Method for analyzing polymer composite material in which filler is dispersed in matrix polymer

Abstract

Provided is a novel analysis method for analyzing the relationship between the dispersion form of a filler in a matrix polymer and various characteristics of a polymer composite material.　This method for analyzing a polymer composite material in which a filler is dispersed in a matrix polymer comprises implementation by a computer of: a step (S1) for creating an analysis model of a polymer composite material in which a filler is dispersed in a matrix polymer; a step (S2) for calculating a feature quantity indicating the dispersion form of a filler particle model in the analysis model; a step (S3) for performing a characteristic simulation for calculating a physical quantity relating to mechanical characteristics or electrical characteristics of the analysis model; and a step (S4) for creating a data set including the physical quantity obtained through the characteristic simulation and the feature quantity indicating the dispersion form of the filler particle model (S4).

Description

A method for analyzing polymer composites in which fillers are dispersed in a matrix polymer

[0001] The present invention relates to a method for analyzing a polymer composite material in which a filler is dispersed in a matrix polymer.

[0002] It is known that the properties of composite materials, which are made by blending fillers with polymers, change depending on the filler's dispersion form (morphology). In the development of such composite materials, computer simulations and machine learning are used to analyze the effect of the filler's dispersion form on the properties of the composite material.

[0003] For example, Patent Document 1 proposes a method for creating a simulation model, a simulation method, and a program that can analyze the contribution of the morphology of a heterogeneous material to its mechanical properties.

[0004] Patent No. 5854067

[0005] However, analysis of the dispersion morphology of fillers has not yet progressed sufficiently, and the reality is that design guidelines utilizing the results of such analysis have not yet been established.

[0006] The present invention has been made in view of the above circumstances, and by providing a new method for analyzing the relationship between the dispersion morphology of a filler in a composite material in which a filler is blended with a polymer and the various properties of the composite material, it promotes the accumulation of knowledge about the dispersion morphology and contributes to the establishment of design guidelines.

[0007] The gist of the present invention is the following [1] to [8]. [1] A method for analyzing a polymer composite material in which a filler is dispersed in a matrix polymer, comprising the steps of: creating an analytical model of the polymer composite material in which a filler is dispersed in a matrix polymer; calculating feature quantities indicative of the dispersion morphology of a filler particle model in the analytical model; performing a characteristic simulation to calculate physical quantities related to mechanical properties or electrical properties of the analytical model; and creating a dataset including the physical quantities related to the mechanical properties or electrical properties obtained by the characteristic simulation and feature quantities indicative of the dispersion morphology of the filler particle model. [2] The method for analyzing a polymer composite material according to [1], wherein the analytical model is an analytical model including a matrix polymer model and a filler particle model, and the step of creating the dataset includes creating a dataset including the physical quantities related to the mechanical properties or electrical properties obtained by the characteristic simulation, feature quantities indicative of the dispersion morphology of the filler particle model, and feature quantities related to the number of filler particle models. [3] The method for analyzing a polymer composite material according to [1] or [2], comprising a step of performing multivariate analysis or machine learning using the dataset to create a predictive model. [4] The method for analyzing a polymer composite material according to any one of [1] to [3], wherein the step of calculating a feature quantity indicating the dispersion morphology of a filler particle model in the analytical model comprises a step of setting vertices based on the coordinates of the filler particle model and calculating a feature quantity based on Delaunay division. [5] The method for analyzing a polymer composite material according to any one of [1] to [4], wherein the step of calculating a feature quantity indicating the dispersion morphology of a filler particle model in the analytical model comprises a step of calculating a feature quantity based on the Iδ index shown in the following formula (1): [Mathematical Formula 1] In the formula (1), q is the number of regions of the analysis model divided equally into a predetermined number of regions, and X jis the number of filler particle models in the jth region. [6] The method for analyzing a polymer composite material according to any one of [1] to [5], wherein the step of calculating a feature quantity indicating the dispersion morphology of the filler particle model in the analytical model includes a step of calculating a feature quantity based on at least one of a radial component and an angular component of a power spectrum obtained by Fourier transform. [7] The method for analyzing a polymer composite material according to any one of [1] to [6], wherein the step of performing a characteristic simulation to calculate a physical quantity related to a mechanical property or an electrical property of the analytical model is a step of performing a characteristic simulation to calculate a physical quantity related to a mechanical property, in which the analytical model is deformed and a physical quantity related to stress of the deformed analytical model is calculated. [8] The method for analyzing a polymer composite material according to any one of [1] to [7], wherein the polymer composite material having a filler dispersed in a matrix polymer is a vibration-damping rubber.

[0008] According to the present invention, a new method for analyzing the dispersion morphology of a filler in a polymer composite material in which the filler is dispersed in a matrix polymer can be provided, which can promote the accumulation of knowledge regarding the dispersion morphology of fillers and contribute to the establishment of design guidelines for polymer composite materials.

