Topological and atomic scale characterization of grain boundary networks in polycrystalline and nanocrystalline materials

Progress in Materials Science 56 (2011) 864–899

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Progress in Materials Science journal homepage: www.elsevier.com/locate/pmatsci

Topological and atomic scale characterization of grain boundary networks in polycrystalline and nanocrystalline materials Mo Li ⇑, Tao Xu School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, GA 30332, United States

a r t i c l e

i n f o

Article history: Accepted 7 January 2011 Available online 1 February 2011

a b s t r a c t Microstructure in polycrystalline materials, either coarse-grained or nano-crystalline, is characterized by the topological structure of grain boundary networks which are composed of an array of complex geometric entities with different dimensions such as grain volume, grain boundary plane, triple junction line, and vertex point. The ensemble of these entities gives rise to statistical properties represented by their distribution functions, means, variances, and correlation functions. Moreover, contrast to Gibbs’ description, on atomic scales these entities are no longer mathematically abstract geometric objects such as simple plane, line or point; rather they possess finite thickness and volumes, as well as certain specific atomic structures and chemistry. While some of these entities can be measured from experiment, a large number of them still remain inaccessible, that includes identification of the full range of topological properties and the structure characterization on atomic scales. In this article, we present algorithms and numerical methods to characterize systematically these entities in grain boundary networks in polycrystalline samples which are either from serial sectioning of real polycrystals or from digital microstructures generated using inverse Monte Carlo methods. The rendered microstructures are represented by the topological and geometric properties such as the grain volume, grain boundary area, triple junction length, and their statistical properties. Most importantly we give the atomic coordinates and label the type of the topological entities to which each atom belongs in the polycrystalline and nano-crystalline materials. Such quantitative characterization, unavailable before, enables detailed and rigorous treatment of

⇑ Corresponding author. E-mail address: [email protected] (M. Li). 0079-6425/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.pmatsci.2011.01.011

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microstructures in a wide range of modeling applications including both atomistic simulation and continuum modeling. Ó 2011 Elsevier Ltd. All rights reserved.

Contents 1.

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Characterizing topological properties from the digitized polycrystalline samples . . . . . . . . . . 1.2. Rendering both topological entities and atomic information of the microstructure attributes Characterization of topological properties of digital microstructures. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Identifying topological entities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Computing topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Digital polycrystals from inverse Monte carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Select input target microstructure attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Construct Voronoi cells using Poisson–Voronoi process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Use inverse Monte Carlo method to optimize the microstructure . . . . . . . . . . . . . . . . . . . . . . . Atomic models of polycrystals and microstructure characterization on atomic scales . . . . . . . . . . . . . 4.1. Atomic models of polycrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Atomic scale characterization of microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Calculating grain boundary area and triple junction length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Other microstructure-related properties of the grain boundary network . . . . . . . . . . . . . . . . . 4.4.1. Atomic number fraction of atoms in the topological entities . . . . . . . . . . . . . . . . . . . . 4.4.2. Configuration energy of the topological entities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3. Factors affecting the grain boundary thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4. Atomic volume of grain boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Microstructure tailoring or optimization in design of polycrystalline materials. . . . . . . . . . . . 5.2. Flexible models for microstructure modeling using continuum and atomistic methods . . . . . 5.3. Compliment simulation tool for experimental rendering of microstructures . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction Polycrystalline materials, including nano-crystalline materials are made of an ensemble of single crystallites that are assembled through various ways, such as nucleation and growth, mechanical deformation, or inert-gas condensation [1–3]. The unique feature of polycrystalline materials is the so-called microstructure that consists of a set of topological entities with different dimensionality, such as the three-dimensional grain cell (GC), two-dimensional grain boundary (GB), one-dimensional triple junction (TJ), and zero-dimensional vertex point (VP). These attributes arise when two or more grains are brought in contact with each other. For example, a grain boundary forms when two crystallites are joined with an interface, a triple junction forms when three crystallites are joined along a line, and a vertex point forms when four or more crystallites are joined at a point. Together these topological entities form the grain boundary network (Figs. 1 and 2). The crystallites in real polycrystals contain atoms or molecules, so are the rest of the topological entities in the grain boundary networks. Therefore, besides the topological properties of the grain boundary networks, there are three additional microstructural attributes, atomic structure, crystallographic property, and statistical property, that describe the atomic structure and the crystallographic properties of the crystallites as well as the statistical account of the topological entities since there are a large number of grains in polycrystals. Collectively, these microstructure attributes contribute to the physical, thermodynamic, transport, and mechanical properties of polycrystalline materials. For a long time the grain boundary network has been described as a topological network connected with mathematically abstract geometric objects like plane, line and point [4]. In real polycrystalline

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Fig. 1. The grain boundary networks identified in a BCC b-Ti polycrystal containing 834 grains [14,22]. The corresponding microstructure is shown in Fig. 2. To avoid obstructing the 3-D view, we only plotted the triple junction lines explicitly; the grain boundaries are left as the transparent planes between the triple junction lines. To enhance visualization, we used the color map for the triple junction lines that change color from red on the top to blue at the bottom of the sample. The total number of grain boundary interfaces in the sample is 4569 and the total number of triple junctions is 7193. The detailed algorithm to characterize the grain boundary network is described in Section 2 in the text.

materials, each grain has a specific crystal structure and orientation; very often crystal defects such as dislocations, stacking faults, twins, or vacancies exist which introduce distortions to the lattice structure. On atomic scales, a grain boundary has finite thickness and certain degree of disorder; oftentimes it has specific atomic structure and chemical composition [5,6]. The thickness and structure are closely related to the crystallographic properties such as misorientation of the two adjacent grains that form a boundary. Similarly, a triple junction line is no longer an abstract line but has finite diameter; the vertex point is not a point but possess finite volume. Although in continuum description, the detailed atomic descriptions are often simplified or even neglected due either to the lack of the knowledge of these microstructure entities or to limited resolution of the length scales intrinsic to the methodology, the importance of the atomic features demands increasing attention. The atomicity present in the topological entities contributes directly to the properties of polycrystalline materials, for example, the well-known impurity-segregation-induced grain boundary embrittlement [7–11], grain boundary motion dominated deformation process in nano-crystalline materials [12], and intergranular fracture [13]. In intergranular fracture, atomicity, or local structure and chemistry change demands special treatment even within the continuum description. At the nanometer scale, the presence of large volume faction of grain boundaries introduces not only a large fraction of atoms not belonging to grain crystallites but also increasing degree of disorder especially at and near grain boundaries. The large amount of disorder and the small dimensions of the grain boundary networks make it doubtful to continue to use the continuum approximation made for the microstructure entities, that is, to ignore the atomicity. One must take

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Fig. 2. The microstructure of a BCC b-Ti polycrystal containing 834 grains obtained through serial sectioning [14,22]. The topological entities characterized of this sample is shown in Fig. 1. The sample dimensions are 448  448  402 lm3.

into consideration of the detailed atomic structure and properties of the topological entities in both atomic scale as well as continuum. However, the current status of characterizing the topological entities and rendering the atomic information of these entities is still in early stage. For coarse-grained polycrystalline materials with the mean grain size in the range of submicron to microns, serial sectioning technique using focused ion beam and other etching methods can reveal detailed 3-D topological structures by putting together the information obtained from the 2-D sections (Fig. 2). However, the available resolution of the section thickness does not allow for characterization on the scales less than a few hundred nanometers [14]. Therefore, presently the characterized topological entities such as grain boundary and triple junction as shown in Fig. 1 are more a reference for the approximate location than the exact characterization of the actual grain boundaries which typically have thickness of about a nanometer or so [15]. Atom probe can give microstructure features with higher resolution [16]. But the requirement of certain materials to withstand the high electrical field and reconstruction of 3-D microstructure from the atoms emitted from the curved surface of the sample used in atom probe put limitations on this powerful approach. Recent development of high-energy X-rays by a synchrotron source opens another window for 3-D microstructure mapping [17]. As compared with the serial sectioning that probes only static microstructure, the hard X-ray tomography allows one to probe not only the dynamic evolution of microstructures in such occasions as in-situ grain growth but also non-destructive evaluation of microstructure changes. However, the method is still limited by the spatial resolution and the software to recognize and characterize microstructure patterns faithfully. Perhaps most difficult is to characterize the topological entities in nano-crystalline materials with mean grain size between a few and hundreds of nanometers [18]. For example, TEM could handle only a limited number of grains, while a reliable answer should be drawn from the ensemble of a sufficiently large number of grains and measurements. In addition, at such small scale, characterization of

