Grain boundaries influence a wide array of properties in polycrystalline materials,1 including diffusivity,2,3,4 conductivity,5,6,7 intergranular cracking,8 corrosion resistance,9, 10 embrittlement11, 12 etc. However, there remain fundamental challenges in our ability to compute the structure–property relationships of individual interfaces and to analyze the influence of a collection of grain boundaries (GBs) on the macroscopic properties of materials.13, 14 Even from a modeling perspective, the ability to develop reliable GB structure–property relationships has been identified as one of the biggest obstacles in developing robust bottom-up models for predicting polycrystalline material behavior.15 In this article, in an attempt to provide a basis for GB structure–property relationships, we present an automated algorithm to compute the three-dimensional (3D) polyhedral unit model for describing the atomistic structure of GBs in fcc metallic systems.

Defects in crystalline materials may be visualized as local disruptions in the symmetric arrangement of atoms. For example, a point defect corresponds to a missing or an extra atom in the lattice and a line defect is the termination of an extra plane of atoms, which locally disrupts the lattice structure. Investigating the distortions caused by these defects and the analysis of atomic packing along these defects has led to a mechanistic understanding of their influence on material properties.16, 17 In a similar vein, it is anticipated that a quantitative understanding of the atomistic structure of GBs will offer fundamental insights into their properties, and will help reduce the complexity of the five-dimensional crystallographic phase space.18 For example, it has been proposed that an analysis of GB structures at the atomistic length scale will help classify interfaces as singular, vicinal, and general.19 Such a classification is expected to lead to a general understanding of the trends by which GB properties change, as the crystallographic parameters of the GBs are varied.20

From a geometrical perspective, the earliest models for interface structures include the “amorphous cement” model21 and the sharp interface model with GB atoms on the coincident sites.22 However, GBs generally exhibit a wide range of structures and the extent of “ordering” depends on the density of coincidence lattice points and the symmetry of the boundary-plane (BPl) orientation. Therefore, the atomic structure of GBs has primarily been visualized as clusters of atoms that form certain geometrical motifs. This model, referred to as the structural unit (SU) model, was first proposed by Bishop and Chalmers23 and has been extended to a variety of tilt configurations by Sutton and Vitek.24, 25 However, as described in ref. 26, the SU model is applicable only for pure tilts along low-index rotation axes.

A more general model for the atomic structure description is the 3D polyhedral unit model, proposed by Ashby et al.,27 where GB atoms are represented by an array of closed packed polyhedral motifs. This model was inspired by Bernal’s seminal effort in classifying the atomic structure of liquids as an arrangement of polyhedral units.28 These polyhedra, which correspond to the canonical Bernal holes, were identified using models of dense random packing of hard spheres.29 The set of canonical holes and the corresponding polyhedra are shown in Fig. S1 of the Supplementary Information. Such a classification of atomic structure through the analysis of voids has also been applied to the structural analysis of mono-atomic metallic glasses.30

Ashby et al.27 have utilized this concept to describe the structure of simulated GBs, which primarily included the [100] [110], and [111] symmetric tilt GBs.31,32,33 Through the analysis of these GB structures, Ashby et al. proposed a set of polyhedra, termed as deltahedra (densely packed polyhedral units with equilateral triangles for faces; shown in Fig. S1), that may be commonly observed in the GBs of fcc and bcc crystal structures. The primary utility of the polyhedral unit model is believed to be in predicting segregation sites for small interstitial solute atoms.34 For example, Zhou et al.35 have recently utilized the polyhedral unit model to predict the segregation sites and compute the energetics of hydrogen segregation in Σ5(310) Nickel GB. The polyhedral unit model is also anticipated to provide a basis for analyzing GB structures of complex crystallography (i.e., high-Σ misorientation, and with both twist and tilt character). However, a major obstacle in using this model is the difficulty associated with automatically identifying polyhedra in a quasi-3D structure of atoms in GBs.

In this article, we describe an automated algorithm to identify the voids and the corresponding polyhedral structure of GBs in fcc metallic systems. In section Polyhedral unit model, we describe the method used to cluster the vertices in a Voronoi network (i.e., the Voronoi-vertices) and describe how the clusters of Voronoi-vertices will result in the identification of the polyhedral unit structure of the GBs. In the Results section, we provide several examples illustrating how the proposed algorithm can successfully describe the structure of symmetric tilt GBs, asymmetric tilt GBs, a symmetric twist GB, a and two mixed character GBs. We leverage a database of hard-sphere packings, provided by Holmes-Cerfon,36 for classifying the geometries of the observed GB polyhedral units.

