±0.6°; the color bar indicates the full range of ±12°.

The fundamental physical parameters determining the structure of a LAGB, according to the formulation of Read and Shockley (3), are the relative orientations of the two adjacent crystals (θ) and the orientation of the boundary itself (ϕ). Analysis in terms of the Read–Shockley parameters measured from the HRTEM image (Fig. 1a; θ = 1.5° and ϕ = 46.5°) gives the average separation distances between identical dislocations along the grain boundary (the predicted separation distance between successive [100] dislocations is D_a = 25 nm, and between successive dislocations is D_c = 33 nm). The dislocation density constitutes the Read–Shockley description of the boundary at the mesoscale. By mapping the local rotation, we seek to define the boundary plane on a near-atomic scale and to study exactly how the transition from one grain to the other occurs.

Results

Phase maps were calculated from the (101) and () lattice fringes, and the displacement field was determined to better than 0.1 Å accuracy and nanometer spatial resolution across the entire field of view. Careful calibration of the microscope's optical distortions was necessary to achieve this accuracy.¶ The rigid-body rotation, ω(r), was obtained by differentiation of the displacement fields, u_x and u_y:

[1]

and provided the orientation of the lattice with respect to position r in the xy plane, i.e., the viewing plane.

Each point in the resulting rotation map (Fig. 1d) defines the local orientation of the lattice with respect to the median lattice (1), which is the reference for our measurements of displacement and rotation. The orientations of the two crystals then are given by equal and opposite rotations of the median lattice (±θ/2). The map reveals several features that are not readily visible in the raw HRTEM image: (i) rotation maxima are localized at the dislocation cores, (ii) dislocations that do not exhibit strain contrast in the HRTEM image appear in the map, (iii) the grain boundary appears wavy, and (iv) the distance over which the rotation takes place is heterogeneous along the grain boundary.

The contours in Fig. 1d reveal a complex corrugation of the boundary as it snakes from dislocation to dislocation. We propose that the exact location of the boundary plane be defined as the 0° contour, where the rotation with respect to the median lattice changes from positive to negative.

An enlargement of the central region of the boundary (Fig. 2a), with the 0° contour in bold, shows variable curvature between dislocations. Based on the positions of the rotation maxima, the separation distances D_a = 26 ± 2nmand D_c = 36 ± 3 nm were determined. In cases of weak dissociation, the average position was used to establish the separation distance. Because D_a and D_c differ, a range of separation distances between adjacent dislocations (D_adj) occurs. Different dislocations also can occur near to or overlapping each other, which may have resulted in restructuring (e.g., the combination of [100] and dislocations into a dislocation). We also measured the radius of curvature, R, of the grain boundary between adjacent dislocations by fitting circles to the 0° contour. R increases linearly with D_adj (linear regression gives R = 0.68 ± 0.03 D_adj; Fig. 3). Where D_adj is large (>20 nm), there is an apparent departure from linearity in the plot. However, the significance of this apparent departure is difficult to assess because measurements of R become increasingly uncertain as the boundary flattens.

Fig. 2.

Enlargement of central region in rotation map (Fig. 1d) and corresponding theoretical rotations. (a) The waviness of the boundary is evident from the 0° contour (bold line), which is the trace of the boundary. Rotation maxima (bright areas) and minima (dark areas) are localized on either side of the 0° contour at the positions of the dislocation cores. (b) The rotation map determined numerically by using anisotropic elastic theory shows a remarkable similarity to the measured rotations. Both the waviness of the boundary and the medium-range rotations (shapes of the contours) are accurately modeled by theory. Contours and color scale are as in Fig. 1d.

Fig. 3.

Graph showing the correlation between radius of curvature (R) of the boundary and separation distance between adjacent dislocations (D_adj) measured from Fig. 1b. Radii corresponding to D_adj < 20 nm fall on a line given by R = 0.68 ± 0.03 D_adj (r² = 0.979). Radii for larger D_adj plot off of this line.

We simulated the displacement field of the boundary by using anisotropic elastic theory to test whether this grain-boundary morphology is compatible with elastic theory. The simulations were generated by summing the displacement fields of the individual dislocations to give the total displacement field of the boundary. Each individual displacement field was calculated by using the bulk elastic constants reported for olivine (13), and the Burgers vectors and positions of the dislocations were determined by analyzing the experimental phase images following the method of Hÿtch and colleagues (7). Dissociated dislocations were treated as individual dislocations with partial Burgers-vector components. For example, a dissociated dislocation is entered as a component at position (x, y₁) and a second component at position (x₂, y₂). This treatment of dissociation does not account for the presence of a stacking fault between the partials. However, because the dissociation distances are small, the influence of stacking faults on the overall displacement field of the boundary is expected to be small and localized between the dislocations. The simulations, like the experimental measurements, account for displacements in the plane normal to the dislocation line.