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misorientation and a full range of 3-D topological property measurement are nearly impossible. In addition, the lack of detailed account of 3-D microstructures and calibration of sample conditions from the limited number of 2-D sections can influence the outcome significantly [19]. Mössbauer spectroscopy [20] has been utilized to study the grain boundary regions in nano-crystalline materials through hyperfine field, but it is still challenging to study individual grain boundaries, not to mention triple junctions and vertices. The presence of various defects and voids in experimentally produced nanocrystalline materials also makes it difficult to study the intrinsic topological properties of microstructures as these structural imperfections and flaws often obscure the intrinsic responses from the topological entities. Computer modeling and simulation, on the other hand, are free of most of those limitation, thus able to characterize the topological entities and render atomic information of the microstructures. The challenge is however the development of new algorithms or methods that could allow for efficient and accurate processing. According to the input of the polycrystalline samples, we can divide the characterization approaches into two categories: 1.1. Characterizing topological properties from the digitized polycrystalline samples As we mentioned above, serial sectioning can provide real 3-D microstructures but with limited resolution at submicron scale. Microstructure from digitized 2-D optical image is also limited by the wavelength of the light used. For these types of inputs, we can use a series of geometric methods to characterize the microstructures, that is, to identify grains, grain boundaries, triple junctions, and vertex points; and most importantly, we can record the positions of these entities in discrete unit such as the positions of the pixels or voxels (Figs. 1 and 2). 1.2. Rendering both topological entities and atomic information of the microstructure attributes In this case, we can use two types of input microstructures. One is the computer generated polycrystals, or ‘‘artificial’’ microstructures (Fig. 3) and another is the digitized microstructures from real polycrystals (Fig. 2). For the former, specifically, we shall introduce a recently developed method that can produce close-to-reality microstructures using Voronoi tessellation and inverse Monte Carlo simulation [21]. The advantage of this approach is the flexibility to produce the digital microstructures as compared with the digitized microstructures from experiment. Using the method we can tune many microstructure attributes such as the grain size distribution, misorientation angle distribution, or atomic structures, which is difficult in the digital microstructures obtained from real polycrystals. In experimentally digitized microstructures using 3-D serial sectioning or 2-D imaging, the basic unit is the volume element called voxel, or pixel if imaging is used. A voxel is a volume element resulting from usually three intersecting layers of materials etched off, for example, in serial sectioning [14]. Usually its position (xi, yi, zi) along with the grain index (GI) are recorded. GI gives each grain a unique label which runs from 1 to N of the total number of grains in the sample. Similarly, the position and grain index of a pixel from a 2-D image are recorded. As shown later, in the atomic model of polycrystals, it is the atomic position and the grain index that are recorded and used as input for microstructure characterization. The output of the microstructure from the characterization process consists of three sets of information: (a) the spatial locations of the grain boundary networks, i.e. the positions of pixels or voxels, of the grain cells, grain boundaries, triple junction lines, and vertex points; (b) the atomic information, if atomicity is considered, of the topological entities, including the positions of the atoms, crystallographic information of the crystallites, and grain boundary misorientations; and (c) the information about the types of the topological entities that either the atoms or the voxels or pixels belong to. These three sets of information are usually represented in terms of the locations or positions of the atoms or voxels or pixels, the grain index or number, and the type of the topological entities that an atom or a voxel or pixel belongs to. In other words, we can express the output in the following array: (numb, xi, yi, zi, GI, type), where numb = i stands for the number of the ith atom or voxel or pixel, (xi, yi, zi) is its Cartesian coordinate, GI is the grain index, and type stands for the type of the topological entity, i.e. grain cell, grain boundary, triple junction and vertex point.

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Fig. 3. The microstructure of a nanocrystalline copper (nc-Cu) containing 50 grains. The grain boundary network is characterized using the central-symmetry parameter (CSP).

With the output, we can tackle many problems in polycrystalline material modeling and design with the ability to track certain specific microstructure entities and their contributions to material properties. For example, we can now calculate the structure change of a single grain boundary or a triple junction line during deformation and its contributions to mechanical properties. Similarly, we can look into diffusivity along a specific triple junction. This development opens the door for many other applications in numerical modeling, image processing, and experimental cross-validation of polycrystalline material microstructures. 2. Characterization of topological properties of digital microstructures 2.1. Identifying topological entities Although the grain boundary surface and triple junction line may not be necessarily flat and straight, crystallites or grains in polycrystalline materials in general resemble polyhedron with varying shape and size (Figs. 2–4). A grain boundary network consists of four types of entities: grain cell, grain boundary, triple junction, and vertex point. A grain boundary plane is where two crystallites meet, a triple junction line is where three crystallites meet, and a vertex point is where four or more crystallites meet. We can utilize the basic geometric properties to characterize the topological entities of the digital microstructures from either real polycrystalline samples or numerically constructed ones.

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Fig. 4. (a) The grain boundary network structure in the 50 grain nc-Cu sample shown in Fig. 3. (b) The grain boundary network structure in a BCC b-Ti polycrystal containing 834 grains obtained through serial sectioning [14,22] (Fig. 2). In both cases, the different topological entities are classified into (a) grain cell voxels/atoms (blue); (b) grain boundary voxels/atoms (green); (c) triple junction voxels/atoms (yellow); (d) vertex voxels/atoms (red).

The input microstructure from either imaging or serial sectioning or X-ray tomography in general consists of a series of pixels or voxels with coordinates (xi, yi, zi) and the grain index which gives a distinctive number to a grain that each pixel or voxel or atom is located within. With these two pieces of information, we can sort the voxels or pixels into different topological entities according to the following rules. So we can give a specific label, type, to each voxel or pixel or atom. (a) Grain cell: If the grain index of a pixel or a voxel and that of its nearest neighbors are the same, the pixel or voxel is inside of a grain cell. We then give an index type = grain cell to the pixel or voxel. (b) Grain boundary: If the grain indices of the nearest neighbors of a pixel or a voxel have two different labels, we then assign an index type = grain boundary for the pixel or voxel. (c) Triple junction: If the grain indices of the nearest neighbors of a pixel or a voxel have three different labels, we then assign an index type = triple junction for the pixel or voxel. (d) Vertex point: If the grain indices of the nearest neighbors of a pixel or a voxel have four or more different labels, we then assign an index type = vertex point for the pixel or voxel. This rule utilizing the simple geometric properties of the four types of topological entities can help us to sort up all the pixels or voxels in the input microstructure and produce a new array for the microstructure, that is, (numb, GI, xi, yi, zi, type). In Fig. 4 we show the characterized microstructure of a 50 grain nanocrystalline copper sample (Fig. 3) and a real polycrystalline sample shown in Fig. 2. The same characterization technique was applied to the b-Ti alloy obtained through serial sectioning (Fig. 2). One can see in Fig. 1 that the grain boundary network with the topological entities is clearly revealed by the interconnected triple junction lines at the vertex points while the grain boundaries are not displayed explicitly in order not to obstruct the view of the full network. To display all four topological entities, we extract one grain with a full view of grain boundaries, triple junctions, and vertex points, along with the grain cell (Fig. 5). We like to emphasize here again that this classification method just gives the (approximate) locations of the topological entities with the best spatial resolution of the size of two neighboring pixels or voxels. Clearly, such estimate is too crude to predict the true dimension or size of, say, a grain bound-

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Fig. 5. The different topological entities associated with a single grain. They are sorted into grain boundary shown by the green colored voxels, triple junction shown by the yellow colored voxels, and vertex point shown by the red colored voxels. Note that due to high resolution, one can distinguish each individual voxel that appears as a cube in the figure. A curved grain boundary can also be seen clearly on the left side of the grain.

ary, or a triple junction which is typically on atomic scales [5,6]. As we show below, different approaches need to be developed to reveal the atomicity of the topological entities. 2.2. Computing topological properties Once all the topological entities are identified and classified by labeling each pixel or voxel with a type index, one can compute the topological properties. In Fig. 6, we show the grain size distributions in the polycrystalline b-Ti alloy. The grain size can be expressed as the volume Vk of each grain labeled with the grain index k. Vk is obtained straightforwardly by summing the volumes of all voxels or pixels within that grain with the same GI and the same topological entity type = grain. Or conventionally, one can express the grain size by the equivalent spherical diameter Rk = [3Vk/4p]1/3 once Vk is computed. Since the input microstructure of real polycrystal samples has surfaces, the grains on the surface are cut or chopped. We need to discard those grains to avoid possible errors in computing the topological properties. Fig. 6 shows two grain size distributions computed with and without the surface grains. Fig. 7 shows the distribution of the triple junction length and Fig. 8 shows the grain boundary area distribution in the same b-Ti alloy. For triple junction length calculation, we can use the straight line connecting the two vertex points at the end of the triple junction to compute the length if the line is nearly straight. For a bend or curved line, improvement can be made if we partition the triple junction line into different parts and take the center of mass of each segment and then compute the lengths of the lines between connecting the centers of mass. The rationale behind this approach is that the finite size of the (cubic or other shaped) voxels introduces steps in the TJ lines with the height ranging from one side-length to several side-length of the voxels such that if one computes the length by joining the centers of the voxels, the length could becomes too long. More detailed method can be found in Ref. [22].

