JP2022149869A - Generating method of filler model - Google Patents

Abstract

To provide a generating method of a filler model capable of appropriately representing an outer surface of a filler.SOLUTION: The generating method includes the steps of: defining a virtual space including an entire filler region; disposing a plurality of particle models inside the virtual space; and deleting the particle model existing outside the filler region and generating a filler model in which the plurality of particle models are disposed inside the filler region. The step of disposing the particle models includes: a step of temporarily disposing the plurality of particle models inside the virtual space; a step of defining a first potential defined in advance between the particle models; a first minimization step of calculating minimization of a potential energy for the plurality of particle models; a step of defining a second potential that applies stronger repulsive force than the first potential between the adjacent particle models after the first minimization step; and a second minimization step of calculating minimization of the potential energy for the plurality of particle models.SELECTED DRAWING: Figure 5

Description

The present invention relates to a method and the like for creating a filler model.

In recent years, various simulation methods (numerical calculations) have been proposed for evaluating the properties of composite materials composed of fillers and polymeric materials using computers for the development of polymeric materials such as rubber compounds. In this type of simulation method, a filler model is created by modeling the filler (see, for example, Patent Document 1 below).

In the method of Patent Document 1 below, a plurality of particle models arranged inside a predetermined filler region are combined with each other to create a filler model including a plurality of aggregates. The filler region is set based on the three-dimensional structure of the actual filler. Inside the filler region, a plurality of particle models are arranged in a crystal lattice such as a face-centered cubic lattice.

Japanese Patent No. 5913260

In the above method, the outer surface of the filler model is formed in regular steps. Such a stepped portion has a problem that a behavior that does not occur in the original filler occurs in the simulation.

The present invention has been devised in view of the actual situation as described above, and a main object of the present invention is to provide a method for creating a filler model that can express the outer surface of the filler more appropriately.

The present invention is a method for creating a filler model for numerical analysis of a filler, comprising a step of inputting a filler region, which is a region where the filler exists, into a computer, and configuring the filler model in the computer. The computer defines a virtual space that includes the entire filler region, and temporarily arranges a plurality of the particle models in the virtual space; defining a predetermined first potential between the particle models; and calculating potential energy minimization based on a molecular mechanics method for a plurality of the particle models based on the first potential. 1 minimization step, after the first minimization step, defining a second potential that exerts a stronger repulsive force than the first potential between the adjacent particle models; and a second minimization step of calculating the minimization of the potential energy for a plurality of the particle models.

In the method for creating a filler model according to the present invention, the step of provisionally arranging a plurality of the particle models may include randomly arranging the particle models based on a condition that allows overlapping of the particle models.

In the method for creating the filler model according to the present invention, the relationship between the first potential and the distance between the adjacent particle models may be defined by an exponential decay function.

In the method for creating the filler model according to the present invention, the second potential may be a Leonard Jones potential cut off by an equilibrium length.

In the method for creating a filler model according to the present invention, after the step of creating the filler model, the computer determines that the filling rate of the particle model inside the filler model is equal to that of the particle model of the outer layer of the filler model. A step of remodeling the filler model so as to be smaller than the filling rate may be further included.

In the method for creating a filler model according to the present invention, the step of modifying the filler model is based on a crystal structure smaller than the filling rate of the outer layer of the filler model, and arranges the particle model inside the filler model. A step may be included.

In the method for creating a filler model according to the present invention, the step of creating the filler model may include a step of combining the adjacent particle models after deleting the particle model protruding from the filler region.

By adopting the above configuration, the filler model creation method of the present invention can be performed with a smaller amount of calculation than the relaxation calculation by the molecular dynamics method or the Monte Carlo method. Only potential energy minimization by the molecular mechanics method can be performed. , the particle model can be arranged in an amorphous structure with a smaller amount of calculation than the conventionally known method. In addition, the method for creating a filler model of the present invention can prevent a regular structure from appearing on the outer surface of the filler model, so that the outer surface of the filler can be expressed more appropriately, and the particle model is densely packed. It is possible to create a filler model with a rigid structure.

In addition, since the amorphous structure created by the above-described procedure has a lower filling factor than the face-centered cubic lattice and the body-centered cubic lattice, the number of particle models can be reduced compared to the conventional method. Therefore, it is possible to save the amount of calculation in the simulation using the created filler model.

It is a perspective view which shows an example of the computer used for the preparation methods of a filler model. It is a flowchart which shows an example of the processing procedure of the preparation method of a filler model. (a) and (b) are conceptual diagrams showing an example of a virtual space in which a primary particle model is arranged. FIG. 4 is a conceptual diagram showing an example of a virtual space filled with amorphous structures of a plurality of particle models; 4 is a flow chart showing an example of a processing procedure of a particle model placement step; FIG. 4 is a conceptual diagram showing an example of a state in which particle models are randomly arranged; FIG. 10 is a conceptual diagram showing an example of a change in the state of a particle model in the particle model placement process; 4 is a graph showing an example of a first potential P1 and a second potential P2; It is a conceptual diagram which shows an example of a filler model. It is a flowchart which shows an example of the processing procedure of the preparation method of other embodiment of this invention. It is a sectional view showing an example of a filler model of other embodiments of the present invention. It is a structural formula of polybutadiene. 4 is a flow chart showing an example of a processing procedure of a simulation method for a polymer material; 1 is a conceptual diagram showing an example of a polymer material model; FIG. It is a conceptual diagram which shows an example of the filler model of a comparative example.