Polyhedral unit model

The identification of the polyhedral unit structure for a set of atoms is non-trivial as there exist infinitely many solutions that partition the space, occupied by a set of points, into polyhedral units. One of the possible solutions is the Delaunay triangulation of the 3D set of atoms, which results in a complete partitioning of the space into (irregular) tetrahedra. The approach we utilize to compute the polyhedral unit structure is to analyze the voids in the GB structure. The property that may be directly linked to the quantification of the void structure is the segregation of small interstitial solute atoms to GBs. There is also considerable evidence that the free-volume plays a crucial role in influencing a variety of properties in metallic glasses.37, 38 As GBs are, in general, more disordered than ordered, we anticipate that a polyhedral unit structure based on the analysis of the void structure will be very useful.

Identification of polyhedral units

The analysis of the cavities in dense random packing of spheres has been well established (e.g., see ref. 39 and references therein). In all of these studies, the cavities are identified by first constructing the Voronoi tessellation of the collection of atoms. The Voronoi polyhedra, VP i corresponding to each atom i, partition the 3D space of atoms into distinct domains such that every point in the polyhedron VP i is closest to atom i than any other atom. Each vertex in the Voronoi network is shared by at least four Voronoi polyhedra (whose atoms construct the Delaunay tetrahedron). The Voronoi vertex is the center of the sphere that circumscribes the Delaunay tetrahedron. Therefore, at each Voronoi vertex (VV) an interstitial atom may be placed and a corresponding radius r vv , of the largest sphere (denoted as VV-sphere in this article) that fits in the void, may be defined. The radius of the VV-sphere depends on the radius of the atoms r a in the system. If R vv is the radius of the circumsphere (or the distance between the Voronoi vertex and the atoms of the Delaunay tetrahedron), then the radius of the VV-sphere is r vv = R vv −r a .

For example, for the Bernal’s canonical holes and their corresponding polyhedra, shown in Fig. S1, the Voronoi vertices and the VV-spheres are illustrated. It is clear from Fig. S1 that there exists an overlap between the different VV-spheres that are present in the canonical Bernal holes. We use this property of the VV-spheres to combine Delaunay tetrahedra to form larger polyhedral units. As discussed earlier, each Voronoi vertex corresponds to a unique tetrahedron. Therefore, the problem of combining tetrahedra may be formulated alternatively as combining (or as the clustering of) Voronoi vertices, where each cluster results in a polyhedral unit.

The clustering algorithm combines the Voronoi vertices whose VV-spheres overlap, i.e., the vertex VVi belongs to a cluster C if there exists another vertex VVj ∈ C, such that \(\Vert {\rm V}\,{{\rm V}^i}-{\rm V}\,{{\rm V}^j}\Vert \le {r_{\rm vv}^i}+{r_{\rm vv}^j}\), where \({r_{\rm vv}^i}\) and \({r_{\rm vv}^j}\) are the radii of the VV-spheres at VVi and VVj, respectively. This algorithm will cluster all the vertices in the Voronoi network into disjoint sets. The polyhedral units are then computed by combining all the Delaunay tetrahedra of the Voronoi vertices present in each cluster.

To demonstrate the clustering technique, we start with a simple example and illustrate how the proposed algorithm will lead to the detection of the well-established octahedra and tetrahedra in a single crystal fcc structure. Figure 1a(i) shows a super-cell of an fcc crystal extracted from an infinite lattice. The first step in the above algorithm is to find the Voronoi vertices and their corresponding radii. If the lattice parameter is denoted by a, the atom radius is given by \({r}_{\rm a}=a/(2 \sqrt2)\). The Voronoi vertices are obtained by constructing the Voronoi tessellation of the fcc lattice and are represented as red stars in Fig. 1a(ii). For each vertex, the radius of the VV-sphere r vv is computed. From these computations, two types of Voronoi vertices are identified. The first type is the vertex with a radius \({r}_{\rm vv}={r}_{\rm a}(\sqrt 2-1)\sim 0.414{r}_{\rm a}\), which corresponds to the octahedral interstitial site in an fcc crystal (e.g., the vertex located at (a/2, a/2, a/2)). The second type is the vertex with \({r}_{\rm vv}={r}_{\rm a}(\sqrt{3/2}-1)\sim 0.225{r}_{\rm a}\), which corresponds to the tetrahedral interstitial site (e.g., the vertex located at (a/4,a/4,a/4)). These two types of Voronoi-spheres are shown in Fig. 1a(iii). The distance between these two vertices is \({r}_{\rm a}\sqrt{3/2}\sim 1.2247{r}_{\rm a}\), which is greater than the sum of the radii of the Voronoi vertices in the octahedral and the tetrahedral site, i.e., \({r}_{\rm a}\sqrt{3/2} >{r}_{\rm a}(\sqrt{3/2}+\sqrt 2-2)\).

