JP2022124925A - Method of generating filler model and method of simulating polymeric material - Google Patents

Abstract

To provide a method of generating a filler model approximating to a micro structure of a filler.SOLUTION: The method of generating a filler model includes: a step S1 of inputting a first parameter being an average primary particle count of an aggregate, to a computer; a step S2 of inputting a space for generating the filler model and a volume fraction of the filler model in the space; and a step S3 of defining a primary particle model resulting from modeling a primary particle. The computer performs a step S4 of binding a plurality of primary particle models to generate a plurality of aggregate models and a step S5 of generating the filler model by arranging the plurality of aggregate models in the space so as to satisfy the volume fraction. The step S4 of generating the plurality of aggregate models includes a step of deciding the total number of aggregate models constituting the filler model and a primary particle count of each aggregate model so as to satisfy the first parameter and a step of binding a plurality of primary particle models constituting each aggregate model on the basis of the decided total number and primary particle count.SELECTED DRAWING: Figure 3

Description

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

Various methods for creating a filler model have been proposed in order to use a computer to evaluate the properties of a composite material composed of a filler and a polymeric material such as a polymer chain.

In the method of Patent Document 1 below, the primary particles of the filler are represented by spherical regions, for example, based on the three-dimensional structure of the actual filler, the primary particles of the filler are represented by spherical regions, and the regions are coarse-grained particles. By filling with , a filler model for coarse-grained molecular dynamics calculation is created.

Japanese Patent No. 6353290

The above method can reproduce the actual macro-dispersed state of the filler. However, the spatial resolution of experimentally obtained three-dimensional structures of fillers is limited. For this reason, for example, the three-dimensional structure (that is, the fractal dimension of the aggregate, the aggregate size distribution, etc.) of the secondary aggregate in which the primary particles are chemically bonded (hereinafter sometimes simply referred to as "aggregate") There is room for further improvement in approximating the microstructure of the filler model to the microstructure of the actual filler such as.

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 approximates the microstructure of the filler.

The present invention is a method for creating a filler model for numerical analysis of a filler, wherein the filler contains at least one aggregate formed by binding primary particles, and the method is a step of inputting a first parameter, which is the average number of primary particles of the aggregate, into a computer, and providing the computer with a space for creating the filler model, and a volume fraction of the filler model with respect to the space and defining, in the computer, a primary particle model modeling the primary particles, wherein the computer combines a plurality of the primary particle models to create a plurality of aggregate models and a step of arranging the plurality of aggregate models in the space so as to satisfy the volume fraction to create the filler model, and creating the plurality of aggregate models, A step of determining the total number of aggregate models constituting the filler model and the number of primary particles of each aggregate model so as to satisfy the first parameter, and based on the determined total number and the number of primary particles and combining a plurality of said primary particle models constituting each said aggregate model.

In the method for creating a filler model according to the present invention, further comprising the step of inputting a second parameter, which is the minimum number of primary particles of the aggregates, into the computer, and the step of creating the plurality of aggregate models includes the above It may further include the step of limiting the occurrence of aggregate models with primary particle numbers smaller than the second parameter.

In the method for creating a filler model according to the present invention, the step of determining the total number of aggregates and the number of primary particles of each aggregate is based on the first parameter, the appearance of the number of primary particles of the aggregates generating a set of random numbers according to a frequency distribution, wherein the frequency distribution is defined as an exponentially decaying function of the number of primary particles.

In the method for creating a filler model according to the present invention, the step of combining the primary particle models includes, as an initial state of the aggregate model, one of the primary particle models without combining all the primary particle models with each other. Randomly selecting two of the aggregate models until the number of primary particles of the aggregate model is equal to the determined number of primary particles, and repeating the operation to combine them into one aggregate model.

In the method for creating the filler model according to the present invention, the step of creating the filler model includes, when arranging the plurality of aggregate models in the space, in order from the aggregate model with a large number of primary particles, the aggregate The models may be arranged randomly so that they do not overlap each other.

In the method for creating a filler model according to the present invention, the step of inputting the first parameter includes a step of inputting the size distribution of the aggregates constituting the filler into the computer, and , and the step of determining the first parameter so that the size distribution of the plurality of aggregate models constituting the filler model approaches.

The present invention comprises the steps of: inputting a cell, which is a virtual space corresponding to a part of a polymer material, into a computer; and inputting a molecular chain model, which models the molecular chains of the polymer material, into the computer. , wherein the computer executes the filler model creation method according to any one of claims 1 to 7 to obtain the filler model; and a step of arranging a filler model, and a step of performing a molecular dynamics calculation on the molecular chain model and the filler model.

In the method of creating a filler model of the present invention, in the step of creating the plurality of aggregate models, the total number of aggregate models constituting the filler model so as to satisfy the first parameter, and each aggregate model determining the number of primary particles; and combining the plurality of primary particle models constituting each aggregate model based on the determined total number and the number of primary particles. Therefore, according to the present invention, by specifying the first parameter, the three-dimensional structure of the aggregate model having a specific aggregate size distribution corresponding to the first parameter can be selectively created.

Furthermore, in the creation method of the present invention, the first parameter that approximates the aggregate size distribution of the actual filler is specified, so that the three-dimensional structure of the aggregate model is created so as to approach the actual filler. can be done. Therefore, according to the present invention, the three-dimensional structure of the aggregate model is controlled so that the aggregate size distribution corresponds to the first parameter, and the microstructure of the filler model is changed to the microstructure of the filler. can be approximated to

1 is a perspective view showing an example of a computer; FIG. (a) is a conceptual diagram showing an example of primary particles. (b) is a conceptual diagram showing an example of an aggregate. It is a flowchart which shows an example of the processing procedure of the preparation method of a filler model. It is a conceptual diagram which shows an example of space. It is a flowchart which shows an example of the processing procedure of an aggregate model preparation process. It is a flowchart which shows an example of the processing procedure of a 1st determination process. It is a figure which shows an example of the set of random numbers. (a) to (d) are conceptual diagrams for explaining the procedure for combining primary particle models. It is a flowchart which shows an example of the processing procedure of an aggregate three-dimensional structure determination process. It is a flow chart which shows an example of a processing procedure of a filler model creation process. It is a conceptual diagram which shows an example of a filler model. 9 is a flow chart showing an example of a processing procedure of a parameter input step according to another embodiment of the present invention; It is a graph which shows an example of _1st distribution P1(S). It is a graph which shows an example of _2nd distribution P2(S,n). It is a graph which shows an example of function p _n (n, m). 4 is a graph showing an example of a _first temporary distribution P(S) and a first distribution P1(S) calculated based on the determined first parameter and second parameter; 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 part of a polymer material model; FIG. 4 is a flow chart showing an example of a processing procedure of a coarse-grained model creation process; 2 is a graph showing frequency distributions of aggregate sizes in Examples and Comparative Examples.

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.

FIG. 1 is a perspective view showing an example of a computer 1. FIG. 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. Note that 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 silica is exemplified. The filler is formed as a higher-order structure (agglomerate) in which aggregates (aggregates) of primary particles are aggregated. FIG. 2(a) is a conceptual diagram showing an example of the primary particles 3. FIG. FIG. 2(b) is a conceptual diagram showing an example of aggregate 4. As shown in FIG.

As shown in FIG. 2(a), the primary particles 3 of this embodiment are approximately configured as spherical solids. When the filler is, for example, silica used in tire tread rubber, the primary particles 3 are spherical particles having a particle size (primary particle size) D1 of about 10 to 40 nm. The three-dimensional structure of the aggregates 4 as shown in FIG. 2(b) is formed during the manufacturing process of the filler. For example, a sintering method, a sedimentation method, or the like is used as a method for producing silica. In these methods, floc-like aggregates 4 are formed by diffusing, aggregating, and combining a large number of primary particles having similar sizes in a gas or solution. A filler (not shown) is formed by aggregation of these aggregates 4 . Note that extremely small aggregates (for example, aggregates of less than 10 nm) 4 and primary particles 3 that have not sufficiently aggregated are removed during the production of the filler. The filler after production is mixed with a polymeric material or the like and becomes dispersed inside these composites. In the production method of the present embodiment, a case in which aggregates are uniformly dispersed without having a higher-order structure is exemplified.

