Polymer material simulation method

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 polymer material. [Background technology]

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

[0003] [Patent Document 1] Japanese Patent Publication No. 2023-062527 Summary of the Invention [Problem to be solved by the invention]

[0004] Generally, there are many different types and lengths of molecular chains in polymeric materials, and a method for evaluating the elasticity of these molecular chains has been sought.

[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 polymeric materials. [Means for solving the problem]

[0006] The present invention is a simulation method for a polymeric material, which includes a step of inputting a molecular chain model that models the molecular chain of the polymeric material into a computer, and the computer executes the steps of: extending the molecular chain model and acquiring a distribution of end-to-end distances of the molecular chain model during extension; acquiring a free energy change that indicates the 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 determine a first relationship that indicates the relationship between the end-to-end distance and the elastic force of the molecular chain model; and converting the end-to-end distance in the first relationship into a strain of the molecular chain model to determine a second relationship that indicates the relationship between the strain and the elastic force. [Effects 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 the polymer material. [Brief explanation of the drawings]

[0008] [Figure 1] FIG. 1 is a perspective view of a computer for executing a simulation method for polymeric materials. [Figure 2] This is the structural formula of butadiene rubber. [Figure 3] 1 is a flowchart showing an example of a processing procedure of a simulation method for a polymer material. [Figure 4] FIG. 10 is a diagram showing a cell in which a molecular chain model is placed. [Figure 5] FIG. 1 is a diagram showing a molecular chain model. [Figure 6] 10 is a flowchart illustrating an example of a processing procedure of a model definition step. [Figure 7] 10 is a flowchart illustrating an example of a processing procedure of a distribution acquisition step. [Figure 8] 10 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] 10 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] 10 is a graph showing the 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 of 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] 10 is a graph showing a second relationship between strain and elastic force of a molecular chain model. [Figure 14] 10 is a graph showing a second relationship showing the relationship between strain and elastic force for the first to seventh molecular chain models. 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] In the polymer material simulation method of this embodiment (hereinafter sometimes referred to as the "simulation method"), the elastic force of the molecular chains of the polymer material is evaluated. The simulation method of this embodiment uses a computer.

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

[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 polymer material. The molecular chain of this 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 currently exist. Furthermore, the polymer 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 makes up butadiene rubber is made up of monomers 3 {-[CH2-CH=CH-CH2]-}, which consist of a methylene group (-CH2-) and a methine group (-CH-), linked together at a certain degree of polymerization. In addition, a methyl group (-CH3) is linked to the end of the polymer material instead of the methylene group (-CH2).

[0015] [Simulation method for polymer materials (first embodiment)] Next, the simulation method of this embodiment will be described. Fig. 3 is a flowchart showing an example of the processing procedure of the simulation method for 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 shown as representatives.

[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 the molecular chain of a polymer material is input into 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 to this. 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 currently exist may be input.

[0017] In step S1 of this embodiment, the molecular chain model 4 is modeled based on either an all-atom model or a United Atom Model. Such an all-atom model and a 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 in the equation of motion in the molecular dynamics calculation described below. That is, parameters such as mass, diameter, charge, and initial coordinates are defined for 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 used to constrain 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] Potentials (not shown) that cause interactions (including repulsive and attractive forces) are defined between adjacent particle models 6, 6 via a bond model 7. These potentials include, for example, a bond potential, a bond angle potential, and a bond dihedral angle potential. These potentials can be defined appropriately based on the description in, for example, a patent document (JP 2018-032077 A). This defines a molecular chain model 4 (in this example, a butadiene rubber model 4A).

[0022] The molecular chain model 4 is configured to include a plurality of monomer models 8 modeled after 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, which may increase the calculation time. On the other hand, if the number of monomer models 8 is smaller than necessary, a molecular chain model 4 with a chain length shorter than the actual molecular chain 2 (shown in FIG. 2) is defined, which may prevent sufficient improvement in calculation accuracy. From this perspective, 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 also 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 a single particle model (not shown). Therefore, in the united atom model, the number of particle models 6 to be calculated can be reduced compared to the all-atom model, thereby enabling a reduction in calculation time.

[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 the 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] Cell Input 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 polymer material, is input to the computer 1 (shown in Fig. 1) (step S21). The cell 5 of this embodiment has at least one 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 the molecular dynamics calculations described below, such that, for example, a part of the molecular chain model 4 that has left one face 11a enters from the other face 11b. 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 used as the potential P1 in this embodiment. With such a potential P1, attractive and repulsive forces can be defined between adjacent particle models 6, 6 without the intervention of a bond model 7. The LJ potential can be appropriately defined based on the description in, 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 molecular dynamics calculations, for example, Newton's equations of motion are applied to the cell 5 for a predetermined time, assuming that the molecular chain model 4 follows classical mechanics. The movement of the particle model 6 at each time is then tracked for each unit time. Such structural relaxation calculations 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 (for example, 1 atm) and temperature (for example, 290 K to 305 K) are kept constant (constant NPT) in cell 5. As a result, in step S24, the initial configuration of the molecular-chain model 4 can be relaxed with high accuracy 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 for each unit time until it is determined that the artificial initial configuration of the molecular chain model 4 has been eliminated. Whether or not the artificial initial configuration has been eliminated can be determined in the same manner as in the past. By relaxing the initial configuration in this way, an equilibrium state molecular chain model 4 is 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 acquired for one selected molecular-chain model 4 from among the plurality of molecular-chain models 4 arranged in the cell 5 shown in Fig. 4, but is not limited to this. For example, the distribution of end-to-end distances R may be acquired for each of the plurality of molecular-chain models 4.