Fig. 1 The Voronoi vertex clustering algorithm for identifying the polyhedral unit structure of a an fcc single-crystal region and b a vacancy point defect is illustrated. In a(i), the unit cell of an fcc lattice is provided and in a(ii), the Voronoi polyhedron of one of the atoms is shown. Also highlighted are the Voronoi vertices, obtained by constructing the Voronoi polyhedra of all the atoms in the system. In a(iii), the VV-spheres that fall within the unit cell are shown. In a(iv), the octahedron and tetrahedron corresponding to the two distinct types of VV-spheres are depicted. In the case of the vacancy, in b(i), the fcc lattice with a vacancy is shown and the atoms surrounding the vacancy are highlighted. In b(ii), the VV-sphere that is located within the vacancy is illustrated and the atoms that have contributed to the creation of this sphere are highlighted in red. These atoms are the same as those surrounding the vacancy in b(i). The polyhedron that is obtained by triangulating the atoms in b(iii) is the cuboctahedron, which is identified as the polyhedral unit model of a vacancy in the fcc lattice Full size image

Therefore, none of the VV-spheres overlap with each other and the vertices will not cluster together. Put differently, each cluster contains only one vertex and two types of polyhedral units (the octahedron and the tetrahedron as shown in Fig. 1a(iv)) are obtained by identifying the atoms that correspond to each of the Voronoi vertices in the lattice. In this example, the Voronoi-vertices at the octahedral and the tetrahedral sites are created by the atoms that make the octahedron and the tetrahedron, respectively.

Additionally, we test the applicability of the proposed clustering algorithm for identifying the polyhedral unit structure of an fcc crystal with a vacancy. For this purpose, we have constructed a simulation box with five aluminum fcc super-cells along each dimension (125 unit-cells in total) and removed a single atom from the center of the simulation box. The vacancy point defect is then relaxed using the conjugate gradient minimization in LAMMPS40 and is shown in Fig. 1b(i). Mishin’s EAM potential41 is used for these computations. The Voronoi vertices within the single crystal region do not overlap and, hence, result in the usual tetrahedra and octahedra. Within the vacancy, a single Voronoi-vertex is observed and the VV-sphere is shown in Fig. 1b(ii). The atoms that correspond to this VV-sphere are those that surround the vacancy and the polyhedron that is obtained by considering all of these atoms together is the cuboctahedron (shown in Fig. 1b(iii)). Therefore, this algorithm automatically identifies the polyhedral unit structure that corresponds to a vacancy in an fcc lattice.

Polyhedral Unit Model for the Symmetric Tilt Σ5 \((0\,\bar{2}\,1)\) GB

To illustrate the applicability of the VV clustering algorithm for determining the polyhedral unit structure of a GB, we start with the analysis of a simple [100] symmetric tilt Σ5\((0\,\bar{2}\,1)\) GB (in section S2, the complete crystallographic information, i.e., the Σ-number, and the Miller indices of the boundary-plane in the two crystals (hkl) 1 and (hkl) 2 , are provided for all the GBs analyzed in this article). This interface is chosen as it illustrates all the steps required for classifying the polyhedral unit model. The atomistic structure of this GB is shown in Fig. 2a. The VV-spheres centered at the Voronoi vertices of the atomistic structure of the GB are shown in Fig. 2b(i, ii). Using the clustering algorithm, the clusters of the Voronoi vertices, whose spheres overlap, are identified. A representative example of the polyhedral units, obtained by combining the Delaunay tetrahedra of all the Voronoi vertices in each cluster, is shown in Fig. 2c(i). For Σ5\((0\,\bar{2}\,1)\) GB, the atomistic structure consists of a stacking of polyhedra, each containing thirteen atoms, as shown in Fig. 2d(i, ii). Therefore, the 13-atom polyhedral unit, shown explicitly in Fig. 2c(i), completely describes the 3D structure of the Σ5\((0\,\bar{2}\,1)\) GB.

