Tuning the visco-elasticity of elastomeric polymer materials via flexible nanoparticles: insights from molecular dynamics simulation

Abstract

Tuning the viscoelasticity of polymeric materials by incorporating nanoparticles (NPs) has received considerable scientific and technological interests. Contrary to increasing the energy dissipation for damping materials, here we direct our attention to study how to decrease the energy dissipation of elastomer nanocomposites (ENCs) under periodic dynamic loading–unloading cycles. Through molecular dynamics simulation, we firstly simulate the pure cis-polybutadiene (cis-PB) system, by calculating the mean-squared end-to-end distance and the radius of gyration as a function of the chain length, the diffusion coefficient of polymer chains as a function of the temperature, the glass transition temperature, the stress–strain curves at different strain rates and temperatures, the tension–recovery and compression–recovery curves at various cross-linking densities. These results validate the accuracy of the united atom model and force-field of cis-PB. Then we show that the incorporation of flexible nanoparticles (NPs) such as graphene nanoribbons and carbon nanotubes can effectively decrease the dynamic hysteresis loss, by taking advantage of the reversible mechanical deformation of the anisotropic NPs. This effect can be further strengthened by the stronger interfacial interaction, higher loading and larger size of this kind of NPs. The underlying reason stems from the synergistic motion between the NPs and their surrounding polymer chains, leading to much smaller internal friction. This work may open up potential opportunities to fabricate high-performance polymer nanocomposites, such as energy-saving ENCs tailored for tire tread.

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1. Introduction

Incorporating nanoparticles (NPs) into polymer matrices is an effective means to modulate the viscoelasticity of polymer nanocomposites (PNCs).^1,2 For instance, Koratkar et al.³ have shown that the epoxy thin film containing dense packing of multi-walled CNTs exhibits strong viscoelastic behavior, the loss factor of which can be increased up to 1400% that of the baseline epoxy, due to frictional sliding between nanotube–nanotube interfaces. However, to achieve this high damping property, the weight fraction of CNTs should reach as high as 50%. Later on, they found that rather than taking advantage of the frictional sliding between nanotube–nanotube interfaces, utilizing the nanotube–polymer interfacial frictional sliding can as well lead to the equivalent damping property, which only needs to introduce 1–2% weight fraction of oxidized CNTs.⁴ Furthermore, raising the temperature can activate or trigger the interfacial sliding at relatively low dynamic strain levels.⁵ These viscoelastic PNCs can find potential applications as room- or high-temperature damping materials for reducing vibrational and acoustic effects in dynamic systems. These researches, however, are all focused on increasing the energy dissipation, and little research work has been directed to decrease or lessen the energy dissipation of PNCs.

We know that elastomer is made of long chains of randomly oriented molecules, and these long chains are subject to entanglement and chemical cross-linking for practical applications.⁶ Elastomer is a typical kind of visco-elastic materials, attributed to the viscous friction between molecules.⁷ For the practical application of automobile tires, nano-sized NPs such as carbon black^8–10 or silica^11–13 are always introduced to improve the mechanical properties through nano-reinforcement, such as the modulus, tensile and tear strength, abrasion resistance and so on. Since the loading of NPs is high, the aggregate of NPs always occur attributed to the weak interaction between NPs and elastomer, strong van der Walls interaction between NPs and the loss of the conformational entropy of polymer chains in the vicinity of NPs.^14,15 In fact, some research work have been carried out to graft NPs with polymer chains to achieve a good dispersion, which critically depends on the grafting density and the grafted chain length, and these physical parameters are always difficult to control in practice.^16–19 One harmful consequence of the aggregation of NPs is that the hysteresis loss during the loading–unloading process is significantly enhanced, attributed to the breakdown and re-agglomeration of the cluster of NPs.²⁰ One famous example is the Payne effect,²¹ namely the non-linear decrease of the storage modulus as a function of the dynamic strain amplitude. This leads to the increase of the rolling resistance and much greater fuel consumption of tires when transporting on the road.²² Some strategies have been adopted to lower the hysteresis loss of elastomer nanocomposites made of tire tread. For example, by replacing carbon black, the silica/silane filler was firstly introduced in the year of 1990 to lower the rolling resistance.²³ Martin et al.²⁴ have used epoxidized natural rubber (ENR) reinforced with silica to obtain lower rolling resistance. Wang et al.²⁵ have synthesized solution polymerized styrene-butadiene rubber (SSBR) with tert-butylchlorodiphenylsilane (TBCSi, large-volume functional groups) at the two ends of macromolecular chains (T-SSBR) through anionic polymerization, which exhibits lower heat build-up than those of SSBR composite attributed to the immobilization of the free chain ends and much smaller internal friction. In addition, the solution polymerized styrene-butadiene rubber with alkoxy silane-functionalization at two ends of macromolecular chains (A-SSBR) is shown to have the same helpful effect in obtaining lower rolling resistance.²⁶