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Fig. 6. The distribution of grain volumes in the BCC b-Ti sample (Fig. 2) which contains 834 grains (red line). After removing the surface grains, the distribution is calculated for the remaining 378 inner grains (black curve). The inner grains do not contain any voxels located on the sample surface. The sample dimensions are 448  448  402 lm3.

The boundary area is calculated by using the vertex points at the edge of each gain boundary to form a polygon; we then triangulate the polygon and compute the areas of the triangles. If a grain boundary is curved, we first find the center of mass of the vertex points on the boundary and then draw a line from the center along the normal of the polygon formed by the vertex points which is outside of the curved boundary. The voxel on the boundary that intersects the line is then used as an anchor to triangulate the curved boundary with the vertex points on the edge of the boundary. So the triangles formed by the voxel and the vertex points are much closer to the curved boundary than by simply using the area formed by the vertex points alone. One can improve the accuracy by repeating the procedure in each triangle previously formed: First finding the center of mass of each triangle, then draw a line along the triangle’s surface normal, then finding the intersecting voxel on the boundary, and finally calculating the areas of the triangles formed by the voxel and the three points forming the triangle. Note that the accuracy will not improve indefinitely if the distance between the two neighboring voxels on the curved boundary is of the same order of magnitude as that of the roughness of the boundary formed by the voxels. One can find more detailed algorithm elsewhere [22]. Fig. 9 shows the Lewis law that relates the grain cell volume to the number of grain boundaries in each grain. As we see, the expected linear relation is not obeyed in the polycrystalline sample. Also the influence of the grains on the sample surface on Lewis law is obvious such as in the average number of boundaries per grain hFi, and other topological relations [22]. 3. Digital polycrystals from inverse Monte carlo Method Although the digital polycrystal microstructures from experiment are realistic, they are in general limited in resolution. One additional shortcoming is that each sample could provide only one instance of the microstructures and obtaining more cases would be costly and time consuming [14]. For simulation and theoretical analysis, more flexible approaches would be valued that can produce digital polycrystals with adjustable microstructure attributes. Although there are several methods

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Fig. 7. The triple junction length distribution in polycrystalline b-Ti (Fig. 2). To avoid miscounting, we discarded the triple junctions that are connected to the sample surface. So only the triple junctions inside the sample are used. The total number of such TJs is 6056.

Fig. 8. (a) The grain boundary area distribution and (b) the distribution of summation of the bond angles on each boundary. The average bond angle is 324.37 degree.

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Fig. 9. The Lewis plots hViF / a(FC  F0) for all grains (black) and only the inner grains without the surface grains (red). The average number of boundaries per grain hFi, is 10.9568 and 12.4418 respectively.

to produce ‘‘multi-grain’’ structures [23,24], Voronoi tessellation is the most widely used one to represent the grain boundary networks [23–26]. Voronoi tessellation mimics homogeneous nucleation and grain growth during microstructure formation by partitioning a space into space-filling, convex polyhedron, or cells [27]. The so-called Poisson–Voronoi polyhedron cell structures are constructed by first placing N random points in a space and then dividing the space into N numbers of polyhedra formed by the planes bisecting the bonds connecting each point and its nearest neighbors. The grain boundary networks created this way is composed of mathematically abstract planes between the cells, triple junction lines and vertex points, which visually resemble the microstructures seen in experiments [27]. However, quantitative measurement of these topological structures reveals that they do not always obey the desired topological properties, such as grain size distribution [27]. We showed, however, that this shortcoming can be eradicated using a constrained Voronoi tessellation (CVT) method [27] in which the experimental grain size distribution is used as a target function and through inverse Monte Carlo optimization the desired microstructures can be produced. In addition, one could also give other sets of microstructure parameters, such as the crystallographic orientation, to each grain cell already built from the CVT. The misorientation angle of each grain boundary interface can be calculated [28,29] and the misorientation distribution can be made to follow any desired distribution by using the same inverse method mentioned starting from the random, or MacKenzie distribution. Furthermore, from the grain boundaries with specific misorientation angles, one can identify special boundaries [5]. So the method combining the CVT and inverse Monte Carlo is a viable way to produce ‘‘artificial’’ digital microstructures that are closer to polycrystals. The algorithm for building digital microstructures using Voronoi tessellation consists of three steps [29,30]: 3.1. Select input target microstructure attributes The most accessible microstructure property is the grain size distribution, p(d), and the misorientation angle distribution, p(h), both of which can be measured with relative convenience [18]. For

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coarse-grained polycrystals, conventional stereological approaches using optical microscope, for example, can provide adequate account of grain size distribution, while for nano-crystalline materials, due to the small grain size, grain size measurement using TEM is limited but sufficient in some cases to give a sketch of the distribution function [3,17]. Misorientation distribution can also be measured very conveniently in coarse-grained polycrystals from the electron backscatter patterns in SEM [31,32]. The process has been automated so high quality orientation image of a large number of grains in coarse-grained polycrystals can be produced. We use these two distribution functions as our target functions to optimize the artificial microstructure generated initially from the Voronoi tessellation. 3.2. Construct Voronoi cells using Poisson–Voronoi process The second step is to place N number of points in a space, usually a cubic box with unit length and the periodic boundary conditions. Following the Voronoi construction by making the bisecting planes between the bond linking each point and its nearest neighbor points, we can construct the polyhedron cells or Voronoi cells [21]. Voronoi construction gives the coordinates of the vertex points from which we can calculate the cell volume or grain size, grain boundary area, triple junction length, and other geometric quantities and topological properties. 3.3. Use inverse Monte Carlo method to optimize the microstructure The constructed artificial polycrystalline sample from Voronoi tessellation is known to have a set of specific geometric and topological properties that in many cases differ from the experimental results of real polycrystals [29,30]. In order to make this method more useful in building polyhedron grain cells with desired properties, we developed an inverse method to optimize the microstructures so that their properties are closer to either the experimental ones or the desired (hypothetical or theoretical) ones.

Fig. 10. A three-dimensional topological structure of the grain boundary network for a polycrystalline sample containing 50 grain cells. The corresponding microstructure is shown in Fig. 3. It is constructed using the constrained Voronoi tessellation method. The points are the centers of each grain cell, lines are triple junction lines, and the faces formed by the lines are grain boundaries. The grain cells are represented by the polyhedra formed by these lines and faces. See Ref. [21] for details.

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The basic algorithm uses Monte Carlo method to generate trial Voronoi cells and sample the most probable ones according to the following criteria: (a) Start the Voronoi construction by placing N points in the box of unit length; generate the initial trial Voronoi cells; (b) Calculate the cell size, di, in terms of cell volume or diameter, for each Voronoi cell and the grain cell size distribution p(d); (c) Move the points randomly with a small displacement to create new trial Voronoi cells; then calculate the cell size, di and p(d); (d) Compute the penalty functions from the previously generated two consecutive Voronoi cells, P v ¼ Mk¼1 wk ½pinput ðdk Þ  pðdk Þ2 , where pinput(dk) is the input grain size distribution function, wk is the weight, k stands for the kth bin (k = 1, M). (e) If v in the new trial Voronoi cells after step (c) is smaller than that in the previous ones, keep the new trial cells (or the configuration); if v is larger, but e-v/a P q, where q is a random number and a is another parameter used in controlling the acceptance rate in the Monte Carlo run, also keep the move; otherwise discard the new trial cells. (f) Continue from step (c) till v goes below a preset value, say, 0.001. An example of the optimized grain boundary networks obtained using the algorithm is shown in Fig. 10. The algorithm is reasonably efficiency while running on a single processor. A more efficient method is proposed recently by Suzudo who employed a parallel evolution algorithm [33]. The computation efficient was shown to increase with the number of processors used. The Voronoi cells created through this process have the grain size distributions computed. Fig. 11 shows some of the grain size distributions obtained with different types of input distributions. One can see that the match is excellent, even for the bi-modal grain size distribution. More importantly, we show that the topological properties of the microstructures with different cell size distributions

Fig. 11. Four different cell volume distributions obtained using the inverse Monte Carlo and Voronoi tessellation: lognormal (with the standard deviation r = 0.3); normal (r = 0.3); default Poisson–Voronoi tessellation which has a fixed variance; and bimodal (va = 0.8, vb = 1.3 are the centers of two normal distributions). The lines are the target input distributions and symbols are the ones obtained by using the inverse method.

