Creation method of numeric analysis model of polymer chain and simulation method of high polymer material

Abstract

To provide a creation method of a numeric analysis model of a cut-off polymer chain.SOLUTION: A method to create a numeric analysis model of a polymer chain comprises the steps of: inputting a high polymer chain model; setting a polymer material model to a cell where the high polymer chain model is arranged; calculating deformation of the polymer material model; and cutting off the high polymer chain model extended according to deformation. The cutting-off process comprises: a first process of extracting a partial chain model from the high polymer chain model; a second process of calculating interterminal distance of the extracted partial chain model and determining whether the partial chain model is extended or not based on the calculated interterminal distance of the partial chain model; and a third process of cutting off the high polymer chain model in at least one combination model included in the partial chain models determined to be extended.SELECTED DRAWING: Figure 14

Description

[Technical Field]

[0001] The present invention relates to a method for creating a model for numerical analysis of a polymer chain and a method for simulating a polymer material. [Background technology]

[0002] Patent Document 1 listed below describes a method for analyzing polymeric materials. In this method, first, a molecular chain model including a plurality of particle models and a bond model that connects adjacent particle models is input. Next, a first distance for cutting between adjacent particle models via the bond model is defined. Then, the polymeric material model including the molecular chain model is deformed, and if the distance between adjacent particle models via the bond model is equal to or greater than the first distance, the bond model between the particle models is deleted. [Prior art documents] [Patent documents]

[0003] [Patent Document 1] Patent No. 6776876 Summary of the Invention [Problem to be solved by the invention]

[0004] The distance between the particle models varies due to the influence of thermal fluctuation noise in molecular dynamics calculations. This makes it difficult to quantitatively determine the bond model to be cut based on the first distance. In other words, the influence of thermal fluctuation noise tends to result in a more random cut. Therefore, there is room for further improvement in creating models for numerical analysis of cut polymer chains.

[0005] The present invention has been devised in view of the above circumstances, and its main object is to provide a method that makes it possible to create a model for numerical analysis of a cut polymer chain. [Means for solving the problem]

[0006] The present invention is a method for creating a numerical analysis model of a polymer chain, comprising the steps of: inputting a polymer chain model based on the polymer chain, the polymer chain model including a plurality of particle models and bond models connecting adjacent particle models to each other, into a computer; arranging the polymer chain model in a cell that is a predetermined virtual space to set a polymer material model; calculating, by the computer, the deformation of the polymer material model; and cutting, by the computer, the polymer chain model that has been elongated due to the deformation. The cutting step comprises the steps of: a first step of extracting, from the polymer chain model, partial chain models that include three or more adjacent particle models and the bond models connecting them; a second step of calculating the end-to-end distances for the extracted partial chain models and determining whether the partial chain models have been elongated based on the calculated end-to-end distances for the partial chain models; and a third step of cutting the polymer chain model at at least one of the bond models included in the partial chain models that have been determined to have been elongated. [Effects of the Invention]

[0007] The method of the present invention for creating a numerical analysis model of a polymer chain employs the above steps, making it possible to create a numerical analysis model of a cut polymer chain. [Brief explanation of the drawings]

[0008] [Figure 1] FIG. 1 is a perspective view showing a computer for executing a method for creating a numerical analysis model of a polymer chain and a method for simulating a polymer material. [Figure 2] FIG. 1 is a diagram showing an example of a structural formula of a polymer chain. [Figure 3] 1 is a flowchart showing the processing steps of a method for creating a model for numerical analysis of a polymer chain. [Figure 4] This is a conceptual diagram of a polymer chain model (Kremer-Grest model). [Figure 5]FIG. 1 is a conceptual diagram showing a cell in which a polymer chain model is arranged. [Figure 6] FIG. 10 is a side view showing a polymer material model before deformation calculation. [Figure 7] FIG. 10 is a side view showing a polymer material model during deformation calculation. [Figure 8] FIG. 10 is a side view showing a polymer material model during deformation calculation in another embodiment. [Figure 9] 10 is a flowchart showing an example of a processing procedure of a cutting step. [Figure 10] 1A and 1B are conceptual diagrams showing examples of partial chain models included in a polymer chain model, where FIG. 1A is a conceptual diagram showing a first partial chain model and a second partial chain model, and FIG. 1B is a conceptual diagram showing a third partial chain model and a fourth partial chain model. [Figure 11] 10 is a flowchart showing an example of a processing procedure of a first step. [Figure 12] 10 is a flowchart showing an example of a processing procedure of a second step. [Figure 13] 10 is a flowchart showing an example of a processing procedure of a third step. [Figure 14] FIG. 11 is a conceptual diagram showing a state in which the first partial chain model and the second partial chain model of FIG. 10(a) are each cut. [Figure 15] FIG. 1 is a conceptual diagram showing a polymer chain model in which interactions are defined. [Figure 16] 1 is a graph showing the molecular weight distribution (differential molecular weight distribution) of a polymer material. [Figure 17] 10 is a flowchart showing an example of a processing procedure of a third step according to another embodiment of the present invention. [Figure 18] 1 is a flowchart showing an example of a processing procedure of a simulation method for a polymer material. [Figure 19] 1 is a graph showing molecular weight distributions (integral molecular weight distributions) of Examples and Comparative Examples. [Figure 20] 1 is a graph showing the magnitude of the storage modulus of Examples and Comparative Examples. [Figure 21] 1 is a graph showing loss tangents of Examples and Comparative Examples. DETAILED DESCRIPTION OF THE INVENTION

[0009] Hereinafter, embodiments of the present invention will be described with reference to the drawings. It should be understood that the drawings include exaggerated representations and representations that differ from the dimensional ratios of actual structures in order to facilitate understanding of the contents of the invention. Furthermore, identical or common elements are designated by the same reference numerals throughout the embodiments, and redundant explanations will be omitted. Furthermore, the specific configurations shown in the embodiments and drawings are for the purpose of understanding the contents of the present invention, and the present invention is not limited to the specific configurations shown in the drawings.

[0010] The method for creating a numerical analysis model of a polymer chain of this embodiment (hereinafter, sometimes referred to as the "creation method") is used to create a numerical analysis model of a polymer chain (hereinafter, sometimes referred to as the "polymer chain model") that has been cut due to deformation or the like of a polymer material.

[0011] Deformation of a polymer material includes, for example, deformation of an unvulcanized polymer material and deformation of a vulcanized polymer material. Deformation of an unvulcanized polymer material includes, for example, deformation during kneading or an extrusion process. Deformation of a vulcanized polymer material includes deformation due to aging or wear. A computer 1 is used in the creation method of this embodiment.

[0012] [computer] FIG. 1 is a perspective view showing a computer 1 for executing a method for creating a numerical analysis model of a polymer chain and a method for simulating a polymer material. The computer 1 of this embodiment includes a main body 1a, a keyboard 1b, a mouse 1c, and a display device 1d. The main body 1a is provided with, for example, a central processing unit (CPU), a ROM, a working memory, a storage device such as a magnetic disk, and disk drive devices 1a1 and 1a2. The storage device has pre-stored therein software and the like for executing the creation method of this embodiment and a method for simulating a polymer material (hereinafter sometimes referred to as the "simulation method") described below.

[0013] [Polymer chain] The polymer chains constitute a polymer material. The polymer material of this embodiment is exemplified as rubber (in this example, cis-1,4 polybutadiene (hereinafter sometimes simply referred to as "polybutadiene")), but is not particularly limited thereto. Furthermore, the polymer material may be composed of, for example, multiple rubbers, may be vulcanized (crosslinked), may not be vulcanized, and may contain fillers (for example, silica or carbon). Furthermore, the polymer material may have a new structure that does not currently exist.

[0014] 2 is a diagram showing an example of the structural formula of the polymer chain 2. In this embodiment, the polymer chain 2 is exemplified as polybutadiene.

[0015] The polymer chain 2 is composed of a plurality of atoms (carbon atoms and hydrogen atoms in this example). The polymer chain 2 in this embodiment is composed of monomers 3 {-[CH2-CH=CH-CH2]-}, each of which is composed of a methylene group (-CH2-) and a methine group (-CH-), linked together at a degree of polymerization n. Both ends of the polymer chain 2 are composed of methyl groups (-CH3) instead of methylene groups (-CH2-).

[0016] [Method for creating a model for numerical analysis of polymer chains (first embodiment)] Next, the creation method of this embodiment will be described. Fig. 3 is a flowchart showing the processing steps of the method for creating a numerical analysis model of the polymer chain 2.

[0017] [Enter polymer chain model] In the creation method of this embodiment, first, a polymer chain model is input into a computer 1 (shown in FIG. 1) based on a polymer chain 2 (shown in FIG. 2) (step S1). In this embodiment, a coarse-grained molecular model (in this example, the Kremer-Grest model) is used as the polymer chain model, but this is not particularly limited and may be, for example, an all-atom model or a united-atom model. Figure 4 is a conceptual diagram of a polymer chain model 5 (Kremer-Grest model 5B).

