Method, system, and program for generating crosslinked polymer models

Abstract

To provide a generation method, a system and a program for a crosslinked polymer model that has a heterogenous crosslinked structure.SOLUTION: A method includes: setting a crosslinking probability to some crosslinkable particles among multiple coarse-grained particles constituting a polymer model that are contiguous in a linear or branched form; generating one or more crosslinking agent particle masses in which the multiple crosslinking agent particles are coagulated; arranging the multiple polymer models and the one or more crosslinking agent particle masses in a virtual space, executing molecular dynamics calculation while the crosslinking agent particle masses are kept coagulated, and executing an equilibration process; decoagulating each of the crosslinking agent particles constituting the crosslinking agent particle masses after the equilibration process; executing molecular dynamics calculation while each of the crosslinking agent particles are decoagulated, and executing a crosslinking reaction process for binding the crosslinkable particles and the crosslinking agent particles at the crosslinking probability that is set to the crosslinkable particles, when the crosslinkable particles of the polymer models approach within a prescribed distance to the crosslinking agent particles.SELECTED DRAWING: Figure 5

Description

Technical Field

[0001] The present disclosure relates to a method, system, and program for generating a crosslinked polymer model.

Background Art

[0002] Various methods for generating models used in molecular simulations using molecular dynamics calculations have been proposed. As a method for generating a crosslinked polymer model in which the crosslinked structure in which a polymer and a crosslinking agent are bonded is non-uniform, for example, in Patent Document 1, a virtual space is divided into a plurality of cells, different crosslinking densities are set for each cell, and the number of crosslinkable particles set for the polymer arranged in the cell is set to a number corresponding to the crosslinking density corresponding to the cell.

[0003] Also, as another method, Patent Document 2 discloses setting a plurality of crosslinkable particles in the molecular chain of the polymer constituting the polymer model, and setting priority crosslinkable particles having a high crosslinking probability among a part of the plurality of crosslinkable particles.

Prior Art Documents

Patent Documents

[0004]

Patent Document 1

Patent Document 2

Summary of the Invention

Problems to be Solved by the Invention

[0005] However, experiments have shown that immediately after mixing a crosslinking agent such as sulfur with a polymer such as rubber, aggregated portions of the crosslinking agent remain, suggesting that the crosslinking agent disperses and crosslinks with the polymer during vulcanization. Therefore, the conventional method described above may not be a suitable model generation method for elucidating the phenomena of polymer models, and it is thought that there may be other approaches to generating heterogeneous crosslinked polymer models.

[0006] This disclosure provides a method, system, and program for generating crosslinked polymer models having heterogeneous crosslinking structures. [Means for solving the problem]

[0007] The method for generating a crosslinked polymer model according to the present disclosure is a method executed by one or more processors, and includes: setting a crosslinking probability for some of the crosslinkable particles among a plurality of coarse-grained particles arranged in a linear or branched manner that constitute a polymer model; generating a mass of crosslinking agent particles by aggregating a plurality of crosslinking agent particles; arranging the plurality of polymer models and one or more of the crosslinking agent particle masses in a virtual space, performing a molecular dynamics calculation while maintaining the aggregation of the crosslinking agent particle masses to perform an equilibration process; releasing the aggregation of each crosslinking agent particle constituting the crosslinking agent particle mass after the equilibration process; performing a molecular dynamics calculation while releasing the aggregation of each of the crosslinking agent particles, and performing a crosslinking reaction process to bond the crosslinkable particles and the crosslinking agent particles at the crosslinking probability set for the crosslinkable particles when the crosslinkable particles of the polymer model approach within a predetermined distance of the crosslinking agent particles. [Brief explanation of the drawing]

[0008] [Figure 1] A block diagram showing the system of this embodiment. [Figure 2] A flowchart illustrating the processes that the system will execute. [Figure 3] A schematic diagram showing the crosslinked polymer model produced in this embodiment. [Figure 4]These are explanatory diagrams regarding the types of monomer coarse-grained particles set for polymer model particles; (a) is an explanatory diagram for the first polymer model, and (b) is an explanatory diagram for the second polymer model. [Figure 5] (a) A diagram showing an example in which crosslinking agent particle aggregates are placed in a virtual space, and (b) A diagram showing an example in which crosslinking agent particles are placed one by one in a virtual space. [Figure 6] This figure shows the breakdown of coarse-grained particles in the polymer models that constitute the cross-linked polymer models (Embodiments 1-4, Comparative Examples 1-4). [Figure 7] A diagram showing the stress-strain curves for Example 1 and Comparative Example 1, and the placement of the crosslinking agent particles 4 in the models for Example 1 and Comparative Example 1. [Figure 8] A diagram showing the stress-strain curves for Example 2 and Comparative Example 2, and the arrangement of the crosslinking agent particles 4 in the models for Example 2 and Comparative Example 2. [Figure 9] A diagram showing the stress-strain curves for Example 3 and Comparative Example 3, and the placement of the crosslinking agent particles 4 in the models for Example 3 and Comparative Example 3. [Figure 10] A diagram showing the stress-strain curves for Example 4 and Comparative Example 4, and the arrangement of the crosslinking agent particles 4 in the models for Example 4 and Comparative Example 4. [Figure 11] An explanatory diagram regarding the radial distribution function. [Figure 12] Graph showing the values ​​of the radial distribution function g(r) for crosslinking agent particles 4 in Examples 1-2 and Comparative Examples 1-2. [Figure 13] Graphs showing the radial distribution function g(r) for crosslinking agent particles 4 in Examples 3-4 and Comparative Examples 3-4. [Figure 14] A graph showing the degree of crosslinking non-uniformity in Examples 1-4. [Figure 15] This diagram shows an example of arranging five aggregates of crosslinking agent particles, each formed by aggregating crosslinking agent particles, in a virtual space. [Modes for carrying out the invention]

[0009] Hereinafter, one embodiment of this disclosure will be described with reference to the drawings.