The aggregate size (diameter) of the aggregate 4 is defined, for example, as twice the radius of inertia, and is experimentally measured based on, for example, a transmission electron microscope or dynamic light scattering method. The aggregate size (diameter) of the aggregate 4 of this embodiment is, for example, about 10 to 1000 nm.

FIG. 3 is a flow chart showing an example of a processing procedure of a method for creating a filler model. In the creation method of this embodiment, first, a first parameter is input to the computer 1 (parameter input step S1). The first parameter m of this embodiment is the average number of primary particles of the aggregates 4 . This average number of primary particles (first parameter m) can be appropriately defined as a real number of 1 or more. For example, based on Paper 1 (D. Boldridge, "Aerosol Science and Technology", vol.44, No.182, 2010), the primary A distribution of the number of particles may be estimated, and the average value of the distribution may be set as the first parameter m. The first parameter is stored in computer 1 .

Next, in the creation method of the present embodiment, the space 6 for creating the filler model and the volume fraction of the filler model with respect to the space 6 are input to the computer 1 (step S2). FIG. 4 is a conceptual diagram showing an example of the space 6. As shown in FIG.

Space 6 can be defined as appropriate. The space 6 can be, for example, a sub-region with an arbitrary shape extracted from a space in which no periodic boundary conditions are defined (i.e. open), an entire region in which periodic boundary conditions are defined, or a periodic boundary condition defined by It may be a partial area having an arbitrary shape (for example, a rectangular parallelepiped, a sphere, etc.) extracted from the defined area. The space 6 in this embodiment is defined as a cube with three pairs of planes 7, 7 facing each other. A periodic boundary condition is defined for each plane 7 . Therefore, the space 6 is treated as one in which the plane 7 on one side and the plane 7 on the opposite side are continuous (connected).

The length La of one side of the space 6 can be set appropriately. The length La of the present embodiment is preferably twice or more the maximum value of the radius of inertia (not shown), which is the amount that indicates the spread of the aggregate model 8 (shown in FIGS. 8A to 8D), which will be described later. . As a result, in the aggregate three-dimensional structure determination step S42 (shown in FIG. 5), which will be described later, the space 6 prevents the occurrence of collisions with itself (image) that is periodically arranged due to the periodic boundary conditions. The aggregate model 8 can be arranged appropriately. Space 6 is stored in computer 1 .

The volume fraction φ of the filler model with respect to the space 6 is appropriately obtained as a real number greater than 0 and 1 or less. The volume fraction φ in the present embodiment is, for example, the amount of filler added x phr (x is a real number larger than 0. "phr" is the weight when the weight of the polymer is 100.) , using the density ρ of the polymer chain and the density ρ _f of the filler, it is obtained by the formula “φ=1/(1+100ρ _f /(xρ))”. For example, when the added amount x of the filler is 50 phr, the density ρ of the polymer chain is 0.85 g/cm ³ , and the density ρ _f of the filler is 2.2 g/cm ³ , the volume fraction φ is 0.162. Become. Volume fraction φ is stored in computer 1 .

Next, in the creation method of the present embodiment, the computer 1 defines a primary particle model 5 (shown in FIG. 8(a)) that models the primary particle 3 (shown in FIG. 2(a)) (step S3 ). The primary particle model 5 is treated as a mass point of the equation of motion in molecular dynamics calculations described later. That is, in the primary particle model 5, parameters such as mass, volume, particle size (diameter) D1, charge or initial coordinates are defined based on the primary particles 3. FIG. In step S3 of this embodiment, only the parameters of the primary particle model 5 are defined, and the primary particle model 5 is not placed in the space 6 (shown in FIG. 4). A primary particle model 5 is stored in the computer 1 .

Next, in the creation method of the present embodiment, the computer 1 combines a plurality of primary particle models 5 to create a plurality of aggregate models 8 (aggregate model creation step S4). In this embodiment, one or more aggregate models 8 are created by combining a plurality of primary particle models 5 (shown in FIG. 8(a)). FIG. 5 is a flow chart showing an example of the processing procedure of the aggregate model creation step S4.

In the aggregate model creation step S4 of the present embodiment, first, the total number of aggregate models 8 constituting the filler model and the number of primary particles of each aggregate model 8 are determined so as to satisfy the first parameter ( First determination step S41). FIG. 6 is a flow chart showing an example of the processing procedure of the first determination step S41.

In the first determination step S41 of the present embodiment, first, the total number n _total of primary particle models 5 (shown in FIG. 8(a)) arranged in the space 6 (shown in FIG. 4) is determined (step S51). . The total number n _total of primary particle models 5 can be determined as appropriate. In the present embodiment, the total number n _total of primary particle models 5 is determined by dividing the value obtained by multiplying the volume of the space 6 by the volume filling rate of the filler by the volume of the primary particle models 5 . If the calculated total number n _total of the primary particle models 5 is not an integer, it may be rounded off, for example. In addition, the total number n _total of primary particle models 5 may be appropriately corrected in consideration of overlapping of primary particle models 5 constituting aggregate model 8 . The total number n _total of primary particle models 5 is stored in computer 1 .

Next, in the first determination step S41 of the present embodiment, a set of random numbers according to the appearance frequency distribution of the primary particle number n of the aggregates 4 is generated based on the first parameter (average number of primary particles) (step S52 ). FIG. 7 is a diagram showing an example of the set 21 of random numbers.

As shown in FIG. 7, the random number set 21 consists of one or more random values 22 . In step S52 of the present embodiment, the random number set 21 is generated such that the total of the random values 22 constituting the random number set 21 is greater than or equal to the total number n _total of the primary particle models 5 determined in step S51. .

In step S52 of the present embodiment, first, the function p _n (n, m) between the number n of primary particles of the aggregate 4 and the first parameter m (m is a real number of 1 or more), which is the average number of primary particles, is defined. be done. A second parameter k, which is the minimum number of primary particles, may be used together with the number of primary particles n and the first parameter m. This second parameter k can be appropriately set as an integer of 1 or more. For example, based on the article 1, the distribution of the number of primary particles of the aggregate 4 may be estimated, and the minimum value of the number of primary particles in the distribution may be set as the second parameter k. If such a second parameter k is used, a function p _n (n,m,k) is defined. These functions p _n (n, m) and p _n (n, m, k) are probability mass functions with the primary particle number n of the aggregate 4 as a random variable. In other words, these functions represent the probability that the random number will be n.

The functions p _n (n,m) and p _n (n,m,k) can be defined as appropriate, but must be normalized. That is, the function p _n (n, m) does not take a negative value no matter what primary particle number n is substituted, and the function p _n The sum of (n, m) (the sum of probability masses) must be one. Note that the function p _n (n, m, k) is the same as the function p _n (n, m).

Function _pn (n, m) and function _pn (n, m, k) are such that the expected value (average number of primary particles of aggregate model 8) is the first parameter m (average number of primary particles of aggregate 4) is preferably defined to be equal to or approximate to .

Furthermore, the functions p _n (n, m) and p _n (n, m, k) are defined to approximate the particle number distribution of the actual filler agglomerate 4 (shown in FIG. 2(b)). It is desirable to The particle number distribution of the filler aggregates 4 can be obtained, for example, by performing a large number of 3D TEM observations on individual aggregates 4 . Note that the particle number distribution of the filler aggregates 4 may be obtained based on empirical rules and assumptions from the results of two-dimensional observation with a transmission electron microscope.

In this embodiment, in order to obtain the particle number distribution of the filler aggregates 4 more easily than the above obtaining method, for example, a function _pn (n, m)={(m−1)/m} ⁿ⁻¹ /m is used. Such a function p _n (n, m) is theoretically estimated based on a simulation using a DLCA (Diffusion-limited cluster aggregation) model or an RLCA (Reaction limited aggregation) model. Here, the number n of primary particles is an integer of 1 or more. The function p _n (n, m) of this embodiment is normalized, and its expected value is equal to the first parameter (average number of primary particles of aggregate 4) m.

