JP6944306B2 - Methods, devices and programs for calculating cross-linked polymer parameters - Google Patents

Description

The present invention relates to a method, an apparatus and a program for calculating parameters (swelling degree, effective molecular chain density) of a crosslinked polymer such as rubber.

If the various parameters that make up the molecular model are set appropriately, the physical properties of the substance can be calculated by molecular dynamics simulation. For example, the physical characteristics of a polymer solution in which a polymer is dissolved in a solvent (for example, osmotic pressure and vapor pressure) and the physical characteristics of a crosslinked polymer such as rubber (for example, SS characteristics and viscoelasticity) can be calculated.

Patent Document 1 describes a method for calculating parameters (swelling degree, effective molecular chain density) of a crosslinked polymer such as rubber.

JP-A-2017-49807

In Patent Document 1, in order to calculate the degree of swelling, the effective molecular chain density, the dependence of the solvent chemical potential in the solution state and the volume fraction, and the interaction parameters between the polymer and the solvent are required. ..

In order to obtain the dependence between the solvent chemical potential in the solution state and the volume fraction, a plurality of crosslinked polymer solvent models having different concentrations are prepared, and the solvent chemical potential in the solution state is calculated for each concentration. The free energy perturbation method is illustrated as a method for calculating the solvent chemical potential in a solution state, but it is necessary to increase the number of particles in the entire system due to the large statistical error. The free energy perturbation method requires a plurality of calculations in which the potential between the polymer particles and the solvent molecules is changed for one concentration. Therefore, the method of Patent Document 1 has a high calculation cost. In addition to Patent Document 1, there is no document that discloses a method for determining the degree of swelling.

The present disclosure has focused on such issues, and an object of the present disclosure is to provide a method, an apparatus, and a program for calculating parameters of a crosslinked polymer at a low calculation cost.

The present invention takes the following measures in order to achieve the above object.

That is, the method for calculating the parameters of the crosslinked polymer of the present invention is
For the preset cross-linked polymer model, the coordinates of each particle constituting the cross-linked polymer model are set based on the calculation results of the molecular dynamics calculation in the equilibrium state under the analysis conditions including the predetermined temperature and pressure. Steps to get and
A step of executing Delaunay tessellation to generate an aggregate of tetrahedra having the coordinates of each particle as vertices, and calculating the volume surrounded by the tetrahedron as the volume in a solvent-free state.
A mixed model in which a predetermined amount of a preset solvent molecular model is added to the crosslinked polymer model is subjected to a solvent based on the calculation result of molecular dynamics calculation in an equilibrium state under analysis conditions including predetermined temperature and pressure. Steps to obtain the coordinates of each molecule that composes the crosslinked polymer model after addition,
Delaunay tessellation is performed to generate an aggregate of tetrahedra whose vertices are the coordinates of each particle constituting the crosslinked polymer model after the addition of the solvent, and after expanding the volume surrounded by the tetrahedron. Steps to calculate as the volume of
It includes a step of calculating the degree of swelling based on the volume after expansion and the volume in the solvent-free state.

In this way, an aggregate of tetrahedra with the coordinates of each particle as the apex is generated by the Delaunay division method, and the volume of the region surrounded by the tetrahedron is regarded as the volume of the crosslinked polymer. The calculation cost can be reduced. Nevertheless, it is effective because the volume calculation accuracy is within the permissible range.
Therefore, it is possible to calculate the parameters of the crosslinked polymer at a lower calculation cost as compared with the conventional case.

The block diagram which shows typically the apparatus which calculates the parameter of a crosslinked polymer. Explanatory drawing about Delaunay tessellation. The flowchart which shows the processing routine which the said apparatus executes.

Hereinafter, an embodiment of the present invention will be described with reference to the drawings.

[Device for calculating parameters of crosslinked polymer]
The device 2 of the present embodiment is a device that calculates parameters (swelling degree, etc.) of the crosslinked polymer by molecular dynamics calculation using the crosslinked polymer model and the solvent molecule model.

