JP2013250242A - Apparatus, method and program for calculating viscoelasticity of polymer - Google Patents

Abstract

PROBLEM TO BE SOLVED: To calculate viscoelastic characteristics by molecular dynamics simulation without using a linear response theory.SOLUTION: A system for modeling polymers is prepared (S2), the system is equilibrated at a predetermined temperature (S3), a frequency is determined (S4), and while applying distortion of amplitude to the system, stress including thermal noise is calculated by molecular dynamics simulation (S5). A distortion-stress cross-correlation function is calculated as a time correlation function from the stress and the distortion (S6), and the amplitude of stress obtained by removing the thermal noise from the distortion-stress cross-correlation function and a phase difference between the stress and the distortion are calculated to obtain an index value concerned with viscoelasticity of polymers (S7).

Description

  The present invention relates to a calculation apparatus, a method and a program for calculating viscoelasticity of a polymer.

  When developing polymer materials such as rubber, it is required to predict physical properties such as viscoelastic properties, and the physical properties of polymer materials are predicted by computer simulation.

  For example, Patent Document 1 discloses obtaining various physical properties (including viscoelastic properties) of a polymer material using phase space information obtained from molecular dynamics simulation of the polymer material and linear response theory. ing. However, the linear response theory is a calculation theory of a physical quantity related to a response to a minute displacement given to a substance, and cannot handle a change in physical quantity depending on the displacement quantity. That is, in the linear response theory, a minute external force is assumed, and only the susceptibility in the linear region can be calculated (for example, R. Kubo and K. Tomita, Journal of Physical Society of Japan, vol. 9, pp. 888-919 (1954), Statistical Mechanics, Donald A. McQuarrie, University Science Books (2000), Chapter 22).

WO97 / 06496

  In view of the above, the present invention provides a polymer viscoelasticity calculation apparatus, a method thereof, and a program thereof capable of calculating viscoelasticity characteristics by molecular dynamics simulation without using linear response theory. Objective.

  A polymer viscoelasticity calculation apparatus according to the present invention includes a system creation unit that creates a system modeling a polymer, an equilibration unit that equilibrates the system at a predetermined temperature, and a frequency of strain applied to the system. A frequency determining unit for determining; a stress calculating unit for calculating stress including thermal noise by molecular dynamics simulation while applying strain of a predetermined amplitude at the frequency to the system; and time correlation from the stress including the thermal noise and the strain A cross-correlation function calculation unit for calculating a strain-stress cross-correlation function as a function, and calculating a stress amplitude obtained by removing the thermal noise from the strain-stress cross-correlation function and a phase difference between the stress and the strain. A first index value calculation unit for obtaining an index value related to the viscoelasticity of the molecule.

  According to the present invention, since the stress response is directly calculated by applying strain to the system, instead of linear response theory, viscoelasticity can be calculated for any strain. In molecular dynamics simulations, stress is strongly influenced by thermal noise, but the influence of thermal noise can be eliminated by calculating the strain-stress cross-correlation function, so viscoelasticity can be calculated accurately. it can.

It is a block diagram of the computing device concerning a 1st embodiment. It is a flowchart of the calculation apparatus which concerns on 1st Embodiment. It is a block diagram of the calculation apparatus which concerns on 2nd Embodiment. It is a flowchart of the calculation apparatus which concerns on 2nd Embodiment. It is a figure which shows an example of a polymer model. 3 is a graph showing a change with time in distortion in Example 1. 3 is a graph showing the time change of stress in Example 1. 3 is a graph showing a change over time in a strain-stress correlation in Example 1. 6 is a graph showing a result of fitting strain-stress correlation in Example 1. 3 is a graph showing a time change of stress autocorrelation in Example 1. 6 is a graph showing a stress autocorrelation fitting result in Example 1. 6 is a graph showing a stress fitting result in Example 1. 6 is a graph showing a temporal change in distortion in Example 2. 6 is a graph showing a time change of stress in Example 2. 6 is a graph showing a change over time in a strain-stress correlation in Example 2. It is a graph which shows the fitting result of the distortion-stress correlation in Example 2. 6 is a graph showing a time change of stress autocorrelation in Example 2. It is a graph which shows the fitting result of the stress autocorrelation in Example 2. 6 is a graph showing a stress fitting result in Example 2. It is a graph which shows the relationship between a frequency and a storage elastic modulus as a viscoelasticity calculation result. It is a graph which shows the relationship between a frequency and tan-delta as a viscoelasticity calculation result.

