JP4491548B2 - Corresponding grain boundary model creation program and operation method - Google Patents

Description

  The present invention relates to a method for generating a corresponding grain boundary model. More specifically, in the field of atomic level material simulation, the cell shape and atomic coordinates of a corresponding grain boundary model having a periodic structure are generated easily and efficiently. It is about how to do.

Conventionally, as a method for generating a model including a crystal interface, for example, as shown in the prior art document, a method of rotating an atomic arrangement of two crystals according to a specified interface orientation and joining at a grain interface is known. (See Non-Patent Document 1). A program having the same function is, for example, Cerius2-Interface Builder (manufactured by Molecular Simulation Inc. (MSI)). As shown in FIG. 1 (a), the technique used there is a method in which the grain interface index and the in-plane reference orientation in the two crystals A and B are changed to six Miller indices X _A , Y _A , Z _A ; X _B , Y _B, specified by Z _B, their corresponding vector parallelepiped and coordinate transformation operation R _a, generates two crystal cell and internal atomic coordinates from R _B, and the interface model by joining in the grain boundary To do.

  The problem in the present invention is the method of generating the corresponding grain boundary model. When generating an interface model with the prior art, it is necessary to specify six Miller indices, but these Miller indices have a one-to-one correspondence with the Σ value, which is the value that first characterizes the corresponding grain boundary. However, it varies in various ways depending on the grain interface index and symmetric element conversion. Therefore, there is a problem that the Miller index to be specified is difficult to understand and is difficult to use.

  In addition, the model generated by the prior art is an isolated system model. When periodic boundary conditions commonly used in continuum material simulations are imposed, atomic alignment mismatches, generation of blank areas, The duplication of atoms occurred. Furthermore, in the prior art, the second grain interface inevitably generated in the case of the periodic model is not considered. For this reason, even if a periodic boundary condition is imposed on the generated model, the correspondence of atoms at the second grain interface is not maintained, which may adversely affect the simulation result.

M. J. Weins et al. (1971), "Computer calclations of the structure and energy of high-angle grain boundaries", Journal of Applied Physics, Vol. 42, pp. 2639-2645

  The object of the present invention is to directly specify the Σ value, which is a quantity characterizing the corresponding grain boundary, and the grain interface orientation, thereby satisfying the periodic boundary condition and corresponding atoms in the second grain interface generated in the model. To provide a corresponding grain boundary creation program capable of easily and efficiently creating a corresponding grain boundary model maintaining the relationship, and an operation method thereof.

The present invention for solving the above-described problems comprises the following technical means.
(1) To create a corresponding grain boundary model,
Means for listing candidates for grain interface orientation on the display screen from the Σ value of the specified corresponding grain boundary,
Means for listing translation vector candidates of a corresponding grain boundary model from a specified grain boundary orientation;
Means for determining the periodic cell shape of the corresponding grain boundary model from the specified translation vector and its size;
Means for calculating the atomic coordinates of the model based on the space group, lattice constants, the crystal structure data comprising data elements and atomic coordinates,
Means for outputting the result to a display screen and a file;
Corresponding grain boundary model creation program to function as
(2) The program according to (1), wherein the grain interface orientation is specified from a finite number of candidates extracted based on the corresponding lattice point density on the grain interface and the symmetry of the grain boundary structure.
(3) The grain interface orientation is decomposed into the product of the basic orientation and the independent symmetric element transformation, specified individually, and facilitates the generation of the homogeneous interface produced by the symmetric transformation (1 ) Or (2).
(4) The computer executes the process of determining the periodic cell shape by matching each side of the periodic cell of the corresponding grain boundary model with the corresponding interstitial vector, whereby the periodicity of the model and the two grains in the model are determined. The program according to (1), which ensures interface equivalence.
(5) The program, which is executed in the space group, lattice constants, one in combination with a program that generates a crystal structure data or al crystal atomic coordinates comprising data elements and atomic coordinates computer the ( 1) The program described.
(6) The program according to (1), wherein the program is input to a computer having a server function and executed on a web browser of a user computer via the Internet.