The function p _n (n,m,k) is defined such that aggregate models 8 with a primary particle number n less than a second parameter (minimum primary particle number) k are eliminated. That is, the function p _n (n, m, k) is changed so that the probability that the number of primary particles n is less than the second parameter k is zero, and when the number of primary particles n is greater than or equal to the second parameter k, standardized. In this embodiment, the function p _n (n, m, k)=0 is used when the number of primary particles n is less than the second parameter k. On the other hand, when the number of primary particles n is greater than or equal to the second parameter k, the function p _n (n, m, k)={(m−1)/m} ^nk /m is used. As a result, in the aggregate model creation step S4 of the present embodiment, it is possible to limit the occurrence of aggregate models 8 in which the number of primary particles is smaller than the second parameter k.

Next, in step S52 of the present embodiment, based on the first parameter (average number of primary particles) m and/or the second parameter (minimum number of primary particles) k, the primary particle number n of aggregates 4 follows the appearance frequency distribution A set of random numbers 21 (shown in FIG. 7) is generated. As described above, in the present embodiment, the random numbers 22 constituting the set 21 of random numbers according to the function p _n (n, m) or the function p _n (n, m, k) are summed up in the primary particle model 5 A set 21 of random numbers is generated by successively generating the total number n _total or more.

As mentioned above, the functions p _n (n,m) and p _n (n,m,k) are probability mass functions. Therefore, in order to generate a set 21 of random numbers according to a certain probability mass function, for example, the range of the number of primary particles n that can be generated as random A probability mass function ( _pn (n, m) or function _pn (n, m , k)) can be realized by using the Walker's Alias Method. The value of the primary particle number n, which is the upper limit of the random number value 22, is preferably at least three times the maximum primary particle number of the aggregates 4 contained in the filler model to be created. A set of random numbers 21 is stored in the computer 1 .

Next, in the first determination step S41 of the present embodiment, is the total of the random number values 22 forming the random number set 21 generated in step S52 equal to the total number n _total of the primary particle models 5 determined in step S51? It is determined whether or not (step S53).

In step S53, if it is determined that the total number of random values 22 is equal to the total number n _total of primary particle models 5 (“Y” in step S53), the number of random values 22 in random number set 21 (the random number generated) is specified as the total number of aggregate models 8 (step S54). Further, each random number 22 forming the set 21 of random numbers is specified as the number of primary particles of each aggregate model 8 (step S55). That is, the i-th (index) random value 22 arbitrarily selected from the plurality of random values 22 forming the set 21 of random numbers is set as the primary particle number of the i-th aggregate model 8 . The determined total number of aggregate models 8 and the number of primary particles of each aggregate model 8 are stored in the computer 1 .

On the other hand, if it is determined in step S53 that the sum of the random numbers 22 is not equal to the total number n _total of the primary particle models 5 (“N” in step S53), the random number set 21 generated in step S52 is discarded. Then (step S56), steps S52 and S53 are performed again. As a result, in the first determination step S41, the sum of the primary particle numbers of all aggregates is determined randomly according to the probability distribution function that satisfies the first parameter (average primary particle number of aggregates) of the primary particle model 5. A set of aggregate models 8 can be reliably determined equal to the total number n _total .

Next, as shown in FIG. 5, in the aggregate model creation step S4 of the present embodiment, the total number of aggregate models 8 determined in steps S54 and S55 and the number of primary particles of each aggregate model 8 , a plurality of primary particle models 5 (shown in FIG. 8(a)) constituting each aggregate model 8 are combined (aggregate three-dimensional structure determination step S42). In the aggregate three-dimensional structure determination step S42 of the present embodiment, the three-dimensional structure of the aggregate model 8 is expressed as the arrangement of the primary particle models 5 inside the aggregate model 8. FIG. A plurality of primary particle models 5 are combined so that the three-dimensional structure of this aggregate model 8 statistically approximates the three-dimensional structure of the actual aggregate 4 (shown in FIG. 2(b)).

A procedure for combining a plurality of primary particle models 5 can be adopted as appropriate. For example, it is conceivable to combine a plurality of primary particle models 5 so as to reproduce an actual aggregate manufacturing process. In such a procedure, first, in a virtual space (not shown) in which a large number of primary particle models 5 and surrounding gas and solvent molecules are arranged, the primary particle models 5 are diffused and aggregated to form a large number of aggregates. A simulation is performed to generate the aggregate model 8 . Then, the total number n _total of the primary particle models 5 of the aggregate model 8 determined in the first determination step S41 from among the numerous aggregate models 8 created by the simulation, and the primary particles of each aggregate model 8 A set of aggregate models 8 is selected to be equal in number.

However, in the method described above, a large number of primary particle models 5 and aggregate models 8 arranged in the virtual space need to be simulated over a long period of time in order to diffuse and aggregate them.

In the aggregate three-dimensional structure determination step S42 of the present embodiment, focusing on the fact that the production process of aggregates such as silica can be approximated by a DLCA (Diffusion-limited cluster aggregation) model, a plurality of primary particle models 5 are combined. (Specific procedures will be described later). As a result, in the present embodiment, compared to the method described above, the aggregate structure ( An aggregate model 8) can be created.

8(a) to (d) are conceptual diagrams explaining the procedure for combining the primary particle models 5. FIG. In this embodiment, not only the coupling between the aggregate model 8 and the primary particle model 5 but also the coupling between the aggregate models 8, 8 is allowed. As shown in FIGS. 8(a) to (c), aggregate model 8 and primary particle model 5, and aggregate models 8, 8 through multi-stage hierarchical coupling, are shown in FIG. 8(d). , a three-dimensional structure of a flocculated aggregate model 8 is generated.

If the fractal dimension of the aggregate is not about 1.8, instead of the DLCA model of the present embodiment, for example, DLA (Diffusion-limited aggregation) model or RLCA (Reaction limited aggregation) model is used. Also good. The DLA model is known to produce aggregates with a fractal dimension of about 2.5. On the other hand, the RLCA model is known to produce aggregates with a fractal dimension of about 2.1. These models may be selected to create fractal-dimensional aggregate structures corresponding to the selected models.

Aggregate model 8 is not a physical model, but is described in, for example, paper 2 (A. V. Filippov et al., Journal of Colloid and Interface Science, vol.229, isuue 1, p.261-273, 2000). It may be created based on procedures. In this procedure, for example, an aggregate model 8 is created by selecting the fractal dimension in the range of 1.0 to 2.5 and combining the primary particle models 5 according to an artificial rule.

In the aggregate three-dimensional structure determination step S42 of the present embodiment, the aggregate model 8 is created by hierarchically combining the primary particle models 5 so that the fractal dimension is about 1.8 based on the DLCA model. . FIG. 9 is a flow chart showing an example of the processing procedure of the aggregate three-dimensional structure determination step S42.

In the aggregate three-dimensional structure determination step S42 of the present embodiment, first, one arbitrary random number value 22 is selected from the random number set 21 (shown in FIG. 7) generated in step S52 (step S61). As mentioned above, the random value 22 is equal to the number of primary particles of aggregate models 8 selected from all aggregate models 8 to be created. Also, the number of random numbers 22 forming the random number set 21 generated in step S52 is equal to the total number of aggregate models 8 to be created.

The random value 22 can be selected appropriately. In step S61 of the present embodiment, the random numbers 22 are selected in the random number set 21 in the order of generation of the random numbers 22 (for example, the order of "index" shown in FIG. 7). The selected random value 22 is stored in computer 1 .

In the aggregate three-dimensional structure determination step S42 of the present embodiment, in subsequent steps S62 to S68 and S70, based on the random value 22 selected in step S61 (that is, the number of primary particles n of the aggregate model 8), A three-dimensional structure of an aggregate model 8 having n primary particles is randomly determined.

Next, in the aggregate three-dimensional structure determination step S42 of the present embodiment, as shown in FIG. , the number of primary particles n) is created (step S62). As described above, in step S62 of the present embodiment, as the initial state of the aggregate model 8, each of the primary particle models 5 is an independent aggregate without all the primary particle models 5 being combined with each other. Defined as Model 8. These aggregate models 8 are arranged in independent virtual spaces (not shown). This ensures that the aggregate models 8 (primary particle models 5) do not overlap each other.