[0018] The polymer chain model 5 of this embodiment is configured to include a plurality of particle models 6 and a bond model 7 that bonds adjacent particle models 6, 6. When the polymer chain model 5 is a Kremer-Grest model 5B as in this embodiment, the polymer chain 2 can be represented using particle models 6 whose number is less than the number of atoms that constitute the polymer chain 2 (shown in FIG. 2 ).

[0019] The particle model 6 of this embodiment is treated as a mass point in the equation of motion in the calculation of structural relaxation based on molecular dynamics, which will be described later. For this reason, parameters such as mass, volume, diameter, and charge are defined for the particle model 6.

[0020] The particle model 6 is a replacement of the monomer 3 of the polymer chain 2 shown in Fig. 2 or a structural unit forming a part of the monomer 3. The total number of particle models 6 constituting one polymer chain model 5 (in this example, the Kremer-Grest model 5B) can be set appropriately (for example, set to about 10 to 5000 (in this example, 5000)) as in the conventional case.

[0021] The bond model 7 in this embodiment is defined by a first potential P1 set between the particle models 6, 6. The first potential P1 in this embodiment can be defined as the sum of an LJ potential and a FENE potential. Details and constants of the LJ potential and the FENE potential are as described in Paper 1 (Kurt Kremer & Gary S. Grest, "Dynamics of entangled linear polymer melts: A molecular-dynamics simulation," J. Chem Phys., Vol. 92, No. 8, April 15, 1990, pp. 5057-5086).

[0022] In step S1 of this embodiment, only the polymer chain model 5 (in this example, the Kremer-Grest model 5B) is set, but this is not limiting. For example, a molecular model (not shown) that models a substance (molecule) other than a polymer chain may be set. Such a molecular model may include, for example, one or more particle models 6 and zero or more bond models 7. Furthermore, a polymer chain model 5 including a bond model (not shown) of a crosslinking point may be set. The polymer chain model 5 is stored in a computer 1 (shown in FIG. 1).

[0023] [Set polymer material model] Next, in the creation method of this embodiment, a polymer material model is set by placing a polymer chain model 5 in a cell, which is a predetermined virtual space (step S2). Fig. 5 is a conceptual diagram showing a cell 10 in which a polymer chain model 5 is placed. In Fig. 5, two polymer chain models 5 are shown as representatives.

[0024] In step S2 of this embodiment, first, a cell 10 is input to a computer (shown in FIG. 1). The cell 10 of this embodiment has at least one pair of surfaces 11, 11 facing each other (in this embodiment, three pairs of surfaces 11, 11 facing each other). The cell 10 of this embodiment is defined as a rectangular parallelepiped or a cube (in this embodiment, a cube).

[0025] A periodic boundary condition is defined for each face 11, 11 of the cell 10. The length L1 of each side of the cell 10 is preferably, for example, at least twice the radius of gyration (not shown), which is an amount indicating the extent of the polymer chain model 5. This can prevent collisions with self-images due to the periodic boundary condition in the structural relaxation calculation based on molecular dynamics, which will be described later.

[0026] Next, in step S2 of this embodiment, a polymer chain model 5 is placed in the cell 10. In this embodiment, the polymer chain model 5 is placed in the cell 10 by the computer 1 (shown in FIG. 1), but this is not particularly limited. For example, the polymer chain model 5 may be placed in the cell 10 by an operator.

[0027] In this embodiment, a plurality of polymer chain models 5 are randomly arranged inside the cell 10. The number of polymer chain models 5 can be appropriately set (for example, set to 10 to 100) based on, for example, the calculation capacity of the computer 1 (shown in FIG. 1), the size of the cell 10, the size of the polymer chain models 5, and the like.

[0028] The procedure for arranging the polymer chain models 5 is not particularly limited. For example, the polymer chain models 5 may be randomly arranged inside the cell 10 based on the Monte Carlo method. For such initial arrangement of the polymer chain models 5, for example, commercially available software (for example, COGNAC included in the Soft Material Integrated Simulator (J-OCTA) manufactured by JSOL Corporation) can be used.

[0029] In step S2 of this embodiment, the polymer chain model 5 is placed in the cell 10, thereby setting a polymer material model 12 that models the polymer material. The above-mentioned molecular model (not shown) or a filler model (not shown) that models the filler may also be placed inside the cell 10. The polymer material model 12 is stored in the computer 1 (shown in FIG. 1).

[0030] [Calculate deformation of polymer material model] Next, in the creation method of this embodiment, the computer 1 (shown in FIG. 1) calculates the deformation of the polymer material model 12 (step S3). In step S3 of this embodiment, the extension of the polymer chain model 5 arranged inside the cell 10 is calculated due to the deformation of the polymer material model 12.

[0031] The deformation calculation of the polymer material model 12 can be performed as appropriate once the elongation of the polymer chain model 5 is calculated. In this embodiment, a periodic strain is applied to the polymer material model 12. FIG. 6 is a side view showing the polymer material model 12 before deformation. In FIG. 6, the area surrounded by a thick black frame in the center is the polymer material model 12 before deformation calculation. FIG. 7 is a side view showing the polymer material model 12 during deformation. FIG. 8 is a side view showing the polymer material model 12 during deformation in another embodiment. In FIGS. 7 and 8, the area surrounded by a thick black frame in the center is the polymer material model 12 during deformation calculation. FIGS. 7 and 8 show an arrangement of multiple polymer material models 12 undergoing deformation calculation. In FIGS. 7 and 8, the area surrounded by a thick black frame in the center is surrounded by eight areas surrounded by thin black frames. Each of these areas surrounded by thin black frames is an image cell into which the polymer material model 12 is copied. These image cells are arranged according to the periodic boundary conditions described below.

[0032] As shown in FIG. 6, in step S3 of this embodiment, the periodic shear deformation of the polymer material model 12 is calculated.

[0033] The periodic boundary conditions used in molecular dynamics calculations of shear deformation include, for example, the Lees-Edwards periodic boundary condition or the parallelepiped periodic boundary condition. In the Lees-Edwards periodic boundary condition, shear deformation can be expressed by, for example, shifting the upper face 16a of the periodic boundary to the right and the lower face 16b to the left in accordance with the magnitude of shear for a rectangular periodic boundary condition. Figure 7 shows the periodic shear deformation of a polymer material model 12 based on the Lees-Edwards periodic boundary condition. Figure 8 shows the shear deformation based on the parallelepiped periodic boundary condition by tilting the shape of the periodic boundary condition itself from a rectangular parallelepiped to a parallelepiped.

[0034] The SLLOD method, for example, is used to solve the time evolution equations for molecular dynamics calculations of shear deformation. Details of the SLLOD method are described in the paper2 (Denis J. Evans & G.P. Morriss, "Nonlinear-response theory for steady planar Couette flow," Phys. Rev. A, 30(3): 1984, pp. 1528-1530). Periodic shear can also be achieved by periodically varying the shear strain γ, for example, according to a sinusoidal wave γ(t) = A·sin(ωt), where A is the maximum strain and ω is the frequency. This type of periodic shear calculation makes it possible to perform simulations that correspond to dynamic mechanical analysis (DMA), which applies periodic shear deformation to polymeric materials.

[0035] The deformation calculation is not limited to the mode in which periodic shear deformation is applied as in this embodiment. For example, based on the procedure described in Japanese Patent No. 6408856, the deformation of the polymer material model 12 may be calculated so that one end (one side surface 11) and the other end (the other side surface 11) of the polymer material model 12 are separated from each other in the Y-axis direction in Fig. 5. For these deformation calculations, for example, the above-mentioned commercially available software or the molecular dynamics calculation program LAMMPS can be used.

[0036] In step S3 of this embodiment, the deformation of the polymer material model 12 is calculated, thereby calculating the elongation of the polymer chain model 5 arranged inside the cell 10. In this embodiment, from the start of the deformation calculation of the polymer material model 12 until a predetermined end time has elapsed, the coordinate values of the particle models 6 that make up the polymer chain model 5 are stored in the computer 1 (shown in FIG. 1 ) for each unit time step (time) of the molecular dynamics calculation. This makes it possible to grasp the polymer chain model 5 that has been elongated in accordance with the deformation of the polymer material model 12 for each unit time step. The end time can be set appropriately depending on the purpose of the analysis of the polymer material.

[0037] [Cutting the extended polymer chain model (cutting process)] Next, in the creating method of this embodiment, the computer 1 (shown in FIG. 1) cuts the polymer chain model 5 that has been extended in accordance with the deformation of the polymer material model 12 shown in FIG. 7 (cutting step S4).

[0038] As in the present embodiment, when a plurality of polymer chain models 5 are provided in the polymer material model 12, the elongated polymer chain models 5 are cut out of these polymer chain models 5. Furthermore, when the polymer chain models 5 elongated in accordance with the deformation of the polymer material model 12 are stored (grasped) for each unit time step, the elongated polymer chain models 5 can be cut out, for example, at any unit time step (for example, the step at the end time of the deformation calculation). Fig. 9 is a flowchart showing an example of the processing procedure of the cutting step S4.