Next, the present invention will be described in more detail.
In the present invention, a processing algorithm is employed in which a model is specified by a Σ value and a grain interface index, and a necessary coordinate conversion operation and cell shape selection are automatically or semi-automatically determined. First, the orientation difference matrix of the two crystals A and B and the corresponding lattice point array in the crystal coordinate system are obtained from the designated Σ value based on the corresponding lattice theory. Next, the corresponding lattice point arrangement on the specified grain interface is examined, and the periodic cell shape of the grain boundary model is determined from the translation vector. Coordinate transformation is performed on the periodic cell and the internal atomic coordinates so that the grain interface is parallel to the reference plane of the model coordinate system, thereby obtaining the periodic cell and atomic coordinates of the corresponding grain boundary model. Thereby, instead of the six Miller indices in the conventional method, the Σ value and the grain interface index, which are specific quantities of the corresponding grain boundary, can be specified, so that the corresponding grain boundary model can be easily generated.

  On the other hand, for the second grain interface generated in the periodic model, it is possible to maintain the correspondence of atoms at the grain interface by matching the basic translation vector of the periodic cell with the vector between corresponding lattice points. In addition, for the possibility of multiple models generation such as grain boundary symmetry element transformation and cell shape, selectable options in the program based on crystal structure symmetry and translation vector orthogonality Extract automatically. As a result, the user can efficiently select and generate a variety of corresponding grain boundary models including the homogeneous interface.

Embodiments of the present invention will be specifically described with reference to the drawings. In the present invention, theoretically, two crystals A and B sandwiching an interface as shown in FIG. At the corresponding grain boundary, the crystal orientations of A and B have a certain rotational relationship. For example, 60 rotations with the cubic [111] as an axis gives a Σ3 corresponding grain boundary. However, there are multiple rotation operations that produce the same Σ value (Reference 2: PH Pumphrey and KM Bowkett, (1971) "Axis / angle pair descriptions of coincidence site lattice grain boundaries", Scripta Metallurgica, Vol. 5, pp. 365. -370), different grain boundary structures occur depending on the anisotropy of the crystal structure. A plurality of rotation operations at the same Σ value can be expressed by a product of one representative rotation operation and its symmetric element transformation. In the prior art document, a representative rotation operation at Σ ≦ 49 is given, and 24 symmetrical elements (E, 9C4, 8C3, 6C2) are known in the cubic (Reference 3: H. Grimmer et al. (1974), "Coincidence-site lattices and complete pattern-shift lattices in cubic crystals", Acta Crystallographica, Vol. A30, pp. 197-207). This representative rotation operation and symmetrical element conversion operation are represented by 3 × 3 coordinate conversion matrices RΣ and R _V , respectively.

Next, regarding the grain interface index, there is no limit on the grain interface in the corresponding lattice theory, so there is an infinite number of possibilities for grain boundary selection and accompanying grain boundary structure changes. However, in reality, there are many low-energy grain boundary grain boundaries with good grain boundary symmetry and consistency that are subject to material simulation. The consistency of grain boundaries can be evaluated by the density of corresponding lattice points on the grain interface (points where the atomic coordinates of two crystals match). Therefore, G _ijk k satisfying the following relationship is searched from the corresponding lattice points obtained by the corresponding lattice theory.


G _ijk = iC _x + jC _y + kC _z , (1a)

G _ijk · F _A = 0 (1b)

Here, C _x , C _y , and C _z are basic translation vectors of the corresponding lattice in the crystal coordinate system, and F _A is the grain interface index of the crystal A. By the way, in the cubic crystal, the Miller index vector expressed by the grain interface index is perpendicular to the grain interface. G _ijk forms a two-dimensional grid on the grain interface. The basic translation vectors of this grid are G _Y and G _{Z in the} counterclockwise direction when viewed from the crystal B side. The corresponding lattice density ρ _G on the grain interface is

ρ _G = │G _Y × G _Z │ ^-1 (2)

Is required. Reference 3 gives C _x , C _y , and C _z in a simple cubic Σ ≦ 49. If the crystal lattice is a body-centered lattice or a face-centered lattice, corresponding lattice points are added, and ρ _G , G _Y , and G _z change.