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

[First Embodiment]
A polymer viscoelasticity calculation apparatus 10 according to the first embodiment includes an input unit 12, a system creation unit 14, a balancing unit 16, a frequency determination unit 18, a stress calculation unit 20, and a cross-correlation, as shown in FIG. A function calculation unit 22, a first index value calculation unit 24, a determination unit 26, and an output unit 28 are included.

  The computing device 10 can also be realized by using, for example, a general-purpose computer having a mouse and a keyboard as basic hardware. That is, the input unit 12, the system creation unit 14, the balancing unit 16, the frequency determination unit 18, the stress calculation unit 20, the cross-correlation function calculation unit 22, the first index value calculation unit 24, the determination unit 26, and the output unit 28 It can be realized by causing a processor mounted on the computer to execute a program. At this time, the computing device 10 may be realized by installing the program in a computer in advance, or may be stored in a storage medium such as a CD-ROM or distributed through the network. This program may be realized by appropriately installing it on a computer.

  Hereinafter, the configuration and functions of the above-described units will be described in order.

[1] Input unit 12
The input unit 12 acquires information on the structure of the polymer to be calculated. The information to be acquired includes various data necessary for creating a polymer model by the system creation unit 14. For example, in the case of a structure of a repeating unit such as a monomer constituting a polymer or a plurality of types of units. Includes the composition ratio, the degree of polymerization, and the number of molecular chains. The data input at the input unit 12 includes the temperature and pressure of the system necessary for the molecular dynamics simulation, the amplitude of strain applied to the system, the frequency range (a plurality of frequencies for calculation), and the like. .

  The polymer to be calculated may be a crystalline polymer or an amorphous polymer, and is not particularly limited, but is preferably an amorphous polymer. Amorphous polymers include, for example, homopolymers or copolymers of diene monomers such as butadiene, isoprene, and chloroprene, acrylic acid, methacrylic acid or esters thereof, vinyl acetate, vinyl chloride, styrene, acrylonitrile, and other vinyls. Examples include homopolymers or copolymers of monomeric monomers, and copolymers of these diene monomers and vinyl monomers, and more preferred are diene rubber materials.

[2] System creation unit 14
The system creation unit 14 creates a system in which a polymer is modeled using the information acquired by the input unit 12.

  Examples of the modeling method include various models capable of performing molecular dynamics simulation, and are not particularly limited. For example, in addition to the all-atom model, a macroscopic model of a polymer such as a bead spring model or a united atom model can be used. In the bead spring model, several monomer units corresponding to the kuhn length are regarded as one bead (segment), and are modeled by connecting with a virtual spring. The united atom model is a virtual atom model in which hydrogen is included in a heavy atom (for example, carbon) and handled as one atom (mass point).

  FIG. 5 shows an example of a polymer bead spring model. Reference 1: Dynamics of entangled linear polymer melts: A molecular-dynamics simulation, Kurt Kremer, Gray S. Grest, Journal of chemical physics, v .92, pp.5057-pp.5086 (1990).

  When the polymer to be modeled is an amorphous polymer, a model system, that is, a system is created at a temperature higher than the glass transition temperature.

[3] Equilibration unit 16
The equilibration unit 16 equilibrates the system at a predetermined temperature. The predetermined temperature is set to a temperature at which viscoelastic characteristics are desired. In the case of an amorphous polymer, a temperature higher than the glass transition temperature is set.

  For example, in the case of an NPT ensemble where the number of particles, pressure, and temperature are constant, if the temperature is set, the system will equilibrate with time development. What is necessary is just to judge that it equilibrated in the stage which passed a certain fixed time.

[4] Frequency determining unit 18
The frequency determination unit 18 determines the frequency of distortion applied to the system. As the frequency, a predetermined frequency for which the viscoelastic characteristic is desired is determined, and is selected from a plurality of frequencies input by the input unit 12. For example, you may select sequentially from the low frequency side of a some frequency, and may select sequentially from the high frequency side.

[5] Stress calculator 20
The stress calculation unit 20 performs a molecular dynamics simulation while giving a strain having a predetermined amplitude at the frequency to the system, and calculates a stress at each time.

  The molecular dynamics simulation is known, for example, as described in Reference Document 2 (Susumu Okazaki, Noriyuki Yoshii, “Basics of Computer Simulation”, 2nd edition, published by Kagaku Dojin, 2011). For example, when a polymer is composed of an atomic model, several potential functions are set between the atoms, the atomic position is gradually shifted according to Newton's equation of motion, and the atomic position changes momentarily in this way. It is molecular dynamics simulation that simulates the situation where reaches the statistical equilibrium state.

Specifically, while maintaining the temperature T of the system constant, distorting the sine wave amplitude X ₀ and frequency ω is represented by the following formula (1) in the system. The number of repetitions of the period giving such periodic distortion is not particularly limited, but is usually about 5 to 50 times.

