Method for simulating polymer material

Abstract

To provide a method for simulation which can evaluate the elastic force of molecular chains of a polymer material.SOLUTION: The present invention relates to a method for simulating a polymer material. The method causes a computer to perform Step S3 of expanding a molecular chain model and acquiring the distribution of the distance between terminals of a molecular chain model when expanding; Step S4 of acquiring a free energy change showing the relation between the distance between terminals and the free energy on the basis of the distribution of the distance between the terminals; Step S5 of differentiating the free energy change with respect to the distance between terminals and determining a first relation showing the relation between the distance between terminals and the elastic force of the molecular chain model; and Step S6 of converting the distance between terminals of the first relation to the strain of the molecular chain model, thereby determining a second relation showing the relation between the strain and the elastic force.SELECTED DRAWING: Figure 3

Description

[Technical field]

[0001] The present invention relates to a method for simulating a polymeric material. [Background technology]

[0002] The following Patent Document 1 describes a simulation method for polymer materials. This method includes a step of inputting a molecular chain model that models the molecular chains of a polymer material, and a step of extending the molecular chain model in a predetermined direction. [Prior art documents] [Patent documents]

[0003] [Patent Document 1] JP 2023-062527 A Summary of the Invention [Problem to be solved by the invention]

[0004] In general, there are many different types and lengths of molecular chains in polymeric materials. There has been a demand for a method to evaluate the elasticity of these molecular chains.

[0005] The present invention has been devised in view of the above circumstances, and has as its main object to provide a simulation method capable of evaluating the elastic force of molecular chains of a polymer material. [Means for solving the problem]

[0006] The present invention is a method for simulating a polymer material, comprising the step of inputting a molecular chain model, which models a molecular chain of the polymer material, into a computer, in which the computer executes the steps of: extending the molecular chain model to obtain a distribution of end-to-end distances of the molecular chain model during extension; obtaining a free energy change indicating a relationship between the end-to-end distance and the free energy based on the distribution of end-to-end distances; differentiating the free energy change with respect to the end-to-end distance to obtain a first relationship indicating a relationship between the end-to-end distance and the elastic force of the molecular chain model; and obtaining a second relationship indicating the relationship between the strain and the elastic force by converting the end-to-end distance of the first relationship into a strain of the molecular chain model. Effect of the Invention

[0007] The polymer material simulation method of the present invention employs the above steps, making it possible to evaluate the elastic force of the molecular chains of a polymer material. [Brief description of the drawings]

[0008] [Figure 1] FIG. 1 is a perspective view of a computer for executing a simulation method for a polymer material. [Diagram 2] This is the structural formula of butadiene rubber. [Diagram 3] 1 is a flowchart showing an example of a processing procedure of a simulation method for a polymer material. [Figure 4] FIG. 13 is a diagram showing a cell in which a molecular chain model is placed. [Diagram 5] FIG. 2 is a diagram showing a molecular chain model. [Figure 6] 13 is a flowchart illustrating an example of a processing procedure of a model definition step. [Figure 7] 13 is a flowchart showing an example of a processing procedure of a distribution acquisition step. [Figure 8] 13 is a graph showing a third relationship, which is the relationship between the end-to-end distance of a molecular chain model and the bias potential of the molecular chain model. [Figure 9] 13 is a graph showing a fourth relationship, which is the relationship between the end-to-end distance of a molecular chain model and the frequency with a bias potential. [Figure 10] 13 is a graph showing a fifth relationship, which is the relationship between the total length of a molecular chain model and the frequency of the total length. [Figure 11] 1 is a graph showing free energy changes in a molecular chain model. [Figure 12] 1 is a graph of a first relationship showing the relationship between the end-to-end distance of a molecular chain model and the elastic force of the molecular chain model. [Figure 13] 11 is a graph showing a second relationship between the strain and the elastic force of the molecular chain model. [Figure 14] 13 is a graph showing a second relationship showing the relationship between the strain and the elastic force of the first to seventh molecular chain models. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0009] Hereinafter, an embodiment of the present invention will be described with reference to the drawings. It should be understood that the drawings include exaggerated expressions and expressions different from the dimensional ratio of the actual structure in order to help understand the contents of the invention. Furthermore, the same or common elements are given the same reference numerals throughout the embodiments, and duplicated explanations are omitted. Furthermore, the specific configurations shown in the embodiments and drawings are for understanding the contents of the present invention, and the present invention is not limited to the specific configurations shown in the drawings.

[0010] In the simulation method for a polymer material of this embodiment (hereinafter sometimes referred to as the "simulation method"), the elastic force of the molecular chains of the polymer material is evaluated. In the simulation method of this embodiment, a computer is used.

[0011] [computer] 1 is a perspective view of a computer 1 for executing a simulation method for a polymeric 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 stores software and the like for executing the simulation method of this embodiment in advance.

[0012] [Polymer materials] The polymer material contains molecular chains (polymers) and may further contain fillers, coupling agents, cross-linking agents, and the like in addition to the molecular chains.

[0013] [Molecular chain] The molecular chain is not particularly limited as long as it constitutes a polymeric material. The molecular chain of the present embodiment is exemplified by a case where it is butadiene rubber, but is not particularly limited. The molecular chain may be, for example, styrene butadiene rubber, or may be a molecular chain that does not exist at present. Furthermore, the polymeric material may contain two or more types of molecular chains.