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Fig. 12. Lewis plots of four Voronoi tessellations with different cell volume distributions shown in Fig. 11: lognormal (r = 0.3); normal (r = 0.3); Poisson–Voronoi Tessellation; and bi-modal (va = 0.8, vb = 1.3).

(Lewis’ rule) are different (Fig. 12), which justifies this new approach. So the default Voronoi construction, although producing visually appealing polyhedron form, does not guarantee the quantitative microstructure properties. More applications can be found with the numerically generated grain boundary networks. The output of the digital microstructure from this procedure has the following format, (numb, xi, yi, zi), where the index i stands for the ith vertex point, (xi, yi, zi) is its coordinate, and numb is the number label for the vertex point. It is different from the output format in the experimentally determined digital microstructure where it is not the vertex points, but the pixel or voxel coordinates that are acquired (since there is no direct way to get the vertex points from the experiment). However as shown below, once we know the vertex points, we can produce other topological entities directly from simple geometric relations. As shown in Fig. 13, the grain boundary area, triple junction length, and other topological relations from the artificial digital microstructure can be obtained from the output. Finally, we should mention that the same procedure can be used for misorientation distribution if we chose the input misorientation angle distribution function p(h) [21,28]. Similarly we can use any microstructure parameter distribution function, if available or desired, to optimize the Voronoi construction for polycrystals. 4. Atomic models of polycrystals and microstructure characterization on atomic scales 4.1. Atomic models of polycrystals The grain boundary network obtained from both the experimental input (Section 2) and digital construction using inverse method (Section 3) provides a microstructure scaffolding. There are no atoms or continuum media occupying these topological entities. In order to use these microstructures in various modeling and simulation, we need to put materials into them. The empty grain cells provide the location for us to place the matter inside, that is, atoms or continuum media in each grain cell must be fit within the borders set by the grain boundaries specified by the grain boundary network.

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Fig. 13. (a) The distributions of the face area P(s) for the four samples with different cell volume distributions shown in Fig. 11: lognormal, normal, PVT and bi-modal distributions. (b) The distribution of the triple edge length distributions.

The general procedure to create atomic models of polycrystals is to introduce atoms into the grain cells. Then relaxation is needed to let the atoms move to lower energy configurations. After relaxation, one obtain not only the more equilibrated atomic configurations but also atomic configurations of the microstructure entities. This procedure follows the same basic principles in creating a single grain boundary pioneered by Vitek and his students [5–7] where two (semi-infinitely long) single crystals, often arranged with specific orientations, are brought together along a plane. Upon contacting, the two crystals as well as individual atoms inside are allowed to move to the minimum energy configurations, so the sample gets relaxed and equilibrated. In polycrystal case, the procedure is more complicated. One first, and foremost, has to ‘‘cut and shape’’ a large number of single crystals into specific shape and size in confirmation of the existing empty grain cells in a digital polycrystal that are to be filled with atoms. The next step is to ‘‘fill’’ atoms into these cells. Concurrent to this step, one may need to install the crystallographic properties such as orientation in each of the crystallites, following certain misorientation or texture distribution. The last step is to relax the whole sample to let each atom seek its minimum energy configuration. (One cannot move the crystallites as a whole easily once they are fitted into the grain cell, which is a disadvantage as compared with the bi-crystal [5–7]. But different polycrystal samples can be created with different spatial locations for the crystallites.) We should mention that if a continuum media is to be filled into the empty cells, such as in finite element modeling (FEM), the procedure is much simplified. In this case, one only needs to ‘‘cut and shape’’ the continuum with specific physical and mechanical properties and insert it into each of the empty grain cells. However, as discussed later, it is much more difficult to generate other topological entities in continuum such as grain boundaries and triple junctions that have finite volume or thickness. As a result, either mathematically abstract planes or lines are used, or some pre-chosen planes or lines with certain thickness and diameter are used in ad hoc fashion to represent grain boundaries and triple junctions [13,34,35].

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There are several ways to introduce atoms into the grain cells [24]. Here we present a more rigorous approach based on random ray-crossing algorithm [36,37] to determine whether a lattice point is located inside or outside of an empty grain cell. First, we select N single crystal lattices with specific lattice structure; and the coordinates of the lattices would share a common origin with the sample of the grain cells to be filled. Then we rotate each single crystal lattice according to the orientation information described in misorientation distribution [29]. In most cases, random orientation is used where 3 N random numbers are generated to create three Euler angles for each grain, (ai bi ci), i = 1. . .N, where N is the total number of grain cells. The specific orientation of the crystalline lattice points in the grain is therefore determined by the rotation matrix corresponding to the Euler angles,

2

cos ai cos bi cos ci  sin ai sin ci

6 4 sin ai cos ci þ cos ai cos bi sin ci  cos ai sin bi

 sin ai cos bi cos ci  sin ai cos bi sin ci þ cos ai cos ci sin ai sin bi

sin bi cos ci

3

7 sinbi sin ci 5: cos bi

ð1Þ

Special orientations such as in texture can be produced if the distributions of certain crystallographic orientations are known. Once the orientation is determined, we then use the ray-crossing method to ‘‘cut and shape’’ the single crystal lattice to fit into a cell: From each lattice point of the single crystal lattice, randomly oriented rays of infinite length are generated. The number of intersections between each random ray and all faces of the Voronoi cell is then collected. If the number of crossings made by the ray for a lattice point is odd, then it is inside the grain cell or polyhedron; if the number of crossing is even, then that lattice point is located outside the cell. Specifically, in order to use the ray-crossing algorithm to determine if a lattice point belongs to a grain cell, we need the solutions of a ray that crosses a grain boundary plane. Since the faces are polygons, triangulation needs to be performed on all faces of the grain cell. We then solve the linear equations for the coordinates of the possible intersections between a ray and a non-overlapping triangle on a face, so the number of ‘‘crossing’’ can be determined. For each grain cell, only the lattice points located inside the polyhedron are kept. The same procedure is applied for every cell and the Cartesian coordinates of those selected lattice points are kept that become a part of the digital representation of the atomic structure of a polycrystalline sample. It is worth to mention in passing that in doing so, one must use the same unit for both the lattice and the polycrystalline sample or Voronoi grain cells: If one uses the reduced unit, say [0.5, 0.5], for the sample with an actual linear length L of a cube, one must use the lattice parameter of the crystal lattice for the grain cells scaled by L, or a = a0/L, where a0 is the actual lattice parameter. Of course, one can obtain the actual positions and lengths for the atoms later by multiplying the coordinates by a0/a. Besides, one should also take caution to deal with the atoms near the grain boundaries between adjacent grains that might have overlapped during the process: Since the ray-crossing method deals with (lattice) points while atoms have a physical size, it is inevitable to have some atoms near certain grain boundaries that are closer than the equilibrium interatomic distance, or they may overlap with each other. Atoms that are too close, say, less than 85% of the nearest neighbor distance, must be removed to avoid unphysical high repulsive potential energy during atomistic simulation to be carried out later. Next, molecular dynamic (MD) simulations are performed to relax the non-equilibrium initial atomic configurations. An example of the atomic configuration of a nanocrystalline Cu obtained using the above method is shown in Fig. 14. The topological grain boundary network created using Voronoi tessellation and inverse Monte Carlo method is featured by straight triple junction lines and flat grain boundaries right after the ‘‘filling’’ is done. They however become distorted or curved in atomic models when relaxation takes place (Fig. 14b). Another way to create atomic polycrystal model is to use the experimentally obtained digital microstructures (Fig. 1) as scaffolding, of course, after we use the characterization methods (Section 2.1) to render the topological entities. The procedure to put atoms, or continuum media as well, is the same as described above since we know the actual coordinates of the cell boundaries and triple junction lines. However, since the boundaries and triple junction lines may be curved, fitting becomes more complicated. An efficient algorithm to approximate the curved boundaries and triple junction

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Fig. 14. The atomic configurations of a nanocrystalline copper generated using the method mentioned in Section 4.1. The individual atoms in the sample can be seen. (a) The atomic configuration after filling the atoms into the empty grain cells, and (b) after relaxation using MD simulation at 300 K and zero applied pressure. The grain boundaries appear to be more wavy or rugged in the relaxed sample. The sample contains 100 grains with the mean grain size of 8.05 nm which follow a lognormal grain size distribution and a random misorientation angle distribution. See Ref. [29] for details.