An embodiment of the present invention will be described below with reference to the drawings.
In the method of creating a filler model of the present embodiment (hereinafter sometimes simply referred to as "creating method"), a model for numerical analysis of a filler (filler model) is created. The filler model created by this creation method is used in a polymer material simulation method (hereinafter, sometimes simply referred to as "simulation method"), which will be described later. In this embodiment, a computer is used for the creation method and the simulation method.

Drawing 1 is a perspective view showing an example of computer 1 used for a creation method of a filler model. A computer 1 includes a main body 1a, a keyboard 1b, a mouse 1c and a display device 1d. The main body 1a is provided with an arithmetic processing unit (CPU), a ROM, a work memory, a storage device such as a magnetic disk, disk drive devices 1a1 and 1a2, and the like. The storage device stores in advance a processing procedure (program) for executing the creation method and simulation method of the present embodiment.

Fillers are, for example, silica, carbon black and alumina. A case where the filler of the present embodiment is spherical is exemplified. The filler is formed as a higher-order structure in which, for example, a plurality of secondary aggregates in which spherical primary particles are bonded to each other are non-uniformly dispersed.

FIG. 2 is a flow chart showing an example of a processing procedure of a method for creating a filler model. In the creation method of the present embodiment, first, the filler region 17, which is the region where the filler exists, is input to the computer 1 (step S1). The filler region 17 corresponds to a region in which a particle model 11, which will be described later, may be arranged in the filler model 10 created by the creation method of the present embodiment.

The shape and the like of the filler region 17 can be appropriately set. The shape of the filler region 17 may be set to the shape of the actual three-dimensional structure of the filler, for example, based on observation results using 3DTEM. In step S1, for example, all sets of partial regions in which the filler is present are selected from a set of partial regions obtained by dividing the three-dimensional structure so that the vertical, horizontal, and height are equally spaced. The entirety may be entered as filler region 17 . In addition, the filler region 17 does not need to be one continuous lump, and may include a higher-order structure such as a plurality of isolated aggregates or aggregates.

As shown in FIG. 3, in the present embodiment, the filler region 17 is represented as having one or more primary particles (primary particle model 5 modeling primary particles) having a certain particle size arranged therein. A case is illustrated. As shown in FIG. 3A, for example, when the filler model 10 is composed of one primary particle model 5, the filler region 17 is defined as one spherical region. On the other hand, as shown in FIG. 3B, for example, when the filler model 10 is composed of a plurality of primary particle models 5, the filler region 17 is an aggregate composed of a plurality of primary particles bonded together. , are defined as multiple isolated arrays.

The filler region 17 of this embodiment is input as a set of the particle size D1 of the primary particle model 5 shown in FIG. 3 and the barycenter position of each primary particle model 5 . At this time, the filler region 17 is defined as a region that includes all the primary particle models 5 in a spherical region where the distance from the center of gravity of each primary particle model 5 is less than half the particle size D1. In the filler region 17, the spherical regions of the primary particle models 5 may overlap each other. In such a case, the primary particle models 5 are regarded as aggregates in which they are combined. If the overlap is too small, after the particle models 11 are arranged in the filler region 17, the particle models 11 and 11 are not adjacent to each other between the primary particle models 5 and 5 that are connected to each other (separately separated). The problem that In order to avoid such a problem, it is desirable that the size (thickness) of overlap between the combined primary particle models 5, 5 is sufficiently larger than the diameter of the particle model 11. FIG.

The filler model 10 of the present embodiment is mainly exemplified by a single primary particle model 5 shown in FIG. 3(a). Therefore, in step S2, one filler region 17 is defined inside the virtual space 6 on which the primary particle model 5 is superimposed. In the filler region 17 of the present embodiment, as shown in FIG. 3A, the particle size D1 of the spherical primary particle model 5 and the barycentric position of the primary particle model 5 are defined. Filler area 17 is stored in computer 1 .

Next, in the creation method of this embodiment, a virtual space for creating a filler model is input to the computer 1 (step S2). 3A and 3B are conceptual diagrams showing an example of the virtual space 6 in which the primary particle model 5 is arranged.

The virtual space 6 of this embodiment is defined to include the entire filler region 17 . Note that the shape and size of the virtual space 6 can be appropriately defined as long as the entire filler region 17 is included. The virtual space 6 is, for example, defined as a rectangular parallelepiped region in which the width of the particle size D1 of the primary particle model 5 is added to the upper and lower limits of the filler region 17 in the x-axis direction, the y-axis direction, and the z-axis direction. may That is, the lengths of the filler region 17 in the x-axis direction, the y-axis direction, and the z-axis direction are set on both sides of the x-axis direction, both sides of the y-axis direction, and both sides of the z-axis direction, and the particle size of the primary particle model 5 A rectangular parallelepiped area expanded by the width of D1 is defined as a virtual space 6. FIG. Further, for example, when a periodic boundary condition is defined in the coordinate system in which the primary particle model 5 of the filler region 17 is arranged, the entire coordinate system in which the periodic boundary condition is defined is defined as the virtual space 6. may At this time, periodic boundary conditions necessary for defining the virtual space 6 may be input together with the input of the filler region 17 .

A case where the virtual space 6 of this embodiment is defined as a cube having three pairs of planes 7, 7 facing each other is exemplified. A periodic boundary condition is defined for each plane 7 . Therefore, the virtual space 6 is handled as if one plane 7 and the opposite plane 7 are continuous (connected). The length (filler length) La of one side of the virtual space 6 can be set as appropriate. The filler length La of the present embodiment is desirably set so as to include the entire filler region 17 and have the smallest possible volume. As a result, the filler region 17 is filled with the particle models 11 (shown in FIG. 4), which will be described later, without gaps. A virtual space 6 is stored in the computer 1 .