[0039] [Extracting partial chain models from polymer chain models (first step)] In the cutting step S4 of this embodiment, first, a partial chain model 8 is extracted from the polymer chain model 5 (first step S41). FIG. 10 is a conceptual diagram showing an example of a partial chain model 8 included in the polymer chain model 5. In FIG. 10, the partial chain model 8 is surrounded by a dashed line or a broken line. FIG. 10(a) is a conceptual diagram showing a first partial chain model 8A and a second partial chain model 8B. FIG. 10(b) is a conceptual diagram showing a third partial chain model 8C and a fourth partial chain model 8D.

[0040] 10 shows a polymer chain model 5 bonded by a bond model 7 at a crosslinking point 14. The polymer chain model 5 in FIG. 10 has a shorter chain length (i.e., is simplified) than the polymer chain model 5 in FIG. 4.

[0041] A partial chain model 8 is configured to include three or more adjacent particle models 6 and a bond model 7 that bonds them. At least one such partial chain model 8 is included in the polymer chain model 5. In the cutting step S4 of this embodiment, it is determined whether or not each partial chain model 8 has been elongated for all partial chain models 8 that make up the polymer chain model 5. Then, at least one bond model 7 included in a partial chain model 8 that is determined to have been elongated is cut.

[0042] In the first step S41 of this embodiment, all partial chain models 8 (shown in FIG. 10 ) including a predetermined number of particle models 6 are extracted from each of the multiple polymer chain models 5 included in the polymer material model 12 shown in FIGS. 5 and 6 . The number of particle models 6 constituting the partial chain model 8 can be appropriately set to three or more. A larger number of particle models 6 constituting the partial chain model 8 has the advantage of being able to more effectively suppress noise due to thermal fluctuation. However, a larger number of particle models 6 has the disadvantages of making the partial chain model 8 longer, making the spatial locality of the cutting position unclear, and increasing the total number of partial chain models 8 and the amount of calculation. For this reason, it is desirable to appropriately adjust the number of particle models 6 constituting the partial chain model 8 depending on the application. For example, the number of particle models 6 constituting the partial chain model 8 may be appropriately set depending on the type of polymer chain 2 shown in FIG. 2 . In this embodiment, the number of particle models 6 constituting the partial chain model 8 can be set to, for example, 5 to 100 (9 in FIG. 10 of this example). FIG. 11 is a flowchart showing an example of the processing procedure of the first step S41.

[0043] Here, if all the polymer chain models 5 included in the polymer material model 12 are linear (i.e., they do not include branches and no bonds between particle models 6, 6 corresponding to crosslinks or the like have been added), all the partial chain models 8 can be extracted using the following procedure without using step S411 (shown in Figure 11) of this embodiment described later.

[0044] First, one end of a pair 9 is selected in order to serve as both ends of the partial chain model 8 to be extracted. The procedure involves first serially numbering the particle models 6 in each polymer chain model 5 from one end to the other, incrementing by one for each particle model 6. To prevent overlapping of particle model numbers among multiple polymer chain models 5, for example, the particle model 6 of the first selected polymer chain model 5 is serially numbered starting with "1," and the particle models 6 of subsequently selected polymer chain models 5 are serially numbered starting with the number of the last particle model 6 of the immediately preceding polymer chain model 5 plus "1." Then, the particle models 6 in each polymer chain model 5 are searched one by one, starting from the one with the smallest number, and the i-th particle model 6 found is selected as one end of the pair 9.

[0045] Next, the other ends of the pair 9 are selected to be both ends of the partial chain model 8 to be extracted. As a procedure for this, first, when the number of particle models 6 constituting the partial chain model 8 is defined as m, it is determined whether the (i+m-1)th particle model 6 exists in the same polymer chain model 5 for the i-th particle model 6 selected as one end of the selected pair 9. If the result of this determination is Yes (i.e., it exists), the (i+m-1)th particle model 6 is selected as the other end of the pair 9. If the result of the determination is No (i.e., it does not exist), the search within the polymer chain model 5 ends.

[0046] Then, all particle models 6 from one end to the other end of the pair 9 (i.e., from the i-th to the (i+m-1)-th) are extracted as partial chain models 8. However, there is a problem that this method cannot be used when all polymer chain models 5 included in the polymer material model 12 include branches or when bonds between particle models 6, 6 corresponding to crosslinks or the like are added. To solve this problem, in this embodiment, the processing procedure of the first step S41 shown in FIG. 11 is used.

[0047] [Calculating the path through bonds between all pairs of particle models] In the first step S41 of this embodiment, first, for all particle models 6 included in the polymer material model 12 shown in Figures 5 and 6, the shortest path route via the bond model 7 between each pair of particle models 6, 6 is calculated for each pair 9 consisting of a pair of particle models 6, 6 (step S411). Step S411 is performed as preparation for efficiently searching for a partial chain model 8. In step S411 of this embodiment, the shortest path route is calculated (identified) only when the number of particle models 6 included in the shortest path route is equal to or less than the number of particle models 6 constituting the partial chain model 8.

[0048] Here, the pair 9 consisting of a pair of particle models 6, 6 is not limited to the case where the particle models 6 are directly bonded by the bond model 7. For such pairs 9, all possible combinations of different particle models 6, 6 can be searched for in the polymer material model 12. Note that for the particle models 6, 6 of the pair 9, the order of the particle models 6 is not distinguished and they are considered to be the same. In other words, the pair 9 consisting of the ith particle model 6 and the jth particle model 6 is equivalent to the pair 9 consisting of the jth particle model 6 and the ith particle model 6.

[0049] The length of the shortest path via the bond models 7 of the pair 9 (hereinafter, sometimes referred to as "the length of the shortest path of the pair 9") can be defined as appropriate. A simple definition is the sum of the bond lengths of the bond models 7 passing through the shortest path. In this embodiment, the above simple definition is simplified, and the total number of particle models 6 on the shortest path is defined as the length of the shortest path of the pair 9. The length of the shortest path of the pair 9 is equal to the result of approximating that all bond lengths of the bond models 7 are equal to the equilibrium bond length, and further dividing the path length obtained from the above simple definition by the equilibrium bond length to make it dimensionless, and then adding 1. This simplification, i.e., approximating that all bond lengths of the bond models 7 are equal to the equilibrium bond length, allows a simple queue to be used instead of a prioritized queue such as a heap in the breadth-first search described below, which has the advantage of reducing the amount of calculation.

[0050] In this embodiment, when a pair 9 exists in which the length of the shortest path of the pair 9 is equal to the number of particle models 6 constituting the partial chain model 8, one or more partial chain models 8 are constructed from a set of particle models 6 on one shortest path via the bond model 7 of that pair 9. If the polymer chain model 5 is linear, exactly one partial chain model 8 is constructed. Alternatively, if the polymer chain model 5 is not linear (i.e., if it contains branches or if two polymer chain models 5, 5 are cross-linked), a loop path may exist in the graph (i.e., the cross-linked network structure of the polymer chain) in which the particle models 6 are nodes (in graph theory) and the bond models 7 are edges (also in graph theory). If such a loop path exists, there may be multiple shortest paths corresponding to one pair 9 described above. In this case, there may be a number of partial chain models 8 equal to the number of shortest paths.

[0051] To calculate the length of the shortest path of the pair 9, for example, an algorithm for solving a shortest path problem in graph theory can be used. Examples of such algorithms include the Warshall-Floyd algorithm and the Dijkstra algorithm. However, when the Warshall-Floyd algorithm is used, the calculation time required is cubed with respect to the total number of particle models 6. For example, when the polymer material model 12 includes a large number of particle models 6 (e.g., hundreds of thousands or more), the calculation amount of the shortest path of the pair 9 becomes unrealistic. To solve this problem, in step S411 of this embodiment, the Dijkstra algorithm is used instead of the Warshall-Floyd algorithm. Furthermore, for pairs 9 whose length of the shortest path of the pair 9 exceeds the total length of the partial chain model 8 (in this embodiment, the number of particle models 6 constituting the partial chain model 8), the search is omitted. As a result, for example, when the number of particle models 6 constituting the partial chain model 8 is sufficiently small, the calculation amount of the shortest path of the pair 9 becomes the first power of the total number of particle models 6, making it possible to significantly reduce the calculation amount.

[0052] Next, a detailed procedure for calculating the length of the shortest path of the pair 9 using the Dijkstra algorithm will be described. In this embodiment, first, one particle model 6 is selected from all particle models 6 included in the polymer material model 12 (for example, in order after serial numbers are assigned to the particle models 6). Next, according to the Dijkstra algorithm, starting from the selected particle model 6s, particle models 6 that can be reached via the bond model 7 are searched for. At this time, for a pair 9 consisting of the selected particle model 6s and the particle model 6 being searched for, the length of the shortest path of the pair 9 is set as the search priority, and the particle model 6 with the smaller priority value (i.e., the pair 9 with the shorter length of the shortest path) is searched for first.