[0014] Figure 2 shows the structural formula of butadiene rubber. The molecular chain 2 that constitutes butadiene rubber is a methylene group (-CH 2 -) and methine group (-CH-), 2 -CH=CH-CH 2 ]-} are linked by the degree of polymerization. Also, the ends of polymer materials have methylene groups (-CH 2 ) to a methyl group (-CH 3 ) are concatenated.

[0015] [Simulation method for polymeric materials (first embodiment)] Next, the simulation method of this embodiment will be described. Fig. 3 is a flowchart showing an example of a processing procedure of the simulation method of a polymer material. Fig. 4 is a diagram showing a cell 5 in which a molecular chain model 4 is arranged. Fig. 5 is a diagram showing a molecular chain model 4. In Fig. 4, two molecular chain models 4 are representatively shown.

[0016] [Enter molecular chain model] In the simulation method of this embodiment, first, a molecular chain model 4 (shown in FIGS. 4 and 5) that models a molecular chain of a polymer material is input to a computer 1 (shown in FIG. 1) (step S1). In this embodiment, a butadiene rubber model 4A that models butadiene rubber shown in FIG. 2 is input as the molecular chain model 4, but is not particularly limited. For example, a styrene butadiene rubber model (not shown) that models styrene butadiene rubber may be input, or a molecular chain model (not shown) that models a molecular chain that does not actually exist at present may be input.

[0017] In step S1 of this embodiment, the molecular chain model 4 is modeled based on either the all-atom model or the United Atom Model. Such all-atom model and United Atom Model can analyze chemical reactions in more detail than, for example, a coarse-grained molecular model (not shown) in which a group of atoms is replaced with a single bead.

[0018] In step S1 of this embodiment, a molecular chain model 4 is modeled based on an all-atom model. As shown in Fig. 5, the molecular chain model 4 of this embodiment includes a plurality of particle models 6 and bond models 7 that connect the particle models 6, 6 together.

[0019] The particle model 6 is treated as a mass point of the equation of motion in the molecular dynamics calculation described later. That is, parameters such as mass, diameter, charge, and initial coordinates are defined in the particle model 6. The particle model 6 of this embodiment includes a carbon particle model 6c that models a carbon atom, and a hydrogen particle model 6h that models a hydrogen atom.

[0020] The bond model 7 is for binding the particle models 6, 6. The bond model 7 of this embodiment includes a main chain 7a and a side chain 7b. The main chain 7a includes, for example, a single bond and a double bond.

[0021] Between the particle models 6, 6 adjacent to each other via the bond model 7, a potential (not shown) that generates an interaction (including repulsive and attractive forces) is defined. These potentials include, for example, a bond potential, a bond angle potential, and a bond dihedral angle potential. These potentials can be appropriately defined based on the description in, for example, a patent document (JP Patent Publication No. 2018-032077). This defines a molecular chain model 4 (in this example, a butadiene rubber model 4A).

[0022] The molecular chain model 4 includes a plurality of monomer models 8 modeled on the monomer 3 shown in FIG. 2. The number of monomer models 8 is set appropriately depending on, for example, the type of molecular chain to be analyzed. If the number of monomer models 8 is larger than necessary, the number of particle models 6 to be calculated increases, and the calculation time may increase. On the other hand, if the number of monomer models 8 is smaller than necessary, a molecular chain model 4 having a chain length smaller than the actual molecular chain 2 (shown in FIG. 2) is defined, and the calculation accuracy may not be sufficiently improved. From this viewpoint, the number of monomer models 8 is preferably set to 3 to 10 (4 in this example).

[0023] In this embodiment, the monomer models 8 constituting the molecular chain model 4 include a first monomer model 8A, a second monomer model 8B, a third monomer model 8C, and a fourth monomer model 8D. The first monomer model 8A and the fourth monomer model 8D are arranged on the terminal side of the molecular chain model 4. The molecular chain model 4 is stored in a computer 1 (shown in FIG. 1).

[0024] In step S1 of this embodiment, the molecular chain model 4 is modeled based on the all-atom model, but the molecular chain model may be modeled based on the united atom model. In such a united atom model, the carbon particle model 6c and the hydrogen particle model 6h of the all-atom model shown in FIG. 5 can be integrated and treated as one particle model (not shown). Therefore, in the united atom model, the particle models 6 to be calculated can be reduced compared to the all-atom model, and the calculation time can be shortened.

[0025] [Define polymer material model (model definition process)] Next, in the simulation method of this embodiment, a polymer material model 10 (shown in FIG. 4) that models a polymer material is defined (model definition step S2). FIG. 6 is a flowchart showing an example of the processing procedure of the model definition step S2.

[0026] Fill Cells In the model definition step S2 of this embodiment, first, as shown in Fig. 4, a cell 5, which is a virtual space corresponding to a part of a polymeric material, is input to a computer 1 (shown in Fig. 1) (step S21). The cell 5 of this embodiment has at least a pair of mutually facing faces 11, 11 (in this embodiment, three pairs of mutually facing faces 11, 11). The cell 5 of this embodiment is defined as a rectangular parallelepiped or a cube (in this embodiment, a cube).

[0027] A periodic boundary condition is defined on each face 11 of the cell 5. This makes it possible to perform calculations in a molecular dynamics calculation described below such that, for example, a part of the molecular chain model 4 that has gone out from face 11a on one side enters from face 11b on the other side. The size of the cell 5 can be set appropriately depending on, for example, the total number of molecular chain models 4 to be placed inside the cell 5. The cell 5 is stored in the computer 1 (shown in FIG. 1).