Graphene and carbon nanotubes, as a new family of carbon-based nanomaterials, have also been incorporated into elastomer matrices to improve the mechanical properties. For instance, through a simple latex co-coagulation method, Wu et al.²⁷ have prepared exfoliated graphene oxide (GO) filled natural rubber (NR) nanocomposites, which shows good fracture and fatigue resistance at low filler content. Yan et al.²⁸ have observed that natural rubber (NR) filled with reduced graphene oxide (rGO) segregated network exhibit both better barrier to oxygen and water vapour permeation and mechanical properties compared to pristine rubber and composites with a uniform dispersion of single rGO platelets. George et al.²⁹ have found the enhancement of the tensile strength, tensile modulus and tear strength by the formation of the segregated multi-wall carbon nanotube. However, most researches have focused on studying the static mechanical properties of elastomer nanocomposites, few researches have been carried out to study the effect of the graphene sheet or carbon nanotube on the dynamic mechanical properties.

As for the mechanical properties of graphene or carbon nanotube, some researches have been performed. For instance, Iijima et al.³⁰ has adopted high resolution electron microscope (HREM) and atomistic simulation of the bending of single and multi-walled carbon nanotubes under mechanical stress, finding that the bending completely recovers up to very large bending angles, although some kinks and highly strained tube regions occur. This is attributed to the remarkable flexibility of the hexagonal network, resisting the bond breaking and bond switching up to very high strain value. Experimentally Tang et al.³¹ studied the compressive responses of carbon nanotube networks. Meanwhile, through tip-induced deformation experiment, Gómez-Navarro et al.³² have observed that the graphene monolayer can bend easily in the elastic regime attributed to the high flexibility, indicated by the unaltered electrical conductivity after multiple deformations. By taking advantage of the reversible mechanical deformation of the graphene sheet and carbon nanotube, we can use them to tune the visco-elasticity of elastomer nanocomposites. In this work, through molecular dynamics simulation, we show that the dynamic hysteresis loss will be greatly reduced by introducing the flexible reinforcing NPs such as graphene-nanoribbon and carbon nanotube, compared to the conventionally used rigid or stiff reinforcing NPs such as carbon black and silica. This trend will be strengthened by increasing the interfacial interaction between elastomer and NPs, together with the weight fraction and the size of NPs. This work is of great significance for fabricating energy-saving rubber products, such as automobile tires.

2. Simulation model and method

We adopt molecular dynamics (MD) simulation,³³ which has been shown to be a powerful and reliable tool in studying the structure–property relation of polymer nanocomposites. Our simulation models are composed of cis-polybutadiene (cis-PB) and flexible carbon nano-structured materials, such as nanotube^34–36 and graphene-nanoribbon.^37,38 As for the carbon nano-structured materials, we use the adaptive intermolecular reactive empirical bond order (AIREBO) potential to describe the carbon nano-structured materials. It involves reactive empirical bond order (REBO) potential³⁹ and Lennard-Jones potential⁴⁰ to describe the intra-layer and inter-layer interaction of carbon atoms, as implemented in the software package LAMMPS.⁴¹ The united atom model is performed to represent the cis-PB polymer chains, in which each carbon atom with the bonded hydrogen atoms are grouped to be a single big atom, as shown in Fig. 1(a).

Fig. 1 (a) United atom models for cis-1,4-polyisoprene (cis-PB); (b) graphene-nanoribbon; (c) nanotube.