Under such conditions, a molecular dynamics simulation is performed to calculate a time-dependent stress response B (t). The stress value obtained by the molecular dynamics calculation includes thermal noise, that is, it is the sum of the strong thermal noise N (t) and the true stress response value σ (t), so that the stress response B (t ) Is represented by the following formula (2).

Here, if a sufficiently long periodical strain is applied, the true stress response value σ (t) starts to vibrate at the frequency ω, so that the stress response B (t) has a stress amplitude of σ _0. And the phase difference between strain and stress is represented by δ and expressed by the following formula (3).

  Note that the amplitude of distortion applied to the system is not particularly limited, and can be, for example, 0.1 to 0.6 times the length of the balanced state in the X-axis direction of the system. That is, when the length of the balanced state is 100, distortion having an amplitude of 10 to 60 can be applied.

[6] Cross-correlation function calculator 22
The cross-correlation function calculation unit 22 calculates a strain-stress cross-correlation function as a time correlation function from the stress and strain including the thermal noise obtained above.

The viscoelasticity of the polymer material model is determined by the storage elastic modulus and the phase difference δ. Here, the storage elastic modulus is expressed by the following formula (4) by the amplitude of strain and the amplitude of stress.

Here, σ ₀ and tan δ can be obtained by fitting to a sine curve using the above equation (3). However, since the stress is strongly influenced by thermal noise in the molecular dynamics simulation, if the stress value B (t) of Equation (3) is used directly, the accuracy becomes worse due to the thermal noise N (t). Therefore, in the present embodiment, a time correlation function of strain and stress response, that is, a strain-stress cross-correlation function is calculated to remove thermal noise. The strain-stress cross-correlation function is given by the following equation (5), where T is the delay time.

  Here, C (t) is a cross-correlation function between distortion and thermal noise, and the absolute value is usually small.

[7] First index value calculation unit 24
The first index value calculation unit 24 calculates the amplitude σ ₀ of the stress from which the thermal noise is removed and the phase difference δ between the stress and the strain from the strain-stress cross-correlation function, thereby indicating an index related to the viscoelasticity of the polymer. Calculate the value. In the present embodiment, the storage elastic modulus G ′ (ω) and tan δ represented by the above formula (4) are obtained as the index values.

In calculating the stress amplitude σ ₀ and the phase difference δ, it is preferable to perform fitting using the above equation (5). That is, the first index value calculation unit 24 uses a strain-stress cross-correlation function to fit a sinusoidal curve from which the cross-correlation function C (t) of strain and thermal noise is removed, whereby the stress amplitude σ ₀ and The phase difference δ is calculated. The sine curve here includes a cosine curve. More specifically, in the present embodiment, the time change of stress is fitted by the least mean square method using the following equation (6).

As a result, since the stress amplitude σ ₀ and the phase difference δ are obtained, the storage elastic modulus G ′ (ω) and tan δ can be calculated from the above equation (4) using the strain amplitude X _0. .

[8] Determination unit 26
The determination unit 26 determines whether or not index values related to viscoelasticity are calculated at all frequencies among the plurality of frequencies set by the input unit 12, and until the calculation is performed at all frequencies, the balancing unit 16, Control is performed so that the processes of the frequency determination unit 18, the stress calculation unit 20, the cross-correlation function calculation unit 22, and the first index value calculation unit 24 are repeated. That is, if calculation is not performed for all frequencies, control is performed so that the above processing is repeated. If calculation is performed for all frequencies, the calculation ends.

[9] Output unit 28
The output unit 28 outputs the viscoelastic characteristics of the polymer obtained as described above. The viscoelastic characteristics can be output by displaying on a display or printing with a printer.

  Next, the operation state of the computing device 10 according to the present embodiment will be described based on the flowchart of FIG.

  In step S1, the input unit 12 acquires information including the molecular structure of the polymer to be calculated.

  Next, in step S2, the system creation unit 14 creates a system that models the polymer using the acquired information. Then, the process proceeds to step S3.

  In step S3, the equilibration unit 16 equilibrates the system at a temperature at which viscoelastic characteristics are desired. Then, the process proceeds to step S4.

  In step S4, the frequency determination unit 18 determines the frequency of distortion applied to the system. The frequency of distortion is selected from a plurality of frequencies input in step S1, and for example, when calculating sequentially from the low frequency side, the lowest frequency is selected first. Then, the process proceeds to step S5.