The virtual space (not shown) of this embodiment is open with no boundaries (that is, no periodic boundary conditions are set and it extends infinitely in any direction). Note that the virtual space is a circumscribed sphere that can surround the aggregate model 8 having the maximum size among the aggregate models 8 that can be created by combining the aggregate models 8 with the length of each side. may be set to a finite value sufficiently larger than the diameter of . If such a virtual space is used, periodic boundary conditions may be defined. These aggregate models 8 are stored in computer 1 .

Next, in the aggregate three-dimensional structure determination step S42 of the present embodiment, two aggregate models 8, 8 are selected from all the aggregate models 8 created in step S62 (step S63). In step S63 of this embodiment, two aggregate models 8, 8 are randomly selected. The selected aggregate model 8 , 8 is stored in computer 1 .

Next, in the aggregate three-dimensional structure determination step S42 of the present embodiment, one of the two aggregate models 8, 8 selected in step S63 has the other aggregate model 8 in the vicinity of the aggregate model 8. Temporarily arranged (step S64). As shown in FIG. 8(c), in step S64 of the present embodiment, the position and orientation of the other aggregate model 8B are randomly determined so as not to come into contact with the other aggregate model 8A.

In step S64, one aggregate model 8A is desirably the aggregate model 8 having a relatively large primary particle number n among the two aggregate models 8, 8 selected. If the two aggregate models 8, 8 have the same number of primary particles, any aggregate model 8 can be selected as one aggregate model 8A.

The initial value of the distance between one aggregate model 8A and the other aggregate model 8B is specified as the distance between the center of gravity of one aggregate model 8A and the center of gravity of the other aggregate model 8B. be. This initial value can be set as appropriate.

In step S64 of the present embodiment, first, among the primary particle models 5 constituting one aggregate model 8A, the center of the primary particle model 5 farthest from the center of gravity of one aggregate model 8A and one aggregate model A first distance to the center of gravity of 8A is determined. Next, of the primary particle models 5 constituting the other aggregate model 8B, the center of the primary particle model 5 farthest from the center of gravity of the other aggregate model 8B and the center of gravity of the other aggregate model 8B distance is required. Then, in step S64, the initial value of the distance between one aggregate model 8A and the other aggregate model 8B is set to 1 of the sum of the first distance, the second distance, and the diameter of the primary particle model 5. The tentative arrangement of the other aggregate model 8B is determined so as to double to double. At this time, the initial value of the orientation of the other aggregate model 8B is randomly selected. The initial value of the direction of the other aggregate model 8B viewed from the one aggregate model 8A is also randomly selected. The temporary arrangement of the other aggregate model 8B is stored in the computer 1. FIG.

Next, in the aggregate three-dimensional structure determination step S42 of the present embodiment, the other aggregate model 8B is randomly moved (step S65). The moving distance of the other aggregate model 8B can be set as appropriate. The moving distance in this embodiment is set to 0.1 to 1.0 times the particle diameter (primary particle diameter) D1 of the primary particle model 5 shown in FIG. 2(a). Moreover, it is desirable that the moving direction of the other aggregate model 8B is selected at random. On the other hand, the orientation of the other aggregate model 8B may be fixed or may be slightly rotated at random. The other aggregate model 8B after movement is stored in the computer 1. FIG.

Next, in the aggregate three-dimensional structure determination step S42 of the present embodiment, it is determined whether or not one aggregate model 8A and the other aggregate model 8B are in contact (step S66). In the present embodiment, one of the distances between the centers of the primary particle models 5 constituting one aggregate model 8A and the primary particle models 5 constituting the other aggregate model 8B is the primary particle model 5 is smaller than the particle size D1, it is determined that one aggregate model 8A and the other aggregate model 8B are in contact.

In step S66, when it is determined that one aggregate model 8A and the other aggregate model 8B are in contact ("Y" in step S66), one aggregate model 8A and the other aggregate model 8A are in contact with each other. A step S67 of combining with the aggregate model 8B is performed. Further, when it is determined that one aggregate model 8A and the other aggregate model 8B are not in contact ("N" in step S66), the next step S70 is performed.

In step S70 of the present embodiment, it is determined whether the distance between one aggregate model 8A and the other aggregate model 8B is more than necessary. The phrase "unnecessarily distant" means that the distance between the center of gravity of one aggregate model 8A and the center of gravity of the other aggregate model 8B is equal to or greater than a predetermined threshold. there is The threshold can be appropriately set if it can be determined whether the aggregate model 8A on one side and the aggregate model 8B on the other side are separated more than necessary. The threshold value of this embodiment is set to 1 to 2 times the initial value of the above distance.

In step S70, when it is determined that the distance between the center of gravity of one aggregate model 8A and the center of gravity of the other aggregate model 8B is equal to or greater than the threshold value ("Y" in step S70), one aggregate It is determined that the center of gravity of the model 8A and the center of gravity of the other aggregate model 8B are separated more than necessary. In this case, the placement of the other aggregate model 8B is discarded, and step S64 and subsequent steps are performed.

On the other hand, in step S70, when it is determined that the distance between the center of gravity of one aggregate model 8A and the center of gravity of the other aggregate model 8B is less than the threshold ("N" in step S70), one It is determined that the aggregate model 8A and the other aggregate model 8B are not spaced apart more than necessary, and that there is a high possibility that they will come into contact with each other. In this case, the steps after step S65 are performed again from the placement of the other aggregate model 8B that was moved in step S65. Thus, in the aggregate three-dimensional structure determination step S42 of the present embodiment, since it is possible to prevent the one aggregate model 8A and the other aggregate model 8B from being separated more than necessary, realistic calculation With time, the aggregate model 8A on one side and the aggregate model 8B on the other side can be reliably brought into contact with each other.

In step S67 of this embodiment, one aggregate model 8A and the other aggregate model 8B are combined to create one aggregate model 8 (shown in FIG. 8(d)). In step S67 of the present embodiment, all the primary particle models 5 (shown in FIG. 8(c)) of the other aggregate model 8B are made to belong to the one aggregate model 8A (shown in FIG. 8(c)). and the other aggregate model 8B is deleted. The new aggregate model 8 is stored in computer 1 .

Next, in the aggregate three-dimensional structure determination step S42 of the present embodiment, it is determined whether or not all aggregate models 8 (primary particle models 5) have been combined (step S68). In step S68, "all aggregate models 8 are combined" means that the primary particle models 5 (for example, shown in FIG. 8(a)) created in step S62 are combined and (that is, the number of primary particles of the aggregate model 8 is the same as the random number selected in step S61) shown in FIG.

If it is determined in step S68 that all aggregate models 8 have been combined (“Y” in step S68), the number of primary particles is set equal to the random value 22 (shown in FIG. 7) selected in step S61. The three-dimensional structure of the obtained aggregate model 8 is specified. The three-dimensional structure of the identified aggregate model 8 is stored in the computer 1, and the next step S69 is performed.

On the other hand, if it is determined in step S68 that not all the aggregate models 8 have been combined (“N” in step S68), steps S63 and subsequent steps are performed again. As a result, all aggregate models 8 (primary particle models 5) created in step S62 can be combined as one aggregate model 8 (shown in FIG. 8(d)). Therefore, in the aggregate three-dimensional structure determination step S42, the three-dimensional structure of the aggregate model 8 set to the number of primary particles equal to the random number 22 selected in step S61 can be reliably created.

Next, in the aggregate three-dimensional structure determination step S42 of the present embodiment, it is determined whether or not the three-dimensional structures of all aggregate models have been determined (step S69). In step S69, ``the three-dimensional structures of all aggregate models 8 have been determined'' means that for all the random values 22 that make up the set 21 of random numbers shown in FIG. This means that all aggregate models 8 (shown in FIG. 8(d)) set to the number have been created.