[0009] Furthermore, according to one embodiment of the present invention, the prediction accuracy of the prediction model can be improved.

[0010] FIG. 1 is a flowchart showing an example of a processing procedure of an analysis method according to an embodiment of the present invention. FIG. 2 is a diagram showing an example of an analytical model used in the analysis method according to an embodiment of the present invention. FIG. 3 is a diagram for explaining an example of step S2a of the analysis method according to an embodiment of the present invention. FIG. 4 is a diagram for explaining an example of step S2a of the analysis method according to an embodiment of the present invention. FIG. 5 is a diagram for explaining an example of an analytical model used in the analysis method according to an embodiment of the present invention. FIG. 6 is a diagram for explaining an example of step S2b of the analysis method according to an embodiment of the present invention. FIG. 7 is a diagram for explaining an example of step S2b of the analysis method according to an embodiment of the present invention. FIG. 8 is a diagram for explaining an example of step S2c of the analysis method according to an embodiment of the present invention. FIG. 9 is a diagram for explaining an example of step S2c of the analysis method according to an embodiment of the present invention. FIG. 10 is a diagram for explaining an example of step S2d of the analysis method according to an embodiment of the present invention. FIG. 11 is a diagram for explaining an example of step S3 of the analysis method according to an embodiment of the present invention. FIG. 12 is a diagram for explaining an example of step S3 of the analysis method according to an embodiment of the present invention. FIG. 13 is a diagram for explaining an example of step S4 of the analysis method according to an embodiment of the present invention.

[0011] The method for analyzing a polymer composite material according to this embodiment (hereinafter sometimes referred to as "the present analysis method") is a method for analyzing a polymer composite material in which a filler is dispersed in a matrix polymer, in which a computer executes the following steps S1 to S4. [Step S1] A step of creating an analytical model of the polymer composite material in which a filler is dispersed in a matrix polymer. [Step S2] A step of calculating feature quantities that indicate the dispersion morphology of a filler particle model in the analytical model. [Step S3] A step of performing a characteristic simulation to calculate physical quantities related to the mechanical properties or electrical properties of the analytical model. [Step S4] A step of creating a dataset that includes the physical quantities obtained by the characteristic simulation and feature quantities that indicate the dispersion morphology of the filler particle model. Hereinafter, embodiments of the present invention will be described in detail. In this specification, "X and / or Y (X and Y are arbitrary configurations)" means at least one of X and Y, and can mean three possibilities: X only, Y only, or X and Y.

[0012] This analysis method is executed by a computer, similar to those used in conventional CAE (Computer Aided Engineering). This analysis method is typically executed by software and hardware working together to execute a pre-stored processing routine in an information processing device such as a personal computer equipped with a central processing unit (CPU), ROM, working memory, storage devices such as a magnetic disk, input devices such as a keyboard and a mouse, and a display device such as a monitor.

[0013] In this analysis method, the composite material to be modeled and analyzed is a composite material in which a filler is dispersed in a polymer. Examples of polymers include rubber, resin, and elastomer, and specific examples include, but are not limited to, ethylene-propylene-diene monomer terpolymer (EPDM), acrylic rubber, urethane rubber, styrene-butadiene-styrene block polymer (SBS), styrene-isobutylene-styrene block polymer (SIBS), styrene-butadiene (SB) copolymer, styrene-isoprene (SI) copolymer, styrene-isoprene-styrene (SIS) copolymer, styrene-ethylene-butylene (SEB) copolymer, and styrene-ethylene-butylene-styrene (SEBS) copolymer. Examples of the copolymer include styrene-ethylene-propylene (SEP) copolymer, styrene-ethylene-propylene-styrene (SEPS) copolymer, hydrogenated copolymers of the above, ethylene-propylene copolymer (EPR), butadiene rubber (BR), isoprene rubber (IR), styrene-butadiene rubber (SBR), liquid isoprene rubber (liquid IR), liquid butadiene rubber (liquid BR), liquid styrene-butadiene rubber (liquid SBR), liquid styrene-isoprene rubber (liquid SI), liquid styrene-ethylene-propylene rubber (liquid SEP), and liquid isoprene-butadiene rubber (liquid IR-BR).

[0014] Examples of the filler include carbon black, silica, talc, calcium carbonate, carbon fiber, carbon nanotubes, etc. Examples of the conductive filler include conductive carbon-based fillers such as conductive carbon black, carbon nanotubes, and graphite.

[0015] Examples of composite materials include materials for vibration-isolating rubber used for vibration isolation in automobiles, materials for various tubes and hoses for automobiles, materials for automobile tires, materials for various sealing members, and the like.

[0016] 1 is a flowchart showing an example of the main processing steps of the present analysis method. This flowchart shows an example of an embodiment. The present analysis method is not limited to the order of the flowchart, etc.

