JP2017129539A - Method for simulating polymer material - Google Patents

Abstract

PROBLEM TO BE SOLVED: To convert the time unit of a coarse grained model into the time unit of a polymer chain in a short time.SOLUTION: A simulation method for analyzing a polymer material having a polymer chain using a computer comprises: a first calculation step S21 of calculating motion using the coarse grained model of the polymer chain to calculate a mean square displacement showing the size of spread of the coarse grained model; a second calculation step S22 of calculating a motion using the total atom model of the polymer chain to calculate a mean square displacement showing the size of diffusion of the total atom model; and a step S23 of converting the time unit of the coarse grained model into the time unit of the polymer chain on the basis of the mean square displacement of the coarse grained model and the mean square displacement of the total atom model.SELECTED DRAWING: Figure 16

Description

  The present invention relates to a method for simulating a polymer material capable of converting a time unit of a coarse-grained model into a time unit of a polymer chain.

  In recent years, various simulation methods (numerical calculations) for evaluating the reaction of polymer materials using a computer have been proposed for the development of rubber compounding. In this type of simulation method, for example, a coarse-grained model obtained by modeling a polymer chain with a plurality of beads is input to a computer. Then, the coarse-grained model is placed in a predetermined space, and molecular dynamics calculation is performed by a computer.

  In molecular dynamics calculation using a coarse-grained model, a time unit different from that of an actual polymer chain is used. For this reason, the following Patent Document 1 proposes a method for converting the time unit of the coarse-grained model into the time unit of the polymer chain.

  In the conversion method disclosed in Patent Document 1 below, first, time correlation functions of the lengths of end-to-end vectors of the coarse-grained model and the all-atom model are respectively obtained. Next, for the coarse-grained model and the all-atom model, time parameters defined as times when the end-to-end vector correlation disappears are obtained. The time unit of the coarse-grained model is converted into the time unit of the polymer chain by fitting the time parameter of the coarse-grained model and the time parameter of the all-atom model.

JP 2014-206913 A

  In the conversion method described in Patent Document 1, when calculating the time correlation function of the length of the end-to-end vector of the all-atom model, the equilibrium calculation is performed until the arrangement of the all-atom model is sufficiently relaxed. In this equilibrium calculation, a very long calculation time is required if a realistic length indicating the physical properties of the polymer chain is used as the chain length of the all-atom model in order to accurately convert the time unit. Therefore, it is not realistic to convert the time unit of the coarse-grained model into the time unit of the polymer chain based on the conversion method of Patent Document 1.

  The present invention has been devised in view of the above circumstances, and provides a method for simulating a polymer material capable of converting a coarse-grained model time unit to a polymer chain time unit in a short time. The main purpose is to do.

  The present invention is a method for analyzing a polymer material having a polymer chain using a computer, wherein the computer performs a motion calculation using a coarse-grained model of the polymer chain, and A first calculation step of calculating a mean square displacement indicating a magnitude of diffusion of the visualization model, wherein the computer performs a motion calculation using the all-atom model of the polymer chain, and the magnitude of the diffusion of the all-atom model A second calculation step of calculating a mean square displacement indicating the time unit of the coarse-grained model based on the mean square displacement of the coarse-grained model and the mean square displacement of the all-atom model Is converted to time units of the polymer chain.

  In the simulation method of the polymer material according to the present invention, the first calculation step includes a step of determining an initial arrangement of the coarse-grained model and a step of calculating a relaxation state of the coarse-grained model initially arranged. And the second calculation step calculates the relaxation state based on the arrangement of the coarse-grained model in the relaxation state obtained in the first calculation step without calculating the structural relaxation of the all-atom model. A step of creating the all-atom model may be included.

  In the simulation method of the polymer material according to the present invention, the coarse-grained model includes a plurality of beads corresponding to monomers of the polymer chain, and the step of creating the relaxed state all-atom model includes the computer However, a monomer model representing the monomer may be assigned to each bead of the coarse-grained model in a relaxed state.

  In the polymer material simulation method according to the present invention, the computer calculates the number of Rouse beads of the all-atom model, and the computer calculates the number of Rouse beads of the coarse-grained model. And the computer calculates the number of beads in the coarse-grained model based on the number of beads in the coarse-grained model and the number of beads in the coarse-grained model based on the number of beads in the coarse-grained model. The method may further include calculating a ratio of the number of monomers in the chain.

  In the method for simulating a polymer material according to the present invention, the third calculation step includes a step of setting a plurality of all-atom models having different numbers of monomers in the computer, and the computer includes all atoms having a different number of monomers. And calculating the number of Rouse beads for each model.

  In the simulation method for a polymer material according to the present invention, the fourth calculation step includes a step of setting a plurality of coarse-grained models having different numbers of beads in the computer, and the computer has a coarse number having different numbers of beads. A step of calculating the number of Rouse beads for each visualization model.

In the polymer material simulation method according to the present invention, the number of _Rouse beads N _Rouse may be calculated by the following equation (1).

here,
L: total length of the coarse-grained model or all-atom model R _g : radius of inertia of the coarse-grained model or all-atom model

  In the simulation method for a polymer material according to the present invention, a computer performs a motion calculation using a coarse-grained model of a polymer chain, and calculates a mean square displacement indicating the magnitude of diffusion of the polymer chain. The second calculation step of calculating the motion using the all-atom model of the polymer chain and calculating the mean square displacement indicating the magnitude of the diffusion of the all-atom model, the mean square displacement of the coarse-grained model, and the total atom A step of converting the time unit of the coarse-grained model into the time unit of the polymer chain based on the mean square displacement of the model is included.

  The mean square displacement indicates the average diffusion size (that is, the mean square of the moving distance) of particles in an arbitrary time width. In order to convert the time unit based on the motion calculation for obtaining the mean square displacement, a transition region where the diffusion behavior of the all-atom model and the diffusion behavior of the coarse-grained model are similar (that is, between molecular chains) Is sufficient to calculate the relaxation time (for example, until the entanglement between the molecular chains of the all-atom model is sufficiently relaxed). There is no need to do it. Therefore, in the polymer material simulation method of the present invention, the time unit of the coarse-grained model can be converted into the time unit of the polymer chain in a short time.

It is a perspective view which shows an example of the computer which performs the simulation method of this invention. It is a structural formula of polybutadiene. It is a conceptual diagram which shows an example of an all-atom model. It is a conceptual diagram which shows an example of a coarse-grained model. It is a flowchart which shows an example of the process sequence of the simulation method of the polymeric material of this embodiment. It is a flowchart which shows an example of the process sequence of the monomer number conversion process of this embodiment. It is a flowchart which shows an example of the process sequence of the 3rd calculation process of this embodiment. It is a conceptual diagram which shows an example of the polymeric material model by which all the atom models are arrange | positioned. It is a conceptual diagram explaining the potential of the all-atom model. It is a graph which shows the relationship between the number of beads of Rouse of all atom model, and the number of monomers. It is a flowchart which shows an example of the process sequence of the 4th calculation process of this embodiment. It is a conceptual diagram of the polymer material model in which the coarse-grained model is arranged. It is a conceptual diagram explaining the potential of a coarse-grained model. It is a graph which shows the relationship between the number of beads of Rouse of a coarse-grained model, and the number of beads. It is the graph which fitted the number of beads of Rouse of the all-atom model, and the number of beads of Rouse of the coarse-grained model. It is a flowchart which shows an example of the process sequence of the time conversion process of this embodiment. It is a flowchart which shows an example of the process sequence of the 1st calculation process of this embodiment. It is a graph which shows the relationship between the mean square displacement of a coarse-grained model, and time width. It is a flowchart which shows an example of the process sequence of the 2nd calculation process of this embodiment. It is a graph which shows the relationship between the mean square displacement of all atom models, and time width. It is a graph which shows the relationship between the mean square displacement which fitted the all-atom model and the coarse-grained model, and the time width.

Hereinafter, an embodiment of the present invention will be described with reference to the drawings.
The polymer material simulation method of the present embodiment (hereinafter sometimes simply referred to as “simulation method”) is a method for analyzing a polymer material having a polymer chain. Examples of the polymer material include rubber, resin, and elastomer.

  FIG. 1 is a perspective view showing an example of a computer that executes the simulation method of the present invention. The computer 1 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, an arithmetic processing unit (CPU), a ROM, a working memory, a storage device such as a magnetic disk, and disk drive devices 1a1 and 1a2. The storage device stores in advance software or the like for executing the simulation method of the present embodiment.

Examples of the polymer material include rubber, resin, and elastomer. In the present embodiment, cis-1,4 polybutadiene (hereinafter sometimes simply referred to as “polybutadiene”) is exemplified as the polymer material. FIG. 2 is a structural formula of polybutadiene. In the polymer chain 2 constituting the polybutadiene, a monomer 5 {— [CH ₂ —CH═CH—CH ₂ ] —} composed of a methylene group (—CH ₂ —) and a methine group (—CH—) is polymerized. Concatenated at a degree. In addition, a methyl group (—CH ₃ ) is linked to the end of the polymer material instead of the methylene group (—CH ₂ ). As the polymer material, a polymer material other than polybutadiene may be used.

