Investigation of the Microscopic Viscoelastic Property for Cross-linked Polymer Network by Molecular Dynamics Simulation
The purpose of this work is to develop a new analytical method for simulating the microscopic mechanical property of the cross-linked polymer system using the coarse-grained molecular dynamics simulation. This new analytical method will be utilized for the molecular designing of the tire rubber compound to improve the tire performances such as rolling resistance and wet traction. First, we evaluate the microscopic dynamic viscoelastic properties of the cross-linked polymer using coarse-grained molecular dynamics simulation. This simulation has been conducted by the coarse-grained molecular dynamics program in the OCTA) (http://octa.jp/). To simplify the problem, we employ the bead-spring model, in which a sequence of beads connected by springs denotes a polymer chain. The linear polymer chains that are cross-linked by the cross-linking agents express the three-dimensional cross-linked polymer network. In order to obtain the microscopic dynamic viscoelastic properties, oscillatory deformation is applied to the simulation cell. By applying the time-temperature reduction law to this simulation result, we can evaluate the dynamic viscoelastic properties in the wide deformational frequency range including the rubbery state. Then, the stress is separated into the nonbonding stress and the bonding stress. We confirm that the contribution of the nonbonding stress is larger at lower temperatures. On the other hand, the contribution of the bonding stress is larger at higher temperatures. Finally, analyzing a change of microscopic structure in dynamic oscillatory deformation, we determine that the temperature/frequency dependence of bond stress response to a dynamic oscillatory deformation depends on the temperature dependence of the average bond length in the equilibrium structure and the temperature/frequency dependence of bond orientation. We show that our simulation is a useful tool for studying the microscopic properties of a cross-linked polymer.Abstract
(a) A snapshot of cross-linked polymer network. The light blue lines indicate the polymer chains. The orange spheres indicate cross-linking point. (b) The distribution of the number of beads (molecular weight) between neighboring cross-linking points in our models. The number of beads between neighboring cross-links has a distribution.
The radial distribution for all monomers at T=0.5,0.9,2.5 [T]. There are few peaks, which become broader with increasing temperature.
The logarithm of the mean square displacement of all beads log(g1(t)) vs log(time[τ]) at T=0.5–3.5 [T]. The increase of the mean square displacement means the increase of the mobility.
The master curve of the mean square displacement obtained by the time-temperature reduction law. The slope of 1/4 implies the reptation time regime.
The temperature dependence of the shift factor and the WLF equation (C1=2.17, C2=1.26). Our shift factor is in excellent agreement with WLF equation except T=0.5 [T].
(a) The frequency dependence of E′, tan δ in various temperatures obtained by simulation. (b) The experimental results for the cross-linked polyisoprene rubber. The conditions are γ0=0.003, ν=0.5–100 Hz, and temperature=233–273 K. Our simulation results indicate a tendency similar to the experimental results.
(a) The frequency dependence of two of the viscoelastic properties shifted according to the time-temperature reduction law, obtained by simulation. The master curve has the wide frequency range of 10−5 to 10−1 [1/τ]. (b) The experimental results shifted according to the time-temperature reduction law for the cross-linked polyisoprene rubber. We consider that the frequency range in our simulation is roughly equivalent to 102–105 Hz.
The time dependence of the distance between the neighboring cross-linking points in the direction of the oscillation and the strain; T=0.5 [T], ν=0.01 [1/τ]. This result is the average data in the same phase.
(a) The time dependence of the bonding stress at T=0.9,1.5,2.5 [T] and strain. (b) The time dependence of the nonbonding stress and strain at T=0.9,1.5,2.5 [T] and strain.
(a) The change of the average bond length and (b) the change of the distribution of the bond length by oscillatory deformation; T=0.9 [T], ν=0.01 [1/τ]. The average bond length and the distribution of the bond length change little by this oscillatory deformation.
The time dependence of the bond orientation order parameter to the z-axis and strain; T=0.9 [T], ν=0.01 [1/τ]. The bond orientation delays from the phase change in strain.
(a) The frequency dependence of the amplitude of the bond orientation S0 shifted according to the time-temperature reduction law. (b) The frequency dependence of the phase delay of the bond orientation δ shifted according to the time-temperature reduction law. These results are shifted by the same shift factor as that of the mean square displacement to obtain the master curves at the standard temperature 0.9 [T].