As shown in FIG. 1, the apparatus 2 includes an initial setting unit 20, a model setting unit 21, a molecular dynamics calculation execution unit 22, a first coordinate acquisition unit 23, a first volume calculation unit 24, and the like. It has a second coordinate acquisition unit 25, a second volume calculation unit 26, and a swelling degree calculation unit 27. The device 2 further includes a crosslink density calculation unit 28, but the crosslink density calculation unit 28 may be omitted if necessary. Each of these parts 20 to 28 is realized in collaboration with software and hardware by the CPU executing a processing routine (not shown) stored in advance in an information processing device such as a personal computer equipped with a CPU, a memory, various interfaces, and the like. Will be done.

The initial setting unit 20 shown in FIG. 1 receives an operation from a user via a known operation unit such as a keyboard or a mouse, sets data on a crosslinked polymer model and a solvent molecule model to be analyzed, and calculates molecular dynamics. Execute various settings such as analysis conditions required for, and store these setting values in the memory. As shown in FIG. 1, the crosslinked polymer model data M10 and the solvent molecule model data M2 are stored in the memory. In the crosslinked polymer model data M10, data in which crosslinked beads are reacted and bonded to the polymer model are set. In the crosslinked polymer model data M10, reactive beads are set in the molecular chain of the uncrosslinked polymer model data, the crosslinked beads are charged, and until all the reactive polymers are in an equilibrium state in which they are bonded to the crosslinked beads. Obtained by computing molecular dynamics calculations. The crosslinked polymer model data M10 may be set with data generated in advance outside the apparatus 2, and is generated by a model generator (not shown) in the apparatus 2 using the uncrosslinked polymer model data and the crosslinked bead model data. It may be a crosslink. In the present embodiment, as an example, for an uncrosslinked polymer model having 100 Kremer-Grest molecular chain models consisting of 200 beads, 10 reactive beads are set at equal intervals in each molecular chain. The cross-linked polymer model data M10 generated by adding 500 cross-linked beads and equilibrating them while undergoing a cross-linking reaction is used. FENE-LJ (Lennard-Jones) is set as the binding potential of the uncrosslinked polymer model, and WCA (LJ potential of only repulsive force) is set as the non-bonding potential. Of course, this is just an example, and other settings are possible. The solvent molecule model data M2 is composed of one molecule, and a known non-binding potential of the solvent is set.

The model setting unit 21 shown in FIG. 1 uses preset crosslinked polymer model data M10 and solvent molecule model data M2 to set a mixed model M13 in which the crosslinked polymer model M10 and the solvent molecule model M2 are mixed. The mixed model data M13 is stored in the memory. For example, the model setting unit 21 adds a predetermined amount (for example, 100,000) of the solvent molecule model M2 to the crosslinked polymer model M10 to generate the mixed model data M13. The amount of the solvent molecule model M2 to be added is not limited to 100,000 and can be changed as appropriate.

The molecular dynamics calculation execution unit 22 shown in FIG. 1 executes the molecular dynamics calculation using the mixed model data M13. Examples of the process performed by the molecular dynamics calculation execution unit 22 include an equilibrium process of the crosslinked polymer model and an equilibrium process of the mixed model. In the equilibrium treatment, the behavior of each molecule is calculated until the volume of the crosslinked polymer model M10 or the mixed model data M13 becomes almost constant (volume change becomes equal to or less than the threshold value), and the polymer volume fraction φi in the equilibrium state is calculated. Is calculated. The polymer volume fraction φi is stored in the mixed model data M13.

The first coordinate acquisition unit 23 shown in FIG. 1 is the equilibrium state calculated by the molecular dynamics calculation execution unit 22 under the analysis conditions including the preset temperature and pressure for the preset crosslinked polymer model M10. Based on the calculation result of the molecular dynamics calculation in, the coordinates of each particle constituting the crosslinked polymer model M10 are acquired. The coordinates are x, y, z coordinates in the three-dimensional space.

The first volume calculation unit 24 shown in FIG. 1 executes Delaunay tessellation to generate an aggregate of tetrahedrons having the coordinates of each particle acquired by the first coordinate acquisition unit 23 as vertices. The volume surrounded by the tetrahedron is calculated as the volume in the solvent-free state (V- _free ). As shown in FIG. 2, the Delaunay division transforms a set of points into a set of tetrahedra.

The second coordinate acquisition unit 25 shown in FIG. 1 refers to the mixed model M13 in which a predetermined amount of the preset solvent molecule model M2 is added to the crosslinked polymer model M10 under the analysis conditions including the predetermined temperature and pressure. Based on the calculation result of the molecular dynamics calculation in the equilibrium state calculated by the molecular dynamics calculation execution unit 22 in the above, the coordinates of each particle constituting the crosslinked polymer model M10 after the addition of the solvent are acquired.