[0010] [Model Generation System] The system 1 (device) of this embodiment generates a crosslinked polymer model. In this embodiment, as an example of the crosslinked polymer model, examples of generating two types of models, SBR (styrene-butadiene rubber) and hydrogenated SBR, are given, but the crosslinked polymer models that can be generated are not limited to these two types. SBR is a copolymer of butadiene and styrene, and the butadiene part crosslinks with a crosslinking agent such as sulfur. When SBR is hydrogenated, hydrogen is added to the double bond of the butadiene part, resulting in an ethylene structure. In this embodiment, for molecular simulation including molecular dynamics calculation, the molecular simulation software "LAMMPS" is used.

[0011] As shown in FIG. 1, the system 1 includes a setting unit 10, a mass generation unit 11, a model placement unit 12, an equilibration processing execution unit 13, an aggregation release unit 14, and a crosslinking reaction processing unit 15. The system 1 may also include a radial distribution function calculation unit 16, a determination unit 17, and a non-uniformity calculation unit 18. Each of these units 10 to 18 is realized by the cooperation of software and hardware when the processor 1a executes the processing routine shown in FIG. 2 stored in advance in a computer equipped with a processor 1a, a memory 1b, various interfaces, etc. In this embodiment, the processor 1a in one device realizes each unit, but it is not limited to this. For example, it may be configured to be distributed using a network and a plurality of processors execute the processing of each unit. That is, one or more processors execute the processing. The memory 1b stores data related to the polymer model 3 and the crosslinking agent particles 4 for generating the crosslinked polymer model (the bonding structure of the particles, the number of particles, the bonding potential or non-bonding potential set for the particles), data related to the generated polymer model, and analysis conditions for executing molecular dynamics calculation (constant pressure-temperature conditions, constant volume-temperature conditions, periodic boundary conditions set in the virtual space, etc.). In this embodiment, in order to generate the first polymer model 21 corresponding to the SBR model and the second polymer model 22 corresponding to the hydrogenated SBR model, these data are stored in the memory 1b.

[0012] As schematically shown in FIG. 3, the generated crosslinked polymer model 2 has a plurality of polymer models 3 and crosslinking agent particles 4. The polymer model 3 has a plurality of coarsened particles 30 connected in a linear or branched manner. The plurality of coarsened particles 30 include crosslinkable particles set to be capable of binding to the crosslinking agent particles 4 and non-crosslinkable particles set to be incapable of binding to the crosslinking agent particles 4. The crosslinking agent particles 4 bind (crosslinking reaction) to the crosslinkable particles among the plurality of coarsened particles constituting the polymer model 3.

[0013] FIG. 4 is an explanatory diagram regarding the types of monomer coarsened particles set for the particles 30 of the polymer model 3. FIG. 4(a) is an explanatory diagram regarding the first polymer model 21, and FIG. 4(b) is an explanatory diagram regarding the second polymer model 22. In FIGS. 4(a) and 4(b), only the molecular chain of one polymer model 3 is schematically illustrated. As shown in FIG. 4(a), the coarsened particles 30 of the polymer model 3 in the first polymer model 21 corresponding to the SBR model represent any of a plurality of types of monomer coarsened particles including the first type monomer coarsened particles B and the second type monomer coarsened particles S. In the figure, various potentials (interactions) for representing the characteristics of the monomers are set for each of the coarsened particles 30 constituting the chain of the polymer model 3. Thereby, since a monomer originally composed of a large number of particles is represented by one coarsened particle 30, the number of particles handled by the computer is reduced and the calculation cost is reduced. The first type monomer coarsened particles B represent butadiene, and the second type monomer coarsened particles S represent styrene. As shown in FIG. 4(b), the coarsened particles 30 of the polymer model 3 in the second polymer model 22 corresponding to the hydrogenated SBR model represent any of a plurality of types of monomer coarsened particles including the first type monomer coarsened particles B, the second type monomer coarsened particles S, and the third type monomer coarsened particles E. In both the first polymer model 21 and the second polymer model 22, the first monomer coarse-grained particle B represents butadiene and can therefore be configured to bind to the crosslinking agent particle 4. Not all of the first monomer coarse-grained particle B needs to be configured to bind to the crosslinking agent particle 4; some of the first monomer coarse-grained particle B may be configured to bind to the crosslinking agent particle 4, while the rest may be configured not to bind to the crosslinking agent particle 4. The second monomer coarse-grained particle S represents styrene and is therefore configured not to bind to the crosslinking agent particle 4. The third monomer coarse-grained particle E represents ethylene and is therefore configured not to bind to the crosslinking agent particle 4.

[0014] Multiple coarse-grained particles 30 constituting polymer model 3 are connected by binding potentials (also called binding interactions) in a linear or branched manner, and a non-binding potential (also called non-binding interaction) is set for each particle 30. A non-binding potential is set for the crosslinking agent particle 4. The binding potential is the potential that acts between the bonding partner, and a FENE+LJ potential can be set. Specifically, the binding potential (U) of the Kremer-Grest model shown in equation (2) bond ) can be set. The unbonded potential is the potential that acts between particles, and the WCA potential (LJ potential with only repulsion) can be set. Specifically, the unbonded potential (U) of the Kremer-Grest model shown in equation (1) non-bond ) can be set. The binding potential to be set for the coarse-grained particles 30 of polymer model 3 is the angular potential (U) shown in equation (3) according to the type of monomer set (butadiene, styrene, ethylene). angle The following parameters are added: ) This reproduces the properties according to the type of monomer set. The potentials set for polymer model 3 and crosslinking agent particle 4 are standardized to K=30.0, R0=1.5, ε=1.0, and σ=1.0 in the following equations (1) and (2). Angular parameter (K) of coarse-grained particle B of type 1 monomer angle ) is 3.1, and the angular parameter (K) of the coarse-grained particles S of the second monomer. angle) is 6.3, and the angle parameter (K) of the third monomer coarse-grained particle E angle The value was set to 4.0. r represents the interparticle distance, and θ represents the angle between bonded particles. Note that the parameters are just examples and are not limited to these.