[0017] <Step S1> This analysis method includes step S1 of creating an analytical model of a polymer composite material in which a filler is dispersed in a matrix polymer. The analytical model according to this analysis method is, like a conventional analytical model, an analytical model that is composed of a plurality of unit elements in a virtual model creation region that can be numerically analyzed by a computer, and is created according to a conventionally known method.

[0018] The model creation region is the entire region for which periodic boundary conditions are defined, a partial region having an arbitrary shape extracted from a region for which periodic boundary conditions are defined, or a partial region having an arbitrary shape extracted from a space for which periodic boundary conditions are not defined, etc. The range of the model creation region is set in advance.

[0019] The analytical model preferably includes a filler particle model and a matrix polymer model, and further includes an interfacial phase model that constitutes the interface between the filler particle model and the matrix polymer model. Examples of the filler particle model include a model of primary particles of the filler. The shape of the filler particle model is generated by approximately modeling shapes such as spheres, oblate spheroids (prolate ellipsoids, oblate ellipsoids), plates, and cylinders, and parameters for specifying dimensions such as radius, diameter, and volume (the size of the area occupied in the model creation area) are set in advance. The filler particle model may also be a model of secondary particles that are aggregates of the filler. For example, the feature quantity related to the number of filler particle models may be the number of primary particles or the number of secondary particles.

[0020] For example, the matrix polymer model can be set as the matrix polymer model in the model creation area excluding the area in which the filler particle model is created. Furthermore, the interface between the filler particle model and the matrix polymer model can be set as the interfacial phase model. As an example of the interfacial phase model, a positional element of a filler particle model, at least one adjacent element of which is the matrix polymer model, can be set as the interfacial phase model. Furthermore, when performing various characteristic simulations described below, additional characteristic parameters, etc., that are required are set. Examples include the elastic modulus and conductivity of each model.

[0021] FIG. 2 is an explanatory diagram schematically illustrating an example of an analytical model. For convenience, FIG. 2 depicts a three-dimensional (e.g., rectangular parallelepiped) model creation region and a three-dimensional (e.g., spherical) filler particle model in two dimensions (the same applies to the following figures). The analytical model is created, for example, by randomly arranging multiple filler particle models at arbitrary coordinates (X, Y, Z) in the model creation region based on a predetermined algorithm or parameters. Depending on the predetermined algorithm and parameters, analytical models with different filler particle model dispersion forms, such as those shown in FIGS. 2( a) to 2(c), are created. For example, by appropriately setting different values ​​for feature values ​​related to the particle size of the filler particle model, feature values ​​related to the number of filler particle models, etc., multiple analytical models with different filler particle model dispersion forms can be created.

[0022] <Step S2> This analysis method includes step S2 of calculating a feature amount indicating the dispersion form of the filler particle model in the analytical model created in step S1.

[0023] (Step S2a: Delaunay division) One embodiment of step S2 is step S2a in which vertices are set based on the coordinates of a filler particle model in the analytical model, and Delaunay division is performed. For example, vertices are set based on the coordinates of the center of gravity of the filler particle model, which is a model of primary particles of the filler, and Delaunay division is performed.

[0024] Step S2a may, for example, involve performing Delaunay triangulation on each coordinate of a filler particle model in a two-dimensional image of the analytical model (a two-dimensional image) and calculating the length of a Delaunay side or / and the area of ​​a Delaunay triangle. An example of step S2a will be described using a schematic diagram. For example, for image data of an analytical model having a dispersion configuration as shown in FIG. 3( a), the coordinates of the centers of gravity of the filler particle models (e.g., primary particle models) in the analytical model are set as vertices, and Delaunay triangulation is performed. By performing Delaunay triangulation, multiple Delaunay triangles as shown in FIG. 3( b) are generated. Similarly, for an analytical model having a dispersion configuration as shown in FIG. 4( a), the coordinates of the centers of gravity of the filler particle models (e.g., primary particle models) in the analytical model are set as vertices, and Delaunay triangulation is performed. By performing Delaunay triangulation, multiple Delaunay triangles as shown in FIG. 4( b) are generated. Note that Delaunay triangulation is performed using any geometric algorithm that connects points in space and divides it into a set of triangles. For example, the Python library "OpenCV" can be suitably used.

[0025] Next, the average value and / or variance of the lengths of the Delaunay sides (lengths of the line segments between vertices) constituting the Delaunay triangles generated by the Delaunay triangulation is calculated, or the average value and / or variance of the areas of the Delaunay triangles generated by the Delaunay triangulation is calculated.

[0026] In step S2a, the primary particles of the filler may be used as a filler particle model, and Delaunay division may be performed with any coordinates, such as the coordinates of the center of gravity of the filler particle model, as vertices. Alternatively, the secondary particles formed by agglomeration of the filler may be used as a filler particle model, and Delaunay division may be performed with any coordinates, such as the coordinates of the center of gravity of the secondary particles, as vertices.