In step S69, when it is determined that the three-dimensional structures of all aggregate models have been determined ("Y" in step S69), the series of processes of the aggregate three-dimensional structure determination step S42 are completed. On the other hand, if it is determined in step S69 that the three-dimensional structures of all aggregate models have not been determined ("N" in step S69), one random value 22 that has not yet been selected is selected in step S61. Then, step S62 and subsequent steps are performed again. As a result, in the aggregate three-dimensional structure determination step S42, for all the random numbers 22 that make up the set 21 of random numbers generated in step S52, the aggregate model 8 is set to the same number of primary particles as those random numbers 22. can be created with certainty. Therefore, in the aggregate three-dimensional structure determination step S42, two aggregate models 8, 8 are randomly selected until the number of primary particles of each aggregate model 8 becomes equal to the number of primary particles determined in the first determination step S41. The operation of selecting and combining them into one aggregate model 8 (new aggregate model 8) can be repeated.

In the aggregate three-dimensional structure determination step S42 of the present embodiment, since a plurality of primary particle models 5 are arranged based on the DLCA model, the aggregate model ( aggregate structures) 8 can be created.

Next, in the creation method of the present embodiment, the computer 1 arranges a plurality of aggregate models 8 (shown in FIG. 8(d)) in the space 6 (shown in FIG. 4) to create a filler model ( Filler model creation step S5). In the filler model creation step S5, a plurality of aggregate models 8 are arranged in the space 6 so as to satisfy the volume fraction obtained in step S2. In the filler model creation step S5 of the present embodiment, when arranging a plurality of aggregate models 8 in the space 6, a plurality of aggregate models 8 are randomly placed in the space 6 so that the aggregate models 8 do not overlap each other. placed. FIG. 10 is a flow chart showing an example of the processing procedure of the filler model creation step S5.

In the filler model creation step S5 of the present embodiment, first, one aggregate model 8 is selected from the aggregate models 8 that have not yet been arranged in the space 6 (step S71). A selection procedure is not particularly limited. In the filler model creation step S5 of the present embodiment, aggregate models 8 are selected in descending order of the number of primary particles. If there are a plurality of aggregate models 8 having the same number of primary particles, they can be selected as appropriate. good too.

Next, in the filler model creation step S5 of the present embodiment, the selected aggregate model 8 is provisionally arranged in the space 6 (step S72). In step S72, the selected aggregate model 8 is rotated in a random orientation, and the aggregate model 8 is provisionally arranged at a random position in the space 6. FIG. The temporarily arranged aggregate model 8 is stored in the computer 1 .

Next, in the filler model creation step S5 of the present embodiment, whether the aggregate model 8 temporarily arranged in the space 6 overlaps with a copy of itself (temporarily arranged aggregate model 8) caused by periodic boundary conditions is determined (step S73). In this embodiment, step S73 and subsequent step S74 are performed only when periodic boundary conditions are set in the space 6 . Therefore, when the periodic boundary condition is not set in the space 6, the steps S73 and S74 are omitted, and the next step S75 is carried out.

In step S73, for each primary particle model 5 of the aggregate model 8 provisionally placed in the space 6, aggregation of all copies of itself caused by periodic boundary conditions (especially all copies of regions adjacent to the space 6) is performed. If the center-to-center distance of the aggregate model 8 with each primary particle model 5 is greater than the particle size D1 of the primary particle model 5 (shown in FIG. 2(a)), it does not overlap with its own copy. is judged. On the other hand, if at least one of the above center-to-center distances is less than or equal to the particle size D1, it is determined that there is overlap.

If it is determined in step S73 that the temporarily arranged aggregate model 8 overlaps with its own copy (aggregate model 8) caused by periodic boundary conditions (“Y” in step S73), the temporarily arranged Aggregate model 8 is discarded (step S74), and step S72 and subsequent steps are performed again.

On the other hand, if it is determined in step S73 that the temporarily arranged aggregate model 8 does not overlap with its own copy (aggregate model 8) caused by the periodic boundary condition (“N” in step S73), then of step S75 is performed. Thereby, in the filler model creation step S5 of the present embodiment, the aggregate model 8 can be provisionally arranged so as not to overlap with a copy of itself caused by periodic boundary conditions.

Next, in the filler model creation step S5 of the present embodiment, it is determined whether or not the temporarily arranged aggregate model 8 overlaps with another aggregate model 8 already arranged in the space 6 (step S75 ). In step S75, for each primary particle model 5 of the temporarily arranged aggregate model 8, any of the distances between the centers of the primary particle models 5 of all the already arranged aggregate models 8 is equal to the primary particle model 5, it is determined that there is no overlap. On the other hand, if at least one of the above center-to-center distances is less than or equal to the particle size D1, it is determined that there is overlap.

In step S75, when it is determined that the temporarily arranged aggregate model 8 overlaps with the arranged aggregate model 8 ("Y" in step S75), the temporarily arranged aggregate model 8 is discarded. (Step S74), Step S72 and subsequent steps are performed again.

On the other hand, if it is determined in step S73 that the provisionally arranged aggregate model 8 does not overlap the arranged aggregate model 8 ("N" in step S75), the next step S76 is performed. Thereby, in filler model creation process S5 of this embodiment, aggregate model 8 can be temporarily arranged so that it may not overlap with aggregate model 8 which has been arranged.

Next, in the filler model creation step S5 of the present embodiment, the position of the temporarily arranged aggregate model 8 is determined as the arrangement position of the aggregate model 8 (step S76). The determined aggregate model 8 is stored in the computer 1 .

Next, in the filler model creation step S5 of the present embodiment, it is determined whether or not all the aggregate models 8 have been placed in the space 6 (step S77). In step S77, when it is determined that all the aggregate models 8 have been arranged in the space 6 ("Y" in step S77), the series of processing of the filler model creation step S5 ends. A filler model 10 is created by arranging these aggregate models 8 . FIG. 11 is a conceptual diagram showing an example of the filler model 10. As shown in FIG.

On the other hand, in step S77 shown in FIG. 10, when it is determined that not all aggregate models 8 are arranged in space 6 ("N" in step S77), in step S71, other aggregate models 8 is selected, and step S72 and subsequent steps are performed again. As a result, in the filler model creation step S5, a plurality of aggregate models 8 are randomly arranged in the space 6 in order from the aggregate model 8 with a large number of primary particles so that the plurality of aggregate models 8 do not overlap each other. can be done.

In the creation method of the present embodiment, the three-dimensional structure of the aggregate model 8 can be created so as to approximate the actual filler based on the procedure of the aggregate three-dimensional structure determination step S42 (shown in FIG. 9). Therefore, according to the creation method of the present embodiment, based on the procedure of the filler model creation step S5 (shown in FIG. 10), as shown in FIG. 11, by placing the aggregate model 8 in the space 6 , the microstructure of the filler model 10 can be approximated to the microstructure of the filler.

In the parameter input step S1 (shown in FIG. 3) of the previous embodiment, the case where the average number of primary particles (first parameter m) is obtained as the average value of the appearance frequency distribution of the number of primary particles of aggregates is exemplified. However, it is not limited to such an aspect. For example, the first parameter m (and second parameter k) is determined so that the aggregate size distribution of the filler model 10 (shown in FIG. 11) approaches some distribution such as the actual filler aggregate size distribution. may

In the parameter input step S1, the procedure for determining the first parameter m (and the second parameter k) as described above can be adopted as appropriate. For example, a plurality of first parameters m (and a plurality of sets including the second parameter k) based on, respectively create a filler model 10 (shown in FIG. 11), the aggregate size distribution of those filler models 10 A first parameter m (and a second parameter k) that can be calculated to approximate the desired distribution may be employed. However, in such a method, it is necessary to create a large number of filler models, and there is a problem that a lot of calculation time is required.

In the parameter input step S1 of this embodiment, the first parameter m (and the second parameter k) as described above is determined in a short time based on the procedure described below. FIG. 12 is a flow chart showing an example of the processing procedure of the parameter input step S1 according to another embodiment of the invention. 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 parameter input step S1 of this embodiment, first, the size distribution of aggregates constituting the filler (hereinafter sometimes simply referred to as "first distribution") is input to the computer 1 (step S81). In the creation method of this embodiment, the filler model 10 is created such that the aggregate size distribution of the filler model 10 approaches the first distribution P ₁ (S).

The first distribution P ₁ (S) is a distribution (frequency distribution) showing the relationship between the aggregate size S and its frequency. Aggregate size S is a quantity relating to the spatial extent of individual aggregates. The aggregate size S in this embodiment is defined as twice the radius of gyration Rg. The first distribution P ₁ (S) is experimentally obtained by, for example, transmission electron microscopic observation of a large number of aggregates, dynamic light scattering method, or the like. FIG. 13 is a graph showing an example of the first distribution P ₁ (S).