[0053] In this embodiment, the total number of particle models 6 on the route is set as the search priority. This is equal to the total number of bonded models 7 to be passed through plus "1". Then, particle models 6 are searched in order of priority from particle models 6 with smaller priority values. Basically, all searchable particle models 6 are searched, and for each searched particle model 6, when the search first arrives, the length of the shortest path of the pair 9 consisting of the selected particle model 6s and the arrived particle model 6f is calculated as the priority at that time. However, as described above, when searching for the arrived particle model 6f, if it becomes clear that the length of the shortest path of the pair 9 consisting of the selected particle model 6s and the arrived particle model 6f exceeds the total length of the partial chain model 8 (in this example, the number of particle models 6 constituting the partial chain model 8), the search for further particle models 6 related to the selected particle model 6s is terminated.

[0054] In the Dijkstra algorithm, a prioritized queue such as a heap can be used to search in descending order of priority. As in this embodiment, only when the length of the combined model 7 is constant, breadth-first search can be used, i.e., a normal queue is sufficient. When the number of data points stored in the queue is N, the amount of calculation required to extract one piece of data from the queue is O(log(N)) when a prioritized queue is used, and O(1) when a normal queue is used. Therefore, the amount of calculation can be reduced when a normal queue is used.

[0055] In step S411 of this embodiment, for all particle models 6 included in the polymer material model 12, the length of the shortest path of all possible pairs 9 consisting of a pair of particle models 6, 6 is calculated (identified) only when the length of the shortest path of those pairs 9 (in this embodiment, the number of particle models 6 included in the shortest path) is less than or equal to the total length of the partial chain model 8 (in this embodiment, the number of particle models 6 constituting the shortest path (9 in FIG. 10 of this example)). This allows the length of the shortest path of the pair 9 to be calculated in a short time. For all calculated pairs 9, the pair of particle models 6, 6 and the length of the shortest path of the pair 9 corresponding to the pair of particle models 6, 6 are stored as a set of data in the computer 1 (shown in FIG. 1).

[0056] The data structure for storing all pairs 9 calculated in step S411 of this embodiment can be selected as appropriate. For example, a pair 9 consisting of an ith particle model 6 and a jth particle model 6 may be stored in the ith row and jth column of a two-dimensional array. However, such a two-dimensional array requires a memory area proportional to the square of the total number N of particle models 6 included in the polymer material model 12. For this reason, if the total number N is very large (e.g., hundreds of thousands or more), there is a problem that the memory capacity is insufficient and execution becomes impossible. To solve this problem, instead of a two-dimensional array, it is preferable to store the pair 9 consisting of the ith particle model 6 and the jth particle model 6 in a hash table using the pair of i and j as a key.

[0057] [Select one pair] Next, in the first step S41 of this embodiment, one pair 9 is selected (step S412). In this embodiment, one pair 9 is selected as a candidate for particle models 6, 6 constituting both ends of one or more partial chain models 8. Such a pair 9 may be selected, for example, randomly from all pairs identified in step S411, or may be selected based on a predetermined criterion. In this embodiment, after serially numbering all particle models 6 included in the polymer material model 12, for the sets of particle models 6, 6 included in the pair 9, when the number of one particle model 6 with a smaller (or equal) serial number is designated as i-th and the number of the other particle model 6 is designated as j-th, all pairs 9 identified in step S411 are sorted in ascending order of i, and, if i is the same value, in ascending order of j, and then each pair 9 is selected in the sorted order.

[0058] Furthermore, if the distance of the shortest path between the particle models 6, 6 of the pair 9 via the bond models 7 is not equal to the number of particle models 6 constituting the partial chain model 8 (9 in FIG. 10 in this example), the pair 9 cannot be the particle models 6, 6 constituting both ends of the partial chain model 8, and is therefore discarded. Note that, among all pairs 9 calculated in step S411 of this embodiment, pairs 9 whose distance of the shortest path is longer than the number of particle models 6 constituting the partial chain model 8 may be included, but pairs 9 whose distance is shorter may be included. Note that such short pairs 9 are not required in step S412 (i.e., the step of selecting as candidates for the particle models 6, 6 constituting both ends of the partial chain model 8) but are required in the next step S413. One selected pair 9 is stored in the computer 1 (shown in FIG. 1).

[0059] [Extract a partial chain model with the selected pair as both ends] Next, in the first step S41 of this embodiment, a partial chain model 8 having the selected pair 9 at both ends is extracted (step S413). In this embodiment, first, for the pair of particle models 6, 6 of the selected pair 9, the particle model 6 with the smaller (or equal) serial number is identified as the i-th particle model 6, and the other particle model 6 is identified as the j-th particle model 6. Then, starting from the i-th particle model 6, particle models 6 that can be reached via the bond model 7 are searched for in descending order of priority, defined below, according to Dijkstra's algorithm, to search for a path to the j-th particle model 6. The priority is defined as the length of the shortest path from the particle model 6 currently being searched to the j-th particle model 6. Note that the smaller the priority value (i.e., the shorter the length of the shortest path) is, the higher the priority is. In this search, the length of the shortest path of the pair 9 from the particle model 6 currently being searched to the j-th particle model 6 may (frequently) be shorter than the number of particle models 6 constituting the partial chain model 8. To determine the priorities of these pairs 9, the lengths of the shortest paths of such short pairs 9 calculated in step S411 of this embodiment as described above are used.

[0060] In this embodiment, the priority is calculated by adding "1" to the total number of bonded models 7 that are passed from the particle model 6 currently being searched to the j-th particle model 6. In step S413 of this embodiment, as in step 411, the length of the bonded models 7 is approximated as constant, which can be substituted by a breadth-first search, thereby reducing the amount of calculation. If these searches find a path from the i-th particle model 6 to the j-th particle model 6, the path is considered to be one of the shortest paths connecting both ends, and the sequence of all particle models 6 on the shortest path including both ends is extracted as a partial chain model 8. At least one such path exists.

[0061] As described above, when the polymer chain model 5 is not only a straight chain but also includes branches, or when the polymer chain models 5, 5 are cross-linked, a loop path may exist in the graph with particle models 6 as nodes and bond models 7 as edges (i.e., the cross-linked network structure of the polymer chain). If the length of the loop path is equal to twice the number of particle models 6 that make up the partial chain model 8, multiple paths may exist. The extracted partial chain model 8 (i.e., the arrangement of all particle models 6 on the shortest path including both ends) is stored in the computer 1 (shown in FIG. 1).

[0062] [Determine whether all pairs have been selected] Next, in the first step S41 of this embodiment, it is determined whether or not all pairs 9 have been selected (step S414). In this embodiment, if it is determined that all pairs 9 have been selected ("Yes" in step S414), the series of processes in the first step S41 ends. On the other hand, if it is determined that all pairs 9 have not been selected ("No" in step S414), steps S412 to S414 are performed again. Note that in step S412, which is performed again, one of the remaining pairs 9 that have not yet been selected is selected in the selected polymer chain model 5.

[0063] In this way, in the first step S41 of this embodiment, all of the extractable partial chain models 8 can be searched for in the polymer material model 12.

[0064] 10(a) is a conceptual diagram showing an example of two polymer chain models 5 bonded by a bond model 7 at a crosslinking point 14. When the number of particle models 6 constituting a partial chain model 8 is set to nine, a first partial chain model 8A and a second partial chain model 8B are extracted as partial chain models 8 that do not pass through the bond model 7 at a crosslinking point 14.

[0065] 10(b) is a conceptual diagram showing an example of a partial chain model 8 via the bond model 7 of the crosslinking point 14. For the same polymer material model 12 as in FIG. 10(a), a third partial chain model 8C and a fourth partial chain model 8D are extracted as partial chain models via the bond model 7 of the crosslinking point 14.

[0066] [Determine whether the partial chain model has been extended (Step 2)] Next, in the cutting step S4 of this embodiment, it is determined whether or not the extracted partial chain model 8 (shown in FIG. 10) has been elongated based on the end-to-end distance R of the partial chain model 8 (second step S42). FIG. 12 is a flowchart showing an example of the processing procedure of the second step S42.

[0067] [Select a partial chain model] In the second step S42 of this embodiment, first, one partial chain model 8 (shown in FIG. 10) is selected (step S421). In this embodiment, one partial chain model 8 is selected from all the partial chain models 8 extracted in the first step S41 (i.e., for each of all the polymer chain models 5 shown in FIG. 5, all the partial chain models 8 (shown in FIG. 10) included in those polymer chain models 5). The partial chain model 8 may be selected randomly from all the partial chain models 8, or may be selected based on a predetermined criterion.

[0068] [Calculate the end-to-end distance of the selected partial chain model] Next, in the second step S42 of this embodiment, the end-to-end distance R of the selected partial chain model 8 is calculated (step S422). The end-to-end distance R represents the magnitude of extension of the partial chain model 8.

[0069] 10, the end-to-end distance R is specified as the magnitude (Euclidean distance) of the vector connecting the particle models 6, 6 arranged at both ends of the partial chain model 8. The larger this end-to-end distance R, the more the partial chain model 8 is elongated.