[0027] Delaunay triangulation was performed on each example of the analytical model (image of the two-dimensional analytical model) shown in Figures 5(a) and 5(b), and the results of calculating the average value and variance of the Delaunay side lengths are shown in Table 1 below.

[0028]

[0029] Furthermore, a correlation was calculated and examined between the results (physical quantities indicating the degree of stress concentration) obtained from a characteristic simulation (simulation predicting the degree of stress concentration) that calculates physical quantities related to mechanical properties using an analytical model and the average value and variance of the Delaunay edge lengths. The results confirmed that the smaller the average value and variance of the Delaunay edge lengths, the better the durability (the less stress concentration there is). Therefore, the average value and variance of the Delaunay edge lengths are effective as feature quantities indicating the dispersion morphology of the filler particle model, which can promote the accumulation of knowledge regarding the dispersion morphology of fillers and contribute to the establishment of design guidelines for polymer composite materials. Furthermore, analysis using a dataset including the feature quantities related to the average value and variance of the Delaunay edge lengths can improve the prediction accuracy of the prediction model.

[0030] The Delaunay division in step S2a may be a two-dimensional Delaunay triangulation performed on a two-dimensional analytical model, or may be a three-dimensional polyhedral division, such as a three-dimensional Delaunay tetrahedron division, performed on a three-dimensional analytical model. The above example is a Delaunay division that divides a two-dimensional plane into triangles whose vertices are coordinates that are discretely distributed on the two-dimensional plane. However, the Delaunay division can be extended to a space division method that applies to point clouds in three-dimensional space. In the Delaunay division extended to three dimensions, the three-dimensional space is divided into simplexes whose vertices are coordinates that are discretely distributed in the three-dimensional space. An example of a simplex in three-dimensional space is a polyhedron such as a tetrahedron. The Delaunay division in three-dimensional space divides the three-dimensional space into tetrahedrons whose vertices are coordinates that are discretely distributed in the three-dimensional space. Therefore, for example, in Delaunay tetrahedron division, the average value and / or variance of the lengths of the sides (lengths of the line segments between vertices) that make up the tetrahedron, or the average value and / or variance of the volume of the tetrahedron, are calculated.

[0031] (Step S2b: Iδ Index) Another embodiment of step S2 includes step S2b of calculating the Iδ index shown in the following formula (1). (In the formula (1), q is the number of regions of the analysis model divided equally into a predetermined number of regions, and X j (j is a natural number) is the number of filler particle models in the jth region)

[0032] The Iδ index is expressed as follows: the number of regions (divisions) is q, the number of filler particle models in the jth region (division) is x j (j is a natural number), it is a value expressed by Equation (1), and when the Iδ index is greater than 1, the distribution pattern of the filler particle model is a concentrated distribution, when Iδ is 1, the distribution pattern of the filler particle model is a Poisson distribution, and when Iδ is less than 1, the distribution pattern of the filler particle model is a uniform distribution.

[0033] FIG. 6 is a schematic diagram illustrating an example of step S2b. FIG. 6 is a schematic image of a two-dimensional analysis model. In step S2b, the image is equally divided into predetermined square regions (sections), and the number of filler particle models having center of gravity coordinates within each region (section) is calculated. For example, referring to FIG. 6( a), the number of divided regions (sections) q is 16, and the number of filler particle models in the region (section) of the first row and first column in FIG. 6( a) is calculated as 2, the number of filler particle models in the region (section) of the first row and second column is calculated as 2, and so on, and the value of Iδ expressed by equation (1) is calculated.

[0034] FIG. 7 shows the results of calculating the Iδ index in step S2b for each of the two-dimensional analytical models shown in FIGS. 5(a) and 5(b).

[0035] Furthermore, the correlation between the results (physical quantity indicating the degree of stress concentration) obtained from a characteristic simulation (simulation predicting the degree of stress concentration) using an analytical model to calculate physical quantities related to mechanical properties was calculated and examined with the Iδ index. As a result, it was confirmed that the smaller the Iδ index (the closer it is to a uniform distribution), the better the durability (the more alleviated the stress concentration). Therefore, the Iδ index is effective as a feature quantity indicating the dispersion morphology of the filler particle model, and can promote the accumulation of knowledge regarding the dispersion morphology of fillers and contribute to the establishment of design guidelines for polymer composite materials. Furthermore, the prediction accuracy of the prediction model can be improved by performing analysis using a dataset including the feature quantity related to the Iδ index.

[0036] The Iδ index in step S2b may be calculated for any two-dimensional analytical model of the analytical model, or may be calculated based on coordinates discretely distributed in a three-dimensional space of the analytical model (three-dimensional). In this case, the region (section) is, for example, a cube, and the number of filler particle models having coordinates in the cube is calculated to calculate the Iδ index shown in formula (1).