  In step S5, the stress calculation unit 20 performs a molecular dynamics simulation while applying a periodic strain having a predetermined amplitude to the system at the frequency determined in step S4 over a predetermined period, and calculates a stress response at each time. To do. The stress obtained by this includes thermal noise as described above.

  Next, in step S6, the cross-correlation function calculation unit 22 calculates the strain-stress cross-correlation function represented by the above formula (5) as a time correlation function from the stress and strain including the thermal noise obtained above. To do. Then, the process proceeds to step S7.

In step S7, the first index value calculation unit 24 calculates the stress amplitude σ ₀ and the phase difference δ from which the thermal noise has been removed from the strain-stress cross-correlation function, and provides the index value relating to the viscoelasticity of the polymer. Calculate storage modulus and tan δ. Then, the process proceeds to step S8.

  In step S8, the determination unit 26 determines whether index values related to viscoelasticity have been calculated at all the frequencies among the plurality of frequencies input in step S1, and if not calculated, the determination unit 26 determines whether the calculation is performed in steps S3, S4, Control is performed so as to repeat the processes of S5, S6, and S7. If calculation has been performed for all frequencies, the calculation ends and the process proceeds to step S9.

  In step S9, the output unit 28 outputs the viscoelastic characteristics of the polymer obtained as described above.

  According to the present embodiment configured as described above, since the stress response is directly calculated by applying strain to the system instead of linear response theory, viscoelasticity can be calculated for any strain, and the strain dependence of viscoelasticity Sex can be simulated. In molecular dynamics simulations, stress is strongly affected by thermal noise, but by calculating the strain-stress cross-correlation function, the effect of thermal noise can be eliminated, so the stress response to a given strain is precise. Thus, accurate viscoelastic characteristics can be obtained.

[Second Embodiment]
As shown in FIG. 3, the polymer viscoelasticity calculation apparatus 10A according to the second embodiment includes an input unit 12, a system creation unit 14, an equilibration unit 16, a frequency determination unit 18, a stress calculation unit 20, and a cross-correlation. A function calculation unit 22, a first index value calculation unit 24, an autocorrelation function calculation unit 30, a second index value calculation unit 32, an evaluation unit 34, a determination unit 26, and an output unit 28 are included. That is, the calculation device 10A according to the second embodiment is obtained by adding an autocorrelation function calculation unit 30, a second index value calculation unit 32, and an evaluation unit 34 to the calculation device 10 according to the first embodiment. Other configurations are common. Hereinafter, the added components will be described in detail.

[10] Autocorrelation function calculator 30
The autocorrelation function calculator 30 calculates a stress autocorrelation function as a time correlation function from the stress including thermal noise obtained by the stress calculator 20. The stress autocorrelation function is given by the following equation (7), where T is the delay time.

  Here, D (t) is an autocorrelation function of thermal noise, and has a strong peak near the origin and rapidly attenuates.

[11] Second index value calculation unit 32
The second index value calculation unit 32 calculates an index value related to the viscoelasticity of the polymer by calculating the stress amplitude σ ₀ from which the thermal noise is removed from the stress autocorrelation function. In this embodiment, the storage elastic modulus G ′ (ω) represented by the above formula (4) is obtained as the index value.

When calculating the amplitude σ _{0 of the} stress, it is preferable to perform fitting using the above equation (7). That is, the second index value calculation unit 32 calculates the stress amplitude σ ₀ by fitting to a sine curve from which the auto-correlation function D (t) of the thermal noise is removed using the stress auto-cross-correlation function. . The sine curve here includes a cosine curve. More specifically, in this embodiment, the time change of stress is fitted by the minimum mean square method using the following formula (8).

As described above, D (t) has a strong peak near the origin and rapidly attenuates. Therefore, fitting is performed from the time when D (t) is sufficiently close to zero, so that the true stress response amplitude σ ₀ can be accurately obtained. It can be calculated well. Since the stress amplitude σ ₀ is obtained in this way, the storage elastic modulus G ′ (ω) can be calculated by the above equation (4) using the strain amplitude X ₀ .

[12] Evaluation unit 34
The evaluation unit 34 uses the index value (hereinafter, first index value) obtained by the first index value calculation unit 24 as the index value (hereinafter, second index value) obtained by the second index value calculation unit 32. In contrast, the accuracy is evaluated. The index value to be compared may be the stress amplitude σ ₀ or the storage elastic modulus G ′ (ω).