  In the simulation method of the present embodiment, an all-atom model obtained by modeling the polymer chain 2 of the polymer material and a coarse-grained model are used. FIG. 3 is a conceptual diagram showing an example of an all-atom model. FIG. 4 is a conceptual diagram illustrating an example of the coarse-grained model.

  As shown in FIG. 3, the all-atom model 3 is expressed by a particle model 7 in which atoms are modeled based on the actual structure of the polymer chain 2 shown in FIG. As shown in FIG. 4, the coarse-grained model 4 includes a plurality of beads (particles) 16 corresponding to the monomer 5 of the polymer chain 2 shown in FIG. Each bead 16 is represented as a sphere having a diameter.

  In the molecular dynamics calculation using the all-atom model 3 (shown in FIG. 3), a time unit based on the actual polymer chain 2 (shown in FIG. 2) is used. On the other hand, in the molecular dynamics calculation using the coarse-grained model 4 (shown in FIG. 4), a time unit different from that of the actual polymer chain 2 is used. Therefore, in order to handle the calculation result of the molecular dynamics calculation of the coarse-grained model 4 as the actual movement of the polymer chain 2, the time unit of the coarse-grained model 4 is changed to the time unit of the polymer chain 2. It is necessary to convert.

  In order to convert the time unit of the coarse-grained model 4 (shown in FIG. 4) into the time unit of the polymer chain 2 (shown in FIG. 2), the time parameter of the coarse-grained model 4 and all atoms It is important to fit the time parameter of model 3 (shown in FIG. 3) with high accuracy. Therefore, the ratio between the number of beads in the coarse-grained model 4 and the number of monomers in the polymer chain 2 is calculated in advance, and the all-atom model 3 and the coarse-grained model 4 are calculated based on the ratio between the number of beads and the number of monomers. Is set so that the calculation result using the all-atom model 3 and the calculation result using the coarse-grained model 4 correspond to each other with high accuracy.

  From this point of view, in the simulation method of the present embodiment, prior to converting the time unit of the coarse-grained model 4 into the time unit of the polymer chain, the number of beads of the coarse-grained model 4 and the polymer chain The ratio of 2 to the number of monomers (that is, the number of monomers of the polymer chain 2 per one bead 16 of the coarse-grained model 4) is calculated. FIG. 5 is a flowchart showing an example of a processing procedure of the polymer material simulation method of the present embodiment.

  As shown in FIG. 5, in the simulation method of the present embodiment, the computer 1 uses the number of beads of the coarse-grained model 4 (shown in FIG. 4) and the number of monomers of the polymer chain 2 (shown in FIG. 2). (Monomer number conversion step S1).

In the monomer number conversion step S1 of the present embodiment, based on the number of Rouse beads of the all-atom model 3 (shown in FIG. 3) and the number of Rouse beads of the coarse-grained model 4 (shown in FIG. 4), The ratio between the number of beads of the visualization model 4 and the number of monomers of the polymer chain 2 is calculated. Here, the number of Rouse beads N _Rouse is one of the parameters of the Rouse model known in polymer physics, and is calculated by the following equation (1).

here,
L: total length of the coarse-grained model or all-atom model R _g : radius of inertia of the coarse-grained model or all-atom model

In the above equation (1), for example, unlike the above-mentioned Patent Document 1 using the ensemble average of the length of the end-to-end vector V1 (shown in FIG. 3) of the all-atom model 3, the inertia radius R _g is used to The number of beads is required. Thereby, in this embodiment, since the influence of the unstable mobility of the particle model 7 on the terminal side of the all-atom model 3 can be reduced, the number of Rouse beads can be obtained with high accuracy.

  In the above formula (1), as the value of the number of beads of Rouse is larger, the all-atom model 3 (shown in FIG. 3) or the coarse-grained model 4 (shown in FIG. 4) becomes a compactly folded shape. On the other hand, the smaller the value of the number of beads of Rouse, the smaller the bend of the all-atom model 3 or the coarse-grained model 4 becomes, and the shape continuously extends linearly. Therefore, the number of Rouse beads is used as a parameter indicating the shape of the all-atom model 3 or the coarse-grained model 4.

  By fitting the number of beads of Rouse of all-atom model 3 (shown in FIG. 3) and the number of Rouse beads of coarse-grained model 4 (shown in FIG. 4), It is known that the ratio between the number of beads of the coarse-grained model 4 and the number of monomers of the polymer chain 2 can be accurately calculated by associating with the visualization model 4 (see, for example, Patent Document 1 above). ). Based on such knowledge, in the simulation method of the present embodiment, the ratio between the number of beads in the coarse-grained model 4 and the number of monomers in the polymer chain 2 (that is, one bead 16 in the coarse-grained model 4). Number of monomers per polymer chain 2). FIG. 6 is a flowchart showing an example of the processing procedure of the monomer number conversion step S1 of the present embodiment.

  In the monomer number conversion step S1 of the present embodiment, first, the computer 1 calculates the number of Rouse beads of the all-atom model 3 (shown in FIG. 3) (third calculation step S11). FIG. 7 is a flowchart illustrating an example of a processing procedure of the third calculation step S11 of the present embodiment.

  In the third calculation step S11 of the present embodiment, first, a plurality of all-atom models 3 having different numbers of monomers are set in the computer 1 (step S111). As shown in FIG. 3, the all-atom model 3 includes a plurality of particle models 7 and a bond 8 that connects the particle models 7 and 7.

  The particle model 7 and the bond 8 are connected based on the unit structure representing the monomer 5 of the polymer chain 2 shown in FIG. The monomer model 11 is connected based on a predetermined number of monomers (that is, molecular weight (degree of polymerization)), whereby the all-atom model 3 is set. In the present embodiment, the monomer models 11 are connected to each other based on a predetermined number of monomers. Thereby, a plurality of all-atom models 3 having different numbers of monomers are set.

  The number of monomers is set within a range in which the relaxation calculation can be completed in a realistic calculation time based on the type of polymer material, the performance of the computer 1 that performs the molecular dynamics calculation described later, and the like. desirable. Note that it is desirable to exclude extremely small values for the number of monomers in order to maintain calculation accuracy. An example of the number of monomers can be selected from 10 to 60. The number of monomers is not limited.

  The particle model 7 is handled as a mass point of a motion equation in a simulation based on molecular dynamics calculation described later. That is, the particle model 7 defines parameters such as mass, diameter, charge, or initial coordinates. The particle model 7 of this embodiment includes a carbon particle model 7C that models the carbon atoms of the polymer chain 2 and a hydrogen particle model 7H that models the hydrogen atoms of the polymer chain 2.

  The bond 8 restrains the particle models 7 and 7. The bond 8 of this embodiment includes a main chain 8A that connects the carbon particle models 7C and 7C, and a side chain 8B that connects the carbon particle model 7C and the hydrogen particle model 7H. The main chain 8A and the side chain 8B are handled as springs in which an equilibrium length and a spring constant are defined, for example.

  The all-atom model 3 includes a bond length that is a bond length between the particle models 7 and 7, a bond angle that is an angle formed by three particle models 7 that are continuous through the bond 8, and a bond that is continuous through the bond 8. In the four particle models 7, the dihedral angles created by the three adjacent particle models 7 are defined. Thereby, the all-atom model 3 has a three-dimensional structure. In the all-atom model 3, the bond length, bond angle, and dihedral angle are changed by receiving an external force or an internal force according to the custom. Thereby, the all-atom model 3 can change the three-dimensional structure.

  The bond length, bond angle, and dihedral angle can be defined by, for example, the potential (GAFF) set based on the paper 1 (J. Comput. Chem. 25, 1157-1174 (2004)). The potential is preferably set according to the structure of the polymer chain 2. Such an all-atom model 3 can be created using material physical property simulation software (for example, J-OCTA manufactured by JSOL Corporation). Each all-atom model 3 is numerical data that can be handled by the computer 1 and is input to the computer 1.

  Next, in the third calculation step S11 of the present embodiment, the computer 1 determines the initial arrangement of each all-atom model 3 having a different number of monomers (step S112). In step S112, a polymer material model in which all atom models 3 are arranged in a predetermined space is set. FIG. 8 is a conceptual diagram showing an example of the polymer material model 10 in which the all-atom model 3 is arranged.

  In step S112 of this embodiment, all the atom models 3 having different numbers of monomers are arranged in the spaces 9 provided independently. Thereby, in step S112, the polymer material model 10 in which the initial arrangement of each all-atom model 3 is determined is defined. In each polymer material model 10, a plurality of all-atom models 3 having the same number of monomers are arranged. The all-atom model 3 is preferably arranged in the space 9 based on the Monte Carlo method.