lines that could set the borders for accurate solution of lattice points inside a grain using ray-crossing method is currently in development [38]. 4.2. Atomic scale characterization of microstructures As we mentioned early, the topological entities in a grain boundary network are formed by assembling grain cells in certain ways, such as sharing an interface, a line, or a point. This leads to specific geometric relations among the grains and the atoms inside. Having recognized these relations, we can sort the digitized microstructure into different topological categories (Section 2.1). Here we will show that we can apply this method, in conjunction with some additional criteria, to sort the atoms in the polycrystals (see Section 4.1) into different topological entities. Let us begin first by labeling each grain before MD relaxation by the grain index, GI, from 1 to N, where N is the total number of grains in the sample. An atom residing in a specific grain also inherits the GI. Then if an atom and its neighboring atoms have the same GI, then they belong to the same grain and thus the atom is classified as a grain atom. If an atom and its neighbors have two different GIs, then it sits at a grain boundary. If an atom and its neighbors have three different GIs, then it sits at a triple junction. If an atom and its neighbors have four or more different GIs, then it sits at a vertex point. As we discuss below, this method only locates the atoms at the middle of each topological entity. Since each of these entities has finite size, extending this method to include more atoms is needed. In order to characterize the topological entities in the relaxed sample, we need to update each atom’s grain index since some atoms on the grain boundary will move during relaxation. So we need to update all atoms’ grain index. We first identify the atoms that belong to the ‘‘nucleus’’ of a grain, which are normally at the center or core of a grain. The core atoms are defined as fcc atoms with at least eight fcc neighbor atoms. The core atoms in a grain are given the same GI as in the unrelaxed structure. Then we proceed to associate the rest of atoms with the core one layer at a time marching outward from the core. The thickness of each layer is the first nearest neighbor distance. After all atoms are covered, the atoms related to the same grain core are given the same grain index. Next,

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the indexed atoms in the relaxed sample are further sorted into four groups: grain atoms, grain boundary interface atoms, triple junction atoms and vertex atoms. This is done as follows: For each atom, we first find its neighbor atoms within a cut-off distance and then count the number of different grain indexes of these neighbor atoms. The cut-off distance is set as one and half first nearest neighbor distance. Using the geometric relations mentioned above, we then identify different types of atoms. For example, atoms and their neighbors in the same grain boundary interface have the same two different GIs; atoms in the same triple junction have the same three different GIs; and atoms in the same vertex have the same four different GIs. Further test using the central-symmetry parameter (CSP) or the translational order parameter (TOP) is employed to see if the atoms are indeed correctly classified. The procedure involves calculating each of the atoms’ CSP or TOP value. If they fall below certain critical value, say CSP = 0.01 for grain boundary, we accept the classification; if not, we extend the cut-off radius, and repeat the process. Fig. 15 shows the atoms that belong to the grain, grain boundary, triple junction, and vertex using the above method. For a better visualization, we only plotted these atoms for a single crystallite. We would like to emphasize that the use of this classification method is not only for better and accurate visualization as done in Fig. 15, but also for numerical calculation of various structure and physical properties associated with those topological entities. For example, we can calculate the number of atoms in each topological entity, analyze their structures, obtain the distributions of number of atoms and potential energies, and finally relate the structure and chemistry to the properties. Using the algorithm, we can obtain the distributions of the number of atoms in grain cores (Fig. 16), GB interfaces (Fig. 17) and TJs (Fig. 18) presented in the relaxed nanocrystalline Cu. Both distributions of the number of atoms in GB and TJ closely resemble the face area and triple junction length distributions of the initial Voronoi structure. Figs. 17 and 18 show a large number of small boundaries existing in the

Fig. 15. Different types of atoms of a grain in a nanocrystalline Cu sample that belong to different microstructural topological entities: Blue – grain atoms, Green – grain boundary atoms, Yellow – triple junction atoms, and Red – vertex atom. The atoms, except the grain atoms, are divided between the neighboring grains and only those shared by this grain are plotted.

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Fig. 16. (a) The grain cell volume distribution of a nanocrystalline Cu sample with the mean grain size hDi = 8.05 nm and (b) the distribution of number of atoms in each grain core.

Voronoi structure and also a large amount of short triple junctions. The number of atoms on these small interfaces and short TJs is very small after filling atoms into the Voronoi cells. These small interface and short TJs will disappear upon MD relaxation. Moreover, grain boundaries are often distorted to relax out the stresses near grain boundaries during MD run. In the initial Voronoi structure, vertices are dimensionless points. However, vertices in the relaxed nanocrystalline Cu samples have finite size and physical volume. Moreover, we calculated the atomic Voronoi volume (AVV) for each atom in the nanocrystalline Cu. The summation of AVV for atoms in the same vertex is taken as the volume of a vertex. The distribution of the total AVV per VP is shown in Fig. 19a as well as the distribution of number of atoms per VP (Fig. 19b). The average number of atoms per VP is close to 18 excluding those small vertices with less than eight atoms. 4.3. Calculating grain boundary area and triple junction length Grain boundary characterization using the above-mentioned method provides the detailed information about atomic configuration of each grain boundary entity, from which we can calculate the grain boundary area, triple junction length, and vertex volume. The enabling quantity is the atomic positions of vertices. Since the grain cells in the network is polyhedron, once we know the positions of the vertices, we can calculate the boundary areas, junction lengths, and vertex volumes. The procedure to calculate the GB interface area and triple junction length was briefly discussed in Section 2.2. Here we describe the detailed algorithm for atomic scale calculation: 1. For each grain with the grain index, i, we obtain the list of its neighbor’s grain indexes, neibGI[numNeib[i]], where numNeib[i] is the number of neighbors of grain, i. The number of neighbors of a grain is the same as the number of GB interfaces of that grain. Atoms on the same GB interface will have the same two GIs in the array diffGIs[numDiffGIs[i]].

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Fig. 17. (a) The face area distribution in a Voronoi cell structure with 100 cells and (b) the distribution of the number of atoms per GB interface in a nanocrystalline Cu sample with the mean grain size hDi = 8.05 nm.

2. For each GB interface, we obtain the list of vertices located on the interface, validVertices[numVerticesOnGBInterface]. Vertices that have more than eight atoms are called valid vertices (small vertices, e.g. vertices with less than eight atoms are discarded). Each vertex has four or more GIs in the array diffGIs[numDiffGIs[i]]. Vertices on the same GB interface should have at least two GIs in common. One is current grain’s GI, i, and the other is on the list neibGI[numNeib[i]]. 3. For each GB interface, search for a closed loop that connects all valid vertices. The two common GIs are sorted into the first two slots in the array diffGIs[numDiffGIs[i]]. When searching for the loop, we only need to look at the last two GIs, namely, gid1 and gid2. Fig. 20 shows an interface between two neighboring grains. Each vertex on the interface is labeled by two GIs. Two vertices are connected by a triple junction line if they have a common GIs in either gid1 or gid2. For example, vertex (18, 37) is connected to vertex (35, 37) as they share a common GI, 37. From the neighboring vertices, we can calculate the triple junction length. 4. Determine the center of mass of atoms on the interface (Point O in Fig. 20) from which we can triangulate the interface. The area of the interface is calculated by summing the areas of the triangles formed between the center of mass and two neighboring vertices connected by a triple junction line. The angle between the two lines connecting a pair of neighboring vertices to the center of P interface, ai, is calculated. For each interface, the accumulated inner angle, a = ai, should be close to 360 degree and is used later for calibration. The GB interface described by Fig. 20 represents the majority of GB interfaces present in the relaxed nanocrystalline Cu samples and their interface areas can be calculated using the above procedure. We call those interfaces ‘‘regular interface’’. However, there are special cases where a perfectly closed loop is hard to find. Fig. 21 shows the atomic configuration of a GB interface between two neighboring grains, grain GI = 06 and 34, where there is a missing link between two vertices labeled as (20, 11) and (33, 44). This may be caused by relaxation of the atoms in the two vertex points that are too close

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Fig. 18. (a) The triple junction length distribution of a Voronoi cell structure with 100 cells; (b) the distribution of number of atoms per triple junction in a nanocrystalline Cu sample with the mean grain size hDi = 8.05 nm.

to each other. The missing link can be identified by counting the number of times that each vertex is used to form a link between two neighboring vertices. In a closed loop (Fig. 20), each junction line should be connected to two neighboring vertices. The missing link is found between two vertices that are used only once. Using this method, the missing link is identified and the total area of the GB interface should be increased by the area of triangle formed by the center of mass and vertices (20, 11) and (33, 44). Another special case is shown in Fig. 22 for a boundary between grains labeled as GI = 06 and 46 where an extra vertex (30, 35) appears. Because of the extra vertex, the area of the GB interface is overestimated and needs to be adjusted. Similarly, we can solve the problem by counting the number of times that each vertex is used to form a triangle. As the vertices (7, 30) and (30, 31) have both been counted three times if we include the extra vertex, we need to remove the area of the triangle formed by these two vertices and the face center. Further investigation reveals that the extra vertex (30, 35) is introduced by the neighboring grain 35 when several atoms in grain 35 are located within about 1.5 times the nearest neighbor distance from atoms in the extra vertex. Finally, the total inner angle, a, is examined for each GB interface and the calculated area is kept only if the angle, a, is in the range of 360°±5°. In the case of a missing link, the angle formed between two vertices connecting the missing link and the center of interface is added. In the case of an extra vertex, the angle formed between two vertices used three times during triangulation and the center of interface is subtracted. Using the above algorithms, we can obtain the grain boundary area and the triple junction length. For the later, once the valid vertex points are identified, we can compute the length using the technique presented in Section 2.2. The difference in atomic scale calculation is that we treat atoms, instead of voxels. Further details between the number of atoms in each grain boundary interface and the calculated area is observed using the above-mentioned procedure. Fig. 23a plots the number of atoms in each