Next, in the creation method of the present embodiment, a particle model (coarse-grained particle model) 11 for constructing a filler model 10 (shown in FIG. 9) is defined in the computer 1 (step S3). The particle model 11 is for placement in the filler region 17 . FIG. 4 is a conceptual diagram showing an example of a virtual space 6 in which a plurality of particle models 11 are filled with an amorphous structure.

A particle model (coarse-grained particle model) 11 is appropriately defined. The particle model 11 of the present embodiment is obtained by coarse-graining the atoms forming the filler. The particle model 11 is treated as a mass point of the equation of motion in molecular dynamics calculations and molecular dynamics calculations, which will be described later. That is, parameters such as mass, volume, electric charge, or initial coordinates are defined in the particle model 11 . In this embodiment, a case where the particle model 11 is defined as a spherical particle having a constant diameter D2 is exemplified.

The diameter D2 can be defined as appropriate. For example, when the filler model 10 is used for simulation of a polymer material, which will be described later, the diameter D2 is preferably set to a value approximately equal to the diameter of the particles used to express the polymer chain. As a result, it is possible to prevent the regular structure from appearing on the surface of the filler on the scale of the particles of the polymer chains in the simulation of the polymer material described later, and to avoid the behavior of the polymer chains that does not occur in the actual filler. It becomes possible. A definition of the particle model 11 is entered into the computer 1 .

Next, in the creation method of the present embodiment, the computer 1 arranges a plurality of particle models 11 inside the virtual space 6 (particle model arrangement step S4). FIG. 5 is a flow chart showing an example of the procedure of the particle model placement step S4.

In the particle model placement step S4 of this embodiment, first, the computer 1 determines the number of particles in the particle model 11 (shown in FIG. 4) (step S41). The number of particles is obtained by multiplying the volume of the virtual space 6 by a predetermined filling factor, dividing the result by the volume of one particle model 11, and rounding the result. The volume of the particle model 11 is obtained by multiplying the cube of the radius (half the diameter D2) by 4π/3. The filling rate can be defined as appropriate. The filling rate of this embodiment is set to, for example, about 60 to 68% (64% in this example) so as to correspond to an amorphous structure. The particle count is stored in computer 1 .

Next, in the particle model arrangement step S4 of the present embodiment, the computer 1 provisionally arranges the plurality of particle models 11 inside the virtual space 6 (step S42). FIG. 6 is a conceptual diagram showing an example of a state in which the particle models 11 are randomly arranged. In FIG. 6, a portion of the virtual space 6 shown in FIG. 4 is shown enlarged. In step S42, a plurality of particle models 11 (in this example, the number of particles obtained in step S41) are randomly arranged, allowing overlap between the particle models 11, 11. FIG. The coordinates of particle model 11 are stored in computer 1 .

By the way, in the random arrangement of the particle models 11 shown in FIG. Therefore, when a filler model having such a structure is used for a molecular dynamics simulation, a strong repulsive force is generated between the particle models 11, 11 overlapping with each other, which may cause calculation errors. Moreover, the voids 20 described above may cause behaviors that do not occur in the original filler, such as voids appearing on the surface of the filler.

In order to solve the above problems, the arrangement of the particle model 11 is adjusted so that the overlap between the particle models 11, 11 and the void 20 inside the virtual space 6 are removed to form an amorphous structure. is desirable. 7A to 7D are conceptual diagrams showing an example of changes in the state of the particle model 11 in the particle model placement step S4. FIG. 7(a) shows part of the state where the particle models 11 and 11 overlap each other and the state where the void 20 (shown in FIG. 6) is generated inside the virtual space 6. FIG.

Next, in the particle model placement step S4 of the present embodiment, a predetermined first potential P1 is defined between the particle models 11, 11 as shown in FIG. 7B (step S43). The first potential P1 can be defined as appropriate. FIG. 8 is a graph showing an example of the first potential P1 and the second potential P2.

The first potential P1 is set so that the distance between the adjacent particle models 11 and 11 approaches equal intervals and no voids 20 (shown in FIG. 6) remain inside the virtual space 6 due to potential energy minimization. It is desirable to adopt a material that can bring the overall density closer to uniformity. For this reason, it is desirable that the first potential P1 has a stronger repulsive force as the center-to-center distance r (shown in FIG. 7A) between the particle models 11, 11 is shorter. In addition, when there is overlap between the particle models 11, 11, it is desirable that the repulsive force between them does not become extremely large. From this point of view, it is desirable that the relationship between the first potential P1 of the present embodiment and the distance r between the adjacent particle models 11, 11 is defined by an exponential decay function. The exponential decay function U _exp (r) of this embodiment is defined by the following equation (1).

ここで、式（１）の変数は、次のとおりである。
ｒ：粒子モデル１１、１１の中心間の距離
Ａ、Ｂ：指数減衰関数Ｕ_exp（ｒ）のパラメータ

Here, the variables in equation (1) are as follows.
r: distance between the centers of the particle models 11, 11 A, B: parameters of the exponential decay function U _exp (r)

The exponential decay function U _exp (r) has both parameters A and B set to positive values so that the closer the center-to-center distance r between the particle models 11, 11 becomes, the stronger the repulsive force becomes. Defined. Also, the exponential decay function U _exp (r) can prevent the repulsive force from becoming extremely large even when there is an overlap between the particle models 11 , 11 . Note that the upper limit of the exponential decay function U _exp (r) is the parameter A even when the particle models 11 and 11 completely overlap each other (that is, the distance r=0). As a result, when calculating the potential energy minimization, it is possible to avoid calculation omissions due to the value of the first potential P1, and it is possible to reliably calculate the potential energy minimization.