[0028] [Place molecular chain model] Next, in the model definition step S2 of this embodiment, molecular chain models 4 are placed inside the cells 5 (step S22). In this embodiment, the molecular chain models 4 are randomly placed inside the cells 5 so that the total number falls within the range described above (in this example, 2 to 500). The placement of the molecular chain models 4 may be performed by the computer 1 (shown in FIG. 1) or by an operator. The cells 5 in which the molecular chain models 4 are placed are stored in the computer 1 (shown in FIG. 1).

[0029] [Define potential] Next, in the model definition step S2 of this embodiment, a potential P1 is defined between adjacent particle models 6, 6 without a bond model 7 in between (step S23). In this embodiment, between adjacent molecular chain models 4, 4, a potential P1 is defined between adjacent particle models 6, 6 without a bond model 7 in between.

[0030] A known LJ potential is adopted for the potential P1 in this embodiment. In such a potential P1, an attractive force and a repulsive force can be defined between adjacent particle models 6, 6 without the bond model 7. The LJ potential can be appropriately defined based on the description of, for example, a patent document (JP 2020-086773 A). The potential P1 is stored in a computer 1 (shown in FIG. 1).

[0031] [Calculate structural relaxation] Next, in the model definition step S2 of this embodiment, the initial arrangement of the molecular chain model 4 is relaxed (step S24). In step S24 of this embodiment, a molecular dynamics calculation is performed by the computer 1 (shown in FIG. 1) on the molecular chain model 4 arranged in the cell 5 shown in FIG.

[0032] In the molecular dynamics calculation, for example, Newton's equation of motion is applied to the cell 5 for a given time, assuming that the molecular chain model 4 follows classical mechanics. Then, the movement of the particle model 6 at each time is tracked for each unit time. Such calculation of structural relaxation can be processed using, for example, COGNAC included in the Soft Material Integrated Simulator (J-OCTA) manufactured by JSOL Corporation.

[0033] In the molecular dynamics calculation of this embodiment, the pressure (e.g., 1 atm) and temperature (e.g., 290 K to 305 K) are kept constant (NPT constant) in cell 5. This allows the initial configuration of molecular chain model 4 to be accurately relaxed in step S24 by approximating the molecular motion of an actual polymer material.

[0034] In step S24 of this embodiment, it is desirable to perform molecular dynamics calculations at each unit time until it is deemed that the artificial initial arrangement of the molecular chain model 4 has been eliminated. Whether or not the artificial initial arrangement has been eliminated can be determined in the same manner as in the past. By relaxing the initial arrangement in this way, the molecular chain model 4 in an equilibrium state can be calculated, and a polymer material model 10 that models the polymer material can be defined. The polymer material model 10 is stored in a computer 1 (shown in FIG. 1).

[0035] [Obtain the distribution of end-to-end distances of molecular chain models (distribution acquisition process)] Next, in the simulation method of this embodiment, the computer 1 (shown in FIG. 1) extends the molecular chain model 4 shown in FIGS. 4 and 5, and acquires the distribution of the end-to-end distances R of the molecular chain model 4 during extension (distribution acquisition step S3).

[0036] In this embodiment, the distribution of end-to-end distances R is obtained for one selected molecular-chain model 4 from among the multiple molecular-chain models 4 arranged in cell 5 shown in Fig. 4, but is not limited to this embodiment. For example, the distribution of end-to-end distances R may be obtained for each of the multiple molecular-chain models 4.

[0037] In the distribution acquisition step S3, the state in which the molecular-chain model 4 is stretched can be calculated as appropriate. In this embodiment, a bias potential (constraint potential) is set for the molecular-chain model 4 in an equilibrium state, and the state in which the molecular-chain model 4 is stretched is calculated.

[0038] [Define multiple windows] 7 is a flowchart showing an example of a processing procedure of the distribution acquisition step S3. In the distribution acquisition step S3 of this embodiment, first, a plurality of windows based on the umbrella sampling method are defined (step S31).

[0039] In the umbrella sampling method, multiple windows are defined along a specific reaction coordinate, and then sampling is performed in each window using molecular dynamics calculations with a bias potential applied. Based on the probability distribution of the reaction coordinate in each window obtained by sampling, the free energy change is calculated using the weighted histogram analysis method (WHAM). A window is a system (a unit of simulation) in which a specific reaction coordinate range is divided into smaller reaction coordinate ranges. In these multiple windows, the extension of the molecular chain model 4 is calculated based on these small reaction coordinates, and the free energy change of the molecular chain model 4 is calculated.

[0040] The reaction coordinate can be set appropriately. The reaction coordinate in this embodiment is specified as the end-to-end distance R (shown in FIG. 5) of the molecular chain model 4 during elongation. The end-to-end distance R is specified as the magnitude of a vector (end-to-end vector not shown) connecting both ends of the molecular chain model 4, as in the conventional case. The end-to-end distance R can quantitatively specify the state in which the molecular chain model 4 is elongating (reacting).

[0041] Fig. 8 is a graph showing a third relationship T3 which is the relationship between the end-to-end distance R of the molecular chain model 4 and the bias potential V of the molecular chain model 4. In Fig. 8, the free energy change C1 (shown in Fig. 11) of the molecular chain model 4 is indicated by a dashed line.

[0042] 8, in step S31 of this embodiment, for the end-to-end distance R of the molecular chain model 4, a range between a predetermined first end-to-end distance R1 and a second end-to-end distance R2 that is greater than the first end-to-end distance R1 is divided into predetermined small intervals (ranges) D. In this way, a plurality of windows 15 are defined.