The rotation map (Fig. 2b) was determined numerically, in the same way as the experimental rotations (Figs. 1d and 2a), from the theoretical displacements by using Eq. 1. Elastic theory successfully predicts both the long-range rotation and nanoscale structural heterogeneity, as is evident from the similarity between the experimental and calculated results (compare Fig. 2 a and b). The accuracy with which elastic theory predicts experimentally determined rotations is remarkable and raises the possibility that waviness is inherent to all olivine LAGBs and, beyond that, whether it applies to all materials.

To address these possibilities, we derived the rigid-body rotation from isotropic elastic theory. For this derivation, isotropic instead of anisotropic equations were used to test the hypothesis that waviness occurs independent of the elastic anisotropy of olivine. Starting with theoretical displacements around an isolated edge dislocation (14), the rotation is obtained according to Eq. 1. The result in cylindrical coordinates is

[2]

The rotation across an infinite array of dislocations is the sum of their in-plane rotation contributions. For a set of identical edge dislocations along the y axis with spacing D, the rotation is

[3]

Here b_x is the magnitude of b in the x direction, which is perpendicular to the boundary (b_y = 0). The derivation of this result (and the following one) is given in the Supporting Appendix, which is published as supporting information on the PNAS web site.

For an alternating array of dislocations (b_x, b_y) and (b_x, –b_y) with uniform spacing D/2, the rotation is given by

[4]

Given our definition of the boundary as the 0° contour, Eq. 3 predicts a flat boundary plane (Fig. 4a), whereas a wavy boundary is predicted by Eq. 4. The waviness of the boundary is given by the position of the 0° contour in the xy plane (x for ω = 0°):

[5]

which approximates a sine wave. The waviness of a mixed array with b_x = b_y is confirmed by the rotation map (Fig. 4b). The waviness therefore results from the alternating dislocations rather than their variable spacings. Furthermore, Eq. 2 and the subsequent boundary rotations are independent of the Poisson constant, which indicates the corrugation of the boundary surface is general for mixed arrays of dislocations, i.e., it is material independent.

Fig. 4.

Rotations across boundaries derived from elastic theory. (a) An array of identical dislocations with uniform spacing D gives a flat boundary plane (see Eq. 3). (b) An array of alternating dislocations (spacing D/2) gives a wavy boundary surface (see Eq. 4). Both arrays accommodate a 1.5° global rotation. Insets show the positions and orientations of the dislocations for each array. ⊤ indicates the orientation of the Burgers vector. These simulated boundaries show that waviness is inherent to mixed arrays of dislocations and is material-independent. Contours and color scale are as in Fig. 1d.

Finally, we determined the average radius of curvature of the boundary assuming that adjacent dislocation cores and maximum amplitude (D ln 2/2π) fall on a circle, in which case:

[6]

The experimental results are for a dislocation array with D_a ≠ D_c. If D = 2D_adj, Eq. 6 gives R = 0.677 D_adj, which compares well with R = 0.68 ± 0.03 D_adj determined from the measured values, even though Eq. 6 is derived for a regularly spaced boundary. Together these results suggest that the linear relationship between curvature and dislocation spacing is general.

Discussion

Many physical properties of polycrystalline materials are controlled by the properties of the boundary network separating constituent grains (1). The mobility of LAGBs is critical for recrystallization and grain growth by subgrain rotation, grain-boundary migration, or both. Mesoscopic modeling is used to simulate crystal plasticity (15), dislocation patterning (16), solidification, recrystallization, and grain growth (17–19). These models directly or indirectly include local crystal orientation and rotation. Therefore, including both waviness and curvature in the description of LAGBs should have many implications for the study of both natural and synthetic materials.

LAGBs are common in deformed olivine. Indeed, the alignment of constituent grains along preferred directions in Earth's mantle, which is facilitated by the presence of LAGBs, is cited as a major source of measured seismic anisotropies (20). The impact of LAGBs on internal friction in solids, including polycrystalline olivine, is of primary interest in experimental seismic-wave attenuation research (21).

In a wider context, engineering of grain boundaries is a way to tune bulk properties in many technologically important materials. For example, LAGBs affect critical current densities in high-T_c superconductors (22), and current transport through LAGBs is shown to strongly decrease with increasing lattice rotation (23).

These unique materials properties result from phenomena operating on length scales from a few microns down to a few tenths of a nanometer, i.e., the length scales of the grain boundaries and their constituent dislocations. We suggest that the waviness of LAGBs could provide the basis for a reexamination of material properties with respect to structure at the nanometer scale.