  More specifically, the evaluation unit 34 compares the first index value and the second index value, and if the difference between the two is within a predetermined range, the accuracy of the viscoelastic property calculated by the first index value calculation unit 24 And control to proceed to the determination step S8. On the other hand, if the difference between the two is not within the predetermined range, it is determined that the accuracy of the viscoelastic property calculated by the first index value calculation unit 24 is not secured, and the process proceeds to the stress calculation step S5, where molecular dynamics simulation is performed. Control is performed so that the stress is calculated again after changing the simulation conditions. That is, in order to increase the accuracy of the simulation, for example, the number of repetitions of the strain period applied to the system is increased, and the stress calculator 20, the cross-correlation function calculator 22, the first until the difference between the two is within a predetermined range. Control is performed so that the processing of the index value calculation unit 24, the autocorrelation function calculation unit 30, and the second index value calculation unit 32 is repeated.

  Here, the predetermined range related to the difference between the first index value and the second index value is not particularly limited. For example, it is a criterion whether the difference between the two is within ± 10% of the first index value. It can be.

  The operation state of the computing device 10A according to the second embodiment is as shown in the flowchart of FIG. 4, and is different from the first embodiment in that steps S10, S11, and S12 are added. Hereinafter, the added steps will be described.

  After the first index value calculation unit 24 calculates the storage elastic modulus G ′ (ω) and tan δ as the first index value in step S7, the autocorrelation function calculation unit 30 is obtained in step S5 in step S10. From the stress including thermal noise, the stress auto-correlation function expressed by the above formula (7) is calculated as a time correlation function. Then, the process proceeds to step S11.

In step S11, the second index value calculation unit 32 calculates the stress amplitude σ ₀ from which the thermal noise is removed from the stress auto-correlation function, and the storage elastic modulus G ′ (ω) as the second index value. Is calculated. Then, the process proceeds to step S12.

  In step S12, the evaluation unit 34 compares the storage elastic modulus as the first index value obtained in step S7 with the storage elastic modulus as the second index value obtained in step S11. If it is within the predetermined range, it is determined that the accuracy of the first index value obtained in step S7 is secured, and the process proceeds to step S8. If it is not within the predetermined range, it is determined that the accuracy is not secured. Proceed to step S5.

  In step S5 that has returned from step S12 in this way, the stress is calculated again after changing the simulation conditions in the molecular dynamics simulation, and the process proceeds to step S6.

  In step S8, the determination unit 26 determines whether or not the first index value obtained in step S7 has been calculated for all frequencies. In step S9, the output unit 28 outputs the first index value obtained as described above as the viscoelastic property of the polymer.

  According to the second embodiment, in addition to the strain-stress cross-correlation function, a stress auto-correlation function is calculated, and the second index value obtained from the stress auto-correlation function is used to obtain the strain-stress cross-correlation function. Since the accuracy of the obtained first index value is evaluated, more accurate viscoelastic characteristics can be obtained.

  In the second embodiment, the determination of the frequency in step S4 is preferably performed by sequentially selecting a plurality of frequencies input in step S1 from the low frequency side. The accuracy of the viscoelastic property calculated by molecular dynamics simulation is generally inferior in noise and lower in accuracy on the lower frequency side. Therefore, if calculation accuracy is ensured on the low frequency side, it is highly possible that calculation accuracy is ensured even on the higher frequency side. For this reason, by calculating from the low frequency side, it is possible to reduce the waste of calculation man-hours by changing the simulation conditions as much as possible.

  In the example illustrated in FIG. 4, when the evaluation unit 34 determines in step S12 that the accuracy is not ensured, the process proceeds to step S5. However, the control may be performed to proceed to step S2. That is, when the difference between the first index value and the second index value is not within the predetermined range, the evaluation unit 34 proceeds to the system creation step S2 and controls to recalculate the stress after reconfiguring the system. Good. For example, since the accuracy can be increased by increasing the number of molecular chains of the polymer model that constitutes the system, the system is increased until the difference between the two is within a predetermined range while increasing the number of molecular chains in this way. Processing of creating unit 14, balancing unit 16, frequency determining unit 18, stress calculating unit 20, cross-correlation function calculating unit 22, first index value calculating unit 24, autocorrelation function calculating unit 30, and second index value calculating unit 32 You may control to repeat.

  In the first and second embodiments, viscoelastic characteristics for a plurality of frequencies are obtained. However, as another embodiment, viscoelastic characteristics for a single frequency may be obtained. In that case, the frequency determination unit 18 determines the frequency acquired by the input unit 12 as the frequency of distortion applied to the system.

  Moreover, in the said 1st and 2nd embodiment, although the viscoelastic characteristic about single amplitude is calculated | required, you may make it obtain | require viscoelastic characteristics about several amplitude as other embodiment. In that case, an amplitude setting unit for setting the amplitude of strain applied to the system is provided, and the stress calculation unit 18 calculates the stress by molecular dynamics simulation while giving the strain of amplitude set by the amplitude setting unit to the system. . Then, it is determined whether or not index values related to viscoelasticity have been calculated with all amplitudes among a plurality of preset amplitudes, and control is performed so as to repeat calculation for acquiring index values until calculation is performed with all amplitudes. A second determination unit may be provided.