[0043] The first end-to-end distance R1 and the second end-to-end distance R2 can be set appropriately. The first end-to-end distance R1 is set, for example, to 0.8 to 1.1 times (0.9 times in this example) the end-to-end distance R of the molecular chain model 4 before elongation (i.e., equilibrium state). The second end-to-end distance R2 is set, for example, to the end-to-end distance of the molecular chain model 4 after elongation from the equilibrium state. This second end-to-end distance R2 is set, for example, according to the type of molecular chain to be evaluated, and is set, for example, to 1.2 to 2.0 times (1.3 times in this example) the end-to-end distance R of the molecular chain model 4 before elongation (equilibrium state).

[0044] The number of divisions into the multiple windows 15 is set appropriately. If the number of divisions is small, the number of samples of the end-to-end distance will be small, which may make it difficult to accurately obtain the free energy change C1. On the other hand, if the number of divisions is large, the number of free energy samples will increase, which may increase the calculation time more than necessary. From this perspective, the number of divisions can be set to 5 to 50.

[0045] The interval D for dividing into the multiple windows 15 can be set based on the first end-to-end distance R1 and the second end-to-end distance R2, and the above-mentioned division number.

[0046] In step S31 of this embodiment, a range from the first end-to-end distance R1 to the second end-to-end distance R2 is divided by an interval D to define a plurality of windows 15 having different ranges of reaction coordinates (end-to-end distance R). The plurality of windows 15 are stored in the computer 1 (shown in FIG. 1).

[0047] [Select one window] Next, in the distribution acquisition step S3 of this embodiment, one window 15 is selected from the multiple windows 15 (step S32). In step S32 of this embodiment, one window 15 for which the distribution of the end-to-end distance R has not been calculated is selected from the multiple windows 15. This window 15 may be selected randomly or based on a predetermined criterion (e.g., ascending order of the magnitude of the reaction coordinate, etc.). The selected window 15 is stored in the computer 1 (shown in FIG. 1).

[0048] [Calculate distribution of end-to-end distances for selected windows] Next, in the distribution acquisition step S3 of this embodiment, the distribution of the end-to-end distance R of the molecular-chain model 4 is calculated in the selected window 15 (step S33). In step S33 of this embodiment, the molecular-chain model 4 shown in FIG. 5 is extended in the selected window 15, and the distribution of the end-to-end distance R of the molecular-chain model 4 during extension is acquired in time series.

[0049] In step S33 of this embodiment, first, a bias potential (constraint potential) is set for the molecular chain model 4 so that the equilibrium molecular chain model 4 (shown in FIG. 5) is extended within the range of the reaction coordinate (end-to-end distance R) of the selected window 15 (shown in FIG. 8). Then, based on the set bias potential, a molecular dynamics calculation is performed to extend the molecular chain model 4.

[0050] In step S33 of this embodiment, it is preferable to extend the molecular chain model 4 based on molecular dynamics calculations based on the density functional tight-binding method. The density functional tight-binding method is a semi-empirical method based on the density functional method. This molecular dynamics calculation based on the density functional tight-binding method makes it possible to analyze chemical reactions based on quantum mechanics. In this embodiment, software capable of molecular dynamics calculations based on the density functional tight-binding method is used. The software is, for example, "BIOVIA Materials Studio DFTB+" manufactured by Dassault Systèmes, "DCDFTBMD" manufactured by CrossAbility Co., Ltd., and the open source program "DFTB+". At least one of such software is stored in a computer 1 (shown in FIG. 1).

[0051] In step S33, a solvent model (not shown) that models the solvent may be placed around the molecular chain model 4 to calculate the extension of the molecular chain model 4. This makes it possible to virtually calculate the extension and elastic force of the molecular chain in the solvent.

[0052] In step S33 of this embodiment, in order to sample many states in which molecular-chain model 4 is extended to a predetermined target range, a bias potential is applied to molecular-chain model 4 based on a third relationship T3, which is the relationship between the end-to-end distance (reaction coordinate) R of molecular-chain model 4 and the bias potential (constraint potential) V acting on molecular-chain model 4, as shown in FIG. 8 .

[0053] 8 shows the third relationship T3 in each of the multiple windows 15. In step S33, a bias potential based on the third relationship T3 is applied to one window 15 selected in step S32 among these windows 15. The third relationship T3 is stored in the computer 1 (shown in FIG. 1).

[0054] Furthermore, in step S33 of this embodiment, a fourth relationship T4 is obtained, which is the relationship between the end-to-end distance R of the molecular chain model 4 and the frequency (trajectory) F1 of the end-to-end distance obtained by molecular dynamics calculation with a bias potential acting on the molecular chain model 4 at the end-to-end distance R. Fig. 9 is a graph showing the fourth relationship T4, which is the relationship between the end-to-end distance R of the molecular chain model 4 and the frequency F1 with the bias potential.

[0055] 9 shows the fourth relationship T4 obtained for each of the multiple windows 15. In step S33, the fourth relationship T4 is obtained for one of these windows 15 that was selected in step S32.

[0056] The fourth relationship T4 is preferably obtained so as to overlap (overflow into) the fourth relationship T4 of the adjacent window 15 on the reaction coordinate (end-to-end distance R). This allows the free energies (described below) calculated at the ends of the reaction coordinates of each window 15 in the adjacent windows 15, 15 to be smoothly combined. The fourth relationship T4 is stored in the computer 1 (shown in FIG. 1).

