Solitary Wave in One-dimensional Buckyball System at Nanoscale

We have studied the stress wave propagation in one-dimensional (1-D) nanoscopic buckyball (C60) system by molecular dynamics (MD) simulation and quantitative modeling. Simulation results have shown that solitary waves are generated and propagating in the buckyball system through impacting one buckyball at one end of the buckyball chain. We have found the solitary wave behaviors are closely dependent on the initial temperature and impacting speed of the buckyball chain. There are almost no dispersion and dissipation of the solitary waves (stationary solitary wave) for relatively low temperature and high impacting speed. While for relatively high temperature and low impacting speed the profile of the solitary waves is highly distorted and dissipated after propagating several tens of buckyballs. A phase diagram is proposed to describe the effect of the temperature and impacting speed on the solitary wave behaviors in buckyball system. In order to quantitatively describe the wave behavior in buckyball system, a simple nonlinear-spring model is established, which can describe the MD simulation results at low temperature very well. The results presented in this work may lay a solid step towards the further understanding and manipulation of stress wave propagation and impact energy mitigation at nanoscale.

it is also important to study the temperature effect on the solitary wave behaviors of 1-D buckyball chains. The understanding of the stress wave propagation behavior at nanoscale discrete system is beneficial to design and tune the dynamical mechanical properties of the nanoscale metamaterials for the current emerging applications of shock wave protection, impact energy mitigation, tunable acoustic devices, etc.