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Fig. 19. (a) The distribution of the total atomic Voronoi volume (AVV) per vertex and (b) The distribution of number of atom per vertex junction in a nanocrystalline Cu sample with the mean grain size hDi = 8.05 nm.

grain boundary interface, nf, against the calculated area, a. A linear relation, a = 1.8629nf  659.2112, is clearly observed. Each point on Fig. 23a represents a GB in a nanocrystalline Cu sample with 50 grains and the mean grain size of 12.17 nm. In calculating the number of atoms on the GBs, we included the atoms on the TJ. The number of atoms in TJs is multiplied by 2/3 since the same TJ on a GB is shared by two neighboring grains. The number of atoms in vertices is divided by two for a similar reason. Extrapolating the linear relation to zero grain boundary area, we see there are a finite number of atoms, between 175 and 350 atoms, which indicates that there exists a physical limit in the number of atoms to form a grain boundary definable in atomic scale. Similarly we plot in Fig. 23b the number of atoms in each triple junction, ntj, against the calculated triple junction length, l. A linear relation between ntj and l, l = 0.2733ntj  1.2979, is clearly seen. Since atoms in vertices connected by the same triple junction are included and the average number of atoms on vertices (>8 atoms) is 17.24, after correcting this, we found that the minimum number of atoms in a TJ is around 18, which indicates the physical limit in the number of atoms to form a triple junction. Taking advantage of the new method, we can test mean grain size effects on grain boundary area and triple junction length. To do so, we made five nanocrystalline Cu samples with different mean grain sizes generated using the same Voronoi cell structure. The mean grain sizes are 6.09, 7.61, 8.70, 10.15, and 12.17 nm respectively. After MD relaxation at 300 K, the same procedure is performed on the five samples. The linear relations are clearly shown in Fig. 24 for the samples with different grain sizes after a simply least square fitting. Once again, the GB interface area approaches zero at the finite number of atoms. Let kf be the ratio between the calculated interface area a and nf, the number of atoms per unit area. The number density per area, qGB, is inverse proportional to kf. kf increases with mean grain sizes, suggesting the number density on GB interfaces is decreasing with increasing mean grain sizes within this range of the grain sizes. Similarly, we obtain the linear relation between the TJ lengths and the number of atoms in each TJ for three samples, shown in Fig. 25. The ratio, kl, between calculated TJ length, l, and the number of atoms per TJ, ntj, increases slightly with the mean grain sizes.

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Fig. 20. The atomic configuration of a GB in a nanocrystalline Cu sample with the mean grain size hDi = 8.05 nm. A closed loop is formed by links (blue color online) sharing common grain indices, gid1 or gid2, which are shown as a pair in the parenthesis. The center of mass for all GB atoms (green atoms) is identified by the letter ‘‘O’’. Yellow atoms are TJ atoms and red atoms are VP atoms.

We applied our atomic scale characterization method on the nanocrystalline Cu samples relaxed at different temperatures from 100 to 500 K. The same 50 grain nanocrystalline Cu sample with a mean grain size of 12.17 nm is relaxed at different temperature using Gibbs ensemble molecular dynamics simulations. The same procedure is applied on the relaxed samples to calculate the GB interface area and triple junction lengths. Fig. 26 shows the linear relation between the calculated interface area, a, and the number of atoms per interfaces, nf, at five different temperatures. The slope, a/nf, decreases as the temperature increases indicating that the number of atom per unit area on GB interfaces decreases with rising temperatures. Fig. 27 shows the linear relation between the calculated TJ length, l, and the number of atoms per TJ, ntj, at different temperatures. Once again, kl increases slightly with rising temperature. The fact that the change in kl is much smaller or less obvious than change in kf shows that TJ atoms are more disordered than GB atoms and therefore less sensitive to temperature change. 4.4. Other microstructure-related properties of the grain boundary network Besides some of the geometric properties presented above, we can compute other microstructurerelated properties, some of which have been sought and still in debate for quite a long time. 4.4.1. Atomic number fraction of atoms in the topological entities Given a microstructure in a polycrystalline material, what is the fraction of the atoms in the grain boundaries, triple junctions, or vertices? For nano-crystalline materials, the answer is usually esti-

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Fig. 21. The atomic configuration of a GB interface between grains labeled as GI = 06 and grain 34. The missing link between (20, 11) and (33, 44) can be identified as two vertices are used only once to form a link. Other vertices are used twice to form a link.

mated based on the assumption that a grain is a sphere or a cube and the grain boundary has a thickness d [39,40]. For both cases, if the length scale of the grain is D and the atomic density per unit volume is assumed roughly the same on the interface and inside the grain, the fraction of atoms on the interface versus that inside the grain is f  3d/D. Therefore, if d is 0.5 nm, or roughly two atomic spacing for most metals, about 50% atoms would be on the interface if D is about 3 nm. However, grains in poly- and nano-crystalline materials rarely have such a simple geometrical shape. It is challenging to estimate the fraction of atoms in different type of GB entities without a predefined GB thickness. With the availability of the classification method of atoms in the topological entities, we now can calculate the fractions straightforwardly. First, we need to have the grain boundary thickness, d. To do this, we need to determine the normal vector of each GB interface. This can be done by taking certain pairs of atoms in the grain boundary and then the cross products of the vectors linking each pair of atoms to the face center are used to obtain the surface normal. We then cut a cylinder along the normal across the interface. The radius of the cylinder is 25 Å and the length of the cylinder is 36 Å. In case the interface area has a length less than 25 Å, we use the actual number of atoms on that boundary. We then slice the cylinder into many thin slices of a thickness of 2 Å each that are parallel to the interface, which will give us the GB slices positioned across the boundary along the direction of the interface normal. We can then calculate the profile of a boundary from the geometrical or thermodynamic properties of atoms in each slice and the statistical averages for all boundaries in the sample. The statistical averages include average centralsymmetry parameter, potential energy, atomic stresses, atomic Voronoi volume, etc. Fig. 28 shows the grain boundary profiles defined by the CSP and AVV. From the profiles, we can determine the mean grain boundary thickness d of about 0.8 nm as determined by the full width at the half-height of the profiles. Using this value in the relation, f  3d/D, we can see that the fraction of atoms at the grain boundary is about 20% for the nc-Cu with the mean grain size of 12.17 nm.

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Fig. 22. Atomic configuration of the GB interface between two grains labeled as GI = 06 and 46. An extra vertex identified as (30, 35). The vertices marked with 2 and 3 are the number of times they are counted in measuring the triangles they form. The dotted-line is the triple junction line after the extra vertex is removed. The correct area is calculated by excluding the extra vertex, (30, 35).

On the other hand, using the classification method we can directly obtain the average atomic number fraction of atoms in grain, grain boundaries, triple junctions, and vertices in the nanocrystalline Cu samples with the mean grain size ranging from 6.09 to 12.17 nm. As shown in Fig. 29a, the fraction of the atoms in grain boundaries is about 15% for the nanocrystalline Cu with the mean grain size at 12.17 nm. As the grain size increases, we see that the fraction of atoms belonging to GB, TJ and VP all decreases, while the fraction of atoms belonging to the grain increases. 4.4.2. Configuration energy of the topological entities It has been a long debate over whether a triple junction has a higher configuration energy than that of a grain boundary. The average potential energies are calculated straightforwardly for grain atoms, GB atoms, TJ atoms and VP atoms and shown in Fig. 29b. The calculation shows that the potential energy of the grain boundary atoms is about the same as that of the triple junction atoms in the nanocrystalline Cu samples with the mean grain size ranging from 6.09 to 12.17 nm. We also see that the average potential energy for the grain atoms decreases as the mean grain size increases, which is consistent with the increasing atomic fraction of grain atoms with the grain size. As shown in Fig. 29c, the degrees of disorder measured by the CSP value in GB, TJ, and VP change in ascending order. Since the CSP measures the inversion symmetry, its correlation with the potential energy change, which is more related to the change of interatomic distance, is not as close and sensitive as one compares Fig. 29b and c. 4.4.3. Factors affecting the grain boundary thickness We briefly discussed the grain boundary thickness calculation using the grain boundary profile analysis of the CSP profile. The thickness of each GB interface is estimated from the full width at the half-height of the CSP profile,

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Fig. 23. (a) The linear relation between the number of atoms per GB interface, nf, and the calculated GB interface area; (b) the linear relation between the number of atoms per TJ, ntj, and the calculated TJ length. The number of grains in the nanocrystalline Cu sample is 50 and the mean grain size is 12.17 nm.