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

[0038] Define multiple windows 7 is a flowchart showing an example of the 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). Furthermore, 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, thereby calculating the free energy change of the molecular chain model 4.

[0040] The reaction coordinate can be set as appropriate. In this embodiment, the reaction coordinate 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 connecting both ends of the molecular chain model 4 (end-to-end vector, not shown), 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] 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, the 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. This defines a plurality of windows 15.

[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 extension (i.e., in the 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 extension from the equilibrium state. This second end-to-end distance R2 is set appropriately, for example, depending on 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 extension (in the 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 end-to-end distance samples 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 result in an unnecessary increase in calculation time. From this perspective, the number of divisions can be set to 5 to 50.

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

[0046] In step S31 of this embodiment, the range from the first end-to-end distance R1 to the second end-to-end distance R2 is divided by the 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 plurality of 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 plurality of windows 15. This window 15 may be selected randomly or may be selected based on a predetermined criterion (e.g., ascending order of the magnitude of the reaction coordinate). The selected window 15 is stored in the computer 1 (shown in FIG. 1).

[0048] [Calculate the distribution of edge 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 molecular chain model 4 (shown in FIG. 5) in an equilibrium state 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 density functional tight-binding methods. The density functional tight-binding method is a semi-empirical technique based on density functional methods. This density functional tight-binding molecular dynamics calculation enables analysis of chemical reactions based on quantum mechanics. In this embodiment, software capable of performing molecular dynamics calculations based on density functional tight-binding methods is used. Examples of software include "BIOVIA Materials Studio DFTB+" from Dassault Systèmes, "DCDFTBMD" from CrossAbility, Inc., and the open-source program "DFTB+." At least one of these software programs 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 allows the extension and elastic force of the molecular chain in the solvent to be calculated in a pseudo manner.

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

[0053] 8 shows the third relationship T3 for each of the multiple windows 15. In step S33, a bias potential based on the third relationship T3 is applied to one of the windows 15 selected in step S32. 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 acquired, 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 (extend into) the fourth relationship T4 of an 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 to be smoothly combined. The fourth relationship T4 is stored in 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 the distribution of end-to-end distances R in the selected window 15.

[0058] [Calculating strain in 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=(L-L0) / L0…(1) where: ε: Distortion of the molecular chain model L0: Total length of the 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 L0 of the molecular chain model 4 before elongation (hereinafter sometimes referred to as the "total length before elongation") is the total length of the molecular chain model 4 in an 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] In the above formula (1), L-L0 represents the extension amount of the molecular chain model 4. The strain ε of the molecular chain model 4 is calculated by dividing this extension amount L-L0 by the total length L before extension.

[0061] The total length L0 before elongation and 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 L0 before elongation and the total length L after elongation are each calculated as the total length of the monomer models 8.

[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 specified as appropriate.

[0063] The first length L1 is specified, for example, by the linear 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 linear distance between the first midpoint 16a and the second midpoint 16b, which 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 linear distance between the second midpoint 16b and the third midpoint 16c, which 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 linear 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 the 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 obtained for each of the multiple windows 15. In step S34, the fifth relationship T5 is obtained for 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 this embodiment, the total length L after elongation can be uniquely identified in the selected window 15 by taking a weighted average with frequency F2 of the total length L after elongation. Then, the uniquely identified total length L after elongation and the total length L before elongation are substituted into the above formula (1), thereby identifying the strain ε of the molecular-chain model 4 in the selected window 15. The identified 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 windows 15 have been selected (step S35). If it is determined that all 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 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 end-to-end distance R has not been calculated) is selected from the multiple windows 15.

[0071] As described above, 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 into 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 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 change in free energy 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 end-to-end distances 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 obtained based on the distribution of the end-to-end distance R obtained in time series (obtained in all of the multiple windows 15). In this embodiment, based on the weighted histogram analysis method of umbrella sampling (WHAM), 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 taken into consideration for all of the windows 15. This makes it possible to calculate the free energy change with respect to the end-to-end distance R. The free energy change in this embodiment is obtained by molecular dynamics calculation based on the density functional tight-binding method, and therefore, unlike when, for example, quantum chemical calculations that perform energy optimization are used, the influence of temperature can be taken into consideration.