  The space 9 corresponds to a microstructure portion of the polymer material to be analyzed. The space 9 of the present embodiment is defined as a cube having three pairs of planes 9S and 9S facing each other. A periodic boundary condition is defined for each plane 9S. Thereby, in the space 9, for example, it is possible to calculate so that a part of the all atom model 3 that has come out from one plane 9Sa enters from the opposite plane 9Sb. Therefore, it can be handled as one plane 9Sa and the opposite plane 9Sb being continuous (connected).

  The number of all atom models 3 arranged in each space 9 is appropriately set based on the number of monomers and the size of the space 9. If the number of all-atom models 3 is small, in the molecular dynamics calculation described later, one end of all-atom model 3 that goes out from one plane 9Sa and enters from the opposite plane 9Sb, The other end of the arranged all-atom model 3 is likely to be entangled, and there is a risk that calculation will be lost. Furthermore, in order to calculate the inertial radius of the all-atom model 3 with high accuracy, it is necessary to prevent entanglement between all-atom models as much as possible at equilibrium. For this reason, as the number of all-atom models 3, it is desirable to select the number that allows the periodic boundary length at equilibrium to be longer than three times the inertial radius at equilibrium. On the contrary, even if the number of all atom models 3 is large, the number of particles of the particle model 7 handled as the mass point of the equation of motion increases, which may require a lot of calculation time. From such a viewpoint, the number of all atomic models 3 arranged in the space 9 is preferably 20 or more, and preferably 200 or less.

The length La of one side of the space 9 can be set as appropriate. The length La of this embodiment is preferably set so that the initial density of the particle model 7 in the system is, for example, 0.001 g / cm ³ . The size of the space 9 is desirably set to a stable volume at 1 atm, for example. In such a space 9, the rotational motion of the all-atom model 3 can be calculated smoothly in the molecular dynamics calculation described later.

FIG. 9 is a conceptual diagram for explaining the potential of the all-atom model 3. In step S112, in each polymer material model 10 in which the initial arrangement of the all-atom model 3 is determined, an interaction potential P1 is defined between the particle models 7 and 7 of the adjacent all-atom models 3 and 3. The interaction potential P1 of this embodiment is an LJ potential U _LJ (r _ij ) and is defined by the following formula (2).

Here, each constant and variable are parameters of Lennard-Jones potential, and are as follows.
r _ij: distance between the particles model r _c: cutoff distance epsilon: the intensity of the LJ potential which is defined between the particles Model sigma: corresponding to the diameter of the particle model The distance r _ij and cutoff distance r _c, each particle It is defined as the distance between the centers of models 7 and 7.

In the interaction potential P1, the repulsive force acting between the particle models 7 and 7 increases as the distance r _ij between the particle models becomes smaller than σ. Further, the interaction potential P1 becomes minimum when the distance r _ij between the particle models becomes σ, and no repulsive force or attractive force acts between the particle models 7 and 7. Furthermore, as the interaction potential P1 is larger than the distance r _ij between the particle models, the attractive force acting between the particle models 7 and 7 works. As described above, the interaction potential P1 can define repulsive force and attractive force according to the distance r _ij between the particle models.

  The interaction potential P1 includes a first potential P1a set between the carbon particle models 7C and 7C, a second potential P1b set between the hydrogen particle models 7H and 7H, and the carbon particle model 7C and the hydrogen particle model 7H. The third potential P1c set between the first and second potentials is included. Each constant in the above formula (2) can be set as appropriate based on the above paper 1. The polymer material model 10 in which the initial arrangement of these all-atom models 3 is determined is input to the computer 1.

  Next, in the third calculation step S11 of the present embodiment, the computer 1 calculates the relaxation state of the initially arranged all-atom model 3 (step S113). In step S113, molecular dynamics calculation is performed for each polymer material model 10 (shown in FIG. 8) for which the initial arrangement of the all-atom model 3 is determined. In the molecular dynamics calculation, for example, Newton's equation of motion is applied on the assumption that all the atomic models 3 arranged in the space 9 for a predetermined time follow classical mechanics. Then, the movements of all the particle models 7 at each time are tracked and stored in the computer 1. The molecular dynamics calculation is performed under the condition that, for example, the number, volume, and temperature of the particle model 7 in the system are constant. Such molecular dynamics calculation can be performed using, for example, the molecular dynamics calculation program LAMMPS.

  Next, in the third calculation step S11 of the present embodiment, the computer 1 determines whether or not the initial arrangement of all the atom models 3 has been sufficiently relaxed (step S114). Note that the criterion for determining the relaxation of the initial arrangement can be appropriately set as long as it can be regarded that the artificial initial arrangement of the all-atom model 3 is sufficiently eliminated. In step S114, when it is determined that the initial arrangement of all atom models 3 has been sufficiently relaxed (“Y” in step S114), the next step S115 is performed. On the other hand, when it is determined that the initial arrangement of all atom models 3 has not been sufficiently relaxed (“N” in step S114), the unit time is set (for example, the simulation time already performed in step S113). Proceeding (step S116), step S113 (molecular dynamics calculation) and step S114 are performed again. Thereby, the initial arrangement of the all-atom model 3 can be surely relaxed.

  Next, in the third calculation step S11 of the present embodiment, the computer 1 determines whether or not the relaxation states of all the all-atom models (that is, all the all-atom models having different numbers of monomers) 3 have been calculated ( Step S115). In step S115, when it is determined that the relaxation calculation of all the all-atom models 3 has been calculated (“Y” in step S115), the next step S117 is performed. On the other hand, when it is determined that the relaxation calculation of all the all-atom models 3 has not been completed ("N" in step S115), the all-atom model 3 whose relaxation state has not been calculated (that is, as shown in FIG. 8). The polymer material model 10) is selected (step S118), and steps S113 to S116 are performed. Thereby, the equilibrium state (relaxed state) of all the all-atom models 3 is calculated.

  Next, in the third calculation step S11 of the present embodiment, the computer 1 calculates the total length L of the all-atom model 3 (step S117). In step S117, based on the method of the above-mentioned Patent Document 1, deformation calculation for forcibly stretching the all-atom model 3 in the main chain direction is performed. Thereby, the total length L of the all-atom model 3 can be obtained. The total length L of the all-atom model 3 is input to the computer 1.

Next, in the third calculation step S11 of the present embodiment, the computer 1 calculates the inertia radius R _g of the all-atom model 3 (step S119). The radius of inertia of all-atom model 3 is described in, for example, Paper 2 (Kurt Kremer & Gary S. Grest, “Dynamics of entangled linear polymer melts: A molecular-dynamics simulation”, J. Chem Phys. Vol. 92, No. 8, 15 April 1990) is calculated from the coordinate value of the particle model 7 of the all-atom model 3 obtained by the molecular dynamics calculation of the step S113 based on the formula 2.5 described in 15 April 1990). The radius of inertia of the all-atom model 3 is input to the computer 1.

Next, in the third calculation step S11 of the present embodiment, the computer 1 calculates the number of Rouse beads of the all-atom model 3 (step S120). As described above, the number of Rouse beads is calculated by the above formula (1) based on the total length L of the all-atom model 3 and the inertia radius R _g of the all-atom model 3. The number of Rouse beads of the all-atom model 3 is input to the computer 1.

  Next, in the third calculation step S11 of the present embodiment, the computer 1 determines whether or not the number of beads of Rouse has been calculated for all all atomic models 3 having different numbers of monomers (step S121). In Step S121, when it is determined that the number of Rouse beads of all the all-atom models 3 has been calculated (“Y” in Step S121), the next fourth calculation step S12 is performed. On the other hand, if it is determined that the number of Rouse beads of all all-atom models 3 has not been calculated (“N” in step S121), another all-atom model 3 for which the number of Rouse beads has not been calculated is selected. (Step S122), Step S117 and Step S119 to Step S121 are performed. Thereby, in 3rd calculation process S11, the number of beads of Rouse is calculated for every all-atom model 3 from which the number of monomers differs. FIG. 10 is a graph showing the relationship between the number of Rouse beads and the number of monomers in the all-atom model 3. In the example of FIG. 10, the number of Rouse beads of the all-atom model 3 having the number of monomers of 20, 30, and 40 is shown. The number of Rouse beads of the all-atom model 3 is input to the computer 1.

  Next, in the monomer number conversion step S1 of the present embodiment, the computer 1 calculates the number of Rouse beads of the coarse-grained model 4 (fourth calculation step S12). FIG. 11 is a flowchart illustrating an example of a processing procedure of the fourth calculation step S12 of the present embodiment.

  In the fourth calculation step S12 of this embodiment, first, a plurality of coarse-grained models 4 with different bead numbers (chain lengths) are set in the computer 1 (step S131). As shown in FIG. 4, the coarse-grained model 4 includes a plurality of beads 16 and a binding chain 17 that connects the beads 16 and 16.

