JP2021057163A - Search method of diffusion path - Google Patents

Abstract

To provide a search method of a diffusion path, capable of effectively searching a diffusion path of an atom in a crystal.SOLUTION: A search method of a diffusion path, includes: an initial structure setting step of setting a position of a plurality of atoms other than a diffusion atom; a reference system calculation step of calculating an average structure of the plurality of atoms by performing a molecular dynamic calculation to which a position has been set in the initial structure setting step with the plurality of atoms; an interaction energy calculation step of obtaining an interaction energy of the diffusion atom and a circumference atom of the diffusion atom when the diffusion atom is arranged at a lattice point of a lattice by setting a three-dimensional lattice to the average structure of the plurality of atoms obtained by the reference system calculation step; a repetition step of obtaining the interaction energy of the diffusion atom and the circumference atom of the diffusion atom when changing the lattice point for arranging the diffusion atom; and a diffusion path search step of searching a diffusion path of the diffusion atom from the diffusion atom obtained by the interaction energy calculation step and the repetition step, and the interaction energy of the diffusion atom and the circumference atom.SELECTED DRAWING: Figure 2

Description

The present invention relates to a method for searching for a diffusion path.

For the purpose of further improving the performance of various materials, the search for new materials and the selection / search of substitution elements that replace some of the elements constituting the substance are being actively carried out.

For example, in Patent Document 1, as a method for selecting a substitution element for a lithium composite oxide, _{when nickel of the lithium composite oxide LiMO 2} (M contains nickel) is replaced with a candidate element, the candidate element is contained in the nickel site. The solid solubility determination process to determine whether or not
Is the adsorption energy Ea of water molecules on the crystal surface when the lithium composite oxide nickel is replaced by the candidate element smaller than the adsorption energy Eb of the water molecules on the crystal surface of the lithium composite oxide before the replacement with the candidate element? Adsorption energy determination process to determine
When it is determined that the candidate element is contained in the nickel site in the solid solubility determination step and Ea <Eb is determined in the adsorption energy determination step, the candidate element is replaced with nickel of the lithium composite oxide. A selection step for selecting an element and a method for selecting a substitution element for a lithium composite oxide having the same are disclosed.

JP-A-2018-140917

As described above, in selecting and searching for new materials and substitution elements, atoms that affect the target reaction, function, etc. move within the crystal of the target substance. , It is preferable to accurately grasp whether it spreads.

However, the elements are often densely packed in the crystal, and it seems that there are only slight gaps between the atoms constituting the crystal. For this reason, when estimating the atom size from the atomic radius or van der Waals radius for the diffused atom, which is the atom for which the diffusion path is to be investigated, the size of a plurality of gaps in the crystal is compared and the diffusion path of the diffused atom is compared. Was difficult to identify.

In addition, due to the electrostatic interaction between the diffused atom and the surrounding atoms, the diffused atom does not always pass through a wide space in the crystal. Therefore, as described above, it is difficult to specify the diffusion path of the diffusion atom from the atomic radius of the diffusion atom and the size of the gap in the crystal.

Therefore, in view of the problems of the prior art, one aspect of the present invention is to provide a method for searching for a diffusion path that can efficiently search for a diffusion path of atoms in a crystal.

According to one aspect of the present invention in order to solve the above problems.
The initial structure setting process for setting the positions of multiple atoms other than the diffusion atom contained in the crystal used for the diffusion path search, and the initial structure setting step.
A reference system calculation step of calculating the average structure of a plurality of the atoms by performing a molecular dynamics calculation using the plurality of the atoms whose positions are set in the initial structure setting step.
When a three-dimensional lattice is set in the average structure of a plurality of the atoms obtained in the reference system calculation step and the diffusion atoms are arranged at the lattice points of the lattice, the diffusion atoms and the surroundings of the diffusion atoms The process of calculating the interaction energy to obtain the interaction energy with the atom,
A repetitive step of obtaining the interaction energy between the diffused atom and the atoms around the diffused atom when the lattice point of the lattice in which the diffused atom is arranged is changed.
It has a diffusion path search step of searching for a diffusion path of the diffusion atom from the interaction energy of the diffusion atom obtained in the interaction energy calculation step and the diffusion atom and atoms around the diffusion atom. A method for searching for a diffusion route is provided.

According to one aspect of the present invention, it is possible to provide a method for searching for a diffusion path that can efficiently search for a diffusion path of atoms in a crystal.

The schematic diagram which shows the initial structure of _{LiMn 2} O ₄ which searched the diffusion path in Example 1. FIG. The schematic diagram which shows the diffusion path of the Li atom of _{LiMn 2} O ₄ obtained in Example 1. FIG. The schematic diagram which shows the diffusion path of the Li atom of _{LiMn 2} O ₄ obtained in the comparative example 1. FIG.