  Moreover, in the said 1st and 2nd embodiment, although the viscoelastic characteristic about single temperature is calculated | required, you may make it obtain | require the viscoelastic characteristic about several temperature as other embodiment. In that case, a temperature setting unit for setting the temperature at which the system is equilibrated is provided, and the equilibration unit 16 equilibrates the system at the temperature set by the temperature setting unit, and performs viscoelasticity by molecular dynamics simulation at the temperature. Calculate the index value for. Then, it is determined whether or not index values related to all viscoelasticities among a plurality of preset temperatures have been calculated, and control is performed to repeat calculation for acquiring index values until calculation is performed at all temperatures. 3 determination units may be provided.

  As described above, according to this embodiment, the viscoelastic property of the polymer material can be predicted by molecular dynamics simulation. In general, in elastomer materials represented by rubber, it is important to control the loss to vibration at a specific frequency, and since this loss physical property is determined by viscoelastic properties, the present invention develops a polymer material such as rubber. Useful when doing.

[Example 1]
Example 1 shows that the viscoelastic properties of the polymer can be accurately calculated according to the above embodiment. For molecular dynamics simulation, the public program LAMMPS [Large-Scale Atomic / Molecular Massively Parallel Simulator: Sandia National Laboratory, USA] was used.

  Based on Reference Document 1, an amorphous polymer model composed of 40 polymer chains of 50 beads was created as a polymer bead spring model system. In this system, the binding potential and the non-bonding potential were as shown in the following formula according to Reference 1.

Here, U _FENE (r) is an extended bond potential, and k and R ₀ indicate the bond strength and the extended distance (the distance between bonded particles no longer increases), respectively. Yes. r represents the distance between particles. U _LJ (r) is a Leonard Jones _nonbonding potential, A is a parameter indicating the strength of interaction, and B is a particle size. In equation (9b), a cutoff distance of 2 ^1/6 B is set, and non-bonding interaction between particles at a larger distance is set to zero.

  The system was then equilibrated at a temperature of 1.0 under a constant pressure of 4.85. Here, the unit of each parameter such as temperature and pressure is the Leonard Jones unit.

A frequency is set to 4.0 × 10 ⁻⁴ [1 / τ], and a molecular dynamics simulation is performed while applying a strain of amplitude 0.2 at the frequency over 10 periods as shown in FIG. Calculated. The distortion was given in the x-axis direction in the coordinate system under simulation. The time change of the stress obtained by the molecular dynamics simulation is as shown in FIG. The unit of stress is given by the Leonard Jones parameters σ (length) and ε (energy). In this calculation, both values were 1.0, that is, σ = 1.0 and ε = 1.0. As shown in FIG. 7, the time change of the stress obtained by the molecular dynamics simulation has a large thermal noise.

  Next, a strain-stress cross-correlation function represented by the above formula (5) was calculated from the stress including the thermal noise. The result is as shown in FIG. 8, and the influence of thermal noise can be largely eliminated by the time change of the strain-stress cross-correlation (hereinafter sometimes referred to as “strain-stress correlation”) (that is, the expression In (5), the cross-correlation function between the distortion represented by C (t) and thermal noise is small. Since the dimension of the strain-stress correlation is strain × stress, the unit is the same as the stress.

Next, when the time change of stress was fitted to a sine curve by the least mean square method using the above formula (6), the result was as shown in FIG. 9, and the following results were obtained. Strain amplitude X ₀ = 0.2, stress amplitude σ ₀ = 0.01088 [ε / σ ³ ], frequency (angular frequency) ω = 2π × 4.0 × 10 ⁻⁴ [1 / τ], position Phase difference δ = 1.98048 [rad], residual average square root / amplitude = 0.027248. The storage elastic modulus calculated from this was 0.054322 [ε / σ ³ ], and tan δ was 2.302728.

Moreover, the stress autocross correlation function represented by the above formula (7) was calculated from the stress including the thermal noise. The results are as shown in FIG. 10, and in the time change of the stress auto-correlation, the influence of the thermal noise could be removed as compared with the time change of the stress itself shown in FIG. Next, when the stress change with time was fitted to a sine curve by the least mean square method using the above equation (8), the result was as shown in FIG. 11, and the following results were obtained. Strain amplitude X ₀ = 0.2, stress amplitude σ ₀ = 0.01086 [ε / σ ³ ], frequency (angular frequency) ω = 2π × 4.0 × 10 ⁻⁴ [1 / τ], remaining Square root of difference average / amplitude = 1.217957. The storage elastic modulus calculated from this was 0.054318 [ε / σ ³ ]. This value was in good agreement with the storage elastic modulus = 0.054232 obtained from the strain-stress relationship.