[0057] In step S33 of this embodiment, the third relationship T3 in the selected window 15 shown in FIG. 8 is stored in the computer 1 (shown in FIG. 1) as a distribution of end-to-end distances R in the selected window 15.

[0058] [Calculating the strain of molecular chain models] In the distribution acquisition step S3 of this embodiment, the strain ε of the molecular-chain model 4 is calculated in the selected window 15 (step S34). The strain is calculated appropriately. In this embodiment, the strain ε of the molecular-chain model 4 is calculated based on the following formula (1). ε = (LL 0 ) / L 0 …(1) Where: ε: Distortion of the molecular chain model L 0 : Total length of molecular chain model before elongation (chain length) L: Total length of the molecular chain model after elongation (chain length)

[0059] In the above formula (1), the total length of the molecular chain model 4 before elongation (hereinafter, sometimes referred to as the “total length before elongation”) L 0 is the total length of the molecular chain model 4 in the equilibrium state. The total length L of the molecular chain model 4 after elongation (hereinafter sometimes referred to as the “total length after elongation”) is the total length of the molecular chain model 4 elongated within the range of the reaction coordinate (end-to-end distance R) of the selected window 15.

[0060] LL in the above formula (1) 0 indicates the amount of elongation of the molecular chain model 4. This amount of elongation LL 0 is divided by the total length L before elongation to calculate the strain ε of the molecular chain model 4.

[0061] Total length before extension L 0 The total length L after elongation can be calculated appropriately based on the structure of the molecular chain model 4 shown in FIG. 5. In this embodiment, the total length L before elongation is calculated as the sum of the lengths of the monomer models 8. 0 and the total length L after elongation are calculated.

[0062] The lengths of the monomer models 8 include a first length L1 of the first monomer model 8A, a second length L2 of the second monomer model 8B, a third length L3 of the third monomer model 8C, and a fourth length L4 of the fourth monomer model 8D. These first length L1 to fourth length L4 can be appropriately specified.

[0063] The first length L1 is specified, for example, by the straight-line distance between the carbon particle model 6c on one end side of the first monomer model 8A and the first midpoint 16a. The first midpoint 16a is specified as the midpoint between the carbon particle model 6c on the other end side of the first monomer model 8A and the carbon particle model 6c on one end side of the second monomer model 8B. The position of the carbon particle model 6c is specified as the central coordinates of the carbon particle model 6c.

[0064] The second length L2 is determined, for example, by the straight-line distance between the first midpoint 16a and the second midpoint 16b. The second midpoint 16b is determined as the midpoint between the carbon particle model 6c on the other end side of the second monomer model 8B and the carbon particle model 6c on one end side of the third monomer model 8C.

[0065] The third length L3 is determined, for example, by the straight-line distance between the second midpoint 16b and the third midpoint 16c. The third midpoint 16c is determined as the midpoint between the carbon particle model 6c on the other end side of the third monomer model 8C and the carbon particle model 6c on one end side of the fourth monomer model 8D.

[0066] The fourth length L4 is determined by the straight-line distance between the third midpoint 16c and the carbon particle model 6c on the other end side of the fourth monomer model 8D.

[0067] Fig. 10 is a graph showing a fifth relationship T5, which is the relationship between the total length L of the molecular chain model 4 and the frequency F2 of the total length L. Fig. 10 shows the fifth relationship T5 acquired in each of the multiple windows 15. In step S34, the fifth relationship T5 is acquired in one of these windows 15 selected in step S32. The fifth relationship T5 is stored in the computer 1 (shown in Fig. 1).

[0068] As shown in Fig. 10, the total length L of the molecular-chain model 4 after elongation changes depending on the magnitude of the bias potential V (shown in Fig. 8). If such a total length L after elongation is used, it is difficult to uniquely identify the strain ε (not shown) of the molecular-chain model 4 in the window 15 selected in step S32. For this reason, it is preferable that the total length L after elongation be uniquely identified in the selected window 15.

[0069] In step S34 of the present embodiment, the total length L after elongation is weighted and averaged by the frequency F2 in the selected window 15, so that the total length L after elongation can be uniquely identified. Then, the uniquely identified total length L after elongation and the total length L before elongation are 0and are substituted into the above formula (1), thereby specifying the strain ε of the molecular chain model 4 in the selected window 15. The specified strain ε is stored in the computer 1 (shown in FIG. 1).

[0070] [Determine if all windows are selected] Next, in the distribution acquisition step S3 of this embodiment, it is determined whether or not all of the windows 15 have been selected (step S35). If it is determined that all of the windows 15 have been selected ("Yes" in step S35), the series of processes in the distribution acquisition step S3 is terminated. On the other hand, if it is determined that all of the windows 15 have not been selected ("No" in step S35), steps S32 to S35 are performed again. Note that in step S32, which is performed again, one unselected window 15 (for which the distribution of the end-to-end distance R has not been calculated) is selected from the multiple windows 15.

[0071] Thus, in the distribution acquisition step S3 of this embodiment, a molecular dynamics calculation with a bias potential applied is performed in all of the multiple windows 15, and the distribution of the end-to-end distance R (the third relationship T3 shown in FIG. 8) can be calculated. As described above, the multiple windows 15 are obtained by dividing the range from the first end-to-end distance R1 before elongation to the second end-to-end distance R2 after elongation at intervals D along the end-to-end distance R, which is the reaction coordinate (i.e., the reaction time series). The distribution of the end-to-end distance R of the molecular chain model 4 elongated from the first end-to-end distance R1 to the second end-to-end distance R2 can be acquired in time series.