The second volume calculation unit 26 shown in FIG. 1 executes Delaunay tessellation to generate an aggregate of tetrahedrons having the coordinates of each particle constituting the crosslinked polymer model M10 after addition of the solvent as vertices. Then, the volume surrounded by the tetrahedron is calculated as _{the volume after expansion (V expansion).}

The swelling degree calculation unit 27 shown in FIG. 1 calculates the swelling degree φ _{saturation based on the volume after expansion} (V expansion) and the volume in the solvent-free state (V- _free ). Specifically, it can be calculated by the formula: swelling degree φ _saturation = volume of crosslinked polymer after _expansion (V expansion) / volume of crosslinked polymer in a solvent-free state (V _{not added).}

Crosslink density calculating unit 28 shown in FIG. 1, by using the interaction parameter χ of polymer and solvent which is set in advance, and a swelling degree φ _saturated, it calculates the effective molecular chain density N _e by Equation (1). χ may be input and set by the user in advance via the operation unit, or may be configured to be calculated and set inside the device 2 based on the description in JP-A-2017-40967.

[Method of calculating parameters of crosslinked polymer]
A method of calculating the parameters of the crosslinked polymer using the apparatus 2 shown in FIG. 1 will be described with reference to FIG.

First, in step ST1, the initial setting unit 20 sets various settings such as the crosslinked polymer model M10 to be analyzed, the solvent molecule model data M2, and the analysis conditions (temperature, pressure, etc.) required for the molecular dynamics calculation. And store these setting values in the memory.

In the next step ST2, the molecular dynamics calculation execution unit 22 executes the molecular dynamics calculation in the equilibrium state under the analysis conditions including the predetermined temperature and pressure for the preset crosslinked polymer model M10. ..

In the next step ST3, the first coordinate acquisition unit 23 acquires the coordinates of each particle constituting the crosslinked polymer based on the calculation result of the molecular dynamics calculation execution unit 22.

In the next step ST4, the first volume calculation unit 24 executes Delaunay tessellation to generate an aggregate of tetrahedra having the coordinates of each particle as vertices, and the volume surrounded by the tetrahedron. Is calculated as the volume in the solvent-free state (V- _free).

In the next step ST5, the molecular dynamics calculation execution unit 22 applies the analysis conditions including the predetermined temperature and pressure to the mixed model M13 in which the preset solvent molecule model M2 is added to the crosslinked polymer model M10 in a predetermined amount. Underneath the molecular dynamics calculations in equilibrium.

In the next step ST6, the second coordinate acquisition unit 25 acquires the coordinates of each particle constituting the crosslinked polymer model after the solvent is added, based on the calculation result of the molecular dynamics calculation execution unit 22.

In the next step ST7, the second volume calculation unit 26 executes Delaunay tessellation to obtain an aggregate of tetrahedrons having the coordinates of each particle constituting the crosslinked polymer model after addition of the solvent as vertices. The generated volume surrounded by the tetrahedron is calculated as _{the volume after expansion (V expansion).}

In the next step ST8, the swelling degree calculation unit 27 calculates the swelling degree φ based on _{the volume after expansion} (V expansion) and the volume in the solvent-free state (V- _free).

In the next step ST9, the crosslinking density calculating unit 28 calculates and the interaction parameter χ solvent polymeric previously set, by using the degree of swelling φ _saturated by the equation (1) an effective molecular chain density N _e do.

The effect of the present disclosure will be described.

(1) Calculation time As the crosslinked polymer model, 10 reactive beads are set at equal intervals in each molecular chain for the uncrosslinked polymer model in which there are 100 Kremer-Grest molecular chain models consisting of 200 beads. Then, 500 crosslinked beads were added, and the crosslinked polymer model data generated by equilibration while undergoing a crosslinking reaction was used. Both the method of Patent Document 1 and the method of the present embodiment have obtained results of substantially the same degree of swelling. The method of Patent Document 1 required about 98 hours with 10 CPUs. On the other hand, the method of this embodiment requires about 12 hours with 8 CPUs. The performance per CPU is the same. Therefore, it can be clearly understood that the calculation cost can be remarkably reduced while maintaining the accuracy according to the method of the present embodiment.