On the other hand, in order to show the effect of the above embodiment, as a comparative example, the viscoelastic characteristics were calculated by fitting as they were without calculating the correlation function for the time change of the stress itself shown in FIG. That is, when the stress over time was fitted by the least mean square method using the following equation (10), the white curve shown in FIG. 12 was obtained. The results are as follows: strain amplitude X ₀ = 0.2, stress amplitude σ ₀ = 0.24086 [ε / σ ³ ], frequency (angular frequency) ω = 2π × 4.0 × 10 ⁻⁴ [1 / τ The phase difference δ = 2.06297 [rad] and the residual average square root / amplitude = 30.79888. As a result, the residual is very large compared to the value to be fitted, and the reliability is poor. The phase difference δ is close to that obtained from the strain-stress cross-correlation function, but the storage elastic modulus has a difference of 10 times or more. This is presumably because a very large noise compared to the response stress adversely affects the fitting accuracy.

[Example 2]
In order to show that the strain dependency of viscoelasticity can be calculated according to the present embodiment, the same calculation as in Example 1 was performed except that the strain amplitude was set to 0.5 in Example 2.

  Specifically, the pressure, temperature, and frequency were set to be the same as in Example 1, and a molecular dynamics simulation was performed while applying a strain of amplitude 0.5 shown in FIG. . The time change of the stress obtained by the molecular dynamics simulation was as shown in FIG.

Next, a strain-stress cross-correlation function represented by the above formula (5) was calculated from the stress including the thermal noise. The result is as shown in FIG. 15, and the influence of thermal noise could be largely eliminated by the time change of the strain-stress cross-correlation. Next, when the time change of stress was fitted to a sine curve by the least mean square method using the above formula (6), it was as shown in FIG. 16, and the following results were obtained. Strain amplitude X ₀ = 0.5, stress amplitude σ ₀ = 0.130025 [ε / σ ³ ], frequency (angular frequency) ω = 2π × 4.0 × 10 ⁻⁴ [1 / τ], position Phase difference δ = 1.734045 [rad], square root of residual average / amplitude = 0.013949. The storage elastic modulus calculated from this was 0.26005 [ε / σ ³ ], and tan δ was 6.071126.

Moreover, the stress autocross correlation function represented by the above formula (7) was calculated from the stress including the thermal noise. The result was as shown in FIG. Further, when the time change of stress was fitted to a sine curve by the least mean square method using the above equation (8), it was as shown in FIG. 18, and the following results were obtained. Strain amplitude X ₀ = 0.5, stress amplitude σ ₀ = 0.123872 [ε / σ ³ ], frequency (angular frequency) ω = 2π × 4.0 × 10 ⁻⁴ [1 / τ], remaining Square root of difference average / amplitude = 0.101784. The storage elastic modulus calculated from this was 0.247745 [ε / σ ³ ]. This value was in good agreement with the storage elastic modulus = 0.26005 obtained from the strain-stress relationship.

On the other hand, in order to show the effect of the above embodiment, as a comparative example, the viscoelastic characteristics were calculated by fitting as they were without calculating the correlation function for the time change of the stress itself shown in FIG. That is, when the time change of stress was fitted by the least mean square method using the above formula (10), a white curve shown in FIG. 19 was obtained. The results are as follows: strain amplitude X ₀ = 0.5, stress amplitude σ ₀ = 0.547941 [ε / σ ³ ], frequency (angular frequency) ω = 2π × 4.0 × 10 ⁻⁴ [1 / τ The phase difference δ = 1.72409 [rad] and the residual average square root / amplitude = 0.60519. Although the residual is smaller than in the case of strain 0.2, it still has a value several tens of times higher than the strain-stress correlation, and is not reliable. The phase difference δ is close to that obtained from the strain-stress cross-correlation function, but the storage elastic modulus has a difference of about 5 times. This is presumably because a very large noise compared to the response stress adversely affects the fitting accuracy.

  According to Examples 1 and 2 described above, as shown in Table 1 below, it is shown that viscoelasticity changes depending on the magnitude of strain applied to the system, that is, viscoelasticity data depending on strain is obtained. It was.