[0070] The end-to-end distance R specifies the overall elongation of the partial chain model 8, and therefore, for example, the influence of thermal fluctuation noise, which is likely to occur in the length of each bond model 7 between particle models 6, 6, is suppressed (reduced). Therefore, by calculating the end-to-end distance R, the elongation of the partial chain model 8 can be determined more appropriately than in conventional methods that calculate the distance between adjacent particle models 6, 6 via a bond model 7. Note that if the number of particle models 6 constituting the partial chain model 8 is "2," this becomes equivalent to the conventional method that uses the length of each individual bond model 7, and the effect of further suppressing the influence of thermal fluctuation noise is lost. For this reason, the number of particle models 6 constituting the partial chain model 8 is preferably "3" or more. The end-to-end distance R is stored in computer 1 (shown in FIG. 1).

[0071] [Determine whether the selected partial chain model has been extended] Next, in the second step S42 of this embodiment, it is determined whether the selected partial chain model 8 has been elongated based on the end-to-end distance R of the partial chain model 8 (step S423). In step S423 of this embodiment, if the end-to-end distance R of the selected partial chain model 8 is equal to or greater than a predetermined first threshold, it is determined that the selected partial chain model 8 has been elongated. The first threshold can be set appropriately based on, for example, the physical properties of the actual polymer chain 2 shown in FIG. 2 (such as the shear stress of the material and the activation energy of the shear chemical reaction).

[0072] In this embodiment, if it is determined that the end-to-end distance R is equal to or greater than the first threshold value ("Yes" in step S423), it is determined that the selected partial chain model 8 has been elongated (step S424). This determination result is stored in computer 1 (shown in FIG. 1) (step S424).

[0073] On the other hand, if it is determined that the end-to-end distance R is less than the first threshold value ("No" in step S423), it is determined that the selected partial chain model 8 is not elongated (step S425). This determination result is stored in computer 1 (shown in FIG. 1).

[0074] In this way, in the second step S42 of this embodiment, by calculating the end-to-end distance R, the influence of the above-mentioned thermal fluctuation noise can be suppressed compared to the conventional method in which the distance between adjacent particle models 6, 6 via the bond model 7 is calculated. Therefore, in this embodiment, it is possible to more appropriately determine whether the partial chain model 8 has been elongated compared to the conventional method.

[0075] [Determine whether all partial chain models have been selected] Next, in the second step S42 of this embodiment, it is determined whether all partial chain models 8 (for each of all polymer chain models 5 shown in FIG. 5, all partial chain models 8 (shown in FIG. 10) included in those polymer chain models 5) have been selected (step S426). In this embodiment, if it is determined that all partial chain models 8 have been selected ("Yes" in step S426), the series of processes in the second step S42 is terminated. On the other hand, if it is determined that all partial chain models 8 have not been selected ("No" in step S426), steps S421 to S426 are performed again. Note that in the re-performed step S421, one of the remaining partial chain models 8 that has not yet been selected is selected.

[0076] In this way, in the second step S42 of this embodiment, it is possible to determine whether each partial chain model 8 (shown in FIG. 10) included in each of all polymer chain models 5 shown in FIG. 2 has been elongated (i.e., broken).

[0077] [Cleavage at the bond model contained in the extended partial chain model (Step 3)] Next, in the cutting step S4 of this embodiment, for the partial chain model 8 determined to have been elongated in the second step S42, the polymer chain model 5 is cut at at least one bond model 7 included in the partial chain model 8 (third step S43). Fig. 13 is a flowchart showing an example of the processing procedure of the third step S43.

[0078] [Select the partial chain model that is judged to be elongated] In the third step S43 of this embodiment, first, one partial chain model 8 is selected from all partial chain models 8 determined to have been elongated in the second step S42 (step S431). The elongated partial chain model 8 may be selected randomly or based on predetermined criteria. If the selected partial chain model 8 has passed through any of the bond models 7 of other partial chain models 8 cleaved in step S433 (described later) of the third step S43, it is determined that at least one bond model 7 included in the selected partial chain model 8 has been cleaved, and steps S432 to S433 may be omitted.

[0079] [Select one bond model included in the selected partial chain model] Next, in the third step S43 of this embodiment, one binding model 7 is selected from the multiple binding models 7 included in the selected partial chain model 8 (step S432). The binding model 7 can be selected as appropriate. In this embodiment, one binding model 7 can be randomly selected from the multiple binding models 7 included in the selected partial chain model 8.

[0080] [Cut at one selected bond model] Next, in the third step S43 of this embodiment, the selected bond model 7 is cleaved (step S433). Figure 14 is a conceptual diagram showing a state in which the first partial chain model 8A and the second partial chain model 8B in Figure 10(a) have been cleaved.

[0081] In step S433 of this embodiment, the selected bond model 7 (shown by a two-dot chain line) is deleted or invalidated. As a result, the partial chain model 8 (polymer chain model 5) is cut at the selected bond model 7. The cut polymer chain model 5 is stored in the computer 1 (shown in FIG. 1).

[0082] In this embodiment, by cutting at one selected bond model 7, as shown in FIG. 14, the path connecting all particle models 6 constituting the partial chain model 8 is severed. As a result, if there is no other partial chain model 8 with the same ends (i.e., if it is not part of a loop structure), the polymer chain model 5 is separated into a first fragment model 13A of the first partial chain model 8A and a second fragment model 13B of the second partial chain model 8B. Furthermore, the polymer chain model 5 is separated into a model in which the second fragment model 13B of the first partial chain model 8A and the first fragment model 13A of the second partial chain model 8B are connected via a crosslink point 14. The fragment model 13 is a collection of particle models 6 connected by bond models 7 other than the selected bond model 7. These models are defined as a new polymer chain model 5.

[0083] On the other hand, for example, in a loop-shaped partial chain model 8 (not shown), when one selected bond model 7 is cut, a path remains that connects all particle models 6 via other bond models 7. In this case, the polymer chain model 5 is not separated into fragment models 13, but becomes a single polymer chain model 5 with one loop cut.

[0084] [Determine whether all partial chain models that are judged to be elongated have been selected] Next, in the third step S43 of this embodiment, it is determined whether all partial chain models 8 determined to have been elongated in the second step S42 have been selected (step S434). In this embodiment, if it is determined that all partial chain models 8 have been selected ("Yes" in step S434), the series of processes in the third step S43 ends. On the other hand, if it is determined that all partial chain models 8 have not been selected ("No" in step S434), steps S431 to S434 are performed again. Note that in the re-performed step S431, one remaining partial chain model 8 that has not yet been selected is selected from all partial chain models 8 determined to have been elongated.

[0085] As described above, in the third step S43 of this embodiment, in all of the partial chain models 8 determined to have been elongated in the second step S42, any of the bond models 7 included in each partial chain model 8 is cut. This makes it possible to create a numerical analysis model (i.e., polymer chain model 5) that approximates a polymer chain that has been cut at a locally or entirely elongated portion in accordance with the deformation of an actual polymer material.

[0086] Define Interaction Next, in the creation method of this embodiment, interactions are defined between particle models 6, 6 of the polymer chain model 5 (step S5). Note that if it is not necessary to define interactions, step S5 may be omitted. Fig. 15 is a conceptual diagram showing the polymer chain model 5 in which interactions are defined.

[0087] The interaction in this embodiment is defined as a second potential P2 defined between particle models 6, 6 of a pair of polymer chain models 5, 5. The second potential P2 can be set appropriately, and for example, an attractive and repulsive force is defined. The second potential P2 in this embodiment can be defined, for example, as an LJ potential. The details of the LJ potential and the procedure for setting constants, etc. are as described in the above-mentioned paper 1. The second potential P2 is stored in a computer 1 (shown in FIG. 1).

[0088] [Calculate structural relaxation] Next, in the creation method of this embodiment, the computer 1 (shown in FIG. 1) calculates the structural relaxation of the polymer chain model 5 based on molecular dynamics (step S6). Note that if the structural relaxation calculation is not necessary, step S6 may be omitted. In the structural relaxation calculation of this embodiment, for example, Newton's equation of motion is applied in the cell 10 for a predetermined time, assuming that the particle model 6 of the polymer chain model 5 follows classical mechanics. Then, the movement of the particle model 6 at each time is tracked for each unit time step.

[0089] In the molecular dynamics calculation, the pressure and temperature are kept constant, or the volume and temperature are kept constant, in the cell 10. As a result, in step S6, the initial configuration of the polymer chain model 5 is relaxed to approximate the molecular motion of an actual polymer material. In this embodiment, this is continued until the initial configuration of the polymer chain model 5 is sufficiently relaxed. As a result, a polymer material model 12 that models the polymer material is created. For example, the above-mentioned commercially available software can be used to calculate the structural relaxation.

[0090] For example, a filler model (not shown) that models a filler to be blended in the polymer material, or a coupling material model (not shown) that models a coupling agent, may be placed inside the cell 10. The polymer material model 12 is stored in a computer 1 (shown in FIG. 1).

[0091] [Method for creating a model for numerical analysis of polymer chains (second embodiment)] In step S422 (shown in FIG. 12) of the second step S42 in the embodiments described above, the first threshold value of the end-to-end distance R shown in FIG. 10 is set based on the physical properties of the actual polymer chain 2 shown in FIG. 2, but is not limited to this. For example, when the entropy elasticity F of the partial chain model 8 is calculated as a function of the end-to-end distance R, the end-to-end distance R such that the entropy elasticity F is equal to a predetermined second threshold value related to stress may be set as the first threshold value.