Hereinafter, embodiments for carrying out the present invention will be described, but the present invention is not limited to the following embodiments, and various modifications and variations to the following embodiments are made without departing from the scope of the present invention. Substitutions can be added.

The diffusion path search method of the present embodiment can have the following steps.
An initial structure setting step for setting the positions of a plurality of atoms other than the diffusion atom contained in the crystal used for the diffusion path search.
A reference system calculation process that calculates the average structure of multiple atoms by performing molecular dynamics calculations using multiple atoms whose positions were set in the initial structure setting process.
Interaction between diffused atoms and atoms around the diffused atoms when a three-dimensional lattice is set for the average structure of multiple atoms obtained in the reference system calculation process and diffused atoms are placed at the lattice points of the lattice. Interaction energy calculation process for energy.
A repetitive process of finding the interaction energy between a diffused atom and the atoms around the diffused atom when the lattice point of the lattice is changed to place the diffused atom.
A diffusion path search step for searching the diffusion path of a diffusion atom from the interaction energy between the diffusion atom obtained in the interaction energy calculation step and the repetition process and the atoms around the diffusion atom.

The inventor of the present invention has diligently studied a method for efficiently searching for the diffusion path of atoms in a crystal.

When the diffusion atom, which is the atom for which the diffusion path is to be investigated, is at a point in the diffusion path, the free energy of the diffusion atom at that point is smaller than the free energy when the diffusion atom is outside the diffusion path. Therefore, the inventor of the present invention has focused on the fact that the diffusion path of a diffusion atom can be searched by searching for a path in which the free energy of the diffusion atom becomes small.

Therefore, the inventor of the present invention further investigated a method for efficiently searching for a path in which the free energy of the diffused atom becomes small. As a result, by treating the interaction between the diffusion atom, which is the atom whose diffusion path is to be investigated, and the surrounding atoms as a perturbation, that is, a correction term, it becomes possible to use a crystal system that does not contain the diffusion atom as a reference system. , We have found that the cost of implementing the molecular dynamics method can be suppressed and the diffusion path can be derived efficiently, and the present invention has been completed.

Each step will be described below.
(Initial structure setting process)
In the initial structure setting step, the positions of a plurality of atoms other than the diffusion atom contained in the crystal used for the diffusion path search can be set. That is, the initial coordinates of the crystal used in the reference system calculation step can be set.

In the initial structure setting step, the specific method for setting the positions of a plurality of atoms other than the diffusion atoms contained in the crystal used for the diffusion path search is not particularly limited. For example, the atomic arrangement of each atom can be set based on the crystal structure of the crystal, which is experimentally obtained or disclosed in the literature, and can be used as the initial structure.
(Reference system calculation process)
In the reference system calculation step, molecular dynamics calculation can be performed using a plurality of atoms whose positions are set in the initial structure setting step. Then, the average structure of a plurality of atoms can be calculated from the result of the molecular dynamics calculation.

Molecular dynamics calculation is a computer simulation method of the physical movement of an atom, and the time evolution of the position of an atom can be obtained by numerically solving Newton's equation of motion. Therefore, by carrying out the reference system calculation step, it is possible to obtain the time-series change of the coordinates of a plurality of atoms set in the initial structure setting step.

In molecular dynamics calculations, information on atom-to-atom interactions is represented by a functional form for describing potential energy and its parameter set (force field).

The type of force field used in the molecular dynamics calculation in the reference system calculation process is not particularly limited, and various force fields can be used. For example, metal / alloy systems such as EAM and MEAM, inorganic compound systems such as Buckingham, BKS, Clay-FF, CVFF_aug, semiconductor systems such as Tersoff, organic compound systems such as PCFF, Compass, MMFF, OPLS-AA, AMBER, CHARMM, UFF and the like can be used. In addition, a force selected from existing force fields such as X-Pol, AMBER polarization force field, CHARMM polarization force field, which is a polarization force field, ReaxFF, which is a reaction force field, and a force field created by oneself as needed. A field can be used.

In the existing force field, the charge of the target atom may not be specified. In that case, RESP (Restrained Electrostatic Potential) charge, AM1-BCC (Bond Charge Direction) charge, and the like can also be used.

The program (software) used for molecular dynamics calculation is also not particularly limited, but is selected from existing programs such as LAMMPS, DL_POLY, Grocass (Groningen Machine for Chemical Simulations), AMBER, CHARMM, and NAMD, and self-made programs. Program can be used.