[0072] Furthermore, in the distribution acquisition step S3 of this embodiment, in addition to the distribution of the end-to-end distance R (the third relationship T3 shown in FIG. 8), a fourth relationship T4 (the relationship between the end-to-end distance R of the molecular-chain model 4 and the frequency F1 of the bias potential) can be acquired in each of the multiple windows 15 as shown in FIG. 9. Furthermore, in the distribution acquisition step S3 of this embodiment, the strain ε (not shown) of the molecular-chain model 4 is uniquely identified in all of the multiple windows 15 shown in FIG. 10 by the step S34 of calculating the strain ε of the molecular-chain model 4.

[0073] [Get free energy change] Next, in the simulation method of this embodiment, the computer 1 (shown in FIG. 1) acquires the free energy change of the molecular chain model 4 based on the distribution of the end-to-end distance R (step S4).

[0074] The distribution of the end-to-end distance R is sampled in time series based on the bias potential, as shown in Fig. 8. In this embodiment, the bias potential V (shown in Fig. 8) is weighted by the frequency F1 (shown in Fig. 9) of the bias potential in association with the end-to-end distance R of the molecular-chain model 4. As a result, in step S4 of this embodiment, the free energy change with respect to the end-to-end distance R of the molecular-chain model 4 is calculated.

[0075] The free energy change indicates the relationship between the end-to-end distance R and the free energy of the elongating molecular chain model 4. In this embodiment, the range of the end-to-end distance for which the free energy change is obtained is set from the first end-to-end distance R1 to the second end-to-end distance R2 shown in FIG.

[0076] The free energy change is appropriately acquired based on the distribution of the end-to-end distance R acquired in time series (acquired in all of the multiple windows 15). In this embodiment, based on the weighted histogram analysis method (WHAM) of the umbrella sampling method, the fourth relationship T4 (i.e., the relationship between the end-to-end distance R of the molecular chain model 4 and the frequency F1 of the bias potential) shown in FIG. 9 regarding the end-to-end distance R is considered for all of the windows 15. This allows the free energy change with respect to the end-to-end distance R to be calculated. The free energy change in this embodiment is acquired by molecular dynamics calculation based on the density functional tight binding method, and therefore, unlike the case where, for example, a quantum chemical calculation that performs energy optimization is used, the effect of temperature can be considered.

[0077] Fig. 11 is a graph showing the free energy change C1 of the molecular chain model 4. In Fig. 11, the free energy E of the molecular chain model 4 increases as the end-to-end distance R of the molecular chain model 4 increases. The free energy change C1 is stored in computer 1 (shown in Fig. 1).

[0078] [Determine the first relationship that shows the relationship between the end-to-end distance of the molecular chain model and the elasticity] Next, in the simulation method of this embodiment, computer 1 (shown in FIG. 1) determines a first relationship T1 that indicates the relationship between end-to-end distance R of molecular chain model 4 and elastic force F of molecular chain model 4 (step S5). In this step S5, the free energy change of molecular chain model 4 shown in FIG. 11 is differentiated with respect to end-to-end distance R to determine first relationship T1.

[0079] 11 is differentiated with respect to the end-to-end distance R, the free energy gradient of the molecular chain model 4 with respect to the end-to-end distance R is obtained. This free energy gradient indicates the magnitude of the force (the force pulling the molecular chain) acting in the direction in which the end-to-end distance R shrinks in the molecular chain model 4. Therefore, the free energy gradient with respect to the end-to-end distance R can be specified as a first relationship T1 that indicates the relationship between the end-to-end distance R of the molecular chain model 4 and the elastic force F of the molecular chain model 4.

[0080] Fig. 12 is a graph of a first relationship T1 showing the relationship between the end-to-end distance of a molecular chain model 4 and the elastic force (free energy gradient) of the molecular chain model 4. In Fig. 12, the elastic force F of the molecular chain model 4 increases as the end-to-end distance R of the molecular chain model 4 increases. In this embodiment, the range of the end-to-end distance R for which the first relationship T1 is obtained is set from the first end-to-end distance R1 to the second end-to-end distance R2 shown in Fig. 8. The first relationship T1 is stored in computer 1 (shown in Fig. 1).

[0081] As described above, the free energy in this embodiment is obtained by molecular dynamics calculation based on the density functional tight binding method, and therefore, unlike quantum chemical calculations that perform energy optimization, the effect of temperature can be taken into consideration. By differentiating the free energy change C1 (shown in FIG. 11) that shows the relationship between such free energy E and the end-to-end distance R, it is possible to obtain the elastic force F at a finite temperature.

[0082] [Derive the second relationship that shows the relationship between the strain and elasticity of the molecular chain model] Next, in the simulation method of this embodiment, computer 1 (shown in FIG. 1) obtains a second relationship T2 that indicates the relationship between the strain ε of molecular-chain model 4 and the elastic force F of molecular-chain model 4 (step S6). In step S6 of this embodiment, the end-to-end distance R of first relationship T1 shown in FIG. 11 is converted into the strain ε of molecular-chain model 4 to obtain the second relationship T2.

[0083] As described above, the strain ε of the molecular-chain model 4 is uniquely specified for each of the multiple windows 15 shown in Fig. 10 in step S34 shown in Fig. 7. On the other hand, since the end-to-end distance R of the molecular-chain model 4 changes depending on the magnitude of the bias potential V (shown in Fig. 8), the end-to-end distance R is not uniquely specified for each of the multiple windows 15 shown in Fig. 12. For this reason, in order to convert the end-to-end distance R into the strain ε, it is preferable that the end-to-end distance R be uniquely specified for each of the multiple windows 15.