[0037] Furthermore, in step S2b, the primary particles of the filler may be used as a filler particle model, and the value of the Iδ index may be calculated based on arbitrary coordinates such as the coordinates of the center of gravity of the filler particle model, or the secondary particles formed by agglomeration of the filler may be used as a filler particle model, and the value of the Iδ index may be calculated based on arbitrary coordinates such as the coordinates of the center of gravity of the secondary particles.

[0038] (Step S2c: Power Spectrum Obtained by Fourier Transform) Another embodiment of step S2 includes, for example, step S2c, which calculates at least one of the radial and angular components of the power spectrum obtained by Fourier transform. FIG. 8 is a diagram for explaining an example of step S2c. A two-dimensional Fourier transform is performed based on image data of an analytical model having the dispersion form shown in FIG. 8(a). By performing the two-dimensional Fourier transform, the power spectrum shown in FIG. 8(b) is generated. FIG. 8(b) is a power spectrum image in which low-frequency components are distributed near the center and high-frequency components are distributed as they spread outward. For the Fourier transform, a known method, such as FFT (Fast Fourier Transform), is used.

[0039] Next, a feature quantity indicating the dispersion pattern is calculated based on the power spectrum distribution, for example, by extracting at least one of the radial distribution and the angular distribution of the two-dimensional power spectrum in the power spectrum image generated based on the analytical model, according to a conventional method.

[0040] For example, from the viewpoint of calculating feature quantities related to the anisotropy of the analysis model, in the step of extracting angular direction components in a predetermined angular range, integrating the power spectrum for each angle, and calculating the average value of the power spectrum in that range, a step of extracting and integrating angular direction components in the predetermined angular range and calculating the average value of the power spectrum in that predetermined angular range, and extracting and integrating angular direction components in another predetermined angular range different from the predetermined angular range, calculating the average value of the power spectrum in that range, and calculating the ratio of the two average values ​​may be used.

[0041] Specifically, for example, referring to FIG. 9 , in a 400 pixel × 400 pixel power spectrum image generated based on an analysis model, the horizontal axis is used as a reference (the right direction of the horizontal axis is 0° C.), and first, angular direction components in the angle range of −45° to (+) 45° and angular direction components in the angle range of (+) 135° to −135° are extracted and integrated to calculate the average value (PA1) of the power spectrum in both ranges. Next, angular direction components in the angle range of (+) 45° to (+) 135° and angular direction components in the angle range of −45° to −135° are extracted and integrated to calculate the average value (PA2) of the power spectrum in both ranges. Finally, the ratio of the two average values ​​(PA1 / PA2) is calculated, thereby making it possible to calculate a feature amount related to anisotropy, which is the ratio between the vertical direction component and the horizontal direction component. 9, of the four triangular regions T1 to T4 defined by diagonal lines intersecting at the center point of the power spectrum image, the power spectra of the angular components in regions T1 and T2 are extracted and integrated to calculate the average value (PA1), the power spectra of the angular components in regions T3 and T4 are extracted and integrated to calculate the average value (PA2), and the ratio of the average values ​​(PA1 / PA2) is calculated. The larger the ratio of the average values ​​(PA1 / PA2), the stronger the anisotropy of the vertical component.

[0042] The correlation between the results (physical quantities indicating the degree of stress concentration) obtained from a characteristic simulation (simulation predicting the degree of stress concentration) that calculates physical quantities related to mechanical properties using an analytical model and the ratio of the average values ​​was calculated and examined. The results confirmed that the weaker the longitudinal component, the better the durability (the less stress concentration there is in response to changes in longitudinal elongation). Therefore, the power spectrum is effective as a feature indicating the dispersion morphology of the filler particle model, which can promote the accumulation of knowledge regarding the dispersion morphology of fillers and contribute to the establishment of design guidelines for polymer composite materials. Furthermore, analysis using a dataset containing the feature quantities related to the power spectrum can improve the prediction accuracy of the prediction model.

[0043] 9, step S2c may be a step of extracting and integrating a power spectrum in a predetermined angular range and a power spectrum in a predetermined radial range, with the horizontal axis as a reference (0° to the right of the horizontal axis) in a power spectrum image generated based on the analysis model. Specifically, step S2c may be a step of extracting and integrating a power spectrum in an angular range of −15° to 15° and a power spectrum in a radial range of 0 pixel to 30 pixel, with the horizontal axis as a reference (0° to the right of the horizontal axis) in a 400 pixel by 400 pixel power spectrum image generated based on the analysis model.