The total force field energy can be expressed as follows:

E_total = E_bond(r) + E_angle(θ) + E_dihedral(φ) + E_non-bonding(r) (1)

The bond stretching (r), bond angle bending (θ) and dihedral angle torsion (φ) are shown below:

E_bond(r) = k_b(r − r₀)² (2)

E_angle(θ) = k_θ(θ − θ₀)² (3)

where K_b and K_θ represent the stiffness constants for the bond length and bond angle potentials, respectively. r₀ and θ₀ are the equilibrium bond length and bond angle, respectively. The variable K_n contains the coefficients of the dihedral potential. We use the standard Lennard-Jones potential function to describe the non-bonded interaction, as shown below:where r denotes the distance between two interaction sites, σ is the distance at which the potential energy becomes zero, and ε is the energy well depth. We have the cutoff distance r_c = 2.5σ (ref. 42) for the cis-PB chains. The force-field parameters of cis-PB listed in Table 1, follow the simulation work carried out by Tsolou et al.⁴³ The atomic structures of graphene-nanoribbon and nanotube are shown in Fig. 1(b) and (c). Note that most graphene-nanoribbon and nanotube contain 160 carbon atoms. We also study the effect of the length of graphene-nanoribbon on the visco-elasticity of nanocomposite, by purely changing the length at a fixed width. The number of carbon atoms in each graphene-nanoribbon varies from 160, 320 to 640, noted by C₁₆₀, C₃₂₀ and C₆₄₀, respectively. The standard Lennard-Jones potential function (eqn (5)) is also used to describe the interaction between carbon nano-structured materials and polymer chains, by setting σ = 4.009 Å, ε = 0.0936 kcal mol⁻¹.

Table 1 The force-field parameters for the cis-PB model. The united atom model of cis-PB is shown in Fig. 1(a)

(a)
Bond stretching	kb (kcal (mol Å^2)^−1)	r0 (Å)
CH2–CH2	331.5	1.54
CH2–CH	384.5	1.50
CH=CH	516.5	1.34

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(b)
Bond angle bending	kθ (kcal mol^−1)	θ0 (deg)
CH2–CH2–CH	57.5	111.65
CH2–CH–CH	44.7	125.89

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(c)
Dihedral angle torsion	k1 (kcal mol^−1)	k2 (kcal mol^−1)	k3 (kcal mol^−1)	k4 (kcal mol^−1)	k5 (kcal mol^−1)	k6 (kcal mol^−1)
CH2–CH=CH–CH2	—	12.1	—	—	—	—
CH2–CH2–CH=CH	0.5165	−0.236	0.2777	0.1315	0.173	0.082
CH–CH2–CH2–CH	−0.444	0.3095	−1.8195	−0.033	−0.1235	−0.095

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(d)
Non-bonding		ε (kcal mol^−1)	σ (Å)
CH2	CH2	0.0936	4.009
CH2	CH	0.1015	3.793
CH	CH	0.1000	3.385

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For most simulated systems, we use the isothermal–isobaric (NPT) ensemble for equilibration and further collecting data through the Nose–Hoover thermostat,⁴⁴ by setting the pressure P = 1 atm and T = 150 K to 413 K. In most cases, we construct our simulated systems at the temperature of T = 298 K. Periodic boundary conditions are employed in all three directions during the simulation. The velocity-Verlet algorithm is used to integrate the equations of motion, with a time step δt = 1 fs. In our simulation, we realize the cross-linking process as follows: after enough equilibration, the permanent cross-linking bonds are imposed in the system, by randomly selecting one pair of beads, which belong to two different chains. If the distance is smaller than 1.20 Å, then a bond modeled by the harmonic potential energy is produced between these two randomly chosen polymer beads. The interfacial chemical links are produced in the same way. The harmonic potential energy (eqn (2)) is introduced to describe these two kinds of cross-linking bonds with the parameters being set as follows: K_b = 331.5 kcal (mol Ų)⁻¹, r₀ = 1.54 Å. Note that we still further equilibrate the system after cross-linking. To carry out the uniaxial tension, we follow our previous approach. The pure and filled systems are deformed by changing the box length to L₀a in the z direction and to L₀a^−1/2 in the x and y directions, which therefore maintains the volume being constant. The interactions between atoms in the basic cell and image atoms across the cell wall serve to transmit the deformation to the atoms in the basic cell. Because Hossain et al.⁴⁵ have studied the stress relaxation behavior of polymer melts with the same model with ours, and good results have been obtained, we set the strain rate [small epsi, Greek, dot above] = (L(t)_z − L_z)/L_z = 0.000001 fs. The average stress σ in the z direction is obtained from the deviatoric part of the stress tensor σ = (1 + μ)(−P_zz + P) ≈ 3(−P_zz + P)/2, where P = Σ_iP_ii/3 is the hydrostatic pressure. The parameter μ stands for the Poisson's ratio. In most cases we get the stress–strain curves in the z direction.