  In the present embodiment, the coarse-grained model 4 is set by connecting the beads 16 based on a predetermined number of beads. In the present embodiment, a plurality of different bead numbers are set. A plurality of coarse-grained models 4 with different bead numbers are set by connecting the beads 16 based on these bead numbers. The number of beads can be appropriately set based on the type of polymer material, the performance of the computer 1 that performs molecular dynamics calculation described later, and the like. The number of beads is preferably selected from 5 to 1000, for example. Note that the number of beads is not limited to this.

  The beads 16 are handled as mass points of the equation of motion in the molecular dynamics calculation. That is, parameters such as mass, diameter, charge, or initial coordinates are defined for the beads 16. Each of these parameters is stored in the computer 1 as numerical information.

  The binding chain 17 is configured as a binding potential that defines an equilibrium length between the beads 16 and 16. Here, the “equilibrium length” is a binding distance between the beads 16 and 16. When the bond distance changes, the original equilibrium length is maintained by the bond chain 17. Thereby, the coarse-grained model 4 can maintain a linear three-dimensional structure. The equilibrium length is defined as the distance between the centers 16a and 16a of the adjacent beads 16 and 16. The bond potential of the bond chain 17 is preferably set based on, for example, the above paper 2. Such a coarse-grained model 4 is numerical data for handling a polymer material by molecular dynamics calculation, and is input to the computer 1.

  Next, in 4th calculation process S12 of this embodiment, the computer 1 determines the initial arrangement | positioning of each coarse-grained model 4 from which the number of beads differs (process S132). In step S132, a polymer material model in which the coarse-grained model 4 is arranged in a predetermined space is set. FIG. 12 is a conceptual diagram showing an example of the polymer material model 18 in which the coarse-grained model 4 is arranged.

  In step S132 of the present embodiment, the coarse-grained models 4 having different numbers of beads are arranged in the spaces 9 provided independently. Thereby, in step S132, the polymer material model 18 in which the initial arrangement of each coarse-grained model 4 is determined is defined. Each polymer material model 18 includes a plurality of all-atom models 3 having the same number of beads. The coarse-grained model 4 is preferably arranged in the space 9 based on the Monte Carlo method.

  The space 9 corresponds to a microstructure portion of the polymer material to be analyzed, and is set similarly to the space 9 shown in FIG. The number of coarse-grained models 4 arranged in each space 9 is appropriately set based on the number of beads and the size of the space 9. Note that if the number of coarse-grained models 4 is small, like the all-atom model 3 (shown in FIG. 8), one end and the other end of the coarse-grained model 4 are likely to be tangled and there is a risk of calculation loss. Furthermore, in order to calculate the inertia radius of the coarse-grained model 4 with high accuracy, it is necessary to prevent the coarse-grained models 4 from being entangled as much as possible at the time of equilibrium. For this reason, as the number of coarse-grained models 4, it is desirable to select a number that allows the period boundary length at equilibrium to be longer than three times the inertial radius at equilibrium. On the contrary, even if the number of coarse-grained models 4 is large, the number of particles of the beads 16 that are handled as the mass points of the equation of motion increases, which may require a lot of calculation time. From such a viewpoint, the number of coarse-grained models 4 arranged in the space 9 is preferably 50 or more, and preferably 2000 or less.

  FIG. 13 is a conceptual diagram for explaining the potential of the coarse-grained model 4. In step S132, in each polymer material model 18, an interaction potential P2 is defined between the beads 16 and 16 of the adjacent coarse-grained models 4 and 4, respectively. The interaction potential P2 of the present embodiment is an LJ (Lennard-Jones) potential and is defined by the above formula (2). In the above equation (2), “between particle models” is replaced with “between beads”.

The intensity of interaction potential P2 epsilon, distance interaction potential P2 acts sigma, and the cutoff distance r _c, for example, being set on the basis of the article 1 is desirable. The polymer material model 18 for which the initial arrangement of the coarse-grained model 4 is determined is input to the computer 1.

  Next, in the fourth calculation step S12 of the present embodiment, the computer 1 calculates the relaxation state of the coarse-grained model 4 that is initially arranged (step S133). In step S133, molecular dynamics calculation is performed on the coarse-grained model 4 initially arranged in the space 9 shown in FIG. In the molecular dynamics calculation, for example, Newton's equation of motion is applied on the assumption that all coarse-grained models 4 arranged in the space 9 for a predetermined time follow classical mechanics. Then, the movement of all the beads 16 at each time is tracked and stored in the computer 1. The molecular dynamics calculation is performed under the condition that the number, volume and temperature of the beads 16 in the system are constant.

  Next, in the fourth calculation step S12 of the present embodiment, the computer 1 determines whether or not the initial arrangement of the coarse-grained model 4 has been sufficiently relaxed (step S134). Note that the criterion for determining the relaxation of the initial arrangement can be appropriately set as long as it is a standard that allows the artificial initial arrangement of the coarse-grained model 4 to be sufficiently excluded. In step S134, when it is determined that the initial arrangement of the coarse-grained model 4 has been sufficiently relaxed (“Y” in step S134), the next step S135 is performed. On the other hand, when it is determined that the initial arrangement of the coarse-grained model 4 has not been sufficiently relaxed (“N” in step S134), the unit time (for example, the simulation time already performed in step S133 is summarized) Proceeding (step S136), step S133 (molecular dynamics calculation) and step S134 are performed again. Thereby, the initial arrangement of the coarse-grained model 4 can be surely relaxed.

  Next, in the fourth calculation step S12 of the present embodiment, the computer 1 determines whether or not the relaxation states of all the coarse-grained models (that is, all the coarse-grained models having different numbers of beads) 4 have been calculated. (Step S135). In step S135, when it is determined that the relaxation states of all the coarse-grained models 4 have been calculated (“Y” in step S135), the next step S137 is performed. On the other hand, when it is determined that the relaxation states of all the coarse-grained models 4 have not been calculated (“N” in step S135), the coarse-grained models 4 whose relaxation states have not been calculated (that is, in FIG. 12). The polymer material model 18) shown is selected (step S138), and steps S133 to S136 are performed. Thereby, the equilibrium state (relaxed state) of all the coarse-grained models 4 is calculated.

  Next, in the fourth calculation step S12 of the present embodiment, the computer 1 calculates the total length L of the coarse-grained model 4 (step S137). The total length L of the coarse-grained model 4 can be obtained by multiplying the distance between the equilibrium nuclei of adjacent beads in the coarse-grained model 4 by the number of beads. The equilibrium internuclear distance is obtained based on the result of the above molecular dynamics calculation. The total length L of the coarse-grained model 4 is input to the computer 1.

Next, in the fourth calculation step S12 of the present embodiment, the computer 1 calculates the inertia radius R _g of the coarse-grained model 4 (step S139). The radius of inertia of the coarse-grained model 4 is, for example, the coordinate value of the bead 16 of the coarse-grained model 4 obtained by the molecular dynamics calculation in step S133 based on Equation 2.5 described in the above paper 2. Calculated from The radius of inertia of the coarse-grained model 4 is input to the computer 1.

Next, in the fourth calculation step S12 of the present embodiment, the computer 1 calculates the number of beads of Rouse of the coarse-grained model (step S140). As described above, the number of Rouse beads is calculated by the above equation (1) based on the total length L of the coarse-grained model 4 and the inertia radius R _g of the coarse-grained model 4. The number of Rouse beads in the coarse-grained model 4 is input to the computer 1.

  Next, in the fourth calculation step S12 of the present embodiment, the computer 1 determines whether or not the number of beads of Rouse has been calculated for all the coarse-grained models 4 having different numbers of beads (step S141). When it is determined in step S141 that the number of Rouse beads of all the coarse-grained models 4 has been calculated (“Y” in step S141), the next step S13 (shown in FIG. 6) is performed. On the other hand, when it is determined that the number of Rouse beads of all the coarse-grained models 4 has not been calculated (“N” in step S141), the coarse-grained model 4 for which the number of Rouse beads has not been calculated is selected. (Step S142), Step S137, Step S139 to Step S141 are performed. Thus, in the fourth calculation step S12, the number of beads of Rouse is calculated for each coarse-grained model 4 having a different number of beads. FIG. 14 is a graph showing the relationship between the number of beads in the coarse-grained model 4 and the number of beads. In the example of FIG. 14, the number of Rouse beads of the coarse-grained model 4 having the number of beads (chain length) of 10, 13, 20, 27, 50, 80, 100, 200, and 500 is shown. The number of Rouse beads in the coarse-grained model 4 is input to the computer 1.