Next, in the parameter input step S1 of this embodiment, the computer 1 provides the aggregate size distribution (hereinafter, sometimes simply referred to as "second distribution") based on the number n of primary particles for a plurality of numbers n of primary particles different from each other. .) are input (step S82).

The second distribution P ₂ (S,n) is the frequency distribution of aggregate sizes S for aggregates having n primary particles. FIG. 14 is a graph showing an example of the second distribution P ₂ (S,n). FIG. 14 shows the frequency distribution of the aggregate size S for aggregates with primary particle numbers n of 6, 8, and 12, respectively.

The range of the primary particle number n can be set as appropriate. The minimum value of the range of the number of primary particles n is desirably one. When the second parameter k is also defined, the minimum value of the range of the number of primary particles n is preferably set to the value of the second parameter k. On the other hand, the maximum value of the range of the number of primary particles n is preferably set to be equal to or greater than the maximum value of the number of primary particles of the aggregate model 8 constituting the filler model 10 to be created. All integer values between such minimum and maximum values are set as primary particle number n.

In order to accurately estimate the second distribution P ₂ (S, n), the number of aggregates experimentally observed or aggregates created in a simulation to be described later for each value of the number n of primary particles The number of three-dimensional structures of the model 8 (number of primary particles n) is preferably 30 or more.

The second distribution P ₂ (S, n) can be obtained as appropriate. The second distribution P ₂ (S,n) is obtained (observed) experimentally, for example, by performing a number of 3DTEM observations on individual aggregates. The second distribution P ₂ (S, n) may be obtained based on empirical rules or assumptions from the results of two-dimensional observation with a transmission electron microscope.

Also, the second distribution P ₂ (S, n) may be estimated from simulation results using a large number of aggregate models, etc., generated. In such a method for estimating the second distribution P ₂ (S, n), first, the three-dimensional structure of the aggregate model 8 based on the individual primary particle number n is randomly determined in the range of the primary particle number n described above. be done. For determination of the three-dimensional structure of such an aggregate model 8, for example, the procedure of steps S61 to S68 and S70 of the aggregate three-dimensional structure determination step S42 shown in FIG. 9 can be used.

Next, in the method of estimating the second distribution P ₂ (S, n), the radius of inertia is obtained based on the determined three-dimensional structure of the aggregate model 8 . The radius of inertia is, for example, for each primary particle model 5 (shown in FIG. 8(c) as an example) constituting the aggregate model 8, the distance from the center of gravity of the aggregate model 8 of the primary particle model 5 is calculated. and the root mean square of those distances is calculated.

Next, in the method of estimating the second distribution P ₂ (S,n), a value twice the radius of inertia is obtained as the aggregate size S for each aggregate model 8 . Then, based on the aggregate size S of each aggregate model 8, a second distribution P ₂ (S, n) is obtained. A second distribution P ₂ (S,n) is input to the computer 1 .

Next, in the parameter input step S1 of this embodiment, the computer 1 determines (a set of) values to be searched for for the first parameters m (and the second parameters k) (step S83). The range of the determined first parameter (average number of primary particles) m can be defined as appropriate. The range of the first parameter m in this embodiment is set to be the same as the range of the primary particle number n of the second distribution P ₂ (S, n).

Values of the plurality of first parameters m are defined as appropriate. For example, in the range of the number of primary particles n of the second distribution P ₂ (S, n), a constant number of first parameters m may be selected at regular intervals or randomly. In this embodiment, all integer values within the primary particle number n of the second distribution P ₂ (S,n) are determined as the values of the plurality of first parameters m.

When the second parameter k is defined, multiple sets of values (m, k) of the first parameter m and the second parameter k are determined. The range of values of the second parameter k is defined as appropriate. The range of the second parameter k in this embodiment is the same as the range of the primary particle number n of the second distribution P ₂ (S, n).

A set (m, k) of values of a plurality of first parameters m and second parameters k is appropriately defined. For example, for each value of the first parameter m, a certain number of second parameters k may be selected at regular intervals or randomly within the range of the number of primary particles n. In this embodiment, the first parameter m is a range of all integers from 1 to n _max (the maximum value of the number of primary particles n), and the second parameter k is a range of all integers from 1 to n _max , By combining these values, a plurality of sets (m, k) of values of the first parameter m and the second parameter k are defined. A set (m, k) of values of a plurality of first parameters m and second parameters k is, for example, (1, 1), (1, 2), ..., (1, n _max ), (2, 1) , (2, 2), ..., (2, _nmax ), ..., ( _nmax , 1), ( _nmax , 2), ..., ( _nmax , _nmax ). The determined sets of values (m, k) of the plurality of first parameters m and second parameters k are stored in the computer 1 .

Next, in the parameter input step S1 of this embodiment, the computer 1 controls individual values (of set), an estimate of the aggregate size frequency distribution (first temporary distribution P(S)) is calculated (step S84). The estimated value of the aggregate size frequency distribution (first temporary distribution P(S)) is defined by the following equation (1).

ここで、
Ｓ：凝集体サイズ
ｎ：凝集体の一次粒子数
ｍ：第１パラメータ（平均一次粒子数）
ｐ_n（ｎ，ｍ）：凝集体の一次粒子数ｎの出現頻度分布の第１パラメータｍについての関数（確率質量関数）
Ｐ₂（Ｓ，ｎ）：第２分布
ここで、第２分布Ｐ₂（Ｓ，ｎ）は、一次粒子数がｎである凝集体４の凝集体サイズＳの頻度分布である。

here,
S: Aggregate size n: Number of primary particles of aggregate m: First parameter (average number of primary particles)
p _n (n, m): function (probability mass function) for the first parameter m of the appearance frequency distribution of the primary particle number n of aggregates
P ₂ (S, n): Second distribution Here, the second distribution P ₂ (S, n) is the frequency distribution of aggregate sizes S of aggregates 4 having n primary particles.

In the above equation (1), the first temporary distribution P(S) is the probability mass function p _n (n, m) for each primary particle number n, the aggregate size multiplied by the second distribution P ₂ (S, n) of all the primary particle numbers n. An example of the first temporary distribution P(S) is shown in FIG.

The function p _n (n, m) is a function representing the frequency at which the number of primary particles is n, and is the same as the function defined in step S52 (shown in FIG. 6) of the previous embodiments. That is, in this embodiment, the probability mass function p _n (n,m)={(m−1)/m} ⁿ⁻¹ /m is used. Note that when the second parameter k is defined, the function _pn (n, m, k) is used instead of the function _pn (n, m). In this embodiment, the function p _n (n,m,k)=0 is used when the number of primary particles n is less than the second parameter k. On the other hand, when the number of primary particles n is greater than or equal to the second parameter k, the function p _n (n, m, k)={(m−1)/m} ^nk /m is used.

FIG. 15 is a graph showing an example of the function p _n (n,m). Note that the function p _n (n, m, k) is based on the function p _n (n, m), and changes the value when the number of primary particles n is less than the second parameter k to 0, and all primary particles It is obtained by normalization so that the sum of the probabilities (frequency) for the number of particles n is 1.

Next, in the parameter input step S1 of this embodiment, the computer 1 calculates the inter-distribution distance between the first temporary distribution P(S) and the first distribution P ₁ (S) shown in FIG. (Step S85). The inter-distribution distance can be defined as appropriate. The inter-distribution distance is desirably defined to be smaller (closer) as the similarity between the two distributions is higher and larger (farther) as the similarity is lower. The Hellinger distance is used for the inter-distribution distance in this embodiment. The Hellinger distance has a minimum value of '0' when the two distributions are most similar and a maximum value of '1' when they are least similar. Otherwise, a real value between 0 and 1 is obtained depending on the similarity of the two distributions. The calculated inter-distribution distance is obtained for each value of the first parameter m (or a set of values of the first parameter m and the second parameter k). The inter-distribution distances are stored in computer 1 .