W CSP ¼ SCSP =HCSP ;

ð2Þ

where SCSP is the area integrated under each CSP profile and HCSP is the peak height of each CSP profile. The former represents the total amount of disorder in the boundary and the later represents the maximum CSP. However, the grain boundary thickness defined by Eq. (2) can only be used to compare samples with the same number of grains, mean grain size and misorientation distribution. Otherwise, there will be obvious changes in the area under the average CSP profiles. For each GB, we obtain SCSP, HCSP and therefore WCSP and their statistical distributions are accumulated over all GB interfaces identified in the sample. To test the temperature effects on the grain boundary thickness, the same nanocrystalline Cu sample of 50 grain with a mean grain size of 12.17 nm are relaxed at 100, 200, 300, 400 and 500 K using MD simulations. From the GB CSP profiles, we can estimate the GB thickness and the temperature effects on the distributions of GB CSP thickness, WCSP, as shown in Fig. 30a. As expected, the average GB thickness hWCSPi, increases with rising temperature shown in Fig. 30b. Next we look into the misorientation effect on grain boundary thickness. As mentioned in the Introduction, the nature and structure of grain boundaries in polycrystalline materials are closely related to the grain boundary misorientation. In general, the higher the misorientation angle is, the more disordered the grain boundary is, and vice versa. But so far, the majority of the nanocrystalline samples used in numerical modeling are those with random misorientations. Using the algorithms (Section 4.1) we generated samples with the average grain boundary misorientation angles at 10.055°, 14.867°, 19.792°, 24.920°, 29.982°, 40.831° and 46.705° respectively. Fig. 31 shows five of the misorientation distributions. Fig. 32 shows the distribution of the structural disorder in the grain boundaries as measured by the distribution of the CSP profiles hHCSPi for the samples with different misorientations. The inset of Fig. 32 shows the mean peak heights HCSP which increases with the mean misorientation angles (hhi.

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Fig. 24. The effects on the calculated GB interface areas by different mean grain sizes:hdi = 6.09, 7.61, 8.70, 10.15, and 12.17 nm. The slope between the calculated GB interface area, a, and the number of atoms per interface, nf, increases by 11.31% as the mean grain size increases from 6.09 nm to 12.17 nm.

Fig. 25. The effects on the calculated TJ lengths by different mean grain sizes: hdi = 6.09, 7.61, 8.70, 10.15, and 12.17 nm. The slope between the calculated TJ length, l, and the number of atoms per TJ, ntj, increase only slightly as the mean grain size increases.

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Fig. 26. The temperature effect on the linear relation between interface area, a, and the number of atoms per interface, nf, for the same 50 grain nanocrystalline Cu sample with a mean grain size of 12.17 nm. The slope, a/nf, decreases as the temperature increases from 100 to 500 K.

One can see that the misorientation greatly affects the disorder of the grain boundary structures as shown by the mean CSP value per face: The mean CSP value for the sample with minimum misorientation distribution moves to smaller values as compared with that of the sample with the random, or MacKenzie distribution, indicating that less disorder is in the sample with the minimum misorientation. Fig. 33a shows the effects on the average CSP profiles of the samples by different misorientation distributions. The average grain boundary thickness estimated from the profiles increases as the mean misorientation angle increases (shown in Fig. 33b). The area integrated under the average CSP profiles represents the total amount of disorder in the grain boundary; it increases initially and then reach a plateau as the misorientation distributions move closer to the Mackenzie distribution (Fig. 33c). These results show that the changes in misorientation distributions not only affect the geometries of the grain boundaries, but also their width, or volume, or density. 4.4.4. Atomic volume of grain boundary Atomic volume at grain boundary plays an important role in transport and mechanical properties of polycrystalline materials. However, accurate assessment of the atomic volume which is closely related to the activation process has remained a challenge. With the new classification method, we calculated the GB atomic Voronoi volume profiles from the nanocrystalline Cu samples relaxed at different temperatures. The average GB AVV profiles at five temperatures are shown in Fig. 34a. From the average GB AVV profile, we can estimate the average AVV at GB hAVVHMi, which increases linearly with temperature (Fig. 34b). The average atomic Voronoi Volume gives us another way to estimate the GB thickness according to the following equations, WGB = hAVVHMi/ha/nfi. Fig. 34c shows WGB increases with increasing temperature. The thickness calculated this way agrees fairly well with the result from the previous calculations using CSP values. The same type of atomic Voronoi volume calculations are performed on five nanocrystalline Cu samples with different mean grain sizes in the nanocrystalline Cu. The average GB AVV profiles are shown in Fig. 35a. As the mean grain size increases, the average GB AVV profiles shift downward

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Fig. 27. The temperature effects on the linear relation between TJ length, l, and the number of atoms per TJ, ntj, for the same 50 grain nanocrystalline Cu sample with a mean grain size of 12.17 nm. The slope, l/ntj, does not change much as the temperature increases from 100 to 500 K.

Fig. 28. The profiles measured with the CSP (filled square) and AVV (filled triangle) across an grain boundary interface between two grains in the nanocrystalline Cu sample with 50 grains and the mean grain size 12.17 nm.

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Fig. 29. (a) The fraction of atoms in grain cell, GB, TJ, and VP in the nanocrystalline Cu with different mean grain sizes; (b) the average potential energies per atom for the same four types of atoms; (c) the average CSP per atoms for the same four types of atoms.

Fig. 30. (a) The distributions of GB thickness, WCSP, at different temperatures in the entire sample; (b) the average GB thickness hWCSPi, estimated from GB profiles, versus temperature. The sample is the nanocrystalline Cu with 50 grains and a mean grain size of 12.17 nm.

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Fig. 31. The distributions of misorientation angles for a nanocrystalline Cu sample with 50 grains and the mean grain size of 12.17 nm. The filled black square represents the distribution with the smallest misorientation with the mean misorientation angle at 10.055°, the filled black upside-down triangle represents the distribution with the maximum misorientation with the mean misorientation angle at 46.705°. The open black circle represents an intermediate distribution. The filled gray triangle represents the random misorientation distribution and the solid line is drawn from the MacKenzie distribution. See Ref. [29] for details.

and the average AVV at the half-width height also decreases as shown in Fig. 35b. The GB obtained using the average AVV is shown in Fig. 35c, which also increases linearly with grain size. Table 1 shows the temperature effects on the ratio between interface area and number of atoms per interface, a/nf, the average peak height from GB CSP profiles hHCSPi, the average GB CSP thickness hWCSPi, the average GB thickness estimated by Eq. (2) hWGBi, and the average atomic Voronoi Volumes at the full half-width height hAVVHMi. 5. Summary The topological structure of a grain boundary network is the fundamental quantity underlying the properties of polycrystalline materials. Quantitative characterization of the topological entities in terms of their spatial location and type is the first step enabling rigorous treatment of the microstructure–property relations. In this work, we used two cases to demonstrate the viability and validity of a series of numerical algorithms and methods that can produce the microstructure quantities. One is the digitized microstructures from a real polycrystalline material using serial sectioning and the second is the reconstructed digital microstructure using the combined method of Voronoi Tessellation and inverse Monte Carlo method. The microstructures from both approaches are represented in different formats, one is expressed by the position of the digitized voxels or pixels plus the grain index, and the other is by the position of the vertex points plus the grain index. Using the classification methods, we can produce the datum set for the microstructures with each topological entities sorted into grain cell, grain boundary, triple junction, and vertex points. Both the spatial location or coordinate and the associated grain indices are explicitly recorded. With the information, we can calculate a full range of topological properties: grain cell volume, grain boundary area and geometric characteristics, triple junction length, and vertex points, as well as their statistical means, variances, correlations, and distribution functions.