In the above equation (1), parameters A and B can be defined as appropriate. In this embodiment, A=100 k _B T and B=4.0/σ are used. where k _B is the Boltzmann constant and T is the temperature. The values of k _B and T are both parameters used for coarse-grained simulation and can be defined as appropriate. In this embodiment, the values of k _B and T may both be defined as one. σ in the parameter B is a size parameter used for coarse-grained simulation. The value of σ is preferably selected in the range of 0.8 to 1.2 times the diameter D2 of the particle model 11 (shown in FIG. 6). In this embodiment, σ is defined as approximately 0.891 times the diameter D2 so that the cutoff distance r _c of the second potential P2, which will be described later, and the diameter D2 are equal. The first potential P1 defined in this way is a long-range force (long-range force), and acts also between the distant particle models 11, 11. FIG. Therefore, when the total number of particle models 11 increases, the amount of calculation tends to increase sharply, so the cutoff distance may be appropriately defined in order to reduce the amount of calculation. A first potential P1 is stored in the computer 1 .

Next, in the particle model placement step S4 of the present embodiment, the computer 1 calculates potential energy minimization based on the molecular mechanics method for a plurality of particle models 11 based on the first potential P1 (first 1 minimization step S44). In the molecular mechanics calculation of this embodiment, for example, classical mechanics is applied to the particle model 11 in the virtual space 6 (shown in FIG. 6). Then the potential energy of the particle model 11 is calculated. Based on this potential energy, an algorithm such as the steepest descent method or the conjugate gradient method is used to minimize the potential energy, and the arrangement of each particle model 11 is adjusted. In the molecular dynamics calculation of this embodiment (and the molecular dynamics calculation described later), the volume and shape of the virtual space 6 are kept constant. Molecular mechanics calculations can be processed using, for example, COGNAC included in JSOL Co., Ltd. Soft Material Comprehensive Simulator (J-OCTA).

In the first minimization step S44, molecular mechanics calculations are performed between the particle models 11, 11 until the potential energy based on the first potential P1 is sufficiently minimized. It should be noted that the criteria for determining potential energy minimization can be appropriately set as long as the criteria are such that changes in the arrangement of the particle model 11 can be considered to be negligibly small. FIG. 7(b) shows the change from the state shown in FIG. 7(a) due to the minimization of the potential energy of the particle model 11 by the first potential P1.

In the first minimization step S44, as shown in FIGS. 7(a) and 7(b), the calculation for minimizing the potential energy based on the first potential P1 results in the void 20 inside the virtual space 6 ( ) can be removed while the distance between the adjacent particle models 11, 11 can be brought closer to a constant value. Specifically, the first potential P1 generates a stronger repulsive force as the distance between the particle models 11, 11 decreases. Therefore, when the potential energy is high (the repulsive force is strong), the particle models 11 and 11 repel (separate from each other), and their overlap becomes small. On the other hand, the area where the particle model 11 does not exist (that is, the gap 20) is filled with the particle model 11 that has moved avoiding repulsion with the adjacent particle model 11. FIG. In this way, the density decreases in the regions where the density of the particle model 11 is high, and increases in the regions where the density of the particle model 11 is low (voids 20). , the distance between the adjacent particle models 11, 11 can be brought closer to a constant value (shown in FIG. 7(b) as an example).

Next, in the particle model placement step S4 of the present embodiment, instead of the first potential P1, a second potential P2 (FIG. 7 (c)) is defined (step S45). In step S45, the second potential P2 is defined for the particle model 11 after the first minimization step S44.

The second potential P2 can be appropriately employed as long as it causes a stronger repulsive force than the first potential P1 to act between the adjacent particle models 11,11. As shown in FIG. 8, the second potential P2 of the present embodiment is obtained by cutting off the Lennard-Jones potential (hereinafter sometimes simply referred to as "LJ potential") at the equilibrium length. (hereinafter sometimes simply referred to as “repulsive LJ potential”) is employed. An LJ potential with cutoff distance r _c is defined by the following equation (2).

ここで、式（２）の変数は、次のとおりである。
ｒ：粒子モデル１１、１１の中心間の距離
ε：ポテンシャルエネルギーの大きさに関するパラメータ
σ：粗視化シミュレーションに用いられる大きさに関するパラメータ
ｒ_c：カットオフ距離ｒ_c

where the variables in equation (2) are as follows.
r: distance between the centers of the particle models 11, 11 ε: parameter relating to magnitude of potential energy σ: parameter relating to magnitude used for coarse-grained simulation r _c : cutoff distance r _c

The details of the LJ potential (repulsive LJ potential) are described in Paper 1 (Kurt Kremer & Gary S. Grest, "Dynamics of entangled linear polymer melts: A molecular-dynamics simulation", J. Chem Phys. vol.92, No.8, 15 April 1990). The second potential P2 (repulsive force LJ potential) can be defined by using both the above equation (2) and the cutoff distance r _c =2 ^1/6 σ (˜1.122σ).

As shown in FIG. 8, the slope of the second potential P2 is greater than the slope of the first potential P1 (the repulsive force is stronger). Therefore, the second potential P2 can exert a stronger repulsive force between the adjacent particle models 11, 11 than the first potential P1. The second potential P2 is stored in computer 1 .