  Next, as shown in FIG. 6, in the monomer number conversion step S1 of the present embodiment, the computer 1 uses the number of beads of the coarse-grained model 4 (shown in FIG. 4) and the polymer chain 2 (shown in FIG. 2). The ratio with the number of monomers of (shown) is calculated (step S13). In step S13, based on the number of Rouse beads of the all-atom model 3 (shown in FIG. 3) and the number of Rouse beads of the coarse-grained model 4, the high per one bead 16 of the coarse-grained model 4 The number of monomers of molecular chain 2 is obtained.

  In step S13, first, based on the nonlinear least square method considering statistical errors, the relationship between the number of Rouse beads and the number of monomers in the all-atom model 3 shown in FIG. 10, and the coarse-grained model shown in FIG. The number of beads of 4 Rouse and the relationship between the number of beads are fitted. FIG. 15 is a graph obtained by fitting the number of Rouse beads of the all-atom model 3 and the number of Rouse beads of the coarse-grained model 4.

  In the fitting, first, the result of the coarse-grained model 4 (shown in FIG. 14) is fitted with the smooth curve (shown by a broken line) shown in FIG. Next, fitting is performed with respect to the ratio of the number of beads of the coarse-grained model 4 and the number of monomers of the all-atom model 3 so that this curve and the result of the all-atom model 3 (shown in FIG. 10) overlap. Thereby, the number of Rouse beads of the all-atom model 3 and the number of Rouse beads of the coarse-grained model 4 are fitted. As a fitting method, for example, in paper 3 (K. Levenberg, Quarterly of Applied Mathematics 2, 164-168 (1944), D. Marquardt, SIAM Journal on Applied Mathematics 11 (2), 431-441 (1963)) The described Levenberg-Marquardt Method can be used. The fitting using the Levenberg-Marquardt method can be easily performed by, for example, graph creation software (for example, gnuplot).

  As statistical errors used for fitting, a statistical error of the inertia radius of the all-atom model 3 and a statistical error of the inertia radius of the coarse-grained model 4 are calculated. The statistical error of the inertia radius of the all-atom model 3 is obtained by dividing the variance of the time average of the inertia radii for each molecular chain by the number of all-atom models arranged in the space 9 for every atom model 3 with different number of monomers. Can be obtained from the square root of that value. Further, the statistical error of the inertial radius of the coarse-grained model 4 indicates that the coarse-grained coarsening model 4 in which the dispersion of the time-average of the inertial radius for each molecular chain is arranged in the space 9 is different for each coarse-grained model 4 having a different number of beads. A value divided by the number of models can be obtained and obtained from the square root of the value.

  In step S13, a ratio between the number of beads of the coarse-grained model 4 (shown in FIG. 4) and the number of monomers of the polymer chain 2 (shown in FIG. 2) is calculated based on the graph shown in FIG. Yes. In the example of the graph shown in FIG. 15, the ratio between the number of beads in the coarse-grained model and the number of monomers in the polymer chain 2 (that is, the number of monomers per one bead 16) is 1.5. is there. The ratio between the number of beads of the coarse-grained model 4 (shown in FIG. 4) and the number of monomers of the polymer chain 2 (shown in FIG. 2) is input to the computer 1.

  Thus, in the monomer number conversion step S1 of the present embodiment, the number of Rouse beads of the all-atom model 3 (shown in FIG. 3) and the number of Rouse beads of the coarse-grained model 4 (shown in FIG. 4) are calculated. By fitting, it is possible to associate the all-atom model 3 and the coarse-grained model 4 having substantially the same shape with each other. Therefore, the ratio between the number of beads of the coarse-grained model 4 and the number of monomers of the polymer chain 2 Can be calculated with high accuracy. Further, in the present embodiment, since a plurality of Rouse beads of the all-atom model 3 and a plurality of Rouse beads of the coarse-grained model 4 are obtained, fitting can be performed with high accuracy.

  Next, in the simulation method of the present embodiment, the computer 1 converts the time unit of the coarse-grained model 4 (shown in FIG. 4) into the time unit of the polymer chain 2 (shown in FIG. 2) (time conversion). Step S2). In the time conversion step S2 of this embodiment, based on the mean square displacement of the coarse-grained model 4 and the mean square displacement of the all-atom model 3 (shown in FIG. 3), the time unit of the coarse-grained model 4 is Converted to the time unit of the polymer chain. The mean square displacement indicates the degree of diffusion of the coarse-grained model 4 beads 16 or the particle model 7 of the all-atom model 3 (that is, the mean square of the moving distance) in an arbitrary time width. It is. Therefore, the mean square displacement includes a time parameter. In the present embodiment, the mean square displacement of the coarse-grained model 4 and the mean square displacement of the all-atom model 3 are calculated for the transition region between the glass region and the rubber region of the polymer material. FIG. 16 is a flowchart illustrating an example of a processing procedure of the time conversion step S2 of the present embodiment.

  In the time conversion step S2 of the present embodiment, first, the computer 1 calculates a mean square displacement indicating the diffusion size of the coarse-grained model 4 (shown in FIG. 4) (first calculation step S21). FIG. 17 is a flowchart illustrating an example of a processing procedure of the first calculation step S21 of the present embodiment.

  In the first calculation step S21 of the present embodiment, first, the computer 1 determines the number of beads (chain length) of the coarse-grained model 4 (step S211). In step S211, the number of beads indicating the physical properties of the actual polymer chain is specified. If the number of beads (chain length) is small, the coarsening model 4 flows before the entanglement between the coarse-grained models 4 occurs, and the physical properties of the polymer chain may not be sufficiently exhibited. Conversely, if the number of beads (chain length) is large, in the second calculation step S22 (shown in FIG. 16), which will be described later, the average period boundary length is about three times the inertia radius of all atom model 3, It is necessary to increase the number of atomic models 3. This increases the number of particle models 7 (shown in FIG. 9) that are handled as mass points of the equation of motion, which may increase the calculation time. For example, in the coarse-grained model 4 set based on the paper 2, it is desirable that the number of beads (chain length) is set to about 80 to 200.

  Next, in the first calculation step S21 of the present embodiment, the computer 1 sets the coarse-grained model 4 (shown in FIG. 4) based on the determined number of beads (step S212). In step S <b> 212, as many beads 16 as the determined number of beads are linked through the binding chain 17 as shown in FIG. 4. Thereby, the coarse-grained model 4 with the determined number of beads is set. The coarse-grained model 4 is input to the computer 1.

  Next, in the first calculation step S21 of the present embodiment, the computer 1 determines the initial arrangement of the coarse-grained model 4 (step S213). In step S213, as shown in FIG. 12, a polymer material model 21 in which the coarse-grained model 4 is arranged in a predetermined space 9 is set. A plurality of coarse-grained models 4 are arranged in the space 9. The coarse-grained model 4 is preferably arranged in the space 9 based on the Monte Carlo method. In addition, about the number of the coarse-grained models 4 arrange | positioned in each space 9, a mutual potential, etc., it sets with the same procedure as monomer number conversion process S1 mentioned above. The polymer material model 21 in which the initial arrangement of the coarse-grained model 4 is determined is input to the computer 1.

  Next, in the first calculation step S21 of the present embodiment, the computer 1 calculates the relaxation state of the coarse-grained model 4 that is initially arranged (step S214). In step S214, molecular dynamics calculation is performed for the coarse-grained model 4 initially arranged in the space 9. The molecular dynamics calculation is performed in the same procedure as the monomer number conversion step S1 described above.

  Next, in the first calculation step S21 of the present embodiment, the computer 1 determines whether or not the initial arrangement of the coarse-grained model 4 has been sufficiently relaxed (step S215). In step S215, when it is determined that the initial arrangement of the coarse-grained model 4 has been sufficiently relaxed (“Y” in step S215), the next step S216 is performed. On the other hand, when it is determined that the initial arrangement of the coarse-grained model 4 has not been sufficiently relaxed (“N” in step S215), the unit time (for example, the simulation time already performed in step S214 is summarized) Proceeding (step S217), step S214 (molecular dynamics calculation) and step S215 are performed again. Thereby, the initial arrangement of the coarse-grained model 4 can be surely relaxed. The coarse-grained model 4 whose initial arrangement is relaxed is input to the computer 1.

  Next, in the first calculation step S21 of the present embodiment, the computer 1 performs motion calculation on the coarse-grained model 4 whose initial arrangement is relaxed (step S216). As the motion calculation of this embodiment, the above-described molecular dynamics calculation is performed. As a result, the coarse-grained model 4 in the relaxed state is further diffused to calculate the time scale of the transition zone (about the time when entanglement occurs, for example, about 1800τ in the case of the coarse-grained model according to the paper 2). In step S216, the coordinate data of the beads 16 of the coarse-grained model 4 are output. The coordinate data is input to the computer 1.