The set environment for performing the molecular dynamics calculation can be, for example, a vacuum or a periodic boundary condition when a solvent is contained.

The numerical integration method for solving Newton's equation of motion when performing molecular dynamics calculations is also not particularly limited, but is selected from, for example, the Verlet method, the velocity Verlet method, the Leap-frog method, the predictor-corrector method, and the like. Method can be used.

The time width for performing the molecular dynamics calculation is not particularly limited, but it is preferable to select it so that the directions in which a plurality of atoms constituting the crystal can move easily can be grasped and the calculation cost can be suppressed. The time width for performing the molecular dynamics calculation can be, for example, 0.5 fs or more and 2 fs or less.

Further, the temperature control method is not particularly limited, and is selected from, for example, a speed scaling method, a Nose-Hoover heat bath method, a Nose-Hoover chain method, a Berendsen heat bath method, an Andersen heat bath method, a Langevin dynamics method, and the like. Method can be used.

The pressure control method under periodic boundary conditions is also not particularly limited, and for example, a method selected from the Berendsen method, the Parinello-Rahman method, and the like can be used.

The cutoff method can be used to calculate long-range interactions such as electrostatic interactions and van der Waals interactions. In particular, the Particle-Mesh Ewald method, the multipole expansion method, or the like can be used for the calculation of the electrostatic interaction under the periodic boundary conditions.

Molecular dynamics calculations in the reference system calculation process include, for example, CPU (Central Processing Unit), RAM (Random Access Memory), various storage media such as hard disks, output devices such as displays, input devices such as keyboards, and various peripheral devices. It can be carried out using a normal computer system equipped with such as. Examples of computer systems include network servers, workstations, personal computers, and the like.

Specifically, for example, the above-mentioned molecular dynamics calculation program is stored in a storage medium or the like, the program is executed by the CPU, and is stored in a storage medium such as a RAM or an input device such as a keyboard. It can be realized by reading the initial structure and conditions input from.

By carrying out the molecular dynamics calculation, the positions of a plurality of atoms set in the initial structure setting step can be changed over time. Therefore, the average structure of a plurality of atoms can be calculated from the change in the position of a plurality of atoms obtained by using the molecular dynamics method in the reference system calculation step. Specifically, for example, with respect to the temporal change of the positions of a plurality of atoms obtained by the molecular dynamics method, the positions of the plurality of atoms are recorded at predetermined time intervals, and the position information of each atom is averaged. The average structure of a plurality of atoms, specifically the average coordinates, can be obtained.
(Interaction energy calculation process)
In the interaction energy calculation step, first, a three-dimensional lattice can be set in the average structure of a plurality of atoms obtained in the reference system calculation step. The three-dimensional lattice to be set is a virtual one, and is provided to determine the position where the diffusion atom is arranged in the interaction energy calculation step and the repetition step described later.

Then, the interaction energy between the diffused atom and the atoms around the diffused atom when the diffused atom is arranged at the lattice point of the set lattice is obtained.

The lattice spacing of the three-dimensional lattice set in the average structure of a plurality of atoms and the shape of the lattice are not particularly limited. It can be arbitrarily selected according to the average structure of multiple atoms and the accuracy required for the desired diffusion system.

Then, among the plurality of lattice points of the set lattice, the diffused atom is arranged at an arbitrary point, and the diffused atom is arranged between the diffused atom and the atom around the diffused atom. The interaction energy (potential energy) can be obtained from the distance.

The interaction energy is the potential energy of the electrostatic interaction and the repulsive force interaction between the diffusion atom and the atoms around the diffusion atom, and the calculation method thereof is not particularly limited. The interaction energy is preferably obtained by the following formula (1) using, for example, the interatomic distance between an atom within a predetermined cutoff radius from the diffusion atom and the diffusion atom.

なお、上記式（１）中のＺ_ｉ、Ｚ_ｊはそれぞれ原子i、原子ｊの実効電荷、eは電気素量、ｒ_ｉｊは原子iと原子jとの間の距離、ｆ_０は定数、ａ_ｉとｂ_ｉは、それぞれ、原子iの大きさと剛性に関連するパラメータ、ａ_ｊとｂ_ｊは、それぞれ、原子ｊの大きさと剛性に関連するパラメータである。
（繰り返し工程）
繰り返し工程では、相互作用エネルギー算出工程で設定した、３次元の格子の格子点のうち、既に相互作用エネルギーを求めた格子点とは異なる格子点に拡散原子を配置し、相互作用エネルギー算出工程の場合と同様にして相互作用エネルギーを算出できる。

In the above equation (1), Z _i and Z _j are the effective charges of the atom i and the atom j, e is the elementary charge, r _ij is the distance between the atom i and the atom j, and f ₀ is a constant. a _i and b _i are each, parameters related to the size and rigidity of the atoms i, a _j and b _j are each parameter related to the size and rigidity of the atom j.
(Repeat process)
In the iterative process, among the three-dimensional lattice lattice points set in the interaction energy calculation step, diffusion atoms are placed at lattice points different from the lattice points for which the interaction energy has already been obtained, and the interaction energy calculation step is performed. The interaction energy can be calculated in the same way as in the case.