Next, in the particle model placement step S4 of the present embodiment, the computer 1 calculates potential energy minimization for a plurality of particle models 11 based on the second potential P2 (second minimization step S46 ). This second minimization step S46 is performed in the same procedure as the first minimization step S44.

In the second minimization step S46 of the present embodiment, prior to calculation of potential energy minimization based on the second potential P2, as shown in FIG. 7B, based on the first potential P1, adjacent The overlap of the particle models 11, 11 is removed or made sufficiently small. Therefore, in the second minimization step S46, it is possible to prevent the repulsive force from becoming extremely large due to the second potential P2 that exerts a stronger repulsive force than the first potential P1. Therefore, in the second minimization step S46, the potential energy minimization of the particle model 11 can be reliably calculated based on the second potential P2, as shown in FIG. 7(c).

In the second minimization step S46 of the present embodiment, molecular mechanics calculation is performed for all particle models 11 until the repulsive force of the second potential P2 becomes sufficiently weak (small). As a result, in the second minimization step S46, it is possible to calculate an arrangement in which the plurality of particle models 11, 11 are removed from overlapping each other, and an arrangement in which the particle models 11 are filled in the virtual space 6 with an amorphous structure is created. be. The amorphous structure is stored in computer 1 .

Next, in the creation method of the present embodiment, the computer 1 deletes the particle model 11 existing outside the filler region 17 shown in FIG. (Step S5). Step S5 is performed after the second minimization step S46.

Conditions for determining whether or not the particle model 11 protrudes from the filler region 17 can be appropriately defined. In the present embodiment, when part or all of the particle model 11 exists outside the filler region 17 , it is determined that the particle model 11 protrudes from the filler region 17 . More strictly, for example, if the shortest distance between the center coordinates of the particle model 11 and the surface of the filler region 17 (shown in FIG. 3) is smaller than the value obtained by dividing the diameter D2 of the particle model 11 by 2, or When the center coordinates of the particle model 11 exist outside the filler region 17 , it is determined that the particle model 11 protrudes from the filler region 17 . In step S5, the filler model 10 in which a plurality of particle models 11 are arranged inside the filler region 17 can be created by deleting the particle models 11 protruding from the filler region 17 . FIG. 9 is a conceptual diagram showing an example of a filler model.

Next, in the creation method of this embodiment, the computer 1 connects the adjacent particle models 11, 11 (step S6). In step S6 of the present embodiment, the particle models 11, 11 adjacent to each other are connected at a predetermined distance (hereinafter sometimes simply referred to as "bonding distance") or less.

The bonding distance can be set as appropriate. The bond distance in this embodiment is set in the range of 1.0 to 1.5 times (1.1 times in this example) the diameter D2 of the particle model 11 (shown in FIG. 6). Then, in step S6 of the present embodiment, the particle models 11, 11 adjacent to each other at a bond distance or less are connected by the bond chain model 8, as shown in FIG. 7(d). Thereby, in process S6, filler model 10 which combined a plurality of particle models 11 is created. A filler model 10 is stored in the computer 1 .

In this embodiment, the particle models 11, 11 are combined using the binding chain model 8 shown in FIG. 7(d), but the present invention is not limited to such a mode. For example, after deleting the particle model 11 protruding from the filler region 17 (shown in FIG. 3) (step S5), the above step S6 is omitted, and the particle model 11, 11 is not connected between the filler model 10 (not shown) may be created.

In the creation method of the present embodiment, unlike the above-mentioned Patent Document 1, the particle model 11 of the filler model 10 can be arranged in an irregular amorphous (amorphous) state. As a result, in the creation method of the present embodiment, as shown in FIG. 9, the outer surface of the filler model 10 is arranged irregularly, preventing it from being formed in a regular stepped shape as shown in FIG. be able to. It should be noted that the outer surface of the actual filler is not formed in a regular stepped pattern, for example, in the case of silica produced by sintering or precipitation. Therefore, in the creation method of the present embodiment, it is possible to create a filler model 10 having a solid structure in which the outer surface of the actual filler is more appropriately expressed and the particle model 11 is densely packed.

The amorphous structure created by the creation method of the present embodiment has a lower filling factor than the face-centered cubic lattice and the body-centered cubic lattice as in Patent Document 1 above. number can be reduced. Therefore, in the simulation using the filler model 10, the calculation amount can be saved.

Next, in the creation method of the present embodiment, after the filler model 10 is created, a force field used for molecular dynamics calculation may be set between the particle models 11, 11 (step S7). As the force field acting between the adjacent particle models 11, 11, the repulsive force LJ potential represented by the above equation (2) and the cutoff distance r _c =2 ^1/6 σ is used. As the force field between the particle models 11 and 11 that are coupled to each other, for example, the coupling potential represented by the following formula (3) described in the paper 1 may be used.

ここで、式（３）の変数は、次のとおりである。
ｒ：粒子モデル１１、１１の中心間の距離
ｋ：ポテンシャルエネルギーの大きさに関するパラメータ
Ｒ₀：伸びきり長

where the variables in equation (3) are as follows.
r: distance between the centers of the particle models 11, 11 k: parameter related to the magnitude of potential energy R ₀ : stretched length

In this embodiment, the distance between the particle models 11 and 11 which the filler model 10 adjoins is created so that all may approach the diameter D2 (shown in FIG. 4). Also, the diameter D2 is set equal to the cutoff distance r _c =2 ^1/6 σ=approximately 1.122σ. Therefore, if the bond potential represented by the above formula (3) is used as it is for the bond chain model 8 shown in FIG. On the other hand, since the equilibrium length of the binding potential is about 0.961σ, relaxation may occur due to molecular dynamics calculations, and the entire filler model may shrink. To avoid such problems, the coupling potential between the particle models 11, 11 may be adjusted such that the equilibrium length of the coupling potential is equal to the cutoff distance r _c . For example, the parameter σ (the term of U _LJ (r)) of the coupling potential between the particle models 11 and 11 and the extended length R ₀ may both be multiplied by 1.168.