The cell shape and coordinate conversion operation of the grain boundary model satisfying the periodic boundary condition are obtained as follows. First, let Gx be a vector from a corresponding lattice point on the grain interface to an arbitrary corresponding lattice point on the crystal B side. Since the corresponding lattice is a superlattice of the crystal lattice of crystals A and B, the parallelepiped made of G _x , G _Y , and G _z has a period that satisfies the periodicity of atomic arrangement on the crystal A, B, and grain interface. Can be used as a cell. In that case, since each vertex of the cell is an atom corresponding position, the correspondence relationship of atoms at the second grain interface generated by periodicity is also maintained.

Next, consider the transformation from the crystal coordinate system to the model coordinate system. In order to satisfy the continuity of the grain interface, the grain interface needs to be parallel to the reference plane of the model coordinate system. The coordinate transformation matrix RA with the grain interface in the yz plane and G _Y in the y direction is

R _A = (F _A ^* | G _Y ^* | G _Z † ^* ) ^-1 (3)

Is required. Here, (a | b | c) is a matrix in which column vectors a, b, and c are arranged horizontally, G _Z † is a projection vector of G _{Z in} the z-axis direction, and * is a unit vector. By this coordinate transformation, the basic translation vectors X, Y, Z in the model coordinate system and the atomic coordinates r _iA , r _{iB of the} crystals A, B are obtained by the following equations.

X = R _A G _X (4a)
Y = R _A G _Y (4b)
Z = R _A G (4c)
r _{i A} = R _A r _i + Δ _A (5a)
r _{i B} = R _{A RΣR V} r _i + Δ _B (5b)

Here, r _i is the i-th atomic coordinate (including the periodic structure) in the crystal coordinate system, and Δ _A and Δ _B are translational components of the crystals A and B with respect to the model coordinate origin. By extracting r _iA and r _iB in a parallelepiped made of X, Y, and Z, a basic periodic structure corresponding to the corresponding grain boundary model of the minimum size can be obtained. In addition, when Δ _A and Δ _B other than zero are specified, the corresponding atom position shifts at the grain interface, so the generation of adjacent atoms near the grain boundary and the loss of structural similarity between the two grain boundaries in the model Problems occur.

In an actual simulation, a model having a size corresponding to the purpose is required. Therefore, the basic periodic structure obtained above is stacked in an integral multiple in each axial direction to obtain a desired size. _{L x} times the crystal A in the -X direction, + X m _x times the crystal B direction, Y, respectively l _y times the Z-direction, multiplied by l _z, adding translation component corresponding to r _iA, the r _iB. At that time, in order to avoid the overlap of atoms at the boundary (including the grain interface), it is determined in advance to which cell the atom on the boundary surface belongs. The translation vectors of the entire corresponding grain boundary model thus formed are (l _x + _mx ) X, l _y Y, and l _z Z. It is also possible to specify real numbers for l _x and m _x , in which case the atomic correspondence at the second grain interface in the model is broken, but it is lost due to Δ _A and Δ _B other than zero. It is also possible to restore the structural similarity between the two grain boundaries.

In summary, the corresponding grain boundary model that satisfies the periodic boundary condition can be generated by the following procedure.
(1) From the designated Σ value, the representative rotation operation matrix R _Σ and the corresponding lattice basic translation vectors C _x , C _y , C _z of the crystal coordinate system are obtained.
(2) The translation vectors G _x , G _Y , G _z in the crystal coordinate system are obtained based on the specified grain interface index and the formula (1a, b).
(3) The basic translation vectors X, Y, and Z of the periodic cell in the model coordinate system are obtained by the equations (4a, b, c).
(4) Specify translational movement components Δ _A and Δ _B of each crystal.
(5) The atomic coordinates r _iA and r _iB in the parallelepiped cell having X, Y, and Z as sides are obtained by the equation (5a, b).
(6) The obtained basic structure is periodically piled up to make a model of a scale suitable for the purpose.

  According to the present invention, 1) it is possible to provide a corresponding grain boundary model creation program that can easily generate a corresponding grain boundary model that satisfies a periodic boundary condition and a method for operating the same without a high level of expertise. It is possible to provide software that generates cell shapes and atomic coordinates of the corresponding grain boundary model, and 3) it is possible to realize a corresponding grain boundary modeling method that enables operation via a network.