In the repeating step, for example, diffusion atoms can be arranged at all the lattice points of the set three-dimensional lattice, and the process can be repeated until the interaction energy is calculated. Further, for example, among the lattice points of the three-dimensional lattice, diffusion atoms can be arranged for each of the selected lattice points, and the process can be repeated until the interaction energy is calculated.
(Diffusion route search process)
In the diffusion path search step, the diffusion path of an atom can be searched from the interaction energy between the diffusion atom obtained in the interaction energy calculation step and the repetition step and the atoms around the diffusion atom.

As described above, in the diffusion path search method of the present embodiment, the diffusion path of a diffusion atom is searched by searching for a path in which the free energy between the diffusion atom and surrounding atoms becomes small. The free energy G (x) has a relationship with the interaction energy V (r) by the following equation (2).

_{exp (-G (x) / k} B T) α <exp (-V (r) / k B T)> ··· (2)
Here, x in equation (2) is the reaction coordinate, r is the position vector of all atoms, k _B is the Boltzmann constant, T is the absolute temperature, and <...> is the ensemble average for degrees of freedom other than x.

Therefore, the interaction energy V (r) can be calculated, and the value corresponding to the free energy G (x) can be obtained by using the above equation (2).

Then, the coordinates of the lattice points on which the diffusion atoms are arranged and the values corresponding to the free energy calculated from the interaction energy using the above equation (2) can be mapped, that is, three-dimensionally plotted. From the obtained map, the value corresponding to the free energy G (x) is small, and a continuous region can be recognized as the diffusion path of the diffusion atom.

According to the diffusion path search method of the present embodiment described above, the interaction between the diffusion atom and the atoms around the diffusion atom is treated as a perturbation, so that a crystal containing no diffusion atom is used as a reference system. It is possible to suppress the calculation cost when executing the molecular dynamics calculation.

In addition, a three-dimensional lattice is set in the average structure of the reference system obtained by molecular dynamics calculation, and free energy is derived from the interaction energy when diffused atoms are arranged at the lattice points, and the diffused atoms in the crystal are derived. The value corresponding to the free energy can be obtained for the entire crystal. As a result, the free energy is small, a continuous region can be recognized as a diffusion path, and the diffusion path can be searched efficiently.

Hereinafter, the present invention will be described in more detail with reference to Examples. However, the present invention is not limited to the following examples.
[Example 1]
The diffusion path of the Li atom in _{LiMn 2} O ₄ was searched by the following procedure.
(Initial structure setting process)
The initial structure of LiMn ₂ O _{4 was set. Specifically, as shown in FIG. 1, a structure 10 of LiMn 2} O ₄ in which a lithium atom 11, a manganese atom 12, and an oxygen atom 13 are arranged is set in the cell. Then, the lithium atom 111 arranged in the center of the cell, which is a diffusion atom, was deleted to obtain an initial structure. In addition, in FIG. 1, it is shown that atoms having the same hatching are atoms of the same type.
(Reference system calculation process)
Next, using molecular dynamics calculation, the time-series changes of the coordinates of the lithium atom 11, the manganese atom 12, and the oxygen atom 13 whose positions were set in the initial structure setting step were obtained.

The molecular dynamics calculation was performed using LAMMPS as software and the force field developed by Suzuki et al. Of Tohoku University. Then, the coordinates of each atom were input and set as the environment setting in the crystal.

In addition, the velocity Verlet method was used as the velocity calculation method when performing the molecular dynamics calculation, and the time width was set to 1 fs. The Nose-Hoover chain method was used as the temperature control method, and the set temperature was set to 300 K.

The Particle-Mesh Ewald method was used for the calculation of long-distance interactions.

Under the above conditions, a molecular dynamics calculation of 10 ns was performed, and the position of each atom was preserved every 1 ps.