If no bond (binding chain model 8) is defined between the particle models 11, 11 (step S6 is omitted), step S7 may also be omitted. In this case, for example, the coordinates of the entire particle model 11 of the filler model 10 may be fixed, and a constraint condition for defining the filler model 10 as a rigid body may be set.

By the way, in the simulation method described later, molecular dynamics calculation is performed for all the particle models 11 constituting the filler model 10 shown in FIG. 9 . Therefore, when the filling rate (number) of the particle model 11 is large, the calculation time tends to increase. Therefore, the filler model 10 may be remodeled into the creation method so that the filling rate of the particle model 11 is reduced. FIG. 10 is a flow chart showing an example of a processing procedure of a creation method according to another embodiment of the present invention. Drawing 11 is a sectional view showing an example of filler model 10 of other embodiments of the present invention. The virtual space is omitted in FIG. In this embodiment, the same reference numerals are assigned to the same configurations as in the previous embodiments, and the description may be omitted.

In the creation method of this embodiment, as shown in FIG. 11, the filling rate of the particle model 11 inside the filler model 10 is smaller than the filling rate of the particle model 11 of the outer layer 21 of the filler model 10, A filler model 10 is modified. As shown in FIG. 10, this remodeling step includes a step S8 of hollowing out the filler model 10, and a filling rate of the outer layer 21 (amorphous structure) of the filler model 10 inside the hollowed out filler model 10. and a step S9 of filling the particle model 11 with a crystal structure smaller than .

In step S8, the particle models 11 inside the outer layer 21 are deleted while maintaining the particle models 11 of the outer layer 21 of the filler model 10, as shown in FIG. In this embodiment, all particle models 11 not included in the outer layer 21 are deleted from the particle models 11 in the virtual space 6 .

The outer layer 21 can be defined accordingly. The outer layer 21 of this embodiment is defined as a region within a predetermined thickness r1 from the outer surface of the filler region 17 toward the inside. The thickness r1 can be defined as appropriate. The thickness r1 in this embodiment is set to 5% to 20% (10% in this example) of the particle size D1 of the primary particle model 5 (shown in FIG. 3(a)).

The determination as to whether or not it is included in the outer layer 21 can be made as appropriate. In this embodiment, if at least part of the particle model 11 is outside (that is, radially inside) of the outer layer 21 , it is determined not to be included in the outer layer 21 . More specifically, when the center coordinates of the particle model 11 exist outside the outer layer 21 (that is, inside in the radial direction), or when the shortest distance between the center coordinates of the particle model 11 and the outer surface of the outer layer 21 is the particle If the diameter D2 (shown in FIG. 4) of the model 11 is smaller than the value divided by 2, it is determined not to be included in the outer layer 21 .

In step S9, the particle model 11 is filled into the inner region of the outer layer 21 of the filler model 10 based on the crystal structure having a lower filling rate than the amorphous structure. The crystal structure can be appropriately adopted as long as it is smaller than the filling rate of the outer layer 21 (for example, 60% to 68%). In this embodiment, a diamond structure (34% fill factor) is employed.

In the production method of this embodiment, the amorphous arrangement of the particle models 11 that make up the outer layer 21 is maintained, so that the outer surface of the filler can be expressed more appropriately as in the previous embodiments. A filler model 10 can be created. Furthermore, in the creation method of this embodiment, while suppressing a decrease in the strength of the entire filler model 10, since it is possible to reduce the filling rate of the particle model 11 inside the filler model 10, in the simulation method described later, the particle model to be calculated 11 can be reduced. Therefore, the creation method of this embodiment makes it possible to shorten the calculation time while maintaining the calculation accuracy in the simulation method (molecular dynamics calculation) described later.

In this embodiment, for the crystal structure filled inside the filler model 10, the adjacent particle models 11 may be connected by the binding chain model 8 (shown in FIGS. 7(d) and 9). Furthermore, the particle model 11 that constitutes the crystal structure and the particle model 11 that constitutes the outer layer 21 may be connected by the binding chain model 8 . Thereby, the number of the particle models 11 inside the filler model 10 can be reduced while suppressing a decrease in the strength of the filler model 10 .

Next, a method of simulating a polymeric material using the filler model 10 obtained by the above-described creation method will be described. In this simulation method, a computer 1 is used to calculate the performance of a polymer material containing fillers and polymer components.

Polymer components include, for example, natural rubber, synthetic rubber and resins. Examples of the polymer component of the present embodiment include cis-1,4 polybutadiene (hereinafter sometimes simply referred to as "polybutadiene"). FIG. 12 is the structural formula of polybutadiene.

The molecular chain 12 constituting polybutadiene is composed of a monomer 13 {-[CH ₂ -CH=CH-CH ₂ ]-} consisting of a methylene group (-CH ₂ -) and a methine group (-CH-). It is configured by connecting with Moreover, instead of the methylene group (--CH ₂ ), a methyl group (--CH ₃ ) is connected to the end of the polymer material. In addition, materials other than polybutadiene may be used as the polymer component. Moreover, the above-mentioned thing is employ|adopted as a filler. FIG. 13 is a flow chart showing an example of a processing procedure of a polymeric material simulation method.