[Example 3]
About the amorphous polymer model which consists of 50 polymer chains of 100 beads produced based on the above-mentioned reference 1, the temperature is 1.0 and the frequency is 1.0 × 10 ^{−1 to} 3.91 × 10 ⁻⁴ [1. / Τ], the same calculation as in Example 1 was performed while applying a strain of amplitude 0.2 in the x-axis direction over 10 periods. The binding potential and non-bonding potential for the amorphous polymer model were as shown in the above formulas (9a) and (9b).

  The results are as shown in Table 2 and FIGS. 20 and 21 below, and the storage elastic modulus and tan δ were calculated for each frequency.

  Although one embodiment of the present invention has been described above, this embodiment is presented as an example and is not intended to limit the scope of the invention. These novel embodiments can be implemented in various other forms, and various omissions, replacements, and changes can be made without departing from the spirit of the invention. These embodiments and modifications thereof are included in the scope and gist of the invention, and are included in the invention described in the claims and the equivalents thereof.

DESCRIPTION OF SYMBOLS 10 ... Calculation apparatus 12 ... Input part 14 ... System preparation part 16 ... Equilibration part 18 ... Frequency determination part 20 ... Stress calculation part 22 ... Cross correlation function calculation part 24 ... 1st index value calculation part 26 ... Determination part 30 ... Self Correlation function calculation unit 32 ... second index value calculation unit 34 ... evaluation unit

Claims (6)

A system creation unit for creating a system modeling a polymer;
An equilibration unit for equilibrating the system at a predetermined temperature;
A frequency determining unit that determines a frequency of distortion applied to the system;
A stress calculation unit for calculating a stress including thermal noise by molecular dynamics simulation while applying strain of a predetermined amplitude at the frequency to the system;
A cross-correlation function calculator for calculating a strain-stress cross-correlation function as a time correlation function from the stress including the thermal noise and the strain;
A first index value calculation unit that calculates an amplitude of stress obtained by removing the thermal noise from the strain-stress cross-correlation function and a phase difference between the stress and strain to obtain an index value related to viscoelasticity of the polymer;
A viscoelasticity calculation apparatus for polymers, comprising:   The first index value calculation unit calculates the amplitude and the phase difference of the stress by fitting to a sine curve obtained by removing the cross-correlation function of strain and thermal noise using the strain-stress cross-correlation function. The viscoelasticity calculation device according to claim 1, wherein   It is determined whether or not the index value related to the viscoelasticity is calculated at all frequencies among a plurality of preset frequencies, and until the calculation is performed at all frequencies, the balancing unit, the frequency determining unit, the stress calculating unit, The viscoelasticity calculation device according to claim 1, further comprising a determination unit that controls to repeat the processes of the cross-correlation function calculation unit and the first index value calculation unit. An autocorrelation function calculation unit for calculating a stress autocorrelation function as a time correlation function from the stress including the thermal noise obtained by the stress calculation unit;
A second index value calculation unit that calculates an amplitude of stress obtained by removing the thermal noise from the stress autocorrelation function to obtain an index value related to viscoelasticity of the polymer;
An evaluation unit that evaluates accuracy by comparing the index value obtained by the first index value calculation unit with the index value obtained by the second index value calculation unit;
The viscoelasticity calculation device according to any one of claims 1 to 3, wherein the viscoelasticity calculation device is provided. A system creation step for creating a system modeling a polymer;
Equilibrating the system at a predetermined temperature; and
A frequency determining step for determining a frequency of distortion applied to the system;
A stress calculation step of calculating stress including thermal noise by molecular dynamics simulation while applying strain of a predetermined amplitude at the frequency to the system;
A cross-correlation function calculating step for calculating a strain-stress cross-correlation function as a time correlation function from the stress including the thermal noise and the strain;
A first index value calculation step of calculating an amplitude value of stress obtained by removing the thermal noise from the strain-stress cross-correlation function and a phase difference between the stress and strain to obtain an index value related to viscoelasticity of the polymer;
A method for calculating the viscoelasticity of a polymer, comprising: A program for calculating viscoelasticity of a polymer,
On the computer,
A system creation function to create a system modeling a polymer;
An equilibration function for equilibrating the system at a predetermined temperature;
A frequency determining function for determining a frequency of distortion applied to the system;
A stress calculation function for calculating stress including thermal noise by molecular dynamics simulation while applying strain of a predetermined amplitude at the frequency to the system;
A cross-correlation function calculation function for calculating a strain-stress cross-correlation function as a time correlation function from the stress including the thermal noise and the strain;
A first index value calculation function for obtaining an index value related to viscoelasticity of the polymer by calculating an amplitude of a stress obtained by removing the thermal noise from the strain-stress cross-correlation function and a phase difference between the stress and the strain;
A viscoelasticity calculation program for polymers.