[0092] The entropy elasticity F is specified as the force with which the stretched partial chain model 8 tries to return to its original state. The larger this entropy elasticity F, the more the partial chain model 8 is stretched.

[0093] The second step S42 of this embodiment preferably includes a step of calculating the entropy elasticity F based on the following equation (1):

[0094]

number

[0095] The above equation (1) was devised as an equation that closely approximates the numerical solution of the entropy elasticity F of the end-to-end distance R for a partial chain model 8 of finite chain length. The above equation (1) includes a rotational degree of freedom term -2(R / L)-1. This rotational degree of freedom term -2(R / L)-1 is used to consider the influence on the entropy elasticity F of the overall three-dimensional rotational degrees of freedom of the partial chain model 8.

[0096] The procedure for deriving the rotational degree of freedom term -2(R / L)-1 will be explained. First, of the pair of particle models 6, 6 arranged at the ends of the partial chain model 8, the position of one particle model 6 as seen from the other particle model 6 always exists on a spherical surface with the particle model 6 as the center and the radius being the ratio R / L of the end-to-end distance R to the total length L. The surface area A of this sphere is 4π(R / L)2, and the contribution of the surface area A to the free energy is expressed as ln(1 / A) = -2ln(R / L) + constant. The first-order differentiation of this with respect to R / L gives -2(R / L)-1, which is the contribution to the entropy elasticity F. This term is exactly equal to the rotational degree of freedom term -2(R / L)-1 in the above equation (1).

[0097] In this way, by including the rotational degree of freedom term −2(R / L)−1 in the above equation (1), the entropy elasticity F of the molecular model when it contracts to become smaller than the equilibrium length can be calculated with high accuracy.

[0098] The above formula (1) includes a first correction term d. This first correction term d is intended to correct the influence of the end of the partial chain model 8 on the entropy elasticity F. When the partial chain model 8 is a coarse-grained molecular model as in this embodiment, the contribution of one of the particle models 6, 6 at the end of the partial chain model 8 to the entropy elasticity F near that particle model 6 tends to be smaller than the entropy elasticity F of the other particle model 6. For this reason, the first correction term d can be considered to be a value close to "-1." Note that the value of the first correction term d may increase or decrease depending on the definition and chain length of the partial chain model 8, and is therefore desirably adjusted appropriately.

[0099] In this way, by including the first correction term d in the above formula (1), the chain length dependence of the entropy elasticity F can be calculated with high accuracy. In particular, near the equilibrium length, the chain length dependence of the entropy elasticity F can be calculated with high accuracy. Note that when it is considered possible to omit the correction of the influence of the end of the partial chain model 8 on the entropy elasticity F (for example, when the chain length is sufficiently long), the first correction term d may be set to "0".

[0100] The above formula (1) includes a second correction term C. This second correction term C is used to correct the effect on the entropy elasticity F due to interactions within the partial chain model 8. This effect on the entropy elasticity F may vary depending on the definition of the partial chain model 8, the chain length of the molecular chain, and other factors. For this reason, it is desirable to adjust the value of the second correction term C as appropriate. This second correction term C can correct the deviation from the entropy elasticity F of an ideal chain model (not shown) in which the partial chain model 8 is represented by a broken line, for example, when the partial chain model 8 is extended beyond the equilibrium length. This allows the entropy elasticity F of the partial chain model 8 to be calculated with high accuracy. Note that if the partial chain model 8 is not extended sufficiently beyond the equilibrium length, C=1 may be used.

[0101] Furthermore, the entropy elasticity of the partial chain model (Kremer-Grest model) 8 tends to increase significantly in the region close to full extension (where the ratio R / L of the end-to-end distance R to the total length L is close to 1) compared to the entropy elasticity of the ideal chain model (not shown). This is because, in the partial chain model 8, the bond angle between the particle models 6, 6 tends to be close to 60° and 120° due to the addition of interactions between the particle models 6, 6, and therefore is less likely to be close to 180°, which is close to full extension, compared to the ideal chain model. Note that this effect can be ignored when the extension of the partial chain model 8 is small. However, when a large strain is applied to the polymer material model 12 or when the polymer material model 12 includes a filler model (not shown), etc., it is important to consider the increase in entropy elasticity F in situations where the partial chain model 8 can be significantly extended.

[0102] From this perspective, it is preferable that the step of calculating the entropy elasticity F (step S422 shown in Figure 12) includes a step of identifying the second correction term C, for example, based on a function of the ratio R / L of the end-to-end distance R of the partial chain model 8 to the total length L of the partial chain model 8.

[0103] In step S422 (shown in FIG. 12) of the second step S42 of this embodiment, the identified second correction term C is substituted into the above formula (1). As a result, in this embodiment, the entropy elasticity F of the partial chain model 8 can be calculated with high accuracy. The entropy elasticity F is stored in the computer 1 (shown in FIG. 1).

[0104] In this embodiment, by calculating the entropy elasticity F based on the end-to-end distance R, for example, when determining whether or not a break has occurred based on the breaking stress, it is possible to more appropriately determine whether or not the partial chain model 8 has been elongated while suppressing the influence of the thermal fluctuation noise described above. Therefore, a numerical analysis model of a polymer chain that has been broken at a locally or entirely elongated portion as an actual polymer material deforms can be created.

[0105] [Method for creating a model for numerical analysis of polymer chains (third embodiment)] As in the third step S43 of the previous embodiments, when one bond model 7 is randomly selected from the multiple bond models 7 included in the partial chain model 8 shown in FIG. 14 and cut, a fragment model 13 with an extremely small molecular weight may be created. Such fragment models 13 are thought to be unlikely to occur in an actual polymer chain 2. This is thought to be because there is little entanglement with surrounding polymer chains 2 near the end of the polymer chain 2, which could become small fragments if cut. In other words, it is presumed that the lack of entanglement near the end speeds up diffusion motion, which in turn promotes stress relaxation of the tension in the polymer chain 2 generated during deformation, making it difficult for strong tension to be generated in the polymer chain 2 near the end, and as a result, it is difficult for the cutting reaction to proceed.

[0106] In the third step S43 of this embodiment, prior to cleavage by the selected binding model 7, it is determined whether the sizes of the fragment models 13, 13 resulting from cleavage are greater than a second threshold value when it is assumed that the selected binding model 7 has been cleaved. If the sizes of the fragment models 13, 13 are greater than the second threshold value, the partial chain model 8 is cleaved by the selected binding model 7. This can prevent the creation of a fragment model 13 with an extremely small molecular weight.

[0107] The second threshold value can be set as appropriate. In this embodiment, the second threshold value is set based on the molecular weight distribution of a polymer material (not shown) including polymer chain 2 (shown in FIG. 2). The molecular weight distribution of the polymer material can be acquired as appropriate. In this embodiment, first, a polymer material (test piece) to be analyzed (for example, during kneading, extrusion, aging, or wear) is manufactured. Then, the molecular weight distribution of the polymer material is acquired based on known gel permeation chromatography (GPC).

[0108] Figure 16 shows the molecular weight distribution (differential molecular weight distribution) of a polymer material. This graph, which plots the logarithm of molecular weight, exhibits a shape similar to a log-normal distribution with a single peak. This shape characteristic is not only observed during manufacturing, but also after polymer chain scission due to processing, aging, and other factors. Although the peak molecular weight generally shifts to a smaller value, the shape itself tends not to change significantly. On the other hand, if we assume that random scission occurs in the polymer chain, the molecular weight distribution would approach a Poisson distribution, and a large peak would be expected at extremely low molecular weights. However, such a low molecular weight peak is not observed in actual polymer chains. Therefore, it is suggested that in reality, random scission that would generate extremely low molecular weight polymer chains 2 is unlikely to occur, and instead, high molecular weight polymer chains 2 are preferentially scissed.

[0109] It is believed that the physical properties of a polymer material are most affected by the polymer chain 2 having a molecular weight W1 that is the peak of the differential molecular weight distribution (i.e., the molecular weight that is most frequently found). From this perspective, the second threshold value can be set based on the molecular weight W1 that is most frequently found in the molecular weight distribution.

[0110] [Predict whether the size of all fragment models is above the second threshold] 17 is a flowchart showing an example of the processing procedure of the third step S43 of another embodiment of the present invention. In the third step S43 of this embodiment, as shown in FIG. 14, when it is assumed that one selected bond model 7 is cut, it is predicted whether or not the sizes of one or two fragment models 13 resulting from the cut are all larger than the second threshold (step S435).

[0111] A single fragment model 13 resulting from cleavage is, for example, a model in which, when a loop-shaped partial chain model 8 (not shown) is cleaved at one selected bond model 7, a path remains that connects all particle models 6 via other bond models 7.

[0112] The two fragment models 13, 13 resulting from the cutting are models in which the path connecting all of the particle models 6 constituting the polymer chain model 5 is interrupted by cutting at one selected bond model 7, as shown in Fig. 14. In this embodiment, a first fragment model 13A and a second fragment model 13B are included.