[0044] A property simulation (simulation predicting the degree of stress concentration) using an analytical model to calculate physical quantities related to mechanical properties was performed. The correlation between the results (physical quantities indicating the degree of stress concentration) and the power spectrum was calculated and examined. The results showed a high correlation between the power spectrum in the angular range of -15° to 15°, with the horizontal axis as the reference (0°C to the right of the horizontal axis), and the power spectrum in the radial range of 0 to 30 pixels. This indicates that the less periodic structure in the longitudinal direction, the better the durability (relative to longitudinal elongation changes). Therefore, the power spectrum is effective as a feature indicating the dispersion morphology of the filler particle model, which can promote the accumulation of knowledge regarding filler dispersion morphology and contribute to the establishment of design guidelines for polymer composite materials. Furthermore, analysis using a dataset containing the feature quantities related to the power spectrum can improve the prediction accuracy of the prediction model.

[0045] Step S2c may be performed on any two-dimensional analytical model of the analytical model, or may be performed on, for example, a three-dimensional analytical model. That is, step S2c may be performed on data of a two-dimensional array structure with the coordinates of the analytical model in the x and y directions as parameters, or on data of a three-dimensional array structure with the coordinates of the three directions, x, y, and z, as parameters.

[0046] (Another embodiment of step S2) Another embodiment of step S2 includes, for example, step S2d of calculating feature amounts based on persistent homology analysis.

[0047] For example, persistent homology analysis (PH) can be performed on image data corresponding to the analytical model, and information including birth time, death time, and birth-death pair can be calculated as feature amounts.

[0048] Simply put, persistent homology analysis (PH) can quantify the connections between shapes present in an image of an analytical model. PH works by gradually expanding a shape and focusing on the "holes" that appear as the shape expands. As the shape expands, the shapes connect and holes "appear." As the shape expands further, the holes "disappear." Furthermore, for holes that were originally present, there is a point when the hole breaks as the shape shrinks. In other words, it can be seen that the hole "appeared" immediately after that. The time of "appearance" is recorded as "birth time" and the time of "disappearance" as "death time," and these can be used as feature quantities.

[0049] In step S2d, for example, a binarized image corresponding to the analytical model is used. Specifically, the black parts in the image are filler particle models, and the white parts are matrix polymer models. When focusing on the filler particle models, the black parts are expanded or contracted to extract feature quantities. When focusing on the matrix polymer models, the white parts are expanded or contracted to extract feature quantities.

[0050] More specifically, for example, persistent homology analysis (PH) is performed using a binarized image as input information, and a diagram (persistence diagram) is generated in which the horizontal axis represents birth time and the vertical axis represents death time, and then vectorization is performed to extract features.

[0051] The binarization process is not particularly limited, and known methods such as adaptive binarization and Otsu's binarization can be used. Among these, adaptive binarization is preferred. In adaptive binarization, local regions of arbitrary size are set within the entire image, and a threshold is calculated for each local region to binarize each image. The threshold may be calculated using an arithmetic mean value or the average value of the weighting (Gaussian distribution) of the local region, and can be set appropriately. Furthermore, it is preferable to perform noise removal processing prior to the binarization process. Known methods can be used for the noise removal process, and there are no particular limitations. Specifically, filtering using a noise cut filter includes, for example, a non-local-mean filter, an average filter, a Gaussian filter, a median filter, a band-pass filter, an open filter, a close filter, and a bilateral filter.

[0052] The noise removal process and binarization process can be performed using, for example, the Python library "OpenCV." For example, for image data (size 4008 pixels x 2672 pixels) corresponding to the analysis model, the following conditions are set: filter strength parameter h = 10, template window size: 11, search window size: 21, and noise removal process (non-local mean denoising). Next, the following conditions are set: block size: 131, constant C (threshold correction): 5, threshold calculation: arithmetic mean, and adaptive binarization process can be performed.

[0053] The persistent homology analysis based on the binarized image can be performed using "HomCloud," a Python library. For example, a persistence diagram can be created using the binarized image obtained above as input. Focusing on the filler particle model, an example of a persistence diagram created using "HomCloud" is shown in FIG. 10. Note that the argument of distance#transform() is signed=True.

[0054] The persistence diagram can be vectorized using, for example, the following formula:

[0055] Note that vectorization is performed using, for example, PI (Persistence Image) in "HomCloud" (see, for example, Henry Adams et al. "Persistence images: a stable vector representation of persistent homology", In: J. Mach. Learn. Res. 18 (2017), Paper No. 8, 35.). The vectorization conditions are, for example, x#range = (-50, 20) and xbins = 70. The parameter values ​​indicated by σ, C, and p in the above equation are, for example, σ = 3.0, C = 0.001, and p = 4. The values ​​of x#range and xbins are conditions for how to divide the persistence diagram, and are appropriately modified if necessary, referring to the maximum and minimum values ​​of all the persistence diagrams to be analyzed.

[0056] Persistent homology is a type of topological data analysis and has been described in detail in various publications (for example, "Fundamentals of Persistent Homology and Applications to Materials Engineering" (JIM Bulletin), Vol. 58, No. 1, 2019: "Statistical Machine Learning for Persistence Diagrams," Institute of Statistical Mathematics, Chuo University, Internet <URL: https: / / www.math.chuo-u.ac.jp / ENCwMATH / EwM70#Fukumizu.pdf>).