Next, in the first calculation step S21 of the present embodiment, the computer 1 calculates the mean square displacement of the coarse-grained model 4 (step S218). The mean square displacement of the present embodiment is calculated based on the coordinate value of the bead 16 for each unit time from the start of calculation in step S216 to the end of calculation. Mean-square displacement (MSD) is an index indicating the amount of diffusion of beads 16 (shown in FIG. 12) of the coarse-grained model 4. The mean square displacement is defined by the following equation (3).
MSD = <| r (t ′) − r (0) | ² > (3)
The symbols are as follows.
r (0): Coordinate value of bead at arbitrary time r (t ′): Coordinate value of bead after time (time width) t ′ from arbitrary time <>: Ensemble average

  As shown in the above formula (3), the mean square displacement is the time after any time from any time of each bead 16 (shown in FIG. 12) for all the coarse-grained models 4 arranged in the space 9 (arbitrary Is the value obtained by ensemble averaging the square of the distance moved in the time width t ′). With such mean square displacement, the average diffusion size of the beads 16 can be grasped.

  The mean square displacement is preferably calculated for a predetermined bead 16 among all the beads 16 (shown in FIG. 12) constituting the coarse-grained model 4. Thereby, since the beads 16 for which the mean square displacement is calculated are limited, the total amount of coordinate data of the beads 16 of the coarse-grained model 4 output in step S216 is reduced, and the calculation time of the mean square displacement is shortened. be able to. Further, as shown in FIG. 4, the mean square displacement is a part of the beads 16 arranged on the center side 4 </ b> C in the longitudinal direction of the coarse-grained model 4 among all the beads 16 constituting the coarse-grained model 4. Preferably it is calculated for 16. Thereby, since the influence of the diffusion of the end sides 4E and 4E in the coarse-grained model 4 where the entanglement of the beads 16 hardly occurs, the mean square displacement in the transition region can be accurately calculated.

  The number of beads 16 for which the mean square displacement is calculated can be set as appropriate. If the number of beads 16 for which the mean square displacement is calculated is small, the number of samples of the beads 16 of the coarse-grained model 4 is insufficient, the statistical error of the mean square displacement increases, and the time of the coarse-grained model 4 is increased. There is a possibility that the unit cannot be accurately converted into the time unit of the polymer chain. Conversely, if the number of beads 16 for which the mean square displacement is calculated is large, there is a possibility that the influence of diffusion on the end sides 4E and 4E where the entanglement of the beads 16 hardly occurs can not be sufficiently removed. From such a viewpoint, the number of beads 16 for which the mean square displacement is calculated is preferably set to 30 to 70% of the number of all beads 16 constituting the coarse-grained model 4. FIG. 18 is a graph showing the relationship between the mean square displacement of the coarse-grained model 4 and the time width. Such an average value of the mean square displacement is input to the computer 1.

  Next, in the first calculation step S21 of the present embodiment, the computer 1 determines whether or not the entanglement between the coarse-grained models 4 starts to occur (step S219). The determination as to whether or not the coarse-grained models 4 have started to be entangled is made, for example, by determining whether the mean square displacement obtained in step S218 is proportional to the time width approximately ½ power (that is, diffusion of the transition region). And a region in which diffusion is slower than approximately ½ power toward a region proportional to approximately ¼ of the time width (that is, a region corresponding to diffusion of the rubber region). It can be determined based on whether or not it contains. Such determination can be appropriately set according to the structure of the coarse-grained model. Therefore, in the first calculation step S21, the motion calculation is ended before the arrangement of the coarse-grained model 4 is significantly changed.

  In step S219, when it is determined that the entanglement between the coarse-grained models 4 has started to occur ("Y" in step S219), the next second calculation step S22 (shown in FIGS. 16 and 19) is performed. . On the other hand, when it is determined that the coarse-grained models 4 are not entangled with each other (“N” in step S219), the unit time is advanced (for example, the simulation time already performed in step S216 is summarized) ( Step S220), step S216, and step S218 are performed. Thereby, in the first calculation step S21, the mean square displacement of the coarse-grained model 4 until the entanglement between the coarse-grained models 4 starts to occur (that is, corresponding to the diffusion in the transition zone) is calculated.

  Next, in the time conversion step S2, the computer 1 calculates a mean square displacement indicating the diffusion magnitude of the all-atom model 3 (second calculation step S22). FIG. 19 is a flowchart illustrating an example of a processing procedure of the second calculation step S22 of the present embodiment.

  In the second calculation step S22 of the present embodiment, first, the computer 1 creates a relaxed all-atom model 3 (step S221). In step S221, the all-atom model 3 in the relaxed state is created based on the arrangement of the coarse-grained model 4 in the relaxed state obtained in step S215 of the first calculation step S21 shown in FIG.

  In step S221, the monomer model 11 (shown in FIG. 3) is assigned to each bead 16 (shown in FIG. 12) of the coarse-grained model 4 in the relaxed state based on reverse mapping. As described above, the monomer model 11 is obtained by connecting the particle model 7 and the bond 8 based on the unit structure that forms the monomer 5 (shown in FIG. 2) of the polymer chain 2. By assigning such a monomer model 11 to each bead 16 of the coarse-grained model 4, the all-atom model 3 in a relaxed state is set. Note that reverse mapping can be easily performed using the material property simulation software.

  In addition, in order to avoid collision between the particle models 7 and 7 after reverse mapping (the distance is too close), reverse mapping is performed at a sufficiently low density so that collision does not occur, and a molecular dynamics calculation program After energy minimization using LAMMPS or the like, it is desirable to repeatedly perform affine deformation and energy minimization step by step until the all-atom model 3 reaches an equilibrium density. In addition, when it is necessary to retain the stereoisomer of the monomer, for example, as in the method described in JP-A-2014-225226, the potential function of the dihedral angle and bond angle related to the stereoisomer is increased, It is desirable to perform energy minimization. Further, the equilibrium density of the all-atom model 3 can be estimated by fitting and extrapolating with a smooth curve (not shown) from the equilibrium density after relaxation calculation of a plurality of all-atom models 3 having a short chain length. desirable.

  In this embodiment, based on the ratio between the number of beads of the coarse-grained model 4 (shown in FIG. 4) calculated in the monomer number conversion step S1 and the number of monomers of the polymer chain 2 (shown in FIG. 2), A monomer model 11 is assigned to each bead 16. Thereby, the all-atom model 3 in a relaxed state can be created with high accuracy. In step S221, the all-atom model 3 in the relaxed state is created based on all the coarse-grained models 4 in the relaxed state arranged in the space 9.

  In step S221, first, when the ratio between the number of beads in the coarse-grained model 4 (shown in FIG. 4) and the number of monomers in the polymer chain 2 (shown in FIG. 2) is not 1, the number of beads is the same as the number of monomers. So that the positions of both ends on the main chain of the coarse-grained model 4 are maintained and the main chain is resampled at equal intervals (ie , Rearrange the total number of beads 16). For example, when the number of monomers of the polymer chain 2 per bead 16 of the coarse-grained model 4 is 1.5 and the number of beads of the coarse-grained model 4 is 80, the corresponding polymer chain Since the number of monomers of 2 is 120 (that is, the number of monomers is 1.5 × the number of beads is 80), the number of beads in the coarse-grained model 4 is resampled to be 80 to 120.

  Next, in step S221, one monomer model 11 is assigned to each bead 16 of the coarse-grained model 4 after resampling by performing reverse mapping using the material physical property simulation software or the like, and the space 9 From this, the coarse-grained model 4 is removed. Thereby, a polymer material model (not shown) in which all the atomic models 3 in a relaxed state are arranged can be obtained.

  Thus, in step S221, the all-atomic model 3 in the relaxed state is calculated without calculating the structural relaxation (that is, molecular dynamics calculation) of the all-atomic model 3 that requires much time compared to the coarse-grained model 4. Therefore, the calculation time can be greatly shortened. The relaxed all-atom model 3 is input to the computer 1.

  Next, in the second calculation step S22 of the present embodiment, the computer 1 performs motion calculation for the all-atom model 3 in the relaxed state (step S222). As the motion calculation of this embodiment, the above-described molecular dynamics calculation is performed. The molecular dynamics calculation is performed in the same procedure as the molecular dynamics calculation performed in the monomer number conversion step S1 described above. Thereby, the magnitude of diffusion in the transition region of the polymer material can be calculated for the all-atom model 3 in the relaxed state. In step S222, coordinate data of the particle model 7 of the all-atom model 3 is output. The coordinate data is input to the computer 1.

  Next, in the second calculation step S22 of the present embodiment, the computer 1 calculates the mean square displacement of the all-atom model 3 (step S223). The mean square displacement of this embodiment is based on the coordinate value of the particle model 7 for each unit time from the start of calculation of step S222 to the end of calculation. Is calculated. Mean-square displacement (MSD) is an index indicating the degree of particle diffusion. The mean square displacement is defined by the above equation (3). With such mean square displacement, the magnitude of diffusion for each time width of each particle model 7 can be grasped. In the above equation (3), “r (0)” is defined as the coordinate value of a certain particle model at an arbitrary time. In addition, “r (t ′)” is defined as the coordinate value of the same particle model after time (time width t ′) from an arbitrary time.