[0113] In step S435, it is predicted whether the sizes of one or two fragment models 13 are both greater than the second threshold. Incidentally, in order to calculate whether the size of the fragment model 13 will be greater than the second threshold, a simple method of calculating the size of the fragment model can be considered. However, with such a method, when calculating the size of the fragment, it is possible that up to all particle models 6 that make up the partial chain model 8 will be searched, which requires a very large amount of calculation and can be very inefficient. For this reason, when it becomes clear that the size of the fragment model 13 will be greater than the second threshold, further search may be terminated.

[0114] The search algorithm may be, for example, a depth-first search, a breadth-first search, a random search, or the like. In this embodiment, a breadth-first search is used. The prediction result of whether the sizes of all the fragment models 13 are greater than the second threshold value is stored in the computer 1 (shown in FIG. 1).

[0115] [Determine whether the size of the fragment model is greater than the threshold] Next, in the third step S43 of this embodiment, it is determined whether the sizes of the predicted fragment models 13 (in this example, the first fragment model 13A and the second fragment model 13B) are both greater than the second threshold (step S436). In step S436, the above determination is made based on the prediction result in step S435.

[0116] If it is determined that the sizes of the fragment models 13 (first fragment model 13A and second fragment model 13B) are both greater than the second threshold value ("Yes" in step S436), the polymer chain model 5 is cut by the selected bond model 7 (conditional cutting step S437).

[0117] On the other hand, if it is determined that the size of either of the fragment models 13 (the first fragment model 13A and the second fragment model 13B) is equal to or smaller than the second threshold ("No" in step S45), steps S432 to S436 are performed again. In step S432, which is performed again, another bond model 7 that has not yet been selected is selected.

[0118] [Conditional cutting process] In the conditional cleavage step S437, for example, the selected bond model 7 is deleted or invalidated, thereby cleaving the polymer chain model 5. The cleaved polymer chain model 5 is stored in the computer 1 (shown in FIG. 1).

[0119] In the third step S43 of this embodiment, prior to cleavage by the selected binding model 7, it is determined whether the sizes of the fragment models 13, 13 resulting from cleavage are greater than a second threshold value when it is assumed that the selected binding model 7 has been cleaved. If the sizes of the fragment models 13, 13 are greater than the second threshold value, the partial chain model 8 is cleaved by the selected binding model 7. This prevents fragment models 13 that are equal to or smaller than the second threshold value from being created, and allows the partial chain model 8 to be cleaved so as to approximate a desired molecular weight distribution.

[0120] [Simulation method for polymer materials] The numerical analysis model of the polymer chain (polymer chain model 5) created by the creation method of the above-described embodiments is used for simulating a polymer material. Next, a method for simulating a polymer material will be described. Fig. 18 is a flowchart showing an example of the processing procedure of the method for simulating a polymer material.

[0121] [Create a polymer chain model] In the simulation method of this embodiment, first, the computer 1 (shown in FIG. 1) creates the polymer chain model 5 shown in FIG. 14 (step S7). In step S7 of this embodiment, a polymer chain model 5 cut by extension accompanying deformation of the polymer material model 12 is created based on the procedure from step S1 to cutting step S4 of the creation method of the previous embodiments (shown in FIG. 3). The polymer chain model 5 is stored in the computer 1 (shown in FIG. 1).

[0122] [Create a polymer material model] Next, in the simulation method of this embodiment, the computer 1 (shown in FIG. 1) creates a polymer material model 12 (shown in FIG. 15) including a polymer chain model 5 (step S8). In step S8 of this embodiment, an interaction (second potential P2) is defined between the particle models 6, 6 of the polymer material model 12 arranged in the cell 10 based on the procedure of step S5 (shown in FIG. 3) of the previous embodiment. Furthermore, in step S8, the structural relaxation of the polymer chain model 5 is calculated based on the procedure of step S6 (shown in FIG. 3) of the previous embodiment. As a result, a polymer material model 12 (shown in FIG. 15) that models the polymer material is created. The polymer chain model 5 and the polymer material model 12 are stored in the computer 1 (shown in FIG. 1).

[0123] [Calculate deformation of polymer material model] Next, in the simulation method of this embodiment, the computer 1 (shown in FIG. 1) calculates the deformation of the polymer material model 12 (shown in FIG. 15) (step S9). The deformation calculation of the polymer material model 12 can be performed as appropriate. For example, based on the procedure described in Japanese Patent No. 6408856, the extension of the polymer material model 12 in the Y-axis direction may be calculated so that one end (one side surface 11) and the other end (the other side surface 11) of the polymer material model 12 move away from each other. Furthermore, a periodic strain may be applied to the polymer material model 12. The above-mentioned commercially available software may be used for such deformation calculation.

[0124] In step S9 of this embodiment, physical quantities (stress, energy loss, etc.) of the polymer material model 12 can be calculated by calculating the deformation of the polymer material model 12. In this embodiment, a polymer chain model 5 cut so as to approximate a desired molecular weight distribution (in this example, the molecular weight distribution of the analysis target shown in FIG. 16) is used, so physical quantities (stress, energy loss, etc.) that significantly affect the molecular weight distribution can be calculated with high accuracy. The deformation calculation results are stored in computer 1 (shown in FIG. 1).

[0125] [Output deformation calculation results] Next, in the simulation method of this embodiment, the computer 1 (shown in FIG. 1) outputs the calculation results of the deformation (step S10). The calculation results can be output as appropriate. For example, the calculation results may be displayed on a display device 1d (shown in FIG. 1) or may be printed on a printer (not shown) or the like. This makes it possible to notify an operator or the like of the calculation results of the deformation of the polymer material model 12.

[0126] [Evaluate the deformation calculation results] Next, in the simulation method of this embodiment, the calculation results of the deformation are evaluated (step S11). The evaluation of the calculation results may be performed by the computer 1 (shown in FIG. 1) or by an operator.

[0127] In step S11 of this embodiment, it is evaluated whether the calculation result of the deformation (in this example, the physical quantity of the polymer material model 12) satisfies a predetermined standard. The standard can be set appropriately depending on the performance required of the polymer material and the product using the polymer material (for example, a tire, etc.).

[0128] In this embodiment, if it is determined that the deformation calculation result satisfies the criteria ("Yes" in step S11), it is determined that the performance of the polymer material model 12 is good. In this case, a polymer material is produced based on the structure of the polymer chain 2 and the various conditions set in the polymer material model 12 (step S12).

[0129] On the other hand, if it is determined that the deformation calculation result does not satisfy the criteria ("No" in step S11), the performance of the polymer material model 12 is evaluated as not being good. In this case, the structure of the polymer chain 2 and the various conditions set in the polymer material model 12 are changed (step S13), and steps S7 to S11 are performed again.

[0130] In this way, in the simulation method of this embodiment, the structure of the polymer chain 2 and the conditions set in the polymer material model 12 are changed until the performance of the polymer material is improved, so that a polymer material with good performance can be efficiently produced.

[0131] Although a particularly preferred embodiment of the present invention has been described in detail above, the present invention is not limited to the illustrated embodiment and can be modified and implemented in various ways. [Example]

[0132] Based on the processing procedure shown in FIG. 3, a polymer chain model (coarse-grained molecular model) was input, and a polymer chain model for numerical analysis was created by cutting the polymer chain model that elongated with the deformation of the polymer material (Example 1, Example 2, Comparative Example 1, Comparative Example 2, and Comparative Example 3).

[0133] In Examples 1 and 2, a cutting step was carried out to cut a polymer chain model that had been elongated due to deformation, based on the processing procedure shown in Fig. 9. In the cutting steps of Examples 1 and 2, a first step was carried out to extract a partial chain model from the polymer chain model, based on the processing procedure shown in Fig. 11. Furthermore, in the cutting steps of Examples 1 and 2, a second step was carried out to calculate the end-to-end distance of the extracted partial chain model, and to determine whether the partial chain model had been elongated based on the end-to-end distance of the partial chain model, based on the processing procedure shown in Fig. 12.

[0134] In the cutting step of Example 1, a third step was performed in which the polymer chain model was cut at the bond model included in the partial chain model determined to be elongated based on the processing procedure shown in FIG. 13 of the first embodiment described in the specification. If the selected partial chain model passed through any of the bond models of other partial chain models cut in the third step, it was determined to have been cut, and no further bond models were cut. On the other hand, in Example 2, the third step was performed based on the processing procedure shown in FIG. 17 of the third embodiment. In Example 2, the size of the fragment model was set to the maximum chain length, and the second threshold was set to 1000. The models of Examples 1 and 2 are equivalent to those created based on the procedure of the second embodiment described in the specification, in which the end-to-end distance corresponding to the value of entropy elasticity F is set to the second threshold so that the end-to-end distance is the same as the end-to-end distance of the partial chain model to which the last-cut bond model belongs.

[0135] In Comparative Example 1, the step of cutting a plurality of bond models constituting the polymer chain model was omitted. In Comparative Example 2, a step of cutting a bond model randomly selected from a plurality of bond models constituting the polymer chain model was performed. In Comparative Example 3, based on the same procedure as in Patent Document 1, when the distance between adjacent particle models via a bond model was equal to or greater than a predetermined first distance, the bond model between the particle models was deleted, thereby cutting the polymer chain model. In Comparative Examples 2 and 3, the step of cutting the bond models was stopped when the number of cut bond models reached 720.