All MD runs are carried out through the large scale atomic/molecular massively parallel simulator (LAMMPS),⁴¹ which is developed by Sandia National Laboratories. For all cases, we make sure that each chain has transported at least 2R_g, where R_g is the root mean square radius of gyration of polymer chains, to obtain well-equilibrated systems. More detailed simulation techniques can be found in our previously published work.⁴⁶

3. Results and discussion

3.1 The properties of cis-PB

To make sure that our simulation model and method are accurate, we calculate the equilibrium conformational properties of the simulated cis-PB system, which is compared with the work from Tsolou et al.⁴³ The mean-square end-to-end distance, 〈R²〉, and mean-square radius of gyration, 〈R_g²〉, for all systems employed in the simulations are presented in Fig. 2(a), and good consistency is achieved. Fig. 2(b) shows the mean square displacement (MSD) of pure cis-PB at different temperatures, showing that the mobility of polymer chains increases with the increase of the temperature. The diffusion coefficients are obtained by measuring the slope of the MSD curves, which is plotted against the inverse of the temperature in Fig. 2(c). The dependence of the diffusion coefficient on the temperature can be well described with the Arrhenius equation. From the slope of the corresponding curves, one can obtain estimates of the apparent activation energy, E^app_a:

Fig. 2 (a) Values of the mean-square end-to-end distance, 〈R²〉, mean-square radius of gyration, 〈R_g²〉 for the simulated cis-PB system at P = 1 atm and T = 413 K. (b) The mean square displacement (MSD) of cis-PB at different temperatures. (c) Dependence of the cis-PB chain center-of-mass self-diffusion coefficient, D, on temperature, T, as obtained from the change of MSD as a function of time at various temperatures.

It is shown that E^app_a (cis-PB) is equal to 3.24 kcal mol⁻¹, which is in accordance with the result of the MD simulation from Tsolou et al.⁴³ In general, the simulated results indicate that our simulation model and technique are both reasonable.

Because of the limitation of the computational efficiency, short polymer chains are always used, and this situation leads to the occurrence of chain end effects. To determine the reasonable chain lengths for cis-PB which were used in the following investigation of elastomer nanocomposites in our study, the density at the temperature equal to 413 K was examined as a function of the chain length for cis-PB. As shown in Fig. 3(a), the density of cis-PB increases with the increase of the chain length. For cis-PB, when the number of carbon atoms reaches 128, the density becomes almost constant. Based on this, we choose 128 carbon atoms for the modeled cis-PB polymer chain. It is known that the glass transition temperature signifies the transition between the rubbery and glassy state. Fig. 3(b) shows the specific volume (reciprocal of the density) as a function of the temperature for pure cis-PB. A distinct kink in the curve indicates the occurrence of the glass transition. The estimated values of T_g are 187 K for pure cis-PB. It is noted that the value of T_g (187 K) of pure cis-PB almost agrees with other MD simulation work (185 K),⁴⁷ and is only a little higher than the experimental value (170 K).⁴⁸ Obviously, this simulated results indicate that our simulation model and technique are nearly close to the real system.

Fig. 3 (a) Density (413 K) as a function of the chain length for cis-PB. (b) Temperature dependence of the specific volume of cis-PB system.

Next, we perform the uniaxial tension to characterize the mechanical behavior, and its snapshot is presented in Fig. 4(a). We examine the effect of the tensile rate such as 10⁸ s⁻¹, 10⁹ s⁻¹ and 10¹⁰ s⁻¹ on the stress–strain behavior by setting the temperature equal to 150 K below its glass transition temperature, as shown in Fig. 4(b). Obviously, an elastic regime occurs at the small strain, which is followed by the yielding point, then the strain softening and the strain hardening happen at the large strain. It is also found that the elastic modulus and the yield stress are both enhanced with the increase of the strain rate. This result is consistent with the results of Hossain et al.⁴⁵ Meanwhile, the influence of the temperature on the stress–strain behavior is also examined for cis-PB in Fig. 4(c), by setting the tensile strain rate equal to 10⁹ s⁻¹. As expected, the stress–strain property decreases with the increase of the temperature. Above T_g, no yielding point occurs for the rubbery state.