Results
In this work, MD simulations are conducted first at various initial system temperatures and impacting speeds to study the stress wave propagation in 1-D buckyball chains. The effects of the initial system temperature and impacting speed on the solitary wave behaviors are explored. Then, taking account of the nonlinear force-displacement relation of inter-buckyballs, a simple nonlinear-spring model is established which can quantitatively describe the MD simulation results at low temperature very well. It is found the nonlinear inter-buckyballs interaction can be approximately described by a power law, but the power index is larger than the value of the macroscopic elastic spheres governed by the Hertz law which shows a stronger nonlinearity of the inter-buckyballs interaction. Based on the nonlinear-spring model, the relation between the solitary wave speed and the wave amplitude is derived which agrees well with MD simulation results.
The nanoscopic system investigated in this work is introduced as follows (see Fig. 1). The 1-D buckyball chain is composed of 51 C 60 buckyballs with space of d 0 = 10.05 Å (the equilibrium distance of two C 60 molecules 26 ) for adjacent buckyballs. The 0 th buckyball, with an initial speed v, serves as the impactor to generate a wave propagating along the chain. The impactor is also a buckyball (C 60 ) so as to prevent the occurrence of shock wave, which may result from a long contact time 3 . The whole system is tightly held by an ideal frictionless rigid lateral wall in order to maintain a 1-D wave propagation along the axis of the chain. Due to the short-range property of inter-buckyballs interaction, only the adjacent buckyballs have van der Waals interaction. And the adjacent buckyballs space equals to the equilibrium distance, so the initial force on each buckyball caused by other molecules is zero, which is similar to the zero precompression cases of the macroscopic counterpart in previous research 1,2,27 .
To investigate the wave propagation behaviors in this system, MD simulations are carried out based on LAMMPS (large-scale atomic/molecular massively parallel simulator) platform 28 . The full atomistic description of the buckyballs is applied in our MD simulations. The carbon-carbon interaction of intra-buckyball is described by the adaptive intermolecular reactive empirical bond order (AIREBO) potential which has been widely used to simulate the buckyball, carbon nanotube and graphene system [29][30][31] , while the carbon-carbon interaction of inter-buckyballs is accounted for by van der Waals interactions, described by a pairwise Lennard-Jones (L-J) potential term 12 6 . Here the inter-buckyballs van der Waals interactions plays the major role in the solitary wave propagation of the buckyball chains and according to Girifalco's work 26 , the parameters in the L-J potential for C 60 -C 60 interactions are chosen as σ = 3.4656 Å and ε = 0.0658 Kcal/mol slightly different from L-J potential parameters for interlayer graphene interactions. The time integration step is set as 1 fs. Firstly, the system is running for equilibrium for 30 ps in the NVT ensemble (the canonical ensemble) so as to set an initial temperature for the system before the impacting. Later, as shown in Fig. 1, an initial speed v is added to the 0 th buckyball to generate the solitary wave in the buckyball chains and the buckyball system is simulated with NVE ensemble (the micro-canonical ensemble) for next 10 ps to study the solitary wave propagation behaviors in the buckyball chains.
Firstly, we have simulated the 1-D buckyball chains at low initial temperature (T initial = 10 K, which is further normalized as T initial,N = T initial /T 0 = 0.034, where T 0 = 293 K) for different impacting speeds v = 500 m/s, 750 m/s and 1000 m/s, which are normalized as v N = 0.210, 0.315 and 0.419 using one-tenth of the wave speed Resultant force histories are extracted from 10th, 25th and 40th buckyballs to represent the wave signals, as shown in Fig. 2(a). At the low initial system temperature the profile of the stress wave is perfected kept and almost no dissipation is observed for the wave propagation of the 50 C 60 buckyballs in our MD simulations. The wave speed and amplitude increase with the impacting speed, which is similar to the solitary wave in macroscopic granular chains governed by Hertz law. However, as shown in Fig. 2(a), the stress wave in the buckyball chains is composed of one symmetrical positive and negative stress pulses, and the width of the wave decreases with increasing the impacting speed, different from the solitary waves in macroscopic granular chains without precompression (the wave width is constant). We attribute these differences of the solitary wave behaviors are caused by the different interactions in buckyball chains (van der Waals interaction) and macroscopic granular chains (Hertz law). It is also necessary to see how two waves behave after interacting with each other. According to Drazen and Johnson 32 , solitary waves remain unchanged after the collision with another solitary wave. To this end, impacts are generated at both ends by giving 0 th and 50 th buckyballs initial speeds of opposite directions (v left and v right respectively). Simulation results show that in both cases whether the stress waves are identical or different, both of the profile and amplitude of the two interacting waves remain unaltered after collision (see Fig. 2(b)), which further confirms the solitary wave propagation in 1-D buckyball chains.
For the C 60 buckyball chains, the thermal fluctuation may be comparable to the solitary wave amplitude. So the temperature effect plays an important role in the solitary wave behaviors of the buckyball chains. While for the macroscopic granular chains the thermal fluctuation is trivial, so that the temperature effect on the solitary wave behaviors has not been studied. The solitary wave behaviors in 1-D buckyball chains for different initial system temperatures and impacting speeds have been studied via MD simulations. Note that the system temperature may increase after the solitary wave propagation due to the dissipation. Generally, the solitary wave behaviors in 1-D buckyball chains can be divided into three regions, as shown in Fig. 3. For relatively low initial system temperature (T initial, N < 0.341) and high impacting speed (v N > 0.419) (the lower right region in Fig. 3), the profile of the solitary waves are perfectly kept and almost no dissipation is observed for the propagation 50 buckyballs in our simulations. We define this region as stationary solitary wave region. In contract, for relatively high initial system temperature (T initial, N > 0.341) and low impacting speed (v N < 0.419) (the upper left region of Fig. 3), the solitary wave is highly distorted by the random disturbance of the temperature effect, so that the profiles of the solitary waves in different buckyballs are completely different. Besides, the solitary waves are dissipated after propagation of several tens of buckyballs. We call this region as the disturbed solitary wave region. In the middle region, the solitary wave is also distorted and dissipated. But, the profile of the solitary wave in each buckyball is similar. We call this region the distorted solitary wave region. Note that the boundaries between the adjacent regions are not exactly clear and Fig. 3 is just a qualitative description of the three regions. From the perspective of molecular movement these phenomena can be explained. The thermal vibration of the molecules increases with temperature, leading to increasingly drastic uncertain disturbance to the wave propagation. The temperature effect is essentially different from the viscosity effect in macroscopic granular chains, though both of the two effects cause the dissipation of the solitary waves. The quantitative description of the temperature effect on the solitary wave behaviors in nanoscale granular systems should be systematically studied in future works.
It is critical to establish a quantitative physical model to describe the stress wave behaviors in 1-D buckyball chains. For simplicity and referring to the nonlinear-spring model used in macroscale granular chains, a nonlinear-spring model is proposed to describe the stress wave behaviors in 1-D buckyball chains. Based on the nonlinear force-displacement relation obtained from the collision of two C 60 buckyballs, the nonlinear spring can be depicted as Here, the force F and displacement δ are normalized as F N = F/F 0 and δ N = δ /d 0 , respectively, where F 0 = 3.24 nN corresponding to the inter-buckyballs force at displacement δ = 0.86 Å. Note that the compression displacement is set as positive value to consist with the definition in macroscopic granular chains. As the van der Waals interaction, there is a tensile force for tensile displacement which is different from the Hertz law in macroscopic granular chains. By fitting to the MD simulation results, the two parameters are k = 112.5 and n = 1.91, as shown in Fig. 4(a). The index n = 1.91 here is larger than the value for the interaction between two elastic solid spheres (n = 1.5) 3 , suggesting a stronger nonlinearity of inter-buckyballs interactions than its macroscopic counterpart. Then, the equation of motion for the ith buckyball can be written as    Fig. 4(a), this nonlinear-spring model agrees well with the MD simulation results. This remarkable consistency reveals that nonlinear van der Waals interactions between adjacent buckyballs play the major role in the solitary wave behaviors of 1-D buckyball chains at low temperature. The nonlinear-spring model accurately captures the nonlinear inter-buckyballs force-displacement