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[0015] The setting unit 10 shown in Figure 1 sets the above potential for each of the coarse-grained particles 30 and crosslinking agent particles 4 that constitute the polymer model 3. Furthermore, as shown in Figures 4(a) and 4(b), the setting unit 10 sets 23.5% of the particles 30 in the polymer model 3 to styrene-containing type 2 monomer coarse-grained particles S. In the case of the SBR model (first polymer model 21), 76.5% of the particles 30 in the polymer model 3 are set to butadiene-containing type 1 monomer coarse-grained particles B. In the case of the hydrogenated SBR model (second polymer model 22), type 1 monomer coarse-grained particles B and type 3 monomer coarse-grained particles E are assigned to a number of particles 30 corresponding to the hydrogenation rate for 76.5% of the total coarse-grained particles in the polymer model 3. The setting unit 10 sets the angular potential according to the type of monomer coarse-grained particle. The specific number of particles is shown below with Figure 6. Furthermore, the setting unit 10 sets a crosslinking probability for some of the crosslinkable particles among the multiple coarse-grained particles 30 that constitute the polymer model 3. In this embodiment, the same crosslinking probability (20%) is set for all of the first monomer coarse-grained particles B that are set to be crosslinkable. In LAMMPS, the bond potential can be set using the `bond_style fene` command, the non-bond potential can be set using the `pair_style lj / cut` command, and the angular potential within the bond potential can be set using the `angle_style cosine` command.

[0016] The aggregate generation unit 11 shown in Figure 1 generates a crosslinking agent particle aggregate 40 by aggregating multiple crosslinking agent particles 4. In this embodiment, 64 crosslinking agent particles 4 are aggregated into a cube shape with 4 sides (4 3 =64). By setting the above-mentioned bonding potential between each of the 64 crosslinking agent particles 4 and adjacent crosslinking agent particles 4, a bonded state (aggregated state) is created.

[0017] The model placement section 12 shown in Figure 1 places multiple polymer models 3 and one or more crosslinking agent particle clusters 40 in the virtual space Ar1. In this embodiment, as shown in Figure 5(a), first, the polymer models 3 are randomly positioned and placed in the virtual space Ar1, and then three crosslinking agent particle clusters 40 are randomly placed in the virtual space Ar1 so as not to overlap with the polymer models 3. Figure 5(b) shows an example in which crosslinking agent particles 4 are placed one by one in the virtual space Ar1. Note that in Figures 5(a) and 5(b), the crosslinking agent particles 4 are displayed, but the polymer models 3 are not. In LAMMPS, the placement of a model in the virtual space can be achieved using the create_atom command, but coordinate placement is also possible by executing an external program other than LAMMPS.

[0018] The equilibration processing unit 13 shown in Figure 1 performs equilibration by executing molecular dynamics calculations while maintaining the aggregation of the crosslinking agent particle aggregate 40 (with the bonding potential between the crosslinking agent particles 4 set). Specifically, the equilibration processing unit 13 repeatedly performs molecular dynamics calculations under predetermined analysis conditions (such as constant pressure and temperature or constant volume and temperature) until an equilibrium state is reached. For example, molecular dynamics calculations can be performed for a predetermined number of steps under constant pressure and temperature conditions, and then molecular dynamics calculations can be performed for a predetermined number of steps under constant volume and temperature conditions. An equilibrium state is, for example, a state in which the fluctuation of the potential energy of the entire model over a certain period of time falls within a certain range. In this embodiment, as an example, the temperature and pressure are set to 1.0 [LJ units (Leonard Jones units)], but this is just an example and can be set to various values. In LAMMPS, specifying constant pressure and temperature can be done with the `fix npt` command, and specifying constant volume and temperature can be done with the `fix nvt` command.

[0019] The deaggregation unit 14 shown in Figure 1 deaggregates each crosslinking agent particle 4 that constitutes the crosslinking agent particle mass 40 in the crosslinked polymer model 2, which has reached an equilibrium state after the parallelization treatment. Specifically, it deaggregates the bond potential set between adjacent crosslinking agent particles 4 that are set for each crosslinking agent particle 4 constituting the crosslinking agent particle mass 40. As a result, when molecular dynamics calculations are performed, the repulsive force of the non-bonding potential set between the aggregated crosslinking agent particles 4 causes the crosslinking agent particles 4 to repel each other, and the dispersion of the crosslinking agent particles 4 begins.

[0020] The crosslinking reaction processing unit 15 shown in Figure 1 performs molecular dynamics calculations with each crosslinking agent particle 4 de-aggregated and then performs the crosslinking reaction process. The crosslinking reaction process is a process in which molecular dynamics calculations are performed to bind the crosslinkable particles (coarse-grained particles B of type 1 monomer) of the polymer model 3 to the crosslinking agent particles 4 at a set crosslinking probability when the crosslinkable particles (coarse-grained particles B of type 1 monomer) of the polymer model 3 approach within a predetermined distance from the crosslinking agent particles 4. In this embodiment, every 10 calculation steps of the molecular dynamics calculation, all combinations of crosslinkable particles (coarse-grained particles B of type 1 monomer) and crosslinking agent particles 4 within a predetermined distance (in this embodiment, within a distance of 1.0) are evaluated, and the crosslinkable particles and crosslinking agent particles 4 of the evaluation are bound at a set crosslinking probability. The crosslinking reaction process is repeated until all crosslinking agent particles 4 are bound. Once all crosslinking agent particles 4 are bound, the crosslinked polymer model 2 is completed. If necessary, equilibration treatment may be performed on cross-linked polymer model 2. In LAMMPS, the process of bonding the crosslinkable particles and the crosslinking agent particles 4 during the crosslinking reaction can be achieved using the fix bond / create command.