(2) Number of Repeats of Equilibrium Treatment In the method described in Patent Document 1, the non-binding potential between the polymer model and the solvent molecule model was changed in 20 ways in order to calculate the solvent chemical potential in the solution state. That is, 20 kinds of equilibration processing are executed under different conditions.
On the other hand, in the method of the present embodiment, only one calculation for reaching equilibrium swelling is sufficient.
Further, in the method of the present embodiment, the χ parameter is not required for the calculation of the degree of swelling, and the calculation cost for deriving the χ parameter can be omitted.

As described above, the method for calculating the parameters of the crosslinked polymer model of the present embodiment is
Coordinates of each particle constituting the cross-linked polymer model M10 based on the calculation result of the molecular dynamics calculation in the equilibrium state under the analysis conditions including the predetermined temperature and pressure for the preset cross-linked polymer model M10. Steps (ST2-3) to obtain
A step of executing Delaunay tessellation to generate an aggregate of tetrahedra whose vertices are the coordinates of each particle, and calculating the volume surrounded by the tetrahedron as the volume in the solvent-free state (V- _free ). (ST4) and
Based on the calculation result of the molecular dynamics calculation in the equilibrium state under the analysis conditions including the predetermined temperature and pressure for the mixed model M13 in which the preset solvent molecule model M2 is added to the crosslinked polymer model M10 in a predetermined amount. , Steps (ST5-6) to acquire the coordinates of each molecule constituting the crosslinked polymer model M10 after the addition of the solvent.
Delaunay tessellation is performed to generate an aggregate of tetrahedra whose vertices are the coordinates of each particle constituting the crosslinked polymer model M10 after addition of the solvent, and the volume surrounded by the tetrahedron is expanded. Step (ST7) to calculate as volume (V _{expansion) and}
The step (ST8) of calculating the swelling degree φ based on the volume after _expansion (V expansion) and the volume in the solvent-free state (V- _{free), and}
including.

The device 2 for calculating the parameters of the crosslinked polymer model of the present embodiment is
Coordinates of each particle constituting the cross-linked polymer model M10 based on the calculation result of the molecular dynamics calculation in the equilibrium state under the analysis conditions including the predetermined temperature and pressure for the preset cross-linked polymer model M10. First coordinate acquisition unit 23 to acquire
Delaunay tessellation is executed to generate an aggregate of tetrahedra with the coordinates of each particle as the apex, and the volume surrounded by the tetrahedron is calculated as _{the volume without solvent (V-free).} Volume calculation unit 24 of 1 and
Based on the calculation result of the molecular dynamics calculation in the equilibrium state under the analysis conditions including the predetermined temperature and pressure for the mixed model M13 in which the preset solvent molecule model M2 is added to the crosslinked polymer model M10 in a predetermined amount. , A second coordinate acquisition unit 25 that acquires the coordinates of each molecule constituting the crosslinked polymer model M10 after the addition of the solvent, and
Delaunay tessellation is performed to generate an aggregate of tetrahedra whose vertices are the coordinates of each particle constituting the crosslinked polymer model M10 after addition of the solvent, and the volume surrounded by the tetrahedron is expanded. A second volume calculation unit 26 calculated as a volume (V _{expansion), and}
The swelling degree calculation unit 27 that calculates the swelling degree φ based on the volume after _expansion (V expansion) and the volume in the solvent-free state (V- _{free addition),}
To be equipped.

In this way, an aggregate of tetrahedra with the coordinates of each particle as the apex is generated by the Delaunay division method, and the volume of the region surrounded by the tetrahedron is regarded as the volume of the crosslinked polymer. The calculation cost can be reduced. Nevertheless, it is effective because the volume calculation accuracy is within the permissible range.
Therefore, it is possible to calculate the parameters of the crosslinked polymer at a lower calculation cost as compared with the conventional case.

Calculated in the present embodiment, the crosslinking density calculating section 28, a preset polymer and interaction parameter of the solvent chi, using said swelling degree φ _saturated by the equation (1) an effective molecular chain density N _e do.

It can be calculated effective molecular chain density N _e.

The program according to this embodiment is a program that causes a computer to execute the above method. By executing this program, it is possible to obtain the effects of the above method.