Next, in the parameter input step S1 of this embodiment, the computer 1 calculates inter-distribution distances for all values of the first parameter m (or pairs of values of the first parameter m and the second parameter k). It is determined whether or not the calculation has been performed (step S86). If it is determined in step S86 that all inter-distribution distances have been calculated ("Y" in step S86), the next step S87 is performed.

On the other hand, in step S86, the condition that all inter-distribution distances have been calculated for all values of the first parameter m (or pairs of values of the first parameter m and the second parameter k) is satisfied. If it is determined that there is no ("N" in step S86), another value of the first parameter m (or a set of values of the first parameter m and the second parameter k) for which the inter-distribution distance has not been calculated is It is selected (step S88), and steps S84 to S86 are performed again. As a result, in the parameter input step S1, for all values of the plurality of first parameters m (or pairs of values of the first parameter m and the second parameter k), the corresponding first temporary distribution P(S), The inter-distribution distance with the first distribution P ₁ (S) can be reliably calculated.

Next, in the parameter input step S1 of the present embodiment, the computer 1 selects from among all the values of the plurality of first parameters m (or the sets of values of the first parameter m and the second parameter k) Select the value of the first parameter m (or the set of values of the first parameter m and the second parameter k) that minimizes the distance, and approximate it to the aggregate size distribution of the actual filler. (or a set of values of the first parameter m and the second parameter k) (step S87). FIG. 16 is a graph showing an example of the first temporary distribution P(S) and the first distribution P ₁ (S) calculated based on the determined first parameter m and second parameter k. The determined value of the first parameter m (or the set of values of the first parameter m and the second parameter k) is stored in the computer 1 .

In this embodiment, steps S2 to S5 shown in FIG. 3 are performed using the first parameter m (or the set of first parameter m and second parameter k) determined in parameter input step S1. As a result, in the first determination step S41 (shown in FIGS. 5 and 6) of this embodiment, the total number of aggregate models 8 capable of reproducing the aggregate size frequency distribution, and the primary particles of each aggregate model 8 number can be determined. Also, in the aggregate three-dimensional structure determination step S42 (shown in FIGS. 5 and 9), an aggregate model 8 in which the particle size of the primary particles of the aggregate 4 and the fractal dimension D _f are reproduced can be created. Furthermore, in the filler model creation step S5 (shown in FIGS. 3 and 10), a filler model 10 is created in which the volume fraction of the actual filler is reproduced. Therefore, in the production method of the present embodiment, the microstructure of the filler model 10 can be accurately approximated to the microstructure of the actual filler.

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. 17 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 Both ends of this molecular chain 12 are composed of methyl groups (--CH ₃ ) instead of methylene groups (--CH ₂ --). 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. 18 is a flow chart showing an example of a processing procedure of a polymeric material simulation method.

In the simulation method of the present embodiment, first, various settings required for creating a coarse-grained model and various physical property values required for coarse-grained simulation are input to the computer 1 (step S6). In step S6 of the present embodiment, first, a cell, which is a virtual space corresponding to a portion of a polymer material (not shown), is input. FIG. 19 is a conceptual diagram showing part of a polymer material model. In FIG. 19, only a part of the coarse-grained filler model 25 and the coarse-grained molecular chain model 15 are representatively shown.

The cell 14 is a calculation target area in the molecular dynamics calculation described later. The cells 14 of this embodiment are set in the same manner as the spaces 6 shown in FIG. At least one coarse-grained filler model 25 and a plurality of coarse-grained molecular chain models 15 are arranged 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 coarse-grained molecular chain model 15 described later. As a result, in the molecular dynamics calculation, the cell 14 prevents collisions with itself (image) that is periodically arranged due to periodic boundary conditions, and considers the spatial extension of the coarse-grained molecular chain model 15. can be properly calculated by Moreover, the length Lb of the present embodiment is desirably twice or more the maximum value of the radius of inertia (not shown), which is an amount indicating the spread of each aggregate of the coarse-grained filler model 25 to be created. As a result, the cells 14 can appropriately arrange the respective aggregate models 8 by preventing collisions between the periodically arranged aggregates and themselves (images) due to periodic boundary conditions.

The size of the cell 14 is set to a stable volume at, for example, 1 atmosphere. 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 step S6 of the present embodiment, a coarse-grained molecular chain model 15 modeling a molecular chain 12 (shown in FIG. 17) of a polymer material is set in the computer 1, and various physical property values thereof are input. be done.

A coarse-grained molecular chain model 15 is represented using a plurality of particles 16 whose number is smaller than the number of atoms constituting the molecular chain 12 (shown in FIG. 17). Linking chains 17 are bound between these particles 16 , 16 . The coarse-grained molecular chain model 15 of the present embodiment is exemplified by a Kremer-Grest model, but is not particularly limited. The coarse-grained 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 coarse-grained 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.

The binding chain 17 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. Further, the binding potential of the binding chain 17 is described, for example, in Paper 3 (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). Such a coarse-grained 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 the present embodiment, the computer 1 creates a coarse-grained model in which the coarse-grained molecular chain model 15 and the coarse-grained filler model 25 are arranged inside the cell 14 (coarse-grained model creation step S7). In the coarse-grained model creating step S7, a polymer material model illustrated in FIG. 19 is created. FIG. 20 is a flow chart showing an example of the processing procedure of the coarse-grained model creation step S7.

In the coarse-grained model creation step S7 of the present embodiment, first, the filler model 10 shown in FIG. 11 is acquired (step S91). The filler model 10 is created based on the processing procedure (shown in FIG. 3) of steps S1 to S5 of the filler model creation method described above. The filler model 10 to be acquired may be created at any time in this step S91, or the filler model 10 created in advance prior to implementation of the simulation method may be used.

Next, in the coarse-grained model creating step S7 of the present embodiment, as shown in FIG. 19, the coarse-grained filler model 25 is created inside the cell 14 (step S92). In step S92, first, the filler model 10 (shown in FIG. 11) input in step S91 is placed inside the cell 14, and the internal region of the filler model 10 is filled with particles (coarse-grained particles) 11. be done. By replacing the filler model 10 with these particles 11, the coarse-grained filler model 25 shown in FIG. 19 is created. By creating such a coarse-grained filler model 25, the size of the particles 16 of the coarse-grained molecular chain model 15 and the size of the particles 11 of the coarse-grained filler model 25 are adjusted to be approximately the same. It becomes possible to By such adjustment, it is possible to create a highly quantitative model in which coarse-grained granularities of molecular chains and fillers are matched.

The coarse-grained filler model 25 is composed of one or more coarse-grained aggregate models 26 . One coarse-grained aggregate model 26 is represented by combining one or more primary particles. Note that the coarse-grained aggregate model 26 in FIG. 19 is configured as one primary particle. Also, the inner region is composed of a large number of particles 11 .

The filling of the particles 11 may be performed, for example, by arranging the particles 11 inside the filler model in a face-centered cubic lattice structure without gaps, as shown in the patent document (Japanese Patent No. 5913260). The size of the particles 11 is appropriately defined. The size of the particles 11 of the present embodiment is exemplified by the same size as the particles 16 of the coarse-grained molecular chain model 15, but is not particularly limited. In step S92, the coarse-grained filler model 25 is not the filler model 10 (shown in FIG. 11) filled with a large number of particles 11, but the particles 11 are the primary particle model 5 of the filler model 10. Identical, ie, the diameter of each primary particle may be treated as spheres equal to the particle size D1 (shown in FIG. 2(a)).

Next, in the coarse-grained model creation step S7 of the present embodiment, adjacent particles 11, 11 of the coarse-grained filler model 25 are joined (step S93). As a result, the rigidity of the coarse-grained filler model 25 can be increased, so deformation of the coarse-grained filler model 25 can be suppressed in the coarse-grained simulation described later.

Between the particles 11, 11 a binding potential is defined accordingly. The binding potential of this embodiment is exemplified by the same binding potential between the particles 16, 16 of the coarse-grained molecular chain model 15, but is not particularly limited. Instead of defining the bond, in the coarse-grained simulation described later, the position of each particle 11 of the coarse-grained filler model 25 may be fixed, or the individual coarse-grained aggregate model 26 may be treated as a rigid body. may be As a result, the coarse-grained filler model 25 is created in the coarse-grained model creating step S7.