Next, the basic operation of the above method will be described.
The processing algorithm of the program of the present invention is shown in FIG. Hereinafter, it demonstrates along the flow.
[Input 1] Start the program and input the crystal structure data (space group, lattice constant, element and atomic coordinates) of the target crystal.
[Process 1] Based on the inputted crystal structure data, all atomic coordinates in the crystal unit cell are calculated, and the independent symmetric element conversion operation R _V is listed.

[Input 2] Input the Σ value of the target corresponding grain boundary. For those having a plurality of correspondence relationships with the same Σ value (such as Σ13), the type is also specified.
[Process 2] Corresponding grids C _x , C _y , and C _z at the specified Σ values are obtained. A set of grain interfaces having a larger corresponding lattice density ρG on the grain interface with respect to this lattice is searched and listed as a candidate. At that time, those which become the same grain interface by symmetrical transformation are excluded from the candidates. In this process, translation vectors G _x , G _Y and G _z are also obtained.

[Input 3] Select the target one from the grain interface pair options listed in Process 2. In addition, the same type / different interface is designated by selecting the symmetry element transformation for the crystals A and B. It is also possible to program to directly input a grain interface index that is not an option.
[Process 3] For the specified grain interface, the options of translation vectors G _x , G _Y , and G _z are examined. If these vectors are orthogonal, they can be left as they are. In the case of non-orthogonal, it is useful in actual simulation if a linear combination of G _Y and G _z that are orthogonal and other options of G _x are listed.

[Input 4] Input the translation amount of two crystals. The basic model with atoms on the grain interface is all zero.
[Input 5] From the translation vector choices listed in the process 3, the one that matches the target model cell shape is selected. Furthermore, cell sizes l _x , l _y , l _z , and _mx are designated according to the scale of simulation.
[Process 4] The atomic coordinates in the model are calculated based on the value specified by the input 2-5.
[Process 5] The atomic arrangement and cell shape obtained in Process 4 are displayed on the screen. If the result is satisfactory, go to input 6. If the model is to be modified, return to input 2-5.

[Input 6] Enter the file name and file format for outputting atomic coordinate data.
[Process 6] Output atomic coordinates, elements, periodic cell shape information, etc., in the format specified in the file with the specified name.

Next, specific means in which processing by predetermined software for basic operation of the above method is performed in cooperation with hardware resources will be described.
[Input 1] The program is started by a programmed computer, and crystal structure data (space group, lattice constant, element and atomic coordinates) of the target crystal is input by the input means and stored in the storage means.
[Process 1] Based on the inputted crystal structure data, all atomic coordinates in the crystal unit cell are calculated by the data processing means, processed by the data exchange means, and the independent symmetric element conversion operation R _V is listed. Crystal atom coordinates are generated and stored in storage means.

[Input 2] The Σ value of the target corresponding grain boundary is input by input means. For those having a plurality of correspondence relationships with the same Σ value (such as Σ13), the type is also specified and stored in the storage means.
[Process 2] Corresponding grids C _x , C _y , C _z at the specified Σ values are obtained and stored in the storage means as corresponding grid point information. A search means searches for a set of grain interfaces having a larger corresponding lattice density ρG on the grain interface than the lattice, reads out data, lists them as candidates, and stores them in the storage means. At that time, those which become the same grain interface by symmetrical transformation are excluded from the candidates. In this process, translation vectors G _x , G _Y and G _z are also obtained.

[Input 3] A target one is selected from the grain interface pair options listed in the process 2 and input by an input means. Further, the same type / different interface is designated by selecting the symmetric element conversion for the crystals A and B, and stored in the storage means. It is also possible to program by an information processing means so that a grain interface index that is not an option can be directly input.
[Process 3] The specified grain interface is searched by the search means for options of translation vectors G _x , G _Y , G _z , and data is read out. If these vectors are orthogonal, they can be left as they are. In the case of non-orthogonal, it is useful in actual simulation if a linear combination of G _Y and G _z that are orthogonal and another option of G _x are listed and stored in the storage means.