Fig. 4 (a) The snapshots showing the cis-PB structure after equilibration, at 50% true strain and at 100% true strain, respectively. (b) Stress–strain response at 150 K for different strain rates for pure cis-PB. (c) Stress–strain response at a rate of 10⁹ s⁻¹ for different temperatures for pure cis-PB.

3.2 cis-PB nanocomposites filled with carbon nanotube or graphene-nanoribbon

To study the nanotube effect on the visco-elasticity and the dynamic hysteresis loss, we pay close attention to the tension–recovery process. In theory, a large permanent set usually indicates great slippage and internal friction between polymer chains, leading to more significant hysteresis loss.⁴⁹ Moreover, we also calculate the hysteresis loss from the tension–recovery stress–strain curve in one cycle in order for quantitative comparison. Firstly, we study the model of pure cis-PB with different crosslink density, as shown in Fig. 5(a). Obviously, with the increase of the number of the cross-linked bonds ranging from M = 200, 300, 350, and 400, corresponding to the chemical coupling density (defined as the ratio of the number of the cross-linked bonds to the total box volume) ρ₁ = 0.0011, 0.0016, 0.0018 and 0.0022 mol cm⁻³, the stress–strain behaviors are enhanced, exhibiting better mechanical reinforcement performance. Meanwhile, it is observed that the permanent set is also gradually decreasing, which can be attributed to the fact that higher crosslink density leads to better synergistic motion of the polymer chains during the tension–recovery process. We also study the compression–recovery process, the result is displayed in Fig. 5(b), indicating that the permanent set gradually decreases with the increase of the cross-linking density.

Fig. 5 For pure cis-PB with different cross-linking density, denoted by ρ₁ (mol cm⁻³): comparison of (a) the tension–recovery curves and (b) the compression–recovery curves for different cross-linking density; comparison of the hysteresis loss (HL) for different cross-linking density during (c) the tension–recovery process and (d) the compression–recovery process.

In order to further quantitatively examine the effect of the cross-linking density on the visco-elasticity, we calculate the hysteresis loss (HL), namely the ratio of the loss energy to the storage energy in each tension–recovery cycle or compression–recovery cycle, as shown in Fig. 5(c) and (d). Obviously, the value of HL decreases with the increase of ρ₁. Comparatively speaking, the decrease of the HL with the increase of the cross-linking density is more prominent for the case of the tension–recovery process. Overall, the effect of the cross-linking density on the change of the HL for the pure cis-PB system is consistent with the experiment result,⁵⁰ which validates that our simulation model and technique are both reasonable and effective.

Later, we study the change of the specific volume as a function of the temperature to signify the glass transition temperature of carbon nano-structured materials filled system, and the results are presented in Fig. 6(a) and (b). Evidently, a small increase in T_g is observed with the increase of the weight fraction of carbon nanotubes and the interfacial chemical coupling density ρ₂ (defined as the ratio of the number of the interfacial links to the surface area of the nanoparticles), attributed to the interfacial interaction to constrain more polymer chains. For graphene filled polymer system, Fig. 6(b) shows that T_g as well increases with the increase of the interfacial chemical coupling density at the fixed weight fraction of graphene. This finding is consistent with the experimental results.^51,52 As analyzed above, the increase of the interfacial chemical coupling density and the weight fraction of carbon fillers can lead to a slight increase of T_g, but all T_g of these systems are below the simulated temperature, which is fixed at 298 K, indicating that all simulated systems are in the visco-elastic state.

Fig. 6 Temperature dependence of the specific volume of (a) cis-PB/CNT nanocomposite system, (b) cis-PB/graphene nanocomposite system. Note that w denotes the weight fraction of the carbon fillers, ρ₂ (mmol m⁻²) represents the interfacial chemical coupling density, and the crosslink density of cis-PB ρ₁ = 0.0011 mol cm⁻³.