[0057] The persistent homology analysis in step S2d may be performed on any two-dimensional analytical model of the analytical model, or may be performed on, for example, a three-dimensional analytical model. That is, the persistent homology processing may be applied to data of a two-dimensional array structure in which the coordinates in the x and y directions of the analytical model are parameters, or the persistent homology processing may be applied to data of a three-dimensional array structure in which the coordinates in the three directions of x, y, and z are parameters.

[0058] In addition, in step S2, it is preferable to use at least one selected from the group consisting of steps S2a, S2b, S2c, and S2d. In addition, in step S2, at least two or more selected from the group consisting of steps S2a, S2b, S2c, and S2d may be used, or three or more may be used.

[0059] <Step S3> This analysis method includes step S3 of performing a characteristic simulation to calculate physical quantities related to mechanical properties such as durability and viscoelasticity of the analytical model, or electrical properties such as conductivity. A conventionally known method can be used as appropriate for the characteristic simulation. For example, using the finite element method (FEM) or the like, the analytical model is divided into predetermined meshes, and values ​​of material constants and material parameters of a filler particle model, a matrix polymer model, and an interfacial phase model are set, and tensile conditions, voltage application conditions, etc. are set to calculate various properties. More specifically, for example, characteristic simulations using the phase-field method described in Japanese Patent No. 7382478 and Japanese Patent No. 7197871 can be used.

[0060] <Step S4> The analysis method according to this embodiment includes, for example, step S4 of creating a data set including, as objective variables, physical quantities obtained by the characteristic simulation and, as explanatory variables, feature quantities indicating the dispersion form of the filler particle model in the analysis model. For example, a data set such as that shown in Table 2 is created for multiple analysis models (A, B, C, etc.) having different dispersion forms.

[0061] In step S4, it is preferable that the step of creating a dataset is a step of creating a dataset including the physical quantities obtained by the characteristic simulation, feature quantities (at least one of 2Sa to 2Sd) indicating the dispersion morphology of the filler particle model, feature quantities related to the particle size of the filler particle model, feature quantities related to the number of filler particle models, and feature quantities related to the thickness of the interfacial phase model. Note that the dataset may also include feature quantities other than those mentioned above. For example, the dataset may include a feature quantity related to the filler filling rate (such as the volume filling rate of the filler particle model in the analytical model).

[0062] <Step S5> The present analysis method preferably includes step S5a of performing multivariate analysis based on the dataset. As the multivariate analysis, various statistical analysis methods can be appropriately used, such as multiple regression analysis, canonical correlation analysis, logistic regression analysis, and quantification theory type 1. Step S5a is preferably a step of performing multivariate analysis using a dataset that includes, as objective variables, physical quantities related to mechanical properties or electrical properties obtained by characteristic simulation, and includes, as explanatory variables, feature quantities indicating the dispersion morphology and feature quantities related to the number of filler particle models, to create a prediction model.

[0063] The analysis method may also include step S5b of performing machine learning based on the dataset. As the machine learning, various techniques can be appropriately used, such as neural networks. Step S5b is preferably a step of performing machine learning using a dataset including physical quantities related to mechanical properties or electrical properties obtained by characteristic simulation, feature quantities indicating the dispersion morphology, and feature quantities related to the number of filler particle models, to create a prediction model.

[0064] [Example 1] A plurality of analytical models were created based on the method according to the embodiment. Specifically, the analytical models were created with multiple levels of particle size (primary particle size) of the filler particle model (three levels: 10 nm, 15 nm, 30 nm), multiple levels of the number of dispersed particles (the number of aggregates in the structure, corresponding to the "number of filler particle models") (three levels: 200, 600, 1000), multiple levels of thickness of the interfacial phase model (three levels: 5 nm, 10 nm, 15 nm), multiple levels of elastic modulus of the interfacial phase model (three levels: 1.5 MPa, 1.5 MPa, 2.5 MPa), and a sample number (N number) of 5, resulting in a total of 405 analytical models. Furthermore, the elastic modulus of the filler particle model was set to 8 MPa for the analytical models.

[0065] Next, a characteristic simulation was performed to calculate physical quantities related to the mechanical properties of the analytical model. Specifically, using phase-field analysis application software, the analytical model was subjected to tensile deformation in the vertical direction (constant strain condition (strain 1.5)), and a stress simulation calculation was performed to obtain stress distribution information for the analytical model. Figure 11 shows an example of the analytical model, and Figure 12 shows the stress distribution information obtained from the analytical model. A histogram was generated based on the obtained stress distribution information (see Figure 13), and the top 5% of stresses (the area to the right of the dashed line in Figure 13) were divided by the average stress. The value obtained above indicates the degree of stress concentration; a smaller value indicates that the stress is less concentrated and more dispersed, indicating better durability.