In the simulation method of this embodiment, first, a cell, which is a virtual space corresponding to a portion of a polymer material (not shown), is input to the computer 1 (step S11). FIG. 14 is a conceptual diagram showing an example of a polymer material model. FIG. 14 shows only a part of the molecular chain model 15 as a representative.

The cell 14 is a calculation target area in the molecular dynamics calculation described later. The cell 14 of this embodiment is set in the same way as the virtual space 6 shown in FIG. At least one filler model 10 and a plurality of molecular chain models 15 are placed inside the cell 14 .

The length Lb of one side of the cell 14 can be set as appropriate. The length Lb of the present embodiment is desirably three times or more the radius of inertia (not shown), which is an amount indicating the spread of the molecular chain model 15, which will be described later. As a result, in the molecular dynamics calculation, the cell 14 prevents the occurrence of collisions with itself (image) that is periodically arranged due to periodic boundary conditions, and appropriately calculates the spatial extension of the molecular chain model 15. be able to. Also, the size of the cell 14 is set to a volume that is stable at 1 atmosphere, for example. Cell 14 thereby allows to define at least a partial volume of polymeric material to be analyzed. Cell 14 is stored in computer 1 .

Next, in the simulation method of the present embodiment, a molecular chain model 15 modeling a molecular chain 12 (shown in FIG. 12) of a polymer material is input to the computer 1 (step S12). The molecular chain model 15 of this embodiment is a coarse-grained molecular model.

Molecular chain model 15 is represented using a plurality of particles 16 whose number is smaller than the number of atoms that constitute molecular chain 12 (shown in FIG. 12). Linking chains (not shown) are bound between these particles 16 , 16 . The molecular chain model 15 of this embodiment is exemplified by a Kremer-Grest model, but is not particularly limited. The molecular chain model 15 may be, for example, a model based on DPD (dissipative particle dynamics method).

Each particle 16 of this embodiment corresponds to, for example, the monomer 13 of the molecular chain 12 shown in FIG. When the molecular chain 12 is polybutadiene, for example, 1.10 monomers 13 are used as structural units, and the structural units are substituted with one particle 16 . As a result, a plurality of (for example, 10 to 5000) particles 16 are set in the molecular chain model 15 of the present embodiment. Particles 16 are treated as mass points in the equation of motion in molecular dynamics calculations. That is, the particles 16 are defined with parameters such as mass, diameter, charge or initial coordinates. Each of these parameters is stored in the computer 1 as numerical information.

A binding chain (not shown) is configured as a binding potential with a defined equilibrium length between the particles 16,16. Here, the "equilibrium length" is defined as the bond distance between the particles 16, 16 at which the value of the bond potential is the smallest. Also, the binding potential of the binding chain (not shown) can be set based on, for example, Paper 1 mentioned above. Such a molecular chain model 15 is numerical data for treating a polymer material in molecular dynamics calculations, and is input to the computer 1 .

Next, in the simulation method of this embodiment, the computer 1 acquires the filler model 10 (step S13). In step S13, at least one filler model 10 is acquired by performing the procedure of the above-described filler model creation method. A filler model 10 is stored in the computer 1 .

Next, in the simulation method of the present embodiment, the computer 1 arranges the molecular chain model 15 and the filler model 10 inside the cell 14 (step S14). In step S<b>7 of the present embodiment, first, the filler model 10 is placed inside the cell 14 . The position at which the filler model is arranged can be set as appropriate. For example, in the electron beam transmission image of the polymer material, the filler model 10 may be arranged in the region where the filler is arranged. Thereby, in step S14, the placement of the filler model 10 can be arranged inside the cell 14 based on the relative position of the filler in the actual polymer material.

Next, in step S14, a plurality of molecular chain models 15 are placed in regions where filler models 10 are not placed. As a result, in step S14, the molecular chain model 15 can be arranged in the cell 14 by approximating the distribution of the actual molecular chain 12. FIG.

Next, in the simulation method of the present embodiment, the attraction and A potential P3 on which the repulsive force acts is defined (step S15). As the potential P3, the LJ potential described in Patent Document 1 is adopted. Each variable of the LJ potential can be appropriately set based on, for example, Paper 1 mentioned above.

Next, in the simulation method of this embodiment, the computer 1 performs molecular dynamics calculations on the molecular chain model 15 and the filler model 10 (step S16). In the molecular dynamics calculation, for example, Newton's equation of motion is applied assuming that the particle model 11 of the filler model 10 and the particle 16 of the molecular chain model 15 follow classical mechanics for a predetermined time for the cell 14 . Then, the movements of the particle model 11 and the particles 16 at each time are tracked for each unit time.

In step S16, first, the structural relaxation of the molecular chain model 15 is calculated. In this case, in the cell 14, the pressure and temperature are kept constant, or the volume and temperature are kept constant. By calculating this structural relaxation, in step S16, a polymer material model 18 that models a polymer material (not shown) is created.

Next, in step S16 of the present embodiment, deformation calculation (e.g., extension deformation) of the created polymer material model 18 is performed. Physical quantities (for example, stress) obtained by deformation calculation of the polymer material model 18 are stored in the computer 1 .

In the simulation method of the present embodiment, since the filler model 10 having a surface structure that approximates the microstructure of the filler is used, the effect of the surface structure on the physical properties of the polymer material (molecular chain model 15) is accurately evaluated. be able to. Therefore, in the simulation method of this embodiment, the physical quantity obtained by the deformation calculation of the polymeric material model 18 can be calculated with high accuracy.