[Input 4] The translational movement amounts of the two crystals are input and stored in the storage means. The basic model with atoms on the grain interface is all zero.
[Input 5] From the translation vector choices listed in the process 3, one that matches the target model cell shape is selected and stored in the storage means. Furthermore, cell sizes l _x , l _y , l _z , and _mx are designated according to the scale of simulation and stored in the storage means.
[Process 4] Based on the value specified by the input 2-5, the atomic coordinates in the model are calculated by the data processing means.
[Process 5] The atomic arrangement and cell shape obtained in Process 4 are displayed on the screen of the display device. If the result is satisfactory, go to input 6. If the model is to be modified, return to input 2-5.

[Input 6] The file name and file format for outputting atomic coordinate data are input by the input means and stored in the storage means.
[Process 6] Output information such as atomic coordinates, element, and periodic cell shape information in the format specified in the specified file.
Next, the stand-alone operation will be described. The program operates by compiling the source code written according to the above algorithm and installing the obtained execution module in the computer.

  Next, the operation on the network will be described. This program is created in a machine-independent form such as a Java (registered trademark) applet, input to the server, and disclosed on the network, so that the operation via the network is performed. Is possible. That is, the user's server can operate the program by the step of specifying an address on the server computer from the web browser of the computer at hand and obtaining a predetermined address. In this case, the installation work on the user's computer and the recompilation work according to the model are not required, and the burden can be greatly reduced. FIG. 3 shows a state in which a Java (registered trademark) applet is actually operating.

  FIG. 4 shows the output result of the silicon Σ15 (1-20) / (2-4 5) asymmetric grain boundary model generated by the program written by the method of the present invention. The input data is as shown in Table 1.

Table 1
Space group Fd3m
Crystal atom coordinates 0
Σ value 15
Grain interface index (A / B) (1-20) / (2-4 5)
Symmetric element (A / B) E / E
Crystal A translational component Δ _A 0
Crystal B translational component Δ _B 0
Translation vectors G _x , G _Y , G _z [3-60], [210], [003] (3-axis orthogonal)
Cell size l _x , l _y
, L _z, m _x 1,2,2,1

  As described above in detail, the present invention relates to a program and an operation method for creating a corresponding grain boundary model. According to the present invention, 1) corresponding grains satisfying the periodic boundary condition without advanced expertise. It is possible to provide a corresponding grain boundary model creation program that can easily generate a boundary model and its operation method, 2) to provide software that generates cell shapes and atomic coordinates of the corresponding grain boundary model having a periodic structure, and 3) a network. It is possible to achieve a corresponding grain boundary modeling method that enables operation via the.

The shape of a grain boundary model (prior art and the present invention) is shown. The processing algorithm in this invention is shown. An example in which a Java (registered trademark) applet is operated on a web browser (how to take model coordinate axes is different from the text description) is shown. An example of an actually generated silicon Σ15 (1-20) / (2-4 5) asymmetric correspondence grain boundary model is shown.

Claims (6)

To create a corresponding grain boundary model,
Means for listing candidates for grain interface orientation on the display screen from the Σ value of the specified corresponding grain boundary,
Means for listing translation vector candidates of a corresponding grain boundary model from a specified grain boundary orientation;
Means for determining the periodic cell shape of the corresponding grain boundary model from the specified translation vector and its size;
Means for calculating the atomic coordinates of the model based on the space group, lattice constants, the crystal structure data comprising data elements and atomic coordinates,
Means for outputting the result to a display screen and a file;
Corresponding grain boundary model creation program to function as   2. The program according to claim 1, wherein the grain interface orientation is designated from a finite number of candidates extracted by the density of the corresponding lattice points on the grain interface and the symmetry of the grain boundary structure.   3. The grain interface orientation is decomposed into a product of a basic orientation and an independent symmetric element transformation, specified individually, and facilitates the generation of a homogeneous interface produced by symmetry transformation. Program.   The computer executes the process of determining the periodic cell shape by matching each side of the periodic cell of the corresponding grain boundary model with the corresponding interstitial vector, so that the periodicity of the model is equivalent to the two grain interfaces in the model. The program according to claim 1, wherein the program is secured. The program, space group, lattice constants, elemental and combined with atomic coordinates crystal structure data or program from which produce crystals atomic coordinates consisting of data which is executed in one computer of claim 1, wherein program.   2. The program according to claim 1, wherein the program is input to a computer having a server function and executed on a web browser of a user computer via the Internet.