In the coarse-grained model creation step S7 of the present embodiment, a coarse-grained filler model including at least one coarse-grained aggregate model 26 modeling the aggregate (filler aggregate) 4 shown in FIG. 25 is created. The coarse-grained filler model 25 is stored in the computer 1 .

Next, in the coarse-grained model creation step S7 of the present embodiment, as shown in FIG. 19, a plurality of coarse-grained molecular chain models 15 are arranged in regions where no coarse-grained filler models 25 are arranged. (step S94). For the placement of the coarse-grained molecular chain model 15, for example, the procedure shown in the patent document (Japanese Patent No. 6353290) may be used. As a result, in the coarse-grained model creation step S7, the coarse-grained molecular chain model 15 can be arranged in the cell 14 so as to approximate the distribution of the actual molecular chain 12 (shown in FIG. 17).

In the coarse-grained model creation step S7 of the present embodiment, the coarse-grained molecular chain model 15 and the coarse-grained filler model 25 are placed inside the cell 14 to create the coarse-grained model 30. be. The coarse-grained model 30 is stored in the computer 1 .

Next, in the simulation method of the present embodiment, the computer 1 performs a coarse-grained simulation (step S8). In step S8, in order to perform a coarse-grained simulation, first, between particles 16, 16 of adjacent coarse-grained molecular chain models 15, 15, and between particles 16 of the coarse-grained molecular chain model 15 and coarse-grained A potential P2 is defined between the particles 11 of the filler model 25 and at which attractive force and repulsive force act. As this potential P2, the LJ potential U _LJ (r _ij ) described in Patent Document 1 is employed. Each variable of the LJ potential U _LJ (r _ij ) can be appropriately set based on, for example, Paper 3 mentioned above.

Next, in step S<b>8 of the present embodiment, the computer 1 performs molecular dynamics calculations on the coarse-grained molecular chain model 15 and the coarse-grained filler model 25 . In the molecular dynamics calculation, for example, Newton's equation of motion is applied assuming that the particles 11 of the coarse-grained filler model 25 and the particles 16 of the coarse-grained molecular chain model 15 follow classical mechanics for a predetermined time for the cell 14. be done. Then, the movements of the primary particle model 5 and the particles 16 at each time are tracked for each unit time.

In the molecular dynamics calculation of this embodiment, the structural relaxation of the coarse-grained 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, a polymer material model 18 modeling a polymer material (not shown) is created in step S8.

Next, in step S8 of the present embodiment, deformation calculation (e.g., elongation 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 coarse-grained filler model 25 having a surface structure that approximates the microstructure of the filler is used, the surface structure affects the physical properties of the polymer material (coarse-grained molecular chain model 15). Influence can be evaluated with high accuracy. 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 the physical properties of the polymer material model 18 are good (step S9). 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 S9, if the physical properties are good (“Y” in step S9), for example, based on the filler and molecular chain 12 (shown in FIG. 17) that constitute the polymer material model 18, the polymer material is manufactured. (Step S10). The produced polymeric material is used, for example, in the production of industrial products such as tires. On the other hand, in step S9, if the physical properties are not good (“N” in step S9), the structure of at least one of the filler and molecular chain 12 is changed (step S11), and steps S6 to S9 are performed again. be. As a result, the simulation method of the present embodiment can manufacture polymer materials and industrial products having good physical properties.

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.

A model for numerical analysis of a filler was created based on the procedure shown in FIG. 3 (Example). In the embodiment, based on the processing procedure shown in FIG. 5, the total number of aggregate models constituting the filler model, and each aggregate model so as to satisfy the first parameter, which is the average number of primary particles of the aggregates. The primary particle number was determined, and based on the determined total number and primary particle number, multiple primary particle models were combined to make up each aggregate model.

On the other hand, in the comparative example, as in the patent document (Patent No. 5913260), the primary particle model was generated in the region where the fillers specified by 3DTEM observation of the rubber material were arranged without considering the first parameter. placed and a filler model was created.

Then, the frequency distribution of aggregate sizes and the fractal dimension D _f were calculated for the filler models of Examples and Comparative Examples. FIG. 21 is a graph showing the frequency distribution of aggregate sizes in Examples and Comparative Examples.

For the actual filler, the first distribution P ₁ (S) shown in FIG. 13 and the fractal dimension D _f were measured (experimental example). Then, the calculation results of Examples and Comparative Examples were compared with the measurement results of Experimental Example. Common specifications are as follows. The results of the tests are shown in Table 1.
Filler: Silica Filler content: 55 phr
Particle size D1 of primary particles (primary particle model): 20 nm
Virtual space length La: 350 nm

As a result of the test, as shown in FIG. 21, the Hellinger distance between the aggregate size frequency distribution of the example and the first distribution P ₁ (S) shown in FIG. ) was smaller than the Hellinger distance between the aggregate size frequency distribution of the comparative example and the first distribution P ₁ (S), and could be approximated to the first distribution P ₁ (S). Furthermore, in the example, the fractal dimension D _f could be approximated to an actual aggregate compared to the comparative example. Therefore, in the example, the microstructure of the filler model could be approximated to the microstructure of the filler compared to the comparative example.

S1 Step of inputting the first parameter S2 Step of inputting the space and volume fraction S3 Step of inputting the primary particle model S4 Step of creating a plurality of aggregate models Step S5 Step of creating a filler model

Claims (7)

A method for creating a filler model for numerical analysis of a filler, comprising:
The filler contains at least one aggregate formed by binding primary particles,
The method includes:
inputting into a computer a first parameter, which is the average number of primary particles of the aggregates;
A step of inputting a space for creating the filler model and a volume fraction of the filler model to the space into the computer;
defining in the computer a primary particle model modeling the primary particles;
the computer
combining a plurality of said primary particle models to create a plurality of aggregate models;
A step of arranging the plurality of aggregate models in the space to create the filler model so as to satisfy the volume fraction,
The step of creating the plurality of aggregate models includes:
A step of determining the total number of aggregate models constituting the filler model and the number of primary particles of each aggregate model so as to satisfy the first parameter;
combining a plurality of the primary particle models constituting each aggregate model based on the determined total number and the number of primary particles;
How to create a filler model. further comprising inputting a second parameter, which is the minimum number of primary particles of the aggregate, into the computer;
The method of creating a filler model according to claim 1, wherein the step of creating a plurality of aggregate models further includes a step of limiting so that an aggregate model having a primary particle number smaller than the second parameter is not generated. The step of determining the total number of aggregates and the number of primary particles of each aggregate includes the step of generating a set of random numbers according to the appearance frequency distribution of the number of primary particles of the aggregates based on the first parameter. ,
The method of creating a filler model according to claim 1 or 2, wherein the appearance frequency distribution is defined as a function in which the number of primary particles is exponentially attenuated. In the step of combining the primary particle models, as an initial state of the aggregate model, each of the primary particle models is not combined with each other, and each of the primary particle models is an independent aggregate model. and
repeating the operation of randomly selecting two aggregate models and combining them into one aggregate model until the number of primary particles of the aggregate model becomes equal to the determined number of primary particles; The method for creating a filler model according to any one of claims 1 to 3, comprising. In the step of creating the filler model, when arranging the plurality of aggregate models in the space, the aggregate models are arranged randomly so that the aggregate models do not overlap each other in order from the aggregate model with the largest number of primary particles. , The method for creating a filler model according to any one of claims 1 to 4. The step of inputting the first parameter includes inputting the size distribution of the aggregates constituting the filler into the computer;
Any one of claims 1 to 5, comprising a step of determining the first parameter so that the size distribution of the plurality of aggregate models constituting the filler model approaches the size distribution that has been input. A method for creating a filler model described in . inputting a cell, which is a virtual space corresponding to a portion of the polymeric material, into a computer;
a step of inputting a molecular chain model modeling the molecular chain of the polymer material into the computer;
the computer
A step of performing the filler model creation method according to any one of claims 1 to 6 to acquire the filler model;
placing the molecular chain model and the filler model inside the cell;
and performing a molecular dynamics calculation on the molecular chain model and the filler model.
A simulation method for polymeric materials.

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