[0136] Next, two cycles of cyclic shear (strain: 0.1) were performed on polymer material models including the polymer chain models of Example 1, Example 2, Comparative Example 1, Comparative Example 2, and Comparative Example 3. The integrated molecular weight distributions of the polymer chain models after scission were then compared. Figure 19 is a graph showing the molecular weight distributions (integral molecular weight distributions) of the Examples and Comparative Examples. Note that the molecular weight distribution of Comparative Example 1 is omitted in Figure 19.

[0137] Furthermore, the storage modulus and loss tangent were compared for the polymer chain model before scission (Comparative Example 1) and the polymer chain model after scission (Examples 2-3 and Comparative Examples 2-3) in the temperature range of T = 0.45 to 1.0. Fig. 20 is a graph showing the magnitude of the storage modulus. Fig. 20 shows the relationship between the logarithm of the storage modulus and temperature. Fig. 21 is a graph showing the loss tangent. Fig. 21 shows the relationship between the loss tangent and temperature. The common specifications are as follows: Number of polymer chain models: 300 (however, in Comparative Example 3, 30) Total number of particle models for one polymer chain model: 5000 Number of crosslinking points: 11,250 (however, in Comparative Example 3, it was 1,125) Number of broken bond models: 7,200 (however, 0 in Comparative Example 1 and 720 in Comparative Example 3) Deformation process: Time step: 0.005τ Period: 100000τ Distortion: 4.5

[0138] As shown in FIG. 20 , the test results showed that, in Examples 1 and 2, the storage modulus in the high-temperature region (near T = 1.0) after the polymer chain model was cut decreased compared to the value before the polymer chain model was cut (Comparative Example 1), similar to the actual polymer material. This is thought to be because thermal fluctuation noise was suppressed and the extension of the partial chain model was appropriately determined, resulting in the creation of a polymer material model whose cutting point was similar to that of the actual polymer material. Although the storage modulus in Example 1 decreased in the high-temperature region, the decrease was slightly smaller than in Example 2. Furthermore, as shown in FIG. 21 , the loss tangent value in the high-temperature region of Example 1 decreased compared to Example 2. This is thought to be because Example 1 contains low-molecular-weight components, as shown in FIG. 19 , and the bonds in the extended molecular chain model were cut into much finer fragments than would occur in an actual polymer chain (i.e., until the number of particle models constituting the partial chain became equal). In contrast, in Example 2, as shown in FIG. 19, low molecular weight components were not included, and the molecular chain model could be cut to approximate the actual molecular weight distribution, and the physical properties (molecular weight distribution, storage modulus, and loss tangent) before and after the actual cutting could be well reproduced.

[0139] On the other hand, in Comparative Example 2, as shown in Fig. 20, the storage modulus in the high temperature region after cutting the polymer chain model decreased compared to the value before cutting the polymer chain model (Comparative Example 1), but the decrease was not sufficient compared to Examples 1 and 2. This is thought to be because Comparative Example 2 contains a large amount of low molecular weight components as shown in Fig. 19, and therefore, unlike Examples 1 and 2, cutting was performed at a bond model selected randomly from the multiple bond models that make up the polymer chain model, resulting in the creation of a polymer material model whose molecular weight distribution deviates from that of the actual polymer material.

[0140] In Comparative Example 3, as shown in FIG. 19, a large amount of low-molecular-weight components is contained, almost similar to Comparative Example 2. Furthermore, as shown in FIG. 20, in Comparative Example 3, although the storage modulus in the high-temperature region is unclear due to a large statistical error caused by the small number of molecular chains in the model, it is lower than the storage modulus before the polymer chain model was cut (Comparative Example 1). However, the decrease is not sufficient compared to Examples 1 and 2. Therefore, since the molecular weight distribution of Comparative Example 3 is very close to that of Comparative Example 2, it is presumed that the results are similar to those of Comparative Example 2. This is thought to be because, in Comparative Example 3, when determining whether to cut the bond model, the influence of thermal fluctuation noise occurs, resulting in cutting that is more random than Examples 1 and 2 (i.e., similar to Comparative Example 2), and a polymer material model was created whose molecular weight distribution deviates from that of an actual polymer material.

[0141] Thus, in Examples 1 and 2, compared to Comparative Examples 1, 2, and 3, it was possible to create a numerical analysis model of a polymer chain that was cut in the same manner as an actual polymer chain.

[0142] [Note] The present invention includes the following aspects.

[0143] [Invention 1] 1. A method for creating a model for numerical analysis of a polymer chain, comprising: inputting a polymer chain model, based on the polymer chain, including a plurality of particle models and bond models that bond adjacent particle models to each other, into a computer; a step of placing the polymer chain model in a cell that is a predetermined virtual space to set a polymer material model; the computer calculating the deformation of the polymer material model; a cutting step in which the computer cuts the polymer chain model that has been extended due to the deformation, The cutting step includes: a first step of extracting a partial chain model including three or more adjacent particle models and the bond model that bonds them in the polymer chain model; a second step of calculating an end-to-end distance for the extracted partial chain model and determining whether or not the partial chain model has been elongated based on the calculated end-to-end distance for the partial chain model; and a third step of cutting the polymer chain model at at least one of the bond models included in the partial chain model determined to be elongated. How to create a model for numerical analysis of polymer chains. [Invention 2] The method for creating a model for numerical analysis of a polymer chain according to the first aspect of the present invention, wherein the second step determines that the partial chain model has been elongated when the distance between the ends of the partial chain model is equal to or greater than a predetermined first threshold. [Invention 3] the third step is a step of selecting one binding model from the plurality of binding models included in the partial chain model determined to be elongated; a step of predicting whether or not the sizes of fragment models (one or two fragment models generated by the cutting) which are a collection of a plurality of particle models connected by other bond models excluding the selected bond model are all larger than a predetermined second threshold when the selected bond model is cut; and a conditional cleavage step of cleaving the partial chain model with the selected bond model when the predicted sizes of the fragment models are all larger than the second threshold. [Invention 4] The method for creating a model for numerical analysis of a polymer chain according to any one of Inventions 1 to 3, wherein the second step includes a step of calculating the entropy elasticity of the partial chain model as a function of the end-to-end distance of the partial chain model based on the following formula (1):

number

[0144] 5 Polymer chain model 7 Bonding Model 8 Partial chain model

Claims

1. 1. A method for creating a model for numerical analysis of a polymer chain, comprising: inputting a polymer chain model, based on the polymer chain, including a plurality of particle models and bond models that bond adjacent particle models to each other, into a computer; a step of placing the polymer chain model in a cell that is a predetermined virtual space to set a polymer material model; the computer calculating the deformation of the polymer material model; a cutting step in which the computer cuts the polymer chain model that has been extended due to the deformation, The cutting step includes: a first step of extracting a partial chain model including three or more adjacent particle models and the bond model connecting them in the polymer chain model; a second step of calculating an end-to-end distance for the extracted partial chain model and determining whether the partial chain model has been elongated based on the calculated end-to-end distance for the partial chain model; and a third step of cutting the polymer chain model at at least one of the bond models included in the partial chain model determined to be elongated. How to create a model for numerical analysis of polymer chains.

2. 2. The method for creating a model for numerical analysis of a polymer chain according to claim 1, wherein the second step determines that the partial chain model has been elongated when the distance between the ends of the partial chain model is equal to or greater than a predetermined first threshold.

3. the third step is a step of selecting one binding model from the plurality of binding models included in the partial chain model determined to be elongated; a step of predicting whether or not the sizes of fragment models (one or two fragment models generated by the cutting) which are a collection of a plurality of particle models connected by other bond models excluding the selected bond model are all larger than a predetermined second threshold when the selected bond model is cut; and a conditional cutting step of cutting the partial chain model with the selected bond model when the predicted sizes of the fragment models are all larger than the second threshold.

4. 2. The method for creating a model for numerical analysis of a polymer chain according to claim 1, wherein the second step includes a step of calculating the entropy elasticity of the partial chain model as a function of the end-to-end distance of the partial chain model based on the following formula (1): [Equation 1] where: F: Entropy elasticity k Ｂ : Boltzmann constant T: absolute temperature N K : Number of Kuhn segments in the partial chain model L －１ (x): Inverse Langevin function R: End-to-end distance of partial chain model L: total length of the partial chain model d: first correction term C: Second correction term

5. 5. The method for creating a model for numerical analysis of a polymer chain according to claim 4, wherein the step of calculating the entropy elasticity includes a step of specifying the second correction term based on a function of a ratio R / L of an end-to-end distance R of the partial chain model to a total length L of the partial chain model.

6. A method for simulating a polymeric material, comprising: a step of creating the polymer chain model by a computer based on the method for creating a polymer chain model for numerical analysis according to any one of claims 1 to 5; a step of generating a polymer material model including the polymer chain model by the computer; the computer calculating the deformation of the polymer material model; and outputting the calculation result of the deformation by the computer. Simulation methods for polymeric materials.

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