Then, the positions of the stored atoms were averaged, and the reference system, that is, the average structure of the plurality of atoms was calculated.
(Interaction energy calculation process)
A lattice with a lattice spacing of 0.2 Å, that is, a three-dimensional lattice in which a plurality of cubic cells with a side length of 0.2 Å are arranged is set in the average structure of a plurality of atoms obtained in the reference system calculation step. .. Then, when lithium, which is a diffusion atom, is arranged at any one lattice point of the lattice, the interaction energy between the lithium atom and the atom located within a predetermined cutoff radius from the lithium atom is described above. It was calculated by the formula (1) of.

When calculating the interaction energy by the above-mentioned equation (1), the parameters developed by Suzuki et al. Of Tohoku University are used. Specifically, f ₀ = 4.1868 [kJ / molÅ], and the values shown in Table 1 below were used as the parameters of each atom (ion).

（繰り返し工程）
相互作用エネルギー算出工程で設定した３次元の格子の格子点のうち、既に拡散原子であるリチウム原子を配置し、相互作用エネルギーを算出した点とは異なる格子点にリチウム原子を１個配置した。そして、係るリチウム原子と、該リチウム原子の周囲の原子との相互作用エネルギーを相互作用エネルギー算出工程の場合と同様にして算出した。

(Repeat process)
Among the lattice points of the three-dimensional lattice set in the interaction energy calculation step, the lithium atom, which is a diffusion atom, was already arranged, and one lithium atom was arranged at a lattice point different from the point where the interaction energy was calculated. Then, the interaction energy between the lithium atom and the atoms around the lithium atom was calculated in the same manner as in the case of the interaction energy calculation step.

Lithium atoms were arranged at all the lattice points in the same manner as described above, except that the lattice points at which the lithium atoms, which are diffusion atoms, were arranged were changed, and the process was repeated until the interaction energy was calculated.
(Diffusion route search process)
From the interaction energy between the diffusion atom and the atoms around the diffusion atom obtained in the interaction energy calculation step and the repetition step, the value corresponding to the free energy was calculated by the above-mentioned equation (2).

Then, the coordinates of the diffusion atom and the value corresponding to the free energy were mapped, and from the obtained map, a continuous region having a small free energy was searched and used as a diffusion path.

The diffusion path of the diffusion atom obtained in the diffusion path search step is shown in FIG.

As shown in FIG. 2, as the diffusion path of the lithium atom 111, the diffusion paths 211 to 214 were found, and the diffusion path is the lithium atom 111 (see FIG. 1), which is a diffusion atom arranged in the center of the cell. ) And the lithium atoms arranged around the lithium atom 111.

The diffusion paths 211 to 214 _{are consistent with the diffusion paths of lithium atoms in LiMn 2} O ₄ reported so far, and the diffusion path search method used in this example is in line with the actual phenomenon. I was able to confirm that.
[Comparative Example 1]
(Initial structure setting process)
_{The initial structure of LiMn 2} O ₄ was set in the same manner as in the case of Example 1. More specifically, in the cell, as shown in FIG. 1, a lithium atom 11, a manganese atom 12, and an oxygen atom 13 sets the structure 10 of the LiMn ₂ O ₄ which is arranged, and the initial structure. In this comparative example, a lithium atom 111, which is a diffusion atom, was also arranged.
(Diffusion route search process)
The diffusion path was searched from the position of each atom and the atomic radius of the lithium atom 111, which is a diffusion atom.

As a result, the diffusion path 31 could be searched as shown in FIG.

As is clear from the comparison with the results of the above Examples, it was not possible to search all the diffusion routes in this Comparative Example. Further, since the diffusion path is searched from the position of each atom and the atomic radius of the lithium atom 111, it takes a lot of time to search for the diffusion path, and the diffusion path is efficiently searched as in the first embodiment. I couldn't.

Claims (1)

The initial structure setting process for setting the positions of multiple atoms other than the diffusion atom contained in the crystal used for the diffusion path search, and the initial structure setting step.
A reference system calculation step of calculating the average structure of a plurality of the atoms by performing a molecular dynamics calculation using the plurality of the atoms whose positions are set in the initial structure setting step.
When a three-dimensional lattice is set in the average structure of a plurality of the atoms obtained in the reference system calculation step and the diffusion atoms are arranged at the lattice points of the lattice, the diffusion atoms and the surroundings of the diffusion atoms The process of calculating the interaction energy to obtain the interaction energy with the atom,
A repetitive step of obtaining the interaction energy between the diffused atom and the atoms around the diffused atom when the lattice point of the lattice in which the diffused atom is arranged is changed.
It has a diffusion path search step of searching for a diffusion path of the diffusion atom from the interaction energy of the diffusion atom obtained in the interaction energy calculation step and the diffusion atom and atoms around the diffusion atom. How to search for a diffusion path.

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