[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] [Derive the first relationship that shows the relationship between the distance between the ends of the molecular chain model and the elastic force] 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 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] When the free energy change of the molecular chain model 4 shown in Figure 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 can be found. 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] 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 to 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 calculations based on the density functional tight-binding method, and therefore, unlike quantum chemical calculations that perform energy optimization, the influence of temperature can be taken into account. By differentiating the free energy change C1 (shown in FIG. 11) that shows the relationship between the 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 elastic force of the molecular chain model] Next, in the simulation method of this embodiment, the computer 1 (shown in FIG. 1) determines a second relationship T2 that indicates the relationship between the strain ε of the molecular-chain model 4 and the elastic force F of the molecular-chain model 4 (step S6). In step S6 of this embodiment, the second relationship T2 is obtained by converting the end-to-end distance R of the first relationship T1 shown in FIG. 11 into the strain ε of the molecular-chain model 4.

[0083] As described above, the strain ε of the molecular-chain model 4 is uniquely identified for each of the multiple windows 15 shown in Fig. 10 in step S34 shown in Fig. 7. On the other hand, 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), and therefore the end-to-end distance R is not uniquely identified 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 identified 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 and 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 by 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 this embodiment, the end-to-end distance R of the first relationship T1 is converted into the strain ε identified in step S34 in each of the multiple windows 15. This determines the second relationship T2 that indicates the relationship between the strain of the molecular-chain model 4 and the elastic force of the molecular-chain model 4.

[0086] Fig. 13 is a graph showing the second relationship T2, which indicates 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] As shown in the above formula (1), the strain ε in the second relationship T2 is expressed as the ratio (dimensionless number) of the extension amount (L-L0) of the molecular chain model 4 to the total length L of the molecular chain model 4 before extension. Unlike the end-to-end distance R, this 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) which specifies the elastic force F relative to the end-to-end distance R, the second relationship T2, which specifies the elastic force F relative to the end-to-end distance R, does not require fitting the end-to-end distance R to 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 force F of various molecular chains of polymeric materials.

[0088] [Evaluate elastic force] 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 standard. The standard can be set appropriately depending on the performance required of the polymer material or a product using the polymer material (for example, the wear resistance of a tire).

[0090] If it is determined that the elastic force F of the molecular chain model 4 satisfies the criteria ("Yes" in step S7), it is determined that the polymer material has the desired performance. In this case, a 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 performed again. As a result, the simulation method of this embodiment can efficiently produce a polymer material having desired performance 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 implemented in various ways. [Example]

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

[0094] In the present example, 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 example, 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 elastic force. Then, in the examples, the end-to-end distance in the first relationship was converted into strain of the molecular chain model for each of the first to seventh molecular chain models to obtain a second relationship showing the relationship between strain and elastic force. Figure 14 is a graph showing the second relationship showing the relationship between strain and elastic force for each of the first to seventh molecular chain models.

[0096] 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 to seventh molecular chain models (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 molecular chain model to seventh molecular chain model). For this reason, in the comparative example, even if the elastic force F was determined for the first to seventh molecular chain models with respect to the varied end-to-end distance R, it was not possible to uniformly evaluate these elastic forces F. Therefore, in the comparative example, to evaluate the elastic forces of the first to seventh molecular chain models, it was necessary to fit the end-to-end distance R to the experimental values ​​of the end-to-end distance.

[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 Figure 14. These strains ε are dimensionless numbers that are not affected by the type or chain length of the first to seventh molecular chains (first to seventh molecular chain models). By obtaining such elastic force F relative 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 experimental values ​​of end-to-end distances.

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

[0100] [Invention 1] A method for simulating a polymeric material, comprising: a step of inputting a molecular chain model obtained by modeling the molecular chain of the polymer material into a computer, The computer a step of extending the molecular chain model and acquiring a distribution of end-to-end distances of the molecular chain model during extension; A step of obtaining 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 calculating a second relationship that indicates 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. Simulation methods for polymeric materials. [Invention 2] The method for simulating a polymeric material according to aspect 1, 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 distances 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; A 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. [Invention 4] The method for simulating a polymer material according to aspect 3, 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 according to the present 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 plurality of windows. [Invention 6] 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 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. [Invention 7] 7. The method for simulating a polymer material according to any one of aspects 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: Obtaining free energy change S5: Process to find the first relation S6: Process for finding the second relationship

Claims

1. A method for simulating a polymeric material, comprising: a step of inputting a molecular chain model obtained by modeling the molecular chain of the polymer material into a computer, The computer a step of extending the molecular chain model and acquiring a distribution of end-to-end distances of the molecular chain model during extension; A step of obtaining 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 calculating a second relationship that indicates 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. Simulation methods for polymeric materials.

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

3. The step of acquiring the distribution of end-to-end distances 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 a 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. 2. 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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