  In order to shorten the calculation time, the mean square displacement is preferably calculated for a predetermined particle model 7 among all the particle models 7 constituting the all-atom model 3. As shown in FIG. 3, the mean square displacement is preferably calculated for a part of the particle models 7 arranged on the center side in the main chain direction of the all-atom model 3. Thereby, since the influence of the terminal side diffusion in which the entanglement of the particle model 7 hardly occurs in the all-atom model 3 can be eliminated, the mean square displacement in the transition region can be calculated with high accuracy.

  The number of particle models 7 for which the mean square displacement is calculated can be set as appropriate. The number of particle models 7 for which the mean square displacement is calculated is 30 to 70 which is the number of all the particle models 7 constituting the all-atom model 3 from the same viewpoint as the number of beads 16 for which the mean square displacement is calculated. It is desirable to set to%. FIG. 20 is a graph showing the relationship between the mean square displacement of the all-atom model 3 and the time width. Note that the graph of FIG. 20 gives different results depending on the calculation conditions such as temperature and pressure set for the all-atom model 3. For this reason, in order to calculate the ratio between the number of beads of the coarse-grained model 4 and the number of monomers of the polymer chain 2 and to convert the time unit with high accuracy, in the monomer number conversion step S1 and the time conversion step S2, It is necessary to unify the calculation conditions of the all-atom model.

  Next, in the second calculation step S22 of the present embodiment, the computer 1 determines whether or not the mean square displacement of the all-atom model 3 corresponds to the diffusion of the transition region (step S224). Whether or not the mean square displacement of the all-atom model 3 corresponds to the diffusion of the transition region is determined based on a region proportional to approximately the 1/2 power of the time width (that is, a region corresponding to the diffusion of the transition region). Judgment is made based on whether or not it is statistically significant. Accordingly, in the second calculation step S22, the motion calculation is ended before the arrangement of the all-atom model 3 changes greatly.

  In step S224, when it is determined that the mean square displacement of all-atom model 3 corresponds to diffusion in the transition region (“Y” in step S224), the next step S23 (shown in FIG. 19) is performed. Is done. On the other hand, when it is determined that the mean square displacement of the all-atom model 3 does not correspond to the diffusion of the transition region (“N” in step S224), the unit time (for example, the simulation time that has been performed in step S222) Proceeding together (step S225), step S222 and step S223 are performed. Thereby, in the second calculation step S22, the mean square displacement of the all-atom model 3 corresponding to the diffusion of the transition region is calculated. Therefore, in the later-described step S23, the mean square displacement of the coarse-grained model 4 (see FIG. 18). And the mean square displacement (shown in FIG. 20) of the all-atom model 3 can be overlapped.

  Next, in the time conversion step S2 of the present embodiment, the computer 1 converts the time unit of the coarse-grained model 4 to the time unit of the polymer chain (step S23). In step S23, based on the mean square displacement of the coarse-grained model 4 and the mean square displacement of the all-atom model 3, the time unit of the coarse-grained model 4 is converted to the time unit of the polymer chain.

  In step S23, first, the relationship between the mean square displacement of the coarse-grained model 4 shown in FIG. 18 and time, and the relationship between the mean square displacement of the all-atom model 3 and time shown in FIG. Fitting is performed based on a nonlinear least square method considering FIG. 21 is a graph showing the relationship between the mean square displacement obtained by fitting the all-atom model 3 and the coarse-grained model 4 and the time width.

  In the fitting, first, the mean square displacement of the coarse-grained model 4 overlaps with the mean square displacement of the all-atom model 3 (that is, corresponds to the diffusion of the transition region), and the range of the time scale (time width) is a smooth curve. Fitting is performed (not shown). Next, fitting is performed for the ratio between the time unit (time width) of the coarse-grained model 4 and the time unit (time width) of the all-atom model 3 so that this curve and the mean square displacement of the all-atom model 3 overlap. Is done. As a fitting method, for example, the Levenberg-Marquardt Method described in the paper 3 is used, and can be easily performed by, for example, the graph creation software.

  Then, from the result of the above fitting, the time unit of the coarse-grained model 4 is associated with the time unit of the coarse-grained model 4 (time width τ) and the time unit of the all-atom model 3 (time width ns). Can be converted into a time unit of a polymer chain. The converted time unit of the coarse-grained model 4 is input to the computer 1.

  As described above, in the time conversion step S2 of the present embodiment, the arrangement of all the atom models 3 is sufficiently relaxed (that is, the entanglement between all the atom models 3 is unwound) as in the conversion method of Patent Document 1 described above. Up to) The time unit of the coarse-grained model 4 can be converted into the time unit of the polymer chain without performing relaxation calculation. The relaxation calculation of Patent Document 1 requires much time compared with the time conversion step S2 of the present embodiment. Therefore, in the time conversion step S2, the time unit of the coarse-grained model 4 can be converted into the time unit of the polymer chain in a short time.

  Moreover, in the present embodiment, the mean square displacement can be fitted in a wide range of time width (that is, a time scale corresponding to the diffusion of the transition region), so that it is defined as the time when the end-to-end vector correlation disappears. Compared with Patent Document 1 in which the time parameter is fitted pinpointly, it can be converted with higher accuracy.

  As described above, in the monomer number conversion step S1, the coarse-grained model 4 (shown in FIG. 4) is obtained by fitting the number of Rouse beads of the all-atom model 3 and the number of Rouse beads of the coarse-grained model 4. ) And the number of monomers of the polymer chain 2 (shown in FIG. 2). Thereby, in the coarse-grained model 4, the plurality of beads 16 corresponding to the monomers of the polymer chain 2 are accurately set in consideration of the chain length dependency of the inertia radius, and thus the chain length dependency is not considered. Compared with the conversion method described in Patent Document 1, the calculation result of the molecular dynamics calculation using the coarse-grained model 4 and the calculation result of the molecular dynamics calculation using the all-atom model 3 can be associated with each other with high accuracy. it can. Therefore, in the time conversion step S2, the time unit of the coarse-grained model 4 can be accurately converted to the time unit of the polymer chain.

  Next, as shown in FIG. 5, in the simulation method of the present embodiment, the computer 1 performs a simulation under various conditions using the coarse-grained model 4 (step S <b> 3). An example of the simulation of step S3 includes, for example, deformation calculation using a relaxed polymer material model (not shown).

  In step S3, based on the ratio between the number of beads of the coarse-grained model 4 (shown in FIG. 4) obtained in the monomer number conversion step S1 and the number of monomers of the polymer chain 2 (shown in FIG. 2), A visualization model 4 is set. Therefore, in step S3, a result approximate to the polymer chain 2 can be obtained by simulation using the coarse-grained model 4.

  Next, in the simulation method of the present embodiment, the computer 1 determines whether the simulation result using the polymer material model (not shown) is good (that is, has a desired performance) (step S4). ). In step S4, the time unit of the coarse-grained model 4 (shown in FIG. 4) calculated in step S3 based on the result obtained in the time conversion step S2 is the polymer chain 2 (shown in FIG. 2). Converted to time unit. Thereby, the calculation results of molecular dynamics calculation under various conditions using the coarse-grained model 4 can be handled as the actual movement of the polymer chain 2.

  If it is determined in step S4 that the simulation result is good (“Y” in step S4), a polymer material is manufactured based on the simulated polymer material model (not shown) (step S5). ). On the other hand, when it is determined that the simulation result is not good (“N” in step S4), the structure and conditions of the polymer chain 2 (shown in FIG. 2) are changed (step S6), and steps S1 to S1 are performed. S4 is performed again. Thus, in the simulation method of the present embodiment, an unknown polymer material having a desired performance can be developed. In addition, the simulation method of the present embodiment enables a highly accurate simulation result by simulation using the coarse-grained model 4 (shown in FIG. 4) without performing the simulation using the all-atom model 3 in step S4. Therefore, the development cost of the polymer material can be reduced.

  As mentioned above, although especially preferable embodiment of this invention was explained in full detail, this invention is not limited to embodiment of illustration, It can deform | transform and implement in a various aspect.

  According to the processing procedure shown in FIG. 5, a monomer number conversion step for obtaining a ratio between the number of beads of the coarse-grained model 4 (shown in FIG. 4) and the number of monomers of the polymer chain 2 (shown in FIG. 2), A time conversion step of converting the time unit of the visualization model into the time unit of the polymer chain was performed (Example).

  In the monomer number conversion step of the example, according to the processing procedure shown in FIG. 6, a third calculation step for calculating the number of Rouse beads for the all-atom model and a fourth calculation for calculating the number of Rouse beads for the coarse-grained model And a step of converting the coarse-grained model beads into the number of monomers of the polymer chain.