To examine the nanotube effect on the hysteresis loss, we directly mixed the nanotube with cis-PB, as shown in Fig. 7(a). The weight fraction of the nanotubes is w = 12.0% and the cross-linking density of cis-PB ρ₁ = 0.0011 mol cm⁻³. Snapshots shown in Fig. 7(b) and (c) display the microscopic deformation of the nanotubes during the tension–recovery process, which shows the reversible elastic deformation of the nanotubes. Since the interfacial interaction strength always has significant influence on the static and dynamic mechanical properties, and in practical situations some interfacial coupling agents are always introduced to strengthen the interfacial interaction.⁵³ Here we study the effect of the number of the interfacial links on the stress–strain curves during the tension–recovery, as shown in Fig. 8(a). Obviously, with the increase of the number of the interfacial links ranging from M = 0, 50, 100, and 150, corresponding to the chemical coupling density ρ₂ = 0, 0.0016, 0.0032 and 0.0048 mmol m⁻², the stress–strain behaviors are enhanced. For instance, the stress at the strain ε = 200% for the system with the interfacial chemical coupling density ρ₂ = 0.0048 mmol m⁻² is almost seven times than that of the system without interfacial link. Meanwhile, it is observed that the permanent set is also gradually decreasing, which is due to the fact that enhanced polymer–filler interaction leads to better synergistic motion of the nanotubes and the polymer chains during the tension–recovery process. In order to analyze the synergistic motion between polymer chains and carbon fillers during the deformation, we characterize the bond orientation of polymer chains for both pure polymer and cis-PB/carbon nanotube systems, as shown in Fig. 8(c). Obviously, with the increase of the interfacial chemical coupling density, the polymer chains tend to get more nearer to the initial state before the deformation during the tension–recovery process, indicating that the carbon fillers with the reversible mechanical deformation can bring the polymer chains to move together, exhibiting a synergistic motion behavior. Additionally, in Fig. 8(b) we also show the stress–strain curves during the compression–recovery for different interfacial chemical coupling densityρ₂. Reasonably, with the increase of ρ₂, the stress–strain behaviors are increasing and the permanent set is decreasing. For better comparison, we also calculate the hysteresis loss, as shown in Fig. 8(d) and (e). It is evident that with the increase of the interfacial chemical coupling density ρ₂, the hysteresis loss decreases significantly for both tension–recovery and compression–recovery cases. For the tension–recovery process, the hysteresis loss becomes 31% when ρ₂ = 0.0048 mmol m⁻², compared to the hysteresis loss equal to 61% when ρ₂ = 0.0. These quantitative comparisons are consistent with the change of the permanent set discussed above.

Fig. 7 Snapshots of nanotubes (w = 12.0%) filled cross-linked polymer system (ρ₁ = 0.0011 mol cm⁻³). (a) The dispersion of the nanotubes in polymer chains. The microscopic deformation of the nanotube during the tension–recovery process (b) in the z direction and (c) in the x direction. Note that the blue dots represent the nanotubes, and the red points denote the polymer chains. For clarity only one nanotube is shown.

Fig. 8 For nanotubes (w = 12.0%) filled cross-linked polymer system (ρ₁ = 0.0011 mol cm⁻³). Comparison of (a) the tension–recovery curves and (b) the compression–recovery curves for different interfacial chemical coupling density, denoted by ρ₂ (mmol m⁻²). (c) The bond orientation of polymer chains for the cis-PB system, cis-PB/CNT system for different interfacial chemical coupling density. Comparison of the hysteresis loss (HL) for nanotubes filled cross-linked polymer system, the effect of the interfacial chemical coupling density on the HL during (d) the tension–recovery process and (e) the compression–recovery process.

Since the nanotubes exhibit some kind of anisotropic characteristics, the stress–strain tension–recovery curves along the x, y and z directions may be different. We show the simulated stress–strain curves in the cases of the tension–recovery and compression–recovery processes in Fig. 9(a) and (b). We want to point out that for both tension–recovery and compression–recovery processes, the stress–strain behavior is the largest in the y direction, and the smallest in the z direction. And the hysteresis loss in the z direction seems to be the largest, and the smallest in the y direction, although the difference is not so much, as shown in Fig. 9(c). The underlying reason is attributed to a relatively more oriented of carbon nanotubes along the y direction, and less oriented along the x, z direction before deformation. In order to check the orientation of the carbon nanotubes along the x, y and z directions before deformation, we use the second-order Legendre polynomials 〈P₂(cos θ)〉 to characterize the orientation as follows:

〈P₂(cos θ)〉 = (3〈cos² θ〉 − 1)/2  (7)

where θ denotes the angle between a given element (two adjoining centroid in the nanotube) and the reference direction. Since carbon nanotube is a cylinder, we divide each nanotube into eight parts, then calculate each centroid of the part, as schematically shown in Fig. 9(e). The results of the orientation of carbon nanotubes along the x, y and z direction are presented in Fig. 9(d). Obviously, from Fig. 9(d), we can observe that compared to the x and z directions, the carbon nanotubes tend to more orientate and align along the y direction. Therefore, the whole carbon nanotube filled rubber system exhibits the greatest mechanical reinforcement and the smallest hysteresis loss in the y direction. However, for the x and z directions, the carbon nanotubes tend to become perpendicular to these two directions, as indicated by Fig. 9(d), leading to less mechanical reinforcement and larger hysteresis loss, compared to those behavior in the y direction. It is noted that in the following deformation processes we focus on figuring out the stress–strain behavior along the z direction.

Fig. 9 For nanotubes (w = 12.0%) filled cross-linked polymer system (ρ₁ = 0.0011 mol cm⁻³) with interfacial coupling density (ρ₂ = 0.0016 mmol m⁻²), comparison of (a) the tension–recovery curves and (b) the compression–recovery curves in the x, y and z directions; (c) comparison of the hysteresis loss (HL) in the x, y and z direction during the tension–recovery process and the compression–recovery process. (d) The orientation of the carbon nanotube in the x, y and z directions before deformation. (e) The sketch map of the way to calculate the orientation of the carbon nanotube. Note that the red sphere denote the centroid of the according part.

In addition to the effect of the interfacial interaction, we also characterize the effect of the nanotubes by changing the weight fraction (w) of the carbon nanotube. The number of nanotubes in the simulated systems varies from 6, 12, 18, and 24, corresponding to the weight fraction of w = 6.4%, 12.0%, 16.9%, and 21.4%, respectively. Correspondingly, we increase the number of the interfacial links proportionally, ranging from M = 25, 50, 75, to 100, keeping the interfacial chemical coupling density equals to 0.0016 mmol m⁻². The simulated result is presented in Fig. 10(a), which shows the stress–strain curves during the tension–recovery. Obviously, with the increase of the weight fraction of the nanotubes, the stress–strain curves are enhanced and the permanent set are decreased significantly. Meanwhile, similar results are observed during the compression–recovery cycle, as shown in Fig. 10(b). To quantitatively characterize the effect of the weight fraction of the nanotubes on the visco-elasticity of elastomeric polymer materials, we calculate the hysteresis loss during the tension–recovery and compression–recovery processes, and the results are presented in Fig. 10(c) and (d). It is noted that the hysteresis loss decreases gradually with the increase of the weight fraction of the nanotubes. We explain this result as follows: in the case of strong interfacial interaction between polymer chains and nanotubes, and good dispersion of nanotubes, the nanotubes will store and release energy during the loading–unloading period, and this reversible deformation and recovery of nanotubes may also bring more surrounding polymer chains to move together, which is believed to reduce the internal friction between polymer chains and decrease the resulting hysteresis loss (HL). Note that the percolation transition of this modeled CNTs corresponds to around 15.6%, as indicated by the stress of the strain equal to 150% dependence of the weight fraction of the CNTs, therefore the change of the weight fraction of the CNTs here incorporates this percolation threshold.

Fig. 10 For nanotubes filled cross-linked polymer system (ρ₁ = 0.0011 mol cm⁻³). Comparison of (a) the tension–recovery curves and (b) the compression–recovery curves for different weight fraction (w). Comparison of the hysteresis loss (HL) for nanotubes filled cross-linked polymer system, the effect of the weight fraction on the HL during (c) the tension–recovery process and (d) the compression–recovery process. Note that the interfacial links increase in proportion with the increase of weight fraction of the nanotube, keeping the interfacial chemical coupling density ρ₂ = 0.0016 mmol m⁻².

Besides the carbon nanotube, graphene-nanoribbon is also another kind of carbon nano-structured materials with good flexibility and reversible mechanical deformation, which could also be introduced to both enhance the mechanical properties and significantly decrease the HL of elastomeric polymer materials. Here we investigate the effect of graphene-nanoribbon adjusting the viscoelasticity of elastomers, and the snapshots of graphene-nanoribbons filled cross-linked polymer system is presented in Fig. 11. It clearly shows that the graphene-nanoribbon is in its bending state. Firstly we draw attention to the effect of interfacial links on the stress–strain curves and permanent set under both tension and compression situations. Fig. 12(a) shows the stress–strain curves during the tension–recovery cycle. Obviously, as the interfacial chemical coupling density increases, the fillers can bring more surrounding polymer chains to move together, the stress–strain behaviors are enhanced significantly and the permanent set decreases gradually. Meanwhile, similar results are observed during the compression–recovery cycle, as shown in Fig. 12(b). Fig. 12(c) and (d) shows the hysteresis loss during the tension–recovery and compression–recovery. It is evident that the hysteresis loss decreases as the increase of the interfacial chemical coupling density.