[0021] [Method for generating cross-linked polymer models] The method for generating a crosslinked polymer model performed by System 1 described above will be explained using Figure 2. Here, an example of generating an SBR model (first polymer model 21) as the polymer model will be given. Memory 1b is pre-set and stored with the specified pressure and temperature used in molecular dynamics calculations. Memory 1b is also pre-set and stored with data related to the polymer model 3 that constitutes the SBR model (number of chain length particles, number of polymers, number of styrene and butadiene particles to be set) and data related to the crosslinking agent particles 4. First, in step ST1, the setting unit 10 performs various settings. Specifically, the setting unit 10 sets the crosslinking probability for some of the crosslinkable particles among the multiple coarse-grained particles 30 that make up the polymer model 3, which are arranged in a linear or branched manner. The setting unit 10 also sets the potential for the coarse-grained particles 30 and crosslinking agent particles 4 that make up the polymer model. In the next step, ST2, the aggregate generation unit 11 generates a crosslinking agent particle aggregate 40 by aggregating multiple crosslinking agent particles 4. In the next step, ST3, the model placement unit 12 places multiple polymer models 3 and one or more crosslinking agent particle clusters 40 into the virtual space Ar1. In the next step, ST4, the equilibration processing unit 13 performs molecular dynamics calculations while maintaining the aggregation of the crosslinking agent particle mass 40 to perform the equilibration process. In the next step, ST5, the de-aggregation unit 14 de-aggregates each crosslinking agent particle that constitutes the crosslinking agent particle mass after the equilibration treatment. In the next step, ST6, the crosslinking reaction processing unit 15 performs molecular dynamics calculations with the aggregation of each crosslinking agent particle released, and when the crosslinkable particles of polymer model 3 approach the crosslinking agent particles 4 within a predetermined distance, it performs a crosslinking reaction process to bond the crosslinkable particles and the crosslinking agent particles 4 with the crosslinking probability set for the crosslinkable particles.

[0022] <Examples 1-4 and Comparative Examples 1-4> Four Examples 1-4 produced by the method described herein and four Comparative Examples 1-4 produced by other methods are described below. There are two patterns depending on whether it is an SBR model or a hydrogenated SBR model, two patterns depending on the number of crosslinkable particles, and two patterns depending on whether the method results in a uniform crosslinking structure (Figure 5(b)) or the method of this disclosure that results in a non-uniform crosslinking structure (Figure 5(a)). Eight patterns of examples were generated by combining these. All eight polymer models are produced by placing 200 polymer models 3, each formed from 200 coarse-grained particles 30 (chain length 200), into the virtual space Ar1. In the "low number of crosslinking particles" example, there are 192 crosslinking particles 4, and as shown in Figure 5(a), three crosslinking particle clusters 40 are placed into the virtual space Ar1. In the "high number of crosslinking particles" example, there are 320 crosslinking particles 4, and as shown in Figure 15, five crosslinking particle clusters 40 are placed into the virtual space Ar1. Figure 6 shows the breakdown of coarse-grained particles 30 of polymer model 3, which constitutes cross-linked polymer model 2.

[0023] Example 1 (SBR, low number of crosslinking agent particles, this method for agglomerating crosslinking agent particles 4 shown in Figure 5(a)) This is an SBR model (first polymer model 21), with 192 crosslinking agent particles 4, which were produced using this method of agglomerating the crosslinking agent particles 4. Approximately 13% of the butadiene (first monomer coarse-grained particles B) were designated as crosslinkable particles, and the remaining butadiene was designated as non-crosslinkable particles. As shown in Figure 6, there were 9457 styrene (second monomer coarse-grained particles S), 26553 non-crosslinkable butadiene particles, and 3990 crosslinkable butadiene particles. The time required for the crosslinking reaction was 24 minutes and 53 seconds.

[0024] Comparative Example 1 (SBR, "Low" number of crosslinking agent particles, Method of arranging crosslinking agent particles 4 individually as shown in Figure 5(b)) As shown in Figure 5(b), the method involves placing the crosslinking agent particles 4 one by one in the virtual space Ar1, and the crosslinking agent particles 4 are not aggregated. Otherwise, it is the same as in Example 1. The time required for the crosslinking reaction treatment was 4 minutes and 47 seconds.

[0025] Example 2 (SBR, "high" number of crosslinking agent particles, this method for agglomerating crosslinking agent particles 4 shown in Figure 5(a)) The number of crosslinking agent particles 4 was 320 particles. Otherwise, it was the same as in Example 1. The time required for the crosslinking reaction treatment was 17 minutes and 35 seconds.

[0026] Comparative Example 2 (SBR, "Many" Crosslinking Agent Particles, Method of Arranging Crosslinking Agent Particles 4 as shown in Figure 5(b) one by one) As shown in Figure 5(b), the method involves placing the crosslinking agent particles 4 one by one in the virtual space Ar1, and the crosslinking agent particles 4 are not aggregated. Otherwise, it is the same as in Example 2. The time required for the crosslinking reaction treatment was 7 minutes and 27 seconds.

[0027] Example 3 (Hydrogenated SBR, "Low" number of crosslinking agent particles, This method involves agglomerating the crosslinking agent particles 4 shown in Figure 5(a)) This is a hydrogenated SBR model (second polymer model 22), with 192 crosslinking agent particles 4, which were produced using this method to aggregate the crosslinking agent particles 4. All butadiene (coarse-grained particles B of type 1 monomer) were set as crosslinkable particles. As shown in Figure 6, there were 9457 styrene (coarse-grained particles S of type 2 monomer), 0 non-crosslinkable butadiene, 1583 crosslinkable butadiene, and 28960 ethylene (coarse-grained particles E of type 3 monomer). The hydrogenation rate was approximately 95%. The time required for the crosslinking reaction was 48 minutes and 42 seconds.

[0028] Comparative Example 3 (Hydrogenated SBR, "Low" number of crosslinking agent particles, Method of arranging crosslinking agent particles 4 individually as shown in Figure 5(b)) As shown in Figure 5(b), the method involves placing the crosslinking agent particles 4 one by one in the virtual space Ar1, and the crosslinking agent particles 4 are not aggregated. Otherwise, it is the same as in Example 3. The time required for the crosslinking reaction treatment was 17 minutes and 29 seconds.

[0029] Example 4 (Hydrogenated SBR, "High" number of crosslinking agent particles, this method for agglomerating crosslinking agent particles 4 as shown in Figure 5(a)) The number of crosslinking agent particles 4 was 320 particles. Otherwise, it was the same as in Example 3. The time required for the crosslinking reaction treatment was 1 hour, 27 minutes, and 59 seconds.