Fig. 2 An illustration of the VV-clustering algorithm for identifying the polyhedral unit structure of the Σ5\((0\,\bar{2}\,1)\) GB. a The minimum energy atomistic structure (at 0 K) of the GB, generated using OVITO,59 is shown along the tilt axis \([\bar{1}\,0\,0]\), where the atoms are colored according to their centro-symmetry parameter. b(i, ii) Atomistic structure of the GB along with the Voronoi vertices and VV-spheres are shown, as viewed along the tilt axis \([\bar{1}\,0\,0]\) and the boundary-plane normal \([0\,\bar{2}\,1]\), respectively. All the Voronoi vertices of overlapping VV-spheres are clustered together and, in c(i) The polyhedral unit obtained by the clustering algorithm is shown. The polyhedron contains thirteen atoms. In d(i, ii), the polyhedral unit model of the Σ5\((0\,\bar{2}\,1)\) along the tilt axis \([\bar{1}\,0\,0]\) and the boundary-plane normal \([0\,\bar{2}\,1]\), respectively, are shown. Since the polyhedral unit is concave and contains more than 12 atoms, it is first split into smaller units by finding the interstitial voids of the 13-atom unit as shown in c(ii). The split units are then compared with the Cerfon-Holmes database and the iterative cluster (poly-tetrahedral unit) is split further into three tetrahedral units, as shown in c(iii). e The matching between the 11-atom and the best-match seed (octadecahedron) is shown. The registration between the highest-distorted tetrahedron and the perfect tetrahedron is also illustrated. The RMSD errors for all the resultant units in the GB are shown in the table. Finally, in f(i, ii), the structure of the GB, represented using the split units (octadecahedra and tetrahedra), is illustrated Full size image

Once the polyhedra are extracted from the GB, it is necessary to classify the units according to their geometry. As will be described in this article, the database of possible geometrical motifs cannot be classified using the simple polyhedral units proposed by Bernal29 or Ashby et al. 27 In order to capture the complexity of the various polyhedral units that may be observed in general GBs, a new database of geometrical motifs, derived from the rigid hard-sphere packings and enumerated by Holmes-Cerfon,36 is utilized. The rigid hard-sphere packings can be classified as either iterative or seed clusters. Seeds are n-atom packings that cannot be constructed out of rigid packings of less than n atoms, i.e., they contain within them an inherently new structure. An exhaustive list of seeds with n = 4, 6, 7, 8, 9, 10, 11 atoms are shown in Figs. S1 and S2 of the Supplementary Information. On the other hand, iterative packings are those that can be obtained by joining smaller seed clusters.

The process of analyzing the observed GB units involves finding the model-unit (from the Holmes-Cerfon database) that best matches the GB polyhedron. The algorithm required for comparing polyhedral units is described in section S3.1. If the best-match unit is iterative, then the unit is split into seed clusters resulting in GB polyhedral units that correspond only to seeds (e.g., refer to Fig. S5). Hence, the seed units constitute the database of canonical models for the observed GB polyhedra. While this process is, in principle, sufficient for classifying all the observed GB polyhedral units, it is not entirely efficient. This is due to the fact that the total number of rigid hard-sphere packings increases as a factorial function (∼2.5(n−5)!) of the number of atoms n (Table S2). Therefore, an additional processing step is introduced to reduce the size of the observed GB polyhedral units with more than 12 atoms.

Very briefly, if the observed polyhedron contains more than 12 atoms and it is convex, then the unit is minimized in LAMMPS and the minimized structure is deemed as the canonical model of the polyhedral unit. However, if the unit is concave, the unit is split into smaller units that define the interstitial sites42 in the polyhedral unit. The interstitial voids correspond to the largest non-overlapping VV-spheres. In order to obtain all the interstitial voids in the unit, the voids are first sorted according to their sizes. Then the largest VV-sphere is picked and clustered with the VV-spheres, remaining in the list, that it overlapped. The cluster with the largest VV-sphere and its overlapping VV-spheres correspond to one interstitial site. The list of VV-spheres is updated by removing the cluster of VV-spheres of the interstitial site obtained in the previous step and the process is repeated until all the VV-spheres are exhausted in the list and all the interstitial sites enumerated. This post-processing step is described in complete detail, with examples, in section S3.2. To summarize, the post-processing step takes advantage of the interstitial sites42 present in the larger concave units and splits them into polyhedra with less than 12 atoms.

Using these additional steps, the 13-atom unit observed in the Σ5\((0\,\bar{2}\,1)\) GB is first split into smaller units containing 11 and 6 atoms each, as illustrated in Fig. 2c(ii). After splitting, the units are compared with the Holmes-Cerfon’s database and the Root Mean-Square Distortion (RMSD) error (as described in section S3.1) is used to quantify the distortion between the observed GB unit and the perfect polyhedron in the database. The 6-atom unit is identified as an iterative poly-tetrahedral cluster that is split further into three tetrahedral units (as shown in Fig. 2c(iii)). The 11-atom unit is found to match with one of the seed clusters in the Holmes-Cerfon database (also known as the Octadecahedron) with an RMSD error of ∼0.41. The comparison between the GB polyhedral units and the seed units is shown in Fig. 2e along with the RMSD errors. In Fig. 2f(i, ii), the polyhedral unit structure of the Σ5\((0\,\bar{2}\,1)\) GB is illustrated using split units, i.e., the octadecahedra and tetrahedra.