Fig. 11 Snapshots of graphene-nanoribbons (w = 12.0%) filled cross-linked polymer system (ρ₁ = 0.0011 mol cm⁻³). The dispersion of the nanotubes in polymer chains. Note that the blue dots represent the graphene-nanoribbons, and the red points denote the polymer chains. For clarity only one graphene-nanoribbon is shown.

Fig. 12 For graphene-nanoribbons (w = 12.0%) filled cross-linked polymer system (ρ₁ = 0.0011 mol cm⁻³). Comparison of (a) the tension–recovery curves and (b) the compression–recovery curves for different interfacial chemical coupling density ρ₂ (mmol m⁻²). Comparison of the hysteresis loss (HL) for graphene-nanoribbons filled cross-linked polymer system, the effect of the interfacial chemical coupling density on the HL during (c) the tension–recovery process and (d) the compression–recovery process.

Lastly we study the effect of the size of the graphene-nanoribbon on the visco-elasticity of elastomer nanocomposite, by purely changing the length at a fixed width. Fig. 13(a) shows that at moderate length of the graphene-nanoribbon, the stress–strain curve becomes the largest, such as the case when the number of carbon atoms is equal to 320. While for the compression–recovery process, the stress–strain behavior is enhanced with the increase of the length of graphene-nanoribbon, as shown in Fig. 13(b). However, for both the tension–recovery and compression–recovery processes, the hysteresis loss (HL) both gradually decrease with the increase of the length of the graphene-nanoribbon, as presented in Fig. 13(c) and (d). Since larger graphene-nanoribbon tends to exhibit much larger deformation, and accordingly more energy storage and release, bringing more surrounding polymer chains to move together.

Fig. 13 For graphene-nanoribbons (w = 12.0%) filled cross-linked polymer system (ρ₁ = 0.0011 mol cm⁻³). Comparison of (a) the tension–recovery curves and (b) the compression–recovery curves for different length of graphene-nanoribbon. Comparison of the hysteresis loss (HL) for graphene-nanoribbons filled cross-linked polymer system, the effect of the length of graphene-nanoribbon on the HL during (c) the tension–recovery process and (d) the compression–recovery process. Note that the interfacial chemical ρ₂ = 0.0032 mmol m⁻².

4. Conclusion

In this work, we adopt united atom molecular dynamics simulation to investigate the pure cis-PB and cis-PB/flexible fillers nanocomposite. The feasibility of our simulation model is verified by the static and dynamic mechanical properties of cis-PB. In addition, nanotube and graphene-nanoribbon are successfully introduced to adjust the visco-elasticity of cis-PB based nanocomposite. Compared with the pure systems, these flexible carbon nano-structured fillers not only strengthen the polymer matrix, but also decrease the permanent set and hysteresis loss. These superior performance could be attributed to the reversible storage and release of the elastic energy of these fillers during the loading–unloading period. Importantly, the introduction of the interfacial chemical links between these flexible carbon nano-structured fillers and polymer chains will further decrease the hysteresis loss, which is attributed to the synergistic motion between these fillers and the surrounding polymer chains. Meanwhile, increasing the weight fraction of fillers exhibits enhanced stress–strain behaviors and smaller hysteresis loss, which fundamentally results from the fact that the fillers could store and release more energy, and bring more sounding polymers to move together. Moreover, the increase of the size of the graphene-nanoribbon as well greatly decreases the hysteresis loss during the tension–recovery and compression–recovery processes. In general, this work could provide some theoretical guidances for fabricating high performance polymer nanocomposites in the dynamic systems requiring to save energy, such as the automobile tires.

Acknowledgements

This work is supported by the National Basic Research Program of China 2015CB654700 (2015CB654704), the Foundation for Innovative Research Groups of the NSF of China (51221002), the National Natural Science Foundation of China (51333004 and 51403015). The cloud calculation platform of BUCT is also greatly appreciated.

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