[0066] On the other hand, based on the method according to the embodiment, the mean value and variance of the Delaunay side length, the Iδ index, and the angular components (T1 and T2) of the power spectrum of each analytical model were calculated.

[0067] Next, multiple regression analysis was performed. Specifically, the particle size of the filler particle model, the number of dispersed particles, the thickness of the interphase model, the elastic modulus of the interphase model, the average value and variance of the Delaunay side lengths, the Iδ index, and the angle component of the power spectrum (the average values ​​of the T1 and T2 regions) were used as explanatory variables. Based on the variable addition / decrement method, the number of dispersed particles, the thickness of the interphase model, the elastic modulus of the interphase model, the average value of the Delaunay side lengths, the Iδ index, and the angle component of the power spectrum (the average values ​​of the T1 and T2 regions) were selected, and multiple regression analysis was performed using the value indicating the degree of stress concentration obtained by stress simulation calculation as the objective variable. The results are shown in Figure 14.

[0068] [Comparative Example 1] On the other hand, in Comparative Example 1, multiple regression analysis was performed without using the feature quantity indicating the dispersion morphology of the filler particle model. That is, the multiple regression analysis was performed by changing the explanatory variables of Example 1. Specifically, in Comparative Example 1, multiple regression analysis was performed without using the average value and variance of the Delaunay side lengths, the Iδ index, and the angle components (T1 and T2) of the power spectrum as explanatory variables. The results are shown in Figure 15.

[0069] 14 and 15, it can be seen that the coefficient of determination is improved by using the Delaunay edge length, Iδ index, and the angular components of the power spectrum (T1 and T2) as explanatory variables. This demonstrates the usefulness of including feature quantities that indicate the dispersion mode in the analysis.

[0070] Although the above examples show specific embodiments of the present invention, the examples are merely illustrative and should not be construed as limiting. Various modifications that are obvious to those skilled in the art are intended to fall within the scope of the present invention.

[0071] According to one embodiment of the present invention, a new method for analyzing the dispersion morphology of a filler in a polymer composite material in which the filler is dispersed in a matrix polymer can be provided, which is useful in that it can promote the accumulation of knowledge regarding the dispersion morphology of the filler and contribute to the establishment of design guidelines for polymer composite materials. Furthermore, according to one embodiment of the present invention, it is also useful in that it can improve the prediction accuracy of a prediction model.

Claims

1. A method for analyzing a polymer composite material in which a filler is dispersed in a matrix polymer, comprising the steps of: creating an analytical model of the polymer composite material in which a filler is dispersed in a matrix polymer; calculating feature quantities that indicate the dispersion morphology of a filler particle model in the analytical model; performing a characteristic simulation that calculates physical quantities related to the mechanical properties or electrical properties of the analytical model; and creating a dataset that includes the physical quantities related to the mechanical properties or electrical properties obtained by the characteristic simulation and feature quantities that indicate the dispersion morphology of the filler particle model.

2. A method for analyzing a polymer composite material according to claim 1, wherein the analytical model is an analytical model including a matrix polymer model and a filler particle model, and the step of creating the data set is a step of creating a data set including physical quantities related to the mechanical properties or electrical properties obtained by the property simulation, feature quantities indicating the dispersion form of the filler particle model, and feature quantities related to the number of filler particle models.

3. The method for analyzing a polymer composite material according to claim 2, further comprising a step of performing multivariate analysis or machine learning using the dataset to create a predictive model.

4. The analysis method according to claim 3, wherein the step of calculating a feature value indicating the dispersion form of the filler particle model in the analysis model includes a step of setting vertices based on the coordinates of the filler particle model and calculating a feature value based on Delaunay division.

5. The method for analyzing a polymer composite material according to claim 3, wherein the step of calculating a feature quantity indicating the dispersion morphology of the filler particle model in the analytical model includes a step of calculating a feature quantity based on the Iδ index shown in the following formula (1): In the formula (1), q is the number of regions of the analysis model divided equally into a predetermined number of regions, and X j is the number of filler particle models in the jth region.

6. A method for analyzing a polymer composite material according to claim 3, wherein the step of calculating a feature quantity indicative of the dispersion form of the filler particle model in the analytical model includes a step of calculating a feature quantity based on at least one of the radial component and the angular component of the power spectrum obtained by Fourier transform.

7. A method for analyzing a polymer composite material according to any one of claims 4 to 6, wherein the step of performing a characteristic simulation to calculate a physical quantity related to the mechanical properties or electrical properties of the analytical model is a step of performing a characteristic simulation to calculate a physical quantity related to the mechanical properties, which is a step of deforming the analytical model and calculating a physical quantity related to stress in the deformed analytical model.

8. The method for analyzing a polymer composite material according to claim 7, wherein the polymer composite material in which a filler is dispersed in a matrix polymer is a vibration-isolating rubber.

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