[0084] In step S6 of this embodiment, first, the end-to-end distance R is uniquely identified in each of the multiple windows 15. In this embodiment, based on the relationship between the end-to-end distance R of the molecular chain model 4 shown in FIG. 9 and the frequency F1 with bias potential, the end-to-end distance R is weighted-averaged by the frequency F1 in each of the multiple windows 15. This allows the end-to-end distance R to be uniquely identified in each of the multiple windows 15. Then, the end-to-end distance R of the identified multiple windows 15 is identified in the end-to-end distance R of the first relationship T1 shown in FIG. 12. This allows the end-to-end distance R to be uniquely identified in each of the multiple windows 15 of the first relationship T1.

[0085] Next, in step S6 of the present embodiment, the end-to-end distance R of the first relationship T1 is converted into the strain ε specified in step S34 in each of the multiple windows 15. This allows the second relationship T2 indicating the relationship between the strain of the molecular-chain model 4 and the elastic force of the molecular-chain model 4 to be determined.

[0086] Fig. 13 is a graph showing a second relationship T2 indicating the relationship between the strain ε and the elastic force F of the molecular-chain model 4. In Fig. 13, the larger the strain ε of the molecular-chain model 4, the larger the elastic force F of the molecular-chain model 4. The second relationship T2 is stored in computer 1 (shown in Fig. 1).

[0087] The strain ε of the second relationship T2 is, as shown in the above formula (1), the extension amount (LL 0 ) to the total length L of the molecular chain model 4 before elongation (a dimensionless number). Unlike the end-to-end distance R, such strain ε is not affected by the type or chain length of the molecular chain 2 (molecular chain model 4). Therefore, unlike the first relationship T1 (shown in FIG. 12) in which the elastic force F relative to the end-to-end distance R is specified, the second relationship T2 in which the elastic force F relative to the strain ε is specified does not require fitting between the end-to-end distance R and the experimental value of the end-to-end distance. Therefore, the simulation method of this embodiment makes it possible to uniformly evaluate (compare) the elastic forces F of various molecular chains of polymer materials.

[0088] [Evaluate elasticity] Next, in the simulation method of this embodiment, the elastic force F (shown in FIG. 13) of the molecular chain model 4 shown in FIG. 5 is evaluated (step S7). The evaluation may be performed by the computer 1 (shown in FIG. 1) or by an operator.

[0089] In step S7 of the present embodiment, it is determined whether or not the elastic force F of the molecular-chain model 4 satisfies a predetermined criterion. The criterion can be set appropriately depending on the performance required for the polymer material or a product using the polymer material (e.g., wear resistance of a tire, etc.).

[0090] If it is determined that the elastic force F of the molecular chain model 4 satisfies the criteria ("Yes" in step S7), the polymer material is determined to have the desired performance. In this case, the polymer material is manufactured based on the structure of the molecular chain 2 (shown in FIG. 1) (step S8).

[0091] On the other hand, if it is determined that the elastic force F of the molecular chain model 4 does not satisfy the criteria ("No" in step S7), the structure of the molecular chain 2 (shown in FIG. 2) is changed (step S9), and steps S1 to S7 are carried out again. As a result, in the simulation method of this embodiment, a polymer material having desired performance can be efficiently manufactured without actually producing a prototype of the polymer material.

[0092] 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 carried out in various forms. EXAMPLES

[0093] The elastic forces of the first molecular chain model to the seventh molecular chain model were evaluated (Example) based on the processing procedure shown in Fig. 3. The first molecular chain model to the seventh molecular chain model are modeled versions of the first molecular chain to the seventh molecular chain, which are different from each other in at least one of the type and chain length.

[0094] In the present embodiment, the end-to-end distance and free energy were obtained in time series for each of the first to seventh molecular chain models based on the processing procedure shown in Fig. 7. Next, in the present embodiment, the free energy change showing the relationship between the end-to-end distance and free energy of the molecular chain models was obtained based on the end-to-end distance and free energy for each of the first to seventh molecular chain models.

[0095] Next, in the examples, the free energy change was differentiated with respect to the end-to-end distance for each of the first to seventh molecular chain models to obtain a first relationship showing the relationship between the end-to-end distance of the molecular chain models and the elastic force. Then, in the examples, the end-to-end distance in the first relationship was converted into the strain of the molecular chain models for each of the first to seventh molecular chain models to obtain a second relationship showing the relationship between the strain and the elastic force. Fig. 14 is a graph showing the second relationship showing the relationship between the strain and the elastic force for each of the first to seventh molecular chain models.

[0096] In addition, for comparison with the examples, only the first relationship showing the relationship between the end-to-end distance of the molecular chain model and the elastic force was obtained for the first molecular chain model to the seventh molecular chain model (Comparative Example). The common specifications are as follows. Number of window divisions: 20 First end-to-end distance R1: 0.9 times the end-to-end distance in equilibrium Second end-to-end distance R2: 1.3 times the end-to-end distance in equilibrium Hamiltonian: GFN1-xTB Time step: 0.5fs

[0097] As a result of the test, in the comparative example, the end-to-end distance R varied due to the influence of the type and chain length of the first to seventh molecular chains (first to seventh molecular chain models). Therefore, in the comparative example, even if the elastic forces F were obtained for the first to seventh molecular chain models with respect to the varied end-to-end distances R, the elastic forces F could not be evaluated uniformly. Therefore, in the comparative example, in order to evaluate the elastic forces of the first to seventh molecular chain models, fitting of the end-to-end distance R with the experimental values ​​of the end-to-end distances was required.