Although the embodiments of the present invention have been described above with reference to the drawings, it should be considered that the specific configuration is not limited to these embodiments. The scope of the present invention is shown not only by the description of the above-described embodiment but also by the scope of claims, and further includes all modifications within the meaning and scope equivalent to the scope of claims.

For example, although the parts 20 to 28 shown in FIG. 1 are realized by executing a predetermined program on the CPU of the computer, each part may be configured by a dedicated circuit.

It is possible to adopt the structure adopted in each of the above embodiments in any other embodiment. The specific configuration of each part is not limited to the above-described embodiment, and various modifications can be made without departing from the spirit of the present invention.

23 ... 1st coordinate acquisition unit 24 ... 1st volume calculation unit 25 ... 2nd coordinate acquisition unit 26 ... 2nd volume calculation unit 27 ... Swelling degree calculation unit 28 ... Crosslink density calculation unit

Claims (5)

The way the computer does
For the preset cross-linked polymer model, the coordinates of each particle constituting the cross-linked polymer model are set based on the calculation results of the molecular dynamics calculation in the equilibrium state under the analysis conditions including the predetermined temperature and pressure. Steps to get and
Delaunay tessellation is performed to generate an aggregate of tetrahedrons having the coordinates of each particle constituting the preset crosslinked polymer model as vertices, and construct the preset crosslinked polymer model. The step of calculating the volume surrounded by the tetrahedron having the coordinates of each particle as the apex as the volume in the solvent-free state, and
A mixed model in which a predetermined amount of a preset solvent molecular model is added to the crosslinked polymer model is subjected to a solvent based on the calculation result of molecular dynamics calculation in an equilibrium state under analysis conditions including predetermined temperature and pressure. Steps to obtain the coordinates of each molecule that composes the crosslinked polymer model after addition,
Delaunay tessellation is executed to generate an aggregate of tetrahedrons having the coordinates of each particle constituting the crosslinked polymer model after the addition of the solvent as vertices, and the crosslinked polymer model after the addition of the solvent is constructed. The step of calculating the volume surrounded by the tetrahedron whose apex is the coordinate of each particle is the volume after expansion, and
A method for calculating parameters of a crosslinked polymer model, which comprises a step of calculating the degree of swelling based on the volume after expansion and the volume in a solvent-free state. The method according to claim 1, wherein the _{effective molecular chain density Ne} is calculated by the formula (1) using the preset interaction parameter χ of the polymer and the solvent and the swelling degree φ _saturation.

For the preset cross-linked polymer model, the coordinates of each particle constituting the cross-linked polymer model are set based on the calculation results of the molecular dynamics calculation in the equilibrium state under the analysis conditions including the predetermined temperature and pressure. The first coordinate acquisition unit to be acquired and
Delaunay tessellation is performed to generate an aggregate of tetrahedrons having the coordinates of each particle constituting the preset crosslinked polymer model as vertices, and construct the preset crosslinked polymer model. A first volume calculation unit that calculates the volume surrounded by a tetrahedron whose apex is the coordinates of each particle as the volume in a solvent-free state, and
A mixed model in which a predetermined amount of a preset solvent molecular model is added to the crosslinked polymer model is subjected to a solvent based on the calculation result of molecular dynamics calculation in an equilibrium state under analysis conditions including predetermined temperature and pressure. A second coordinate acquisition unit that acquires the coordinates of each molecule that constitutes the crosslinked polymer model after addition, and
Delaunay tessellation is executed to generate an aggregate of tetrahedrons having the coordinates of each particle constituting the crosslinked polymer model after the addition of the solvent as vertices, and the crosslinked polymer model after the addition of the solvent is constructed. A second volume calculation unit that calculates the volume surrounded by the tetrahedron whose vertices are the coordinates of each particle as the expanded volume.
An apparatus for calculating parameters of a crosslinked polymer model, comprising a swelling degree calculation unit for calculating the swelling degree based on the volume after expansion and the volume in a solvent-free state. And interaction parameter χ of polymer and solvent which is set in advance, by using the said swelling degree φ _saturated comprises a cross-linking density calculation unit for calculating an effective molecular chain density N _e by Equation (1), in claim 3 The device described.

A program that causes a computer to execute the method according to claim 1 or 2.

Images