Editorial Type:
Article Category: Research Article
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Online Publication Date: 07 Jan 2020

Mechanisms of Mechanical Behavior of Filled Rubber by Coarse-Grained Molecular Dynamics Simulations

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Page Range: 78 – 106
DOI: 10.2346/tire.20.160117
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ABSTRACT:

We reproduced mechanical behaviors, such as the reinforcement effect, hysteresis, and stress softening, of filled rubber under cyclic deformations using coarse-grained molecular dynamics simulations. We measured polymer density distribution in the nonload equilibrium state and conformational changes in polymer chains during deformation for dispersed and aggregated filler structures. We found that the polymer–filler attractive interactions increase the polymer density in the vicinity of fillers and decrease the polymer density in the other regions. The polymer bonds that connect polymer particles away from fillers are extended when the polymer density decreases. This alteration increases the modulus of the polymer phase, and the reinforcement effect appears. For aggregated filler structures, the polymer chains interacting with adjacent fillers act as a bridge between these fillers and increase the modulus, especially when the strain is low.

To test the mechanisms of hysteresis and stress softening, we measured the changes in the polymer paths. A polymer path is the minimal path of polymer networks between two fillers; in other words, it is the “bridge” that connects two fillers. We found that the polymer paths increase in length, especially during primary loading, because of polymer adsorption/desorption on the filler surface to adjust the change of filler positions. It was also found that the influence of the filler structure diminishes in the first loading. During subsequent unloading, a long path does not become a short path again but will be folded even though the filler distance reduces. Hence, the change in the polymer paths in the second cycle is smaller than that in the first cycle because the polymer path is just unfolded. We confirmed the hysteresis and stress-softening result from these conformational changes. In this article, we also discuss the recovery mechanism for stress softening and the history dependence.

Keywords: coarse-grained molecular dynamics; filled rubber; cyclic deformation; hysteresis; interaction; polymer network; Mullins effect; dispersed/aggregated filler
FIG. 1
FIG. 1

Example of the measured stress–strain curves of a carbon black–filled rubber and an unfilled rubber at elongation rate of 10 mm/min.

FIG. 2
FIG. 2

Ratio of the square of the radius of gyration Rg2 and the square of the end-to-end distance of the molecule.

FIG. 3
FIG. 3

Cross–linking bond distribution as a function of the distance from the filler surface.

FIG. 4
FIG. 4

Microstructure of filled rubber: (a) image observed by electron microscope, (b) binarized image of (a).

FIG. 5
FIG. 5

Filler model for coarse-grained molecular dynamics: (a) dispersed structure model, (b) aggregated structure model.

FIG. 6
FIG. 6

Simulation procedure.

FIG. 7
FIG. 7

Stress–strain curves of an unfilled rubber and filled rubbers obtained by coarse-grained molecular dynamics simulation. Polymer networks of all models are cross-linked. (a) All models. (b) Unfilled models and filled models with weak interaction (ɛ = 1).

FIG. 8
FIG. 8

Polymer density distribution of the unfilled rubber and the filled rubber with ɛ = 15 in a nonload state.

FIG. 9
FIG. 9

Dependence of polymer density on the strength of the interaction between filler and polymer.

FIG. 10
FIG. 10

Dependence of the bond force and bond orientation on the strength of the interaction.

FIG. 11
FIG. 11

Schematic view of the polymer path (called the bridged filler path; BFP). The red lines are the polymer path. The black lines are the polymer network. The yellow region, the black region, and the blue region are the matrix, the filler, and the interaction region between the filler and the polymer, respectively.

FIG. 12
FIG. 12

Fraction of BFP as a function of the nominal strain during the first loading.

FIG. 13
FIG. 13

Decomposed stress–strain curves into the bond stress and the nonbond stress.

FIG. 14
FIG. 14

Stress–strain curves of the aggregated structure and the dispersed structure.

FIG. 15
FIG. 15

Distance between the filler particles in the aggregated models.

FIG. 16
FIG. 16

Schematic view of the bridged filler area (BFA). The shaded area is the BFA. The black region and the blue region are the filler and the interaction regions between the filler and the polymer, respectively.

FIG. 17
FIG. 17

Area of BFA as a function of the nominal strain. Comparison of the BFA with the nominal stress for the same strains.

FIG. 18
FIG. 18

Matching percentage of BFP for the aggregated structure and the dispersed structure as a function of nominal strain: percentage of BFP consisting of the same bonds of every strain at 0.05.

FIG. 19
FIG. 19

Fraction of the short BFP consisting of four bonds or fewer.

FIG. 20
FIG. 20

Average bond force of bonds constituting BFP.

FIG. 21
FIG. 21

Fraction of BFP during the first loading, first unloading, and second loading as a function of the nominal strain.

FIG. 22
FIG. 22

Conformational change of BFP under cycle deformation. The green portion and the line are the polymer particle and the bond constituting the BFP in the nonloaded state, respectively. The purple portion and the lines are the BFP at the end of the first loading.

FIG. 23
FIG. 23

Decomposed stress–strain curves into the bond stress and the nonbond stress: (a) first cycle, (b) second cycle.

FIG. 24
FIG. 24

Stress–strain curves of the aggregated structure and the dispersed structure: (a) first cycle, (b) second cycle.

FIG. 25
FIG. 25

Fraction of BFP during cyclic deformation as a function of nominal strain: (a) dispersed structure, (b) aggregated structure.

FIG. 26
FIG. 26

Matching percentage of BFP: (a) first loading, (b) second loading.

FIG. 27
FIG. 27

Bond force of BFP: (a) first loading, (b) second loading.

FIG. 28
FIG. 28

Distance from the current filler position to initial position.

FIG. 29
FIG. 29

Effect of repulsive relaxation: (a) stress–strain curves, (b) fraction of BFP.

FIG. 30
FIG. 30

Representative behavior of the Mullins effect.

FIG. 31
FIG. 31

Results of cyclic deformation simulation: (a) stress–strain curves, (b) fraction of BFP.

FIG. 32
FIG. 32

Snapshots of filler structures. The upper snapshots are the initial structures, and the lower snapshots are the deformed states stretched to a strain of 1.0. (a) Floating type. (b) Connected type.

FIG. 33
FIG. 33

Stress–strain curves.

FIG. 34
FIG. 34

Stress–strain curves.

Contributor Notes

2 Corresponding author. Email: takashi.kojima02@y-yokohama.com
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