[0098] On the other hand, in the examples, the end-to-end distance R in the first relationship was converted to strain ε for the first to seventh molecular chain models, and a second relationship showing the relationship between strain ε and elastic force was obtained as shown in FIG. 14. These strains ε are dimensionless numbers that are not affected by the types and chain lengths of the first to seventh molecular chains (first to seventh molecular chain models). By obtaining such elastic force F with respect to strain ε, in the examples, the elastic forces of the first to seventh molecular chain models could be uniformly evaluated without the need for fitting to the experimental values ​​of the end-to-end distance.

[0099] [Note] The present invention includes the following aspects.

[0100] [Invention 1] A method for simulating a polymeric material, comprising the steps of: inputting a molecular chain model of the molecular chain of the polymer material into a computer; The computer, Stretching the molecular chain model to obtain a distribution of end-to-end distances of the molecular chain model during stretching; A step of acquiring a free energy change indicating a relationship between the end-to-end distance and free energy based on the distribution of the end-to-end distances; A step of differentiating the free energy change with respect to the end-to-end distance to obtain a first relationship that indicates a relationship between the end-to-end distance and the elastic force of the molecular chain model; and determining a second relationship that indicates a relationship between the strain and the elastic force by converting the end-to-end distance of the first relationship into a strain of the molecular chain model. Simulation methods for polymeric materials. [Invention 2] The method for simulating a polymeric material according to the first aspect of the present invention, wherein the step of acquiring the free energy change includes a step of calculating the free energy change based on an umbrella sampling method. [Invention 3] The step of acquiring the distribution of the end-to-end distance includes a step of defining a plurality of windows based on the umbrella sampling method, the windows being divided at predetermined intervals between a first end-to-end distance and a second end-to-end distance greater than the first end-to-end distance of the molecular chain model; A method for simulating a polymer material according to the present invention 2, comprising a step of calculating the end-to-end distance in each of the plurality of windows. [Invention 4] The method for simulating a polymeric material according to the third aspect of the present invention, wherein the step of acquiring the free energy change includes a step of calculating the free energy change based on a weighted histogram analysis method of the umbrella sampling method. [Invention 5] obtaining the distribution of end-to-end distances includes calculating the distortion in each of the plurality of windows; A simulation method for a polymer material described in invention 3 or 4, wherein the step of determining the second relationship includes a step of determining the second relationship by converting the end-to-end distance of the first relationship into the strain in each of the multiple windows. [Invention 6] A method for simulating a polymer material according to any one of claims 1 to 5, wherein the step of acquiring the distribution of the end-to-end distances includes a step of extending the molecular chain model based on a molecular dynamics calculation based on a density functional tight binding method. [Invention 7] 7. The method for simulating a polymer material according to any one of claims 1 to 6, wherein the step of inputting the molecular chain model comprises modeling the molecular chain model based on either an all-atom model or a united-atom model. [Explanation of symbols]

[0101] S3: A process of acquiring the end-to-end distance and free energy of the molecular chain model in time series. S4: A step of acquiring free energy change S5: Process to find the first relationship S6: Process to find the second relationship

Claims

1. A method for simulating a polymeric material, comprising the steps of: inputting a molecular chain model of the molecular chain of the polymer material into a computer; The computer, Stretching the molecular chain model to obtain a distribution of end-to-end distances of the molecular chain model during stretching; A step of acquiring a free energy change indicating a relationship between the end-to-end distance and free energy based on the distribution of the end-to-end distances; A step of differentiating the free energy change with respect to the end-to-end distance to obtain a first relationship that indicates a relationship between the end-to-end distance and the elastic force of the molecular chain model; and determining a second relationship indicating a relationship between the strain and the elastic force by converting the end-to-end distance of the first relationship into a strain of the molecular chain model. Simulation methods for polymeric materials.

2. The method for simulating a polymer material according to claim 1 , wherein the step of acquiring the free energy change includes the step of calculating the free energy change based on an umbrella sampling method.

3. The step of acquiring the distribution of the end-to-end distance includes a step of defining a plurality of windows based on the umbrella sampling method, the windows being divided at predetermined intervals between a predetermined first end-to-end distance of the molecular chain model and a second end-to-end distance that is greater than the first end-to-end distance; The method for simulating a polymer material according to claim 2 , further comprising the step of: calculating the end-to-end distance in each of the plurality of windows.

4. The method for simulating a polymer material according to claim 3 , wherein the step of acquiring the free energy change includes the step of calculating the free energy change based on a weighted histogram analysis method of the umbrella sampling method.

5. obtaining the distribution of end-to-end distances includes calculating the distortion in each of the plurality of windows; 4. The method for simulating a polymer material according to claim 3, wherein the step of determining the second relationship includes a step of determining the second relationship by converting the end-to-end distance of the first relationship into the strain in each of the plurality of windows.

6. The method for simulating a polymer material according to claim 1 , wherein the step of acquiring the distribution of end-to-end distances includes a step of extending the molecular chain model based on molecular dynamics calculations based on a density functional tight-binding method.

7. 7. The method for simulating a polymer material according to claim 1, wherein the step of inputting the molecular chain model comprises modeling the molecular chain model based on either an all-atom model or a united-atom model.

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