Next, in the simulation method of the present embodiment, it is determined whether or not the physical properties of the polymer material model 18 are good (step S17). Determination as to whether the polymer material is good or not is appropriately carried out based on the strength, durability, etc. required of the polymer material.

In step S17, if the physical properties are good (“Y” in step S17), for example, based on the filler and molecular chain 12 (shown in FIG. 13) that constitute the polymer material model 18, the polymer material is manufactured. (step S18). The produced polymeric material is used, for example, in the production of industrial products such as tires. On the other hand, if the physical properties are not good in step S10 (“N” in step S17), the structure of at least one of the filler and molecular chain 12 is changed (step S19), and steps S11 to S17 are performed again. be. As a result, the simulation method of the present embodiment makes it possible to manufacture a polymer material having good physical properties and an industrial product using the polymer material.

Although the particularly preferred embodiments of the present invention have been described in detail above, the present invention is not limited to the illustrated embodiments and can be modified in various ways.

Based on the procedure shown in FIG. 2, models for numerical analysis of fillers were created (Example 1, Example 2 and Comparative Example).

In Examples 1 and 2, according to the processing procedure shown in FIG. process was performed. Next, in Examples 1 and 2, a second minimization step of calculating potential energy minimization for a plurality of particle models based on a second potential that exerts a stronger repulsive force than the first potential. was carried out. Then, in Examples 1 and 2, the step of deleting the particle model protruding from the filler region composed of one spherical primary particle was performed.

In Example 1, the step of combining adjacent particle models at a predetermined distance or less was performed. FIG. 9 is a conceptual diagram showing an example of a filler model of Example 1. FIG.

On the other hand, in Example 2, based on the processing procedure shown in FIG. 10, the filler model so that the filling rate of the particle model inside the filler model is smaller than the filling rate of the primary particle model in the outer layer of the filler model. A process of refurbishment was carried out. In Example 2, the crystal structure of the particle model inside the filler model is a diamond structure. Then, in Example 2, a step of combining adjacent particle models at a predetermined distance or less was performed. FIG. 11 is a conceptual diagram showing an example of a cross section of the filler model of Example 2. FIG.

In the comparative example, as in Patent Document 1 above, after a plurality of particle models are arranged in a crystal lattice such as a face-centered cubic lattice, a particle model protruding from a filler region composed of a single spherical primary particle. was deleted to create a filler model. FIG. 15 is a conceptual diagram showing an example of a filler model of a comparative example.

Then, for the filler models of Example 1, Example 2 and Comparative Example, based on the processing procedure shown in FIG. . Then, the calculation time required for the simulation was obtained. Common specifications are as follows.

Virtual space length La: 108σ
Filler volume fraction: 30%
Unit time: 0.006τ
Structural relaxation calculation:
Number of steps: 1,000,000 steps

As a result of the test, the outer surface of the filler model of Examples 1 and 2 can be prevented from being formed in a regular stepped shape like the comparative example, and the outer surface of the filler can be expressed more appropriately. I was able to create a filler model. In the simulation using such filler models of Examples 1 and 2, the behavior occurring in the original filler could be reproduced with higher accuracy than the simulation using the filler model of the comparative example.

In addition, in Example 2, the filling rate of the particle model inside the filler model can be made smaller than the filling rate of the particle model in the outer layer of the filler model. It was possible to reduce the calculation time of the example simulation to 88%.

S42 Step of temporarily arranging a plurality of particle models S44 First minimization step S46 Second minimization step

Claims (7)

A method for creating a filler model for numerical analysis of a filler, comprising:
A step of inputting a filler region, which is a region where the filler is present, into a computer;
and defining a particle model for constructing the filler model in the computer,
the computer
defining a virtual space that includes the entire filler region;
arranging a plurality of the particle models inside the virtual space;
deleting the particle model existing outside the filler region and creating the filler model in which a plurality of the particle models are arranged inside the filler region;
The step of arranging the particle model includes:
temporarily arranging the plurality of particle models in the virtual space;
defining a first predetermined potential between the particle models;
a first minimization step of calculating a potential energy minimization based on a molecular mechanics method for a plurality of the particle models based on the first potential;
After the first minimization step, defining a second potential that exerts a stronger repulsive force than the first potential between the adjacent particle models;
a second minimization step of calculating the minimization of the potential energy for a plurality of the particle models based on the second potential;
How to create a filler model. The method of creating a filler model according to claim 1, wherein the step of temporarily arranging a plurality of the particle models is based on a condition that allows the particle models to overlap each other, and the particle models are randomly arranged. The method of creating a filler model according to claim 1 or 2, wherein the relationship between the first potential and the distance between the adjacent particle models is defined by an exponential decay function. The method for creating a filler model according to any one of claims 1 to 3, wherein the second potential is a Leonard Jones potential cut off by an equilibrium length. After the step of creating the filler model, the computer determines that the filling rate of the particle model inside the filler model is smaller than the filling rate of the particle model of the outer layer of the filler model, the filler model The method for creating a filler model according to any one of claims 1 to 4, further comprising the step of modifying the. The step of modifying the filler model includes the step of arranging the particle model inside the filler model based on a crystal structure smaller than the filling rate of the outer layer of the filler model, according to claim 5. Of the filler model How to make. Creating a filler model according to any one of claims 1 to 6, wherein the step of creating the filler model includes a step of combining the adjacent particle models after deleting the particle model protruding from the filler region. Method.

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