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The next challenge is to build atomicity into the digital microstructures already obtained. To this end, we developed several algorithms to cut, shape, and rotate the crystallites and fill them into the grain boundary network scaffolding so specific atomic structure, crystallographic properties, and geometric properties of each individual crystallites or grains can be reconstructed. Following annealing of the initial atomic structure, we can classify the atoms into different topological entities. Once again, this information is expressed as the spatial location of each atom, its type of topological entity, and the grain index. Such rendered microstructure on atomic scale enables us to establish the connections between each of the topological entities and properties including geometric properties (grain volume, shape, triple junction length, vertex volume, etc.), thermodynamic properties (potential energy, fraction of atoms in each topological entity, atomic Voronoi volume, etc.), and structural properties (grain boundary thickness, misorientation, temperature and grain size effects, etc.). We used a nanocrystalline Cu as an example since the nano-crystalline materials with the mean grain size below about 100 nm is the only polycrystalline system that we can deal with on atomic scale under the current computing capability. The results obtained from the calculation using the rendered atomic scale information about the microstructures are revealing, several of which have not been accessible before, including the grain boundary area and triple junction length, the atomic fraction and potential energy of grain boundary and triple junction, and detailed grain boundary thickness change with temperature, misorientation angle, and the mean grain size. Although there are still many open issues left in further improving the methods in both topological and atomic scale to characterize the polycrystal microstructures, we can envision many exciting and possibly enabling applications of the numerical methods developed here in the following areas: 5.1. Microstructure tailoring or optimization in design of polycrystalline materials Experimental realization of different microstructures are in general laborious and costly, and the desired optimal microstructure design is often hard to obtain (and often unknown). Using the charac-

Fig. 32. (a) The distributions of peak height HCSP of CSP profiles versus the mean misorientation angles (hhi). The nanocrystalline Cu samples contain 50 grains and the same topological network structure. (b) The inset is the mean peak heights hHCSPi obtained from the CSP distribution versus the mean misorientation angles (hhi). See Ref. [29] for details.

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Fig. 33. (a) The average grain boundary CSP profiles for the seven nanocrystalline Cu samples with different mean misorientation angles. We use a threshold value, CSPC, to determine the GB thickness. (b) The average grain boundary thickness TCSP increases with the mean misorientation angle; (c) the relation between the total area under the average CSP profiles, SCSP, or the amount of disorder, and mean misorientation angle. See Ref. [29] for details.

Fig. 34. (a) The averaged GB AVV profile over all GBs in the sample at T = 100, 200, 300, 400 and 500 K; (b) the average halfmaxima height at the AVV distribution, AVVHM, versus temperature; (c) the GB thickness determined by, WGB = hAVVHMi/ha/nfi, increases with increasing temperature.

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Fig. 35. (a) The average GB AVV profile at hDi=6.09, 7.61, 8.70, 10.15, and 12.17 nm; (b) the average AVV at the half-width height of the distribution, AVVHM, which decreases slightly with increasing grain sizes; (c) the GB thickness determined by, WGB = hAVVHMi/ha/nfi, decreases with increasing grain size.

Table 1 Temperature effects on the ratio between interface area and number of atoms per interface, a/nf, the average peak height from GB CSP profiles, hHCSPi, the average GB CSP thickness, hWCSPi, the average GB thickness estimated by Eq. (2), hWGBi, and the average atomic Voronoi Volumes, hAVVGBi. T(K)

a/nf

hHCSPi

hWCSPi, Å

hWGBi, Å

hAVVHMi, Å3

100 200 300 400 500

1.9740 1.9173 1.8629 1.7996 1.7507

0.10191 0.10298 0.10327 0.10434 0.10488

5.565 5.837 6.194 6.732 7.373

6.291 6.500 6.722 6.998 7.219

12.161 12.219 12.272 12.341 12.395

terization methods developed here, one can manipulate or tailor the microstructures in computers, in conjunction with the calculations or modeling of the properties (see below), to search for the best microstructure for specific properties and applications. As one has already seen, it is relatively easy to change all microstructure attributes using the methods presented here. For each of the realizations of the microstructure, one can carry out either continuum or atomistic modeling to check the goodness of the trial microstructures for the desired properties. 5.2. Flexible models for microstructure modeling using continuum and atomistic methods As we have already seen, a characterized digital topological network and its atomic scale rendition provide input for polycrystalline material modeling, either on continuum or atomic scales. It is often emphasized that as realistic as possible microstructures are needed to produce reliable answers though modeling, which can only be achieved using the realistic digital microstructures from experiment. However, such thinking may not be the whole story: an equally valuable/important application

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of the ‘‘artificial’’ digital microstructures is in theoretical analysis where being able to deal with a wide range of microstructures is not only necessary but essential. In addition, the flexibility allows us to try and optimize the microstructure to search for the best or desired materials. 5.3. Compliment simulation tool for experimental rendering of microstructures Another potential application of the numerical methods is to build a simulation tool to compliment experimental characterization of microstructures. Like in many other imaging or spectroscopy process, identifying certain structural features that are either outside of the resolution of the instrument, or overcoming certain limitations in specimen or preparation method require compliment simulation tools that can provide some ‘simulated’’ images or spectra based on the best and educated guess. The same applies to microstructure characterization. For example, triple junctions are hard to be accurately characterized, and the worst is that the vertex points can be easily missed when the layer thickness of the material removed during serial sectioning is too coarse. How to reinstall those hard-toidentify microstructure entities and their attributes may need the help from the numerical methods developed here. Acknowledgements M.L. would like to dedicate this paper to Vasek Vitek on the occasion of his 70th birthday. The authors are grateful for the financial support for this work provided by the National Science Foundation (NSF) under the contract number NSF-CMS-0411227, the Air Force Office of Scientific Research (AFOSR) MURI program under the contract number F49620-02-0382, and an NERI-C grant under the contract number DEFG07-14891. References [1] Meijering JL. Interface area, edge length, and number of vertices in crystal aggregates with random nucleation. Philips Res Rep 1953;8:270–90. [2] Fultz B, Frase HN. Grain boundaries of nanocrystalline materials – their widths, compositions and internal structures. Hyperfine Interact 2000;130:81. [3] Weertman JR. In: Koch CC, editor. Nanostructured materials: processing, properties and applications. Norwich, NY: William Andrews Publishing; 2002. p. 397. [4] Gibbs JW. The Collected Works of J. WIllard Gibbs. New York: Longmans Green and Company; 1928. [5] Sutton AP, Vitek V. Coincidence site lattice and dsc dislocation network model of high angle grain-boundary structure. Scripta Metall 1980;14:129–32. [6] Wang GJ, Vitek V. Relationships between grain-boundary structure and energy. Acta Metall 1986;34:951–60. [7] Wang GJ, Sutton AP, Vitek V. Atomistic studies of grain-boundary segregation. J Metals 1982;34:40. [8] Latanisi Rm, Opperhau H. Intergranular embrittlement of nickel by hydrogen – effect of grain-boundary segregation. Metall Trans 1974;5:483–92. [9] Rao PR. Grain boundary segregation in metals. Curr Sci 1997;73:580–92. [10] Was GS. Grain-boundary chemistry and intergranular fracture in austenitic nickel-base alloys – a review. Corrosion 1990;46:319–30. [11] Duscher G, Chisolm MF, Alber U, Ruhle M. Bismuth-induced embrittlement of copper grain boundaries. Nat Mater 2004;3:621. [12] Van Swygenhoven H, Derlet PM, Hasnaoui A. Atomistic modeling of strength of nanocrystalline metals. Adv Eng Mater 2003;5:345–50. [13] Crocker AG, Flewitt PEJ, Smith GE. Computational modelling of fracture in polycrystalline materials. Int Mater Rev 2005;50:99–124. [14] Spanos G, Rowenhorst DJ, Lewis AC, Geltmacher AB. Combining serial sectioning, EBSD analysis, and image-based finite element modeling. MRS Bull 2008;33:597–602. [15] Budai J, Gaudig W, Sass SL. Measurement of grain-boundary thickness using X-ray-diffraction techniques. Philos Mag A – Phys Condens Matter Struct Defect Mech Prop 1979;40:757–67. [16] Seidman DN, Stiller K. An atom-probe tomography primer. MRS Bull 2009;34:717–24. [17] Poulsen HF, Jensen DJ, Vanghan GBM. Three-dimensional X-ray diffraction microscopy using high-energy X-rays. MRS Bull 2004;29:166–79. [18] Van Swygenhoven H, Weertman JR. Deformation in nanocrystalline metals. Mater Today 2006;9:24–31. [19] Dobrich KM, Rau C, Krill CE. Quantitative characterization of the three-dimensional microstructure of polycrystalline Al–Sn using X-ray microtomography. Metall Mater Trans a – Phys Metall Mater Sci 2004;35A:1953–61. [20] Herr U, Jing J, Birranger R, Gonser U, Gleiter H. Investigation of nanocrytalline iron materials by Mossbauer spectroscopy. Appl Phys Lett 1987;50:472. [21] Xu T, Li M. Topological and statistical properties of a constrained Voronoi tessellation. Philos Mag 2009;89:349–74.

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