In the third calculation step of the example, all atom models having different numbers of monomers (in the present embodiment, the first all atom model to the third all atom model) are respectively arranged in space according to the processing procedure shown in FIG. The number of Rouse beads was calculated for every atomic model. FIG. 10 is a graph showing the relationship between the number of Rouse beads and the number of monomers in the all-atom model. In the molecular dynamics calculation, the initial density of the space was set to 0.001 g / cm ³ , the temperature was set to 360 K, and the pressure was set to 1 atm. In addition, the number of monomers (chain length) of each all-atom model, the number of all-atom models arranged in space, and the like are as follows.
First all-atom model:
Number of monomers: 20, number in space: 50, relaxation time: 20 ns
Second all-atom model:
Number of monomers: 30, number in space: 75, relaxation time: 50 ns
Third all-atom model:
Number of monomers: 40, number in space: 75, relaxation time: 100 ns

In the fourth calculation step of the example, a plurality of coarse-grained models having different numbers of beads (in the present embodiment, the first coarse-grained model to the ninth coarse-grained model) are space-in accordance with the processing procedure shown in FIG. The number of coarse beads Rouse beads was calculated. FIG. 14 is a graph showing the relationship between the number of beads of Rouse and the number of beads in the coarse-grained model. The number of beads (chain length) of each coarse-grained model, the number of coarse-grained models arranged in space, and the like are as follows.
First coarse-grained model:
Number of beads: 10, Number of beads in space: 1000, Relaxation time: 15000τ
Second coarse-grained model:
Number of beads: 13, number of beads in space: 1000, relaxation time: 15000τ
Third coarse-grained model:
Number of beads: 20, number of beads in space: 1000, relaxation time: 15000τ
Fourth coarse-grained model:
Number of beads: 27, number in space: 1000, relaxation time: 15000τ
Fifth coarse-grained model:
Number of beads: 50, number of beads in space: 200, relaxation time: 1500000τ
Sixth coarse-grained model:
Number of beads: 80, number in space: 100, relaxation time: 3000000τ
Seventh coarse-grained model:
Number of beads: 100, number in space: 100, relaxation time: 3000000τ
Eighth coarse-grained model:
Number of beads: 200, number in space: 100, relaxation time: 4500000τ
9th coarse-grained model:
Number of beads: 500, number of beads in space: 150, relaxation time: 3000000τ

  In the process of converting the coarse-grained model beads of the example into the number of monomers of the polymer chain, the relationship between the number of Rouse beads and the number of monomers of the all-atom model based on the nonlinear least square method, and the coarse-grained model The number of beads in Rouse and the relationship between the number of beads are fitted, and the ratio between the number of beads in the coarse-grained model and the number of monomers in the polymer chain 2 (that is, the number of monomers per one bead 16) is calculated. It was done. The number of monomers per bead was 1.383 ± 0.014.

  In the time conversion step of the embodiment, according to the processing procedure shown in FIG. 16, the first calculation step for calculating the mean square displacement indicating the diffusion size of the coarse-grained model and the diffusion size of the all-atom model are shown. A second calculation step for calculating the mean square displacement and a step for converting the time unit of the polymer chain were performed.

  In the first calculation step of the embodiment, according to the processing procedure shown in FIG. 17, the step of calculating the relaxation state of the initially arranged coarse-grained model, and the motion calculation using the coarse-grained model of the relaxed state, And calculating a mean square displacement. The number of beads (chain length) in the coarse-grained model was determined to be 80 from the number of monomers per one bead. The relaxation time was 15000τ. FIG. 18 is a graph showing the relationship between the mean square displacement of the coarse-grained model and the time width.

  In the second calculation step of the embodiment, in accordance with the processing procedure shown in FIG. 19, a step of creating a relaxed state all-atom model based on the arrangement of the relaxed state coarse-grained model obtained in the first calculation step; The process of calculating the mean square displacement was performed by calculating the motion using the all-atom model in the relaxed state. The temperature in the motion calculation (molecular dynamics calculation) was set to 360K. FIG. 20 is a graph showing the relationship between the mean square displacement and the time width of the all-atom model.

  In the step of converting to the time unit of the polymer chain in the example, based on the nonlinear least square method, the relationship between the mean square displacement of the coarse-grained model and time, and the mean square displacement of the all-atom model and time. The relationship was fitted and the time unit of the coarse-grained model was converted to the time unit of the polymer chain. As a result, the time unit 1τ of the coarse-grained model was converted to the time unit 44.9 ps of the polymer chain.

  For comparison, the time unit of the coarse-grained model was converted to the time unit of the polymer chain based on the method of Patent Document 1 (Comparative Example). In the comparative example, when calculating the time correlation function of the length of the end-to-end vector of the all-atom model, a deformation calculation for forcibly extending the all-atom model in the main chain direction was performed. In the comparative example, when calculating the time correlation function of the length of the end-to-end vector of the all-atom model, the equilibrium calculation was performed until the arrangement of the all-atom model was sufficiently relaxed. The details of the conversion method are as described in Patent Document 1. The computer used in the examples and comparative examples is 24 CPUs (2 nodes).

  As a result of the test, in the example, the number of days required to convert the time unit of the coarse-grained model into the time unit of the polymer chain was 15 days. On the other hand, when the number of days required for conversion was estimated for the method of the comparative example, it was several years. Therefore, the simulation method of the present invention can convert the time unit of the coarse-grained model into the time unit of the polymer chain in a short time.

S21 1st calculation process S22 2nd calculation process S23 The process of converting the time unit of a coarse-grained model into the time unit of a polymer chain

In the polymer material simulation method according to the present invention, the coarse-grained model includes a plurality of beads corresponding to monomers of the polymer chain, and the computer calculates the number of Rouse beads of the all-atom model. A third calculation step, wherein the computer calculates the number of Rouse beads of the coarse-grained model, and the computer calculates the number of Rouse beads of the all-atom model and the coarse-grained model The method further includes a step of calculating a ratio between the number of beads of the coarse-grained model and the number of monomers of the polymer chain based on the number of beads of Rouse, and the third calculation step includes: A plurality of all-atom models having different numbers, and the computer calculating the number of beads of the Rouse for every all-atom models having different numbers of monomers. The fourth calculation step includes a step of setting a plurality of coarse-grained models having different numbers of beads in the computer, and the computer calculates the number of Rouse beads for each of the coarse-grained models having different numbers of beads. And a calculating step.

Claims (7)

A method for analyzing a polymer material having a polymer chain using a computer,
A first calculation step in which the computer performs a motion calculation using the coarse-grained model of the polymer chain and calculates a mean square displacement indicating a magnitude of diffusion of the coarse-grained model;
A second calculation step in which the computer performs a motion calculation using the all-atom model of the polymer chain, and calculates a mean square displacement indicating a magnitude of diffusion of the all-atom model; and A step of converting a time unit of the coarse-grained model into a time unit of the polymer chain based on a mean square displacement of the generalized model and a mean square displacement of the all-atom model. Molecular material simulation method. The first calculation step includes a step of determining an initial arrangement of the coarse-grained model, and a step of calculating a relaxation state of the coarse-grained model initially arranged.
The second calculation step calculates the all-atomic model in the relaxed state based on the arrangement of the coarse-grained model in the relaxed state obtained in the first calculation step without calculating the structural relaxation of the all-atomic model. The method for simulating a polymer material according to claim 1, comprising a step of creating The coarse-grained model includes a plurality of beads corresponding to monomers of the polymer chain,
3. The method for simulating a polymer material according to claim 2, wherein the step of creating the all-atomic model in the relaxed state is such that the computer assigns a monomer model representing the monomer to each bead of the coarse-grained model in the relaxed state. . A third calculation step in which the computer calculates the number of Rouse beads of the all-atom model;
A fourth calculating step in which the computer calculates the number of Rouse beads of the coarse-grained model;
And the computer uses the number of Rouse beads of the all-atom model and the number of Rouse beads of the coarse-grained model to calculate the number of beads of the coarse-grained model and the number of monomers of the polymer chain. The method for simulating a polymer material according to claim 1, further comprising a step of calculating a ratio. The third calculation step includes a step of setting a plurality of all-atom models having different numbers of monomers in the computer,
5. The method for simulating a polymer material according to claim 4, further comprising a step of calculating the number of beads of the Rouse for every atomic model having a different number of monomers. The fourth calculation step includes a step of setting a plurality of coarse-grained models having different numbers of beads in the computer;
6. The polymer material simulation method according to claim 4, further comprising a step of calculating the number of beads of the Rouse for each coarse-grained model in which the number of beads is different. 7. The method for simulating a polymer material according to claim 4, wherein the number of Rouse beads N _Rouse is calculated by the following formula (1).

here,
L: total length of the coarse-grained model or all-atom model R _g : radius of inertia of the coarse-grained model or all-atom model