[0030] Comparative Example 4 (Hydrogenated SBR, "Many" Crosslinking Agent Particles, Method of Arranging Crosslinking Agent Particles 4 Individually as Shown in Figure 5(b)) As shown in Figure 5(b), the method involves placing the crosslinking agent particles 4 one by one in the virtual space Ar1, and the crosslinking agent particles 4 are not aggregated. Otherwise, it is the same as in Example 4. The time required for the crosslinking reaction treatment was 34 minutes and 13 seconds.

[0031] Regarding the time required for the crosslinking reaction treatment, Examples 1 to 4 took longer than the corresponding Comparative Examples 1 to 4. This is thought to be due to the increased time required to simulate the process in which aggregated portions of the crosslinking agent particles disperse while remaining aggregated. Furthermore, the significantly longer time required for the crosslinking reaction treatment in Example 4 (hydrogenated SBR) compared to the corresponding Example 2 (SBR) is thought to be because the vulcanization rate slows down with hydrogenation.

[0032] <Tensile test simulation and resulting stress-strain curve> To confirm whether the created polymer model exhibits qualitative behavior, a tensile test simulation was performed. Since a known method described in Japanese Patent Publication No. 2017-96871 was used, a detailed explanation is omitted. Specifically, the polymer model is placed in a virtual space, and the virtual space is gradually stretched in the z-axis direction under constant volume and temperature conditions, and the elongation ratio and stress are calculated. Because it is not possible to stretch to the desired elongation ratio in one go, a process of slight stretching and relaxation (equilibrium treatment) is repeatedly performed. For example, the model is stretched to approximately 1.0139 times, then a relaxation treatment is performed, and the process of stretching again to 1.0139 times is repeated. By repeating the process of stretching to approximately 1.0139 times in the z-axis direction 100 times, elongation ratios from 1 to 7 can be obtained, and the stress and strain at each point in time can be obtained. The average value of the obtained stress was acquired for each of the x, y, and z axes. This allows for the acquisition of a stress-strain curve showing the relationship between the elongation ratio and stress. Because the bond potential of the generated polymer model is the FENE+LJ potential, the rubber model does not break even when stretched. To identify the point at which fracture begins, the obtained stress is separated into an energy-elastic component and an entropy-elastic component. The energy-elastic component increases due to the stretching of polymer model 3. The strain at which the energy-elastic component begins to increase is defined as the fracture initiation point. In the stress-strain curves shown in Figures 7-10, the fractured state is indicated by a dotted line.

[0033] Figure 7 shows the stress-strain curves for Example 1 and Comparative Example 1, and the arrangement of crosslinking agent particles 4 in the models for Example 1 and Comparative Example 1. Figure 8 shows the stress-strain curves for Example 2 and Comparative Example 2, and the arrangement of crosslinking agent particles 4 in the models for Example 2 and Comparative Example 2. Comparing Figures 7 and 8, it can be seen that the elongation until fracture of the model with a non-uniform crosslinking structure is lower than the elongation until fracture of the model with a uniform crosslinking structure. Figure 9 shows the stress-strain curves for Example 3 and Comparative Example 3, and the arrangement of crosslinking agent particles 4 in the models for Example 3 and Comparative Example 3. Figure 10 shows the stress-strain curves for Example 4 and Comparative Example 4, and the arrangement of crosslinking agent particles 4 in the models for Example 4 and Comparative Example 4. Comparing Figures 9 and 10, it can be seen that in the hydrogenated SBR model (second polymer model 22), models with a large number of crosslinking particles are less likely to generate heterogeneous models, and as a result, models closer to a homogeneous model are obtained.

[0034] [Method and system for calculating the degree of crosslinking heterogeneity] As shown in Figures 8-10, determining whether a crosslinked structure is uniform or non-uniform tends to be a subjective assessment based on visual inspection of the distribution of crosslinking agent particles 4. Therefore, this paper provides a method for calculating the degree of crosslinking non-uniformity.

[0035] The radial distribution function calculation unit 16 shown in Figure 1 calculates the radial distribution function g(r) for the crosslinking agent particles 4 for the crosslinked polymer model 2. The radial distribution function g(r) is a function that represents the distribution of particles at a distance r from a given particle, as shown in equation (4) and Figure 11. n(r) represents the number of particles in the range from distance r to distance (r+dr) (the range between the outer and inner circles in Figure 11). r represents the inter-particle distance, and ρ represents the density of the model. In Figure 11, particles are shown as solid circles, and circles with radius r and radius (r+dr) centered on a given particle are shown as dashed-dotted circles.

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[0036] Figure 12 is a graph showing the values ​​of the radial distribution function g(r) for crosslinking agent particles 4 in Examples 1-2 and Comparative Examples 1-2, calculated by the radial distribution function calculation unit 16. Figure 13 is a graph showing the radial distribution function g(r) for crosslinking agent particles 4 in Examples 3-4 and Comparative Examples 3-4, calculated by the radial distribution function calculation unit 16. Looking at Figures 12 and 13, it can be seen that in Comparative Examples 1-4, where the crosslinking agent particles 4 are considered to be relatively uniformly dispersed, the radial distribution function g(r) rises from 0 to 1 as the interparticle distance r increases, and then maintains a value of around 1. On the other hand, in Examples 1-4, where the crosslinking agent particles 4 are considered to be relatively non-uniformly dispersed, the value of the radial distribution function g(r) rises from 0 to a peak value Pv greater than 1 as the interparticle distance r increases, and then decreases with a certain slope (approximate) until the value of the radial distribution function g(r) converges to 1. Furthermore, comparing Examples 1 to 4, it can be seen that the greater the non-uniformity, the larger the angle of the decreasing slope (downward to the right). Therefore, we believe that the magnitude of the slope can be used as a value representing the degree of non-uniformity of the bridge cross-linking. As an example, the radial distribution function g(r) of gases, liquids, and solids in which particles are arranged in a grid does not exhibit the characteristic of decreasing with a certain slope from the peak value Pv until the value of the radial distribution function g(r) converges to 1, as seen in the polymer model with the heterogeneous cross-linking structure described above. Therefore, we believe that the above findings are not known.

[0037] The determination unit 17 shown in Figure 1 utilizes the new findings described above and determines whether the value of the radial distribution function g(r) decreases as the interparticle distance r increases, based on the peak value Pv that appears in the value of the radial distribution function g(r). If it is determined that the condition that the value of the radial distribution function g(r) decreases as the interparticle distance r increases is not met based on the peak value Pv, it is determined that the crosslinking structure of the target crosslinked polymer model cannot be said to be heterogeneous.

[0038] The heterogeneity calculation unit 18 shown in Figure 1, when it determines that the value of the radial distribution function g(r) decreases as the interparticle distance r increases from the peak value Pv, calculates a value based on the slope of the decreasing value of the radial distribution function g(r) from the peak value Pv as the heterogeneity of crosslinking in the crosslinked polymer model. In this embodiment, only Examples 1 to 4 satisfy the condition that the value of the radial distribution function g(r) decreases as the interparticle distance r increases from the peak value Pv, while Comparative Examples 1 to 4 do not. For Examples 1 to 4, the linear approximation formula [g(r)=ar+b] was fitted to the values ​​of the radial distribution function g(r) in the range of interparticle distance r=5 to 15, and the slope a and intercept b were calculated. The least squares method was used for fitting. Since the slope a is a negative and small value, it was converted to an absolute value and multiplied by 100, and the result (showing the heterogeneity of crosslinking for Examples 1 to 4) is shown as a graph in Figure 14. Figure 14 shows that the value on the vertical axis represents the degree of heterogeneity. The closer the value on the vertical axis is to 0, the more uniform the crosslinking is, and the larger the value on the vertical axis, the more heterogeneous it is. Examples 1 to 4 are models with more heterogeneous crosslinking structures compared to Comparative Examples 1 to 4. Among them, Example 1 is the most heterogeneous, and the degree of heterogeneity decreases (i.e., becomes more uniform) in the order of Example 3, Example 2, and Example 4, which can be seen numerically, making it possible to quantify the degree of heterogeneity.

[0039] Regarding the range of interparticle distances for calculating the slope, the minimum value is where the value of the radial distribution function g(r) is the peak value Pv. The maximum value may be half the length of one side of the virtual space Ar1 (in this embodiment, periodic boundary conditions are set, and one side of the cube virtual space Ar1 is 40, so half of that is 20), and more preferably, where the value of the radial distribution function g(r) becomes 1 after the peak value. In other words, the range of interparticle distances for calculating the slope is preferably a range where the value of the radial distribution function g(r) is greater than or equal to the peak value Pv and the value of the radial distribution function g(r) is greater than 1. In this embodiment, in Examples 1, 3, and 4, the value of the radial distribution function g(r) is 1 or greater at an interparticle distance r=15, and in Examples 1 to 4, the interparticle distance r=5 is greater than the peak value Pv of the radial distribution function g(r), so the range common to Examples 1 to 4 was set to r=5 to 15. This is just one example; the important thing is that the range over which the slope of the radial distribution function g(r) is evaluated is the same across the multiple models being compared.

[0040] [1] As described above, the method for generating the crosslinked polymer model 2 of this embodiment is a method executed by one or more processors and may include: setting a crosslinking probability for some of the crosslinkable particles among a plurality of linear or branched coarse-grained particles 30 constituting the polymer model 3; generating a crosslinking agent particle mass 40 by aggregating a plurality of crosslinking agent particles 4; arranging the plurality of polymer models 3 and one or more crosslinking agent particle masses 40 in a virtual space Ar1, performing a molecular dynamics calculation while maintaining the aggregation of the crosslinking agent particle mass 40 to perform an equilibration process; releasing the aggregation of each crosslinking agent particle 4 constituting the crosslinking agent particle mass 40 after the equilibration process; performing a molecular dynamics calculation with each crosslinking agent particle 4 released, and performing a crosslinking reaction process in which the crosslinkable particles of the polymer model 3 and the crosslinking agent particles 4 are bonded at the crosslinking probability set for the crosslinkable particles when the crosslinkable particles of the polymer model 3 approach the crosslinking agent particles 4 within a predetermined distance.

[0041] Thus, in the equilibrium state where the polymer model 3 and the crosslinking agent particle aggregate 40 are mixed, the crosslinking agent particles 4 are in an aggregated aggregate state. Subsequently, since the crosslinking reaction treatment is carried out with the aggregated crosslinking agent particles 4 released, it is thought that a heterogeneous crosslinked structure can be obtained that simulates the dispersion during vulcanization while the aggregated parts of the crosslinking agent remain in the experiment.

[0042] [2] In the method for generating the crosslinked polymer model described in [1] above, the same crosslinking probability may be set for all crosslinkable particles. This makes it possible to obtain heterogeneous crosslinked structures without having to set different crosslinking probabilities for each crosslinkable particle. Furthermore, it simplifies the setup process for generating crosslinked polymer models and eliminates arbitrary settings.

[0043] [3] In the method for producing a crosslinked polymer model described in [1] or [2] above, the polymer model 3 may include a first monomer coarse-grained particle B that can bind to the crosslinking agent particle 4, and a second monomer coarse-grained particle S that cannot bind to the crosslinking agent particle 4. Polymer model 3 can be represented using a crosslinkable first-type monomer and a non-crosslinkable second-type monomer.

[0044] [4] In the method for producing a crosslinked polymer model described in [3] above, polymer model 3 may further include coarse-grained particles E of a third monomer that cannot be bound to crosslinking agent particles 4, wherein coarse-grained particles B of a first monomer represent butadiene, coarse-grained particles S of a second monomer represent styrene, and coarse-grained particles E of a third monomer represent ethylene, and polymer model 3 is a model representing hydrogenated SBR.

[0045] [5] The method for calculating the degree of crosslinking heterogeneity in a crosslinked polymer model may include: generating a crosslinked polymer model 2 after all crosslinkable particles have bonded to any of the crosslinking agent particles 4 by performing the crosslinked polymer model generation method described in any of [1] to [4] above; calculating the value of the radial distribution function g(r) for the crosslinking agent particles 4 for the generated crosslinked polymer model 2; determining from the peak value Pv that appears in the value of the radial distribution function g(r) whether the value of the radial distribution function g(r) decreases as the interparticle distance r increases; and, if it is determined from the peak value Pv that the value of the radial distribution function g(r) decreases as the interparticle distance r increases, calculating a value [|slope| × 100] based on the slope of the decreasing value of the radial distribution function g(r) from the peak value Pv as the degree of crosslinking heterogeneity in the crosslinked polymer model 2.

[0046] Conventionally, one might visually determine whether the crosslinking is uniform or non-uniform by visually inspecting the placement of crosslinking agent particles that constitute the model located in the virtual space Ar1. However, this method makes it possible to obtain the degree of non-uniformity of the crosslinking in the crosslinked polymer model as a numerical value, which is useful.

[0047] [6] The system for generating a crosslinked polymer model may include: a setting unit 10 that sets a crosslinking probability for some of the crosslinkable particles among a plurality of linear or branched coarse-grained particles 30 constituting the polymer model 3; a clump generation unit 11 that generates a clump of crosslinking agent particles 40 by aggregating a plurality of crosslinking agent particles 4; an equilibration processing execution unit 13 that places a plurality of polymer models 3 and one or more clumps of crosslinking agent particles 40 in a virtual space Ar1 and performs an equilibration process while maintaining the aggregation of the crosslinking agent particle clumps 40; an aggregation release unit 14 that releases the aggregation of each crosslinking agent particle 4 constituting the crosslinking agent particle clumps 40 after the equilibration process; and a crosslinking reaction processing unit 15 that performs molecular dynamics calculations with each crosslinking agent particle 4 released and performs a crosslinking reaction process to bond the crosslinkable particles and the crosslinking agent particles 4 at the crosslinking probability set for the crosslinkable particles when the crosslinkable particles of the polymer model 3 approach within a predetermined distance of the crosslinking agent particles 4.

[0048] [7] The program may also involve one or more processors performing the following actions: setting a crosslinking probability for some of the crosslinkable particles among a plurality of linearly or branchedly linked coarse-grained particles 30 constituting the polymer model 3; generating a crosslinking agent particle mass 40 by aggregating a plurality of crosslinking agent particles 4; placing a plurality of polymer models 3 and one or more crosslinking agent particle masses 40 in the virtual space Ar1, performing a molecular dynamics calculation while maintaining the aggregation of the crosslinking agent particle mass 40 to perform an equilibration process; releasing the aggregation of each crosslinking agent particle 4 constituting the crosslinking agent particle mass 40 after the equilibration process; and performing a molecular dynamics calculation with each crosslinking agent particle 4 released, and executing a crosslinking reaction process in which the crosslinkable particles of the polymer model 3 and the crosslinking agent particles 4 are bonded together at the crosslinking probability set for the crosslinkable particles when the crosslinkable particles of the polymer model 3 approach the crosslinking agent particles 4 within a predetermined distance.

[0049] Although embodiments of this disclosure have been described above with reference to the drawings, it should be understood that the specific configurations are not limited to these embodiments. The scope of this disclosure is indicated not only by the description of the embodiments above but also by the claims, and further includes all modifications within the meaning and scope equivalent to the claims.

[0050] The structures adopted in each of the above embodiments can be adopted in any other embodiment. The specific configuration of each part is not limited to the embodiments described above, and various modifications are possible without departing from the spirit of this disclosure.

[0051] (A) In the above embodiment, SBR and hydrogenated SBR were given as examples of crosslinked polymer models, but are not limited thereto. Also, although multiple types (3 types) of monomer coarse-grained particles were set for polymer model 3 of the crosslinked polymer, the polymer model 3 may be composed of one type of monomer coarse-grained particle.

[0052] (B) In the above embodiment, the same crosslinking probability (20%) is set for all Type 1 monomer coarse-grained particles B that are set as crosslinkable particles, but the crosslinking probability to be set may differ depending on the particle. Also, the crosslinking probability to be set can be changed as appropriate as long as it is not 0%, for example it may be set to 100%.

[0053] (C) The proportion of the multiple coarse-grained particles 30 constituting the polymer model 3 that are set to be crosslinkable particles can be appropriately changed depending on what you want to achieve in the simulation. Therefore, all of the coarse-grained particles 30 in the polymer model 3 may be crosslinkable particles, or some of the coarse-grained particles 30 may be crosslinkable particles.

[0054] (D) In ​​the above embodiment, when arranging the polymer model 3 and the crosslinking agent particle mass 40 in the virtual space Ar1, the polymer model 3 is placed first, and then the crosslinking agent particle mass 40 is placed, but the order can be changed as appropriate.

[0055] (E) In the above embodiment, the method for calculating the degree of crosslinking heterogeneity in the crosslinked polymer model is applied to the crosslinked polymer model generated by the above method for generating the crosslinked polymer model, but is not limited thereto. For example, it can be performed on a crosslinked polymer model generated by another method. Also, if only the generation of a crosslinked polymer model is required, the radial distribution function calculation unit 16, the determination unit 17, and the degree of heterogeneity calculation unit 18 shown in Figure 1 can be omitted.

[0056] (F) In the above embodiment, 64 crosslinking agent particles 4 are aggregated into a cubic shape to form a crosslinking agent particle mass 40, but the embodiment is not limited to this. The number of crosslinking agent particles 4 constituting one crosslinking agent particle mass 40 can be arbitrarily changed. Furthermore, the shape in which multiple crosslinking agent particles 4 are aggregated is not limited to a cubic shape, but can be changed to any shape. Moreover, although the crosslinking agent particle mass 40 is formed by aggregating multiple crosslinking agent particles 4 with a binding potential, the embodiment is not limited to the setting of the binding potential. For example, multiple crosslinking agent particles 4 in an aggregated state can be set as a rigid body, and the rigid body setting can be released when the aggregation is released.

[0057] For example, the execution order of operations, procedures, steps, and stages in the devices, systems, programs, and methods shown in the claims, specifications, and drawings can be implemented in any order, as long as the output of a previous process is not used in a later process. Even if the flow in the claims, specifications, and drawings is described using terms such as "first," "next," etc., for convenience, it does not mean that the execution must be in that order.

[0058] Each component shown in Figure 1 is implemented by executing a predetermined program on one or more processors, but each component may be configured with dedicated memory or dedicated circuitry. In the above embodiment, each component of System 1 is implemented on a single computer's processor 1a, but each component may be distributed and implemented on multiple computers or in the cloud. In other words, the above method may be executed on one or more processors.

[0059] System 1 includes a processor 1a. For example, the processor 1a may be a central processing unit (CPU), a microprocessor, or other processing unit capable of executing computer executable instructions. System 1 also includes memory 1b for storing data of System 1. In one example, memory 1b includes computer storage media, including RAM, ROM, EEPROM, flash memory or other memory technologies, CD-ROM, DVD or other optical disc storage, magnetic cassette, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other media that can be used to store desired data and that System 1 can access. [Explanation of symbols]

[0060] 10...Setting unit, 11...Agglomeration unit, 13...Equilibrium treatment execution unit, 14...Agglomeration release unit, 15...Crosslinking reaction treatment unit, 2...Crosslinked polymer model, 3...Polymer model, 30...Coarse-grained particles, 4...Crosslinking agent particles, 40...Crosslinking agent particle aggregate, Ar1...Virtual space, B...Type 1 monomer coarse-grained particles, S...Type 2 monomer coarse-grained particles, E...Type 3 monomer coarse-grained particles.

Claims

1. A method performed by one or more processors, Setting a crosslinking probability for some of the crosslinkable particles among the multiple coarse-grained particles that make up the polymer model, which are arranged in a linear or branched manner, Setting a binding potential between each crosslinking agent particle to create a crosslinking agent particle aggregate by aggregating multiple crosslinking agent particles, The plurality of polymer models and one or more of the crosslinking agent particle aggregates are placed in a virtual space, and molecular dynamics calculations are performed while maintaining the aggregation of the crosslinking agent particle aggregates to perform an equilibrium process. The aggregation is released by releasing the binding potential set for each crosslinking agent particle constituting the crosslinking agent particle mass after the equilibration treatment, The molecular dynamics calculation is performed with the aggregation of each of the aforementioned crosslinking agent particles released, and when the crosslinkable particles of the polymer model approach the crosslinking agent particles within a predetermined distance, a crosslinking reaction process is performed to bond the crosslinkable particles and the crosslinking agent particles at a crosslinking probability set for the crosslinkable particles. A method for generating a cross-linked polymer model, including the above.

2. A method for generating a crosslinked polymer model according to claim 1, wherein the same crosslinking probability is set for all of the crosslinkable particles.

3. The method for producing a crosslinked polymer model according to claim 1 or 2, wherein the polymer model comprises coarse-grained particles of a first monomer that can bind to the crosslinking agent particles, and coarse-grained particles of a second monomer that cannot bind to the crosslinking agent particles.

4. The polymer model further comprises coarse-grained particles of a third monomer that cannot bind to the crosslinking agent particles, A method for producing a crosslinked polymer model according to claim 3, wherein the first type monomer coarse-grained particles represent butadiene, the second type monomer coarse-grained particles represent styrene, the third type monomer coarse-grained particles represent ethylene, and the polymer model is a model representing hydrogenated SBR.

5. The method for generating a crosslinked polymer model described in claim 1 is performed to generate the crosslinked polymer model after all of the crosslinkable particles have bonded to any of the crosslinking agent particles, The generated crosslinked polymer model is used to calculate the value of the radial distribution function for the crosslinking agent particles, From the peak value appearing in the value of the radial distribution function, it is determined whether or not the value of the radial distribution function decreases as the interparticle distance increases. When it is determined that the value of the radial distribution function decreases as the interparticle distance increases from the peak value, a value based on the slope of the radial distribution function decreasing from the peak value is calculated as the degree of crosslinking heterogeneity in the crosslinked polymer model. A method for calculating the degree of crosslinking heterogeneity in a crosslinked polymer model, including the above.

6. A setting unit that sets the crosslinking probability for some of the crosslinkable particles among a plurality of coarse-grained particles arranged in a linear or branched manner that constitute a polymer model, A clump generating unit that generates a clump of crosslinking agent particles by setting a bonding potential between each crosslinking agent particle and agglomerating multiple crosslinking agent particles, An equilibration processing execution unit that places the plurality of polymer models and one or more of the crosslinking agent particle aggregates in a virtual space and performs an equilibration process while maintaining the aggregation of the crosslinking agent particle aggregates, The aggregation release unit releases the aggregation by releasing the binding potential set for each crosslinking agent particle constituting the crosslinking agent particle mass after the equilibration treatment, A crosslinking reaction processing unit performs molecular dynamics calculations with each of the aforementioned crosslinking agent particles in a state of agglomeration, and when the crosslinkable particles of the polymer model approach the crosslinking agent particles within a predetermined distance, it performs a crosslinking reaction process to bond the crosslinkable particles and the crosslinking agent particles with a crosslinking probability set for the crosslinkable particles. A system for generating cross-linked polymer models, equipped with the necessary components.

7. Setting a crosslinking probability for some of the crosslinkable particles among the multiple coarse-grained particles that make up the polymer model, which are arranged in a linear or branched manner, Setting a binding potential between each crosslinking agent particle to create a crosslinking agent particle aggregate by aggregating multiple crosslinking agent particles, The plurality of polymer models and one or more of the crosslinking agent particle aggregates are placed in a virtual space, and an equilibration process is performed while maintaining the aggregation of the crosslinking agent particle aggregates. The aggregation is released by releasing the binding potential set for each crosslinking agent particle constituting the crosslinking agent particle mass after the equilibration treatment, The molecular dynamics calculation is performed with the aggregation of each of the aforementioned crosslinking agent particles released, and when the crosslinkable particles of the polymer model approach the crosslinking agent particles within a predetermined distance, a crosslinking reaction process is performed to bond the crosslinkable particles and the crosslinking agent particles at a crosslinking probability set for the crosslinkable particles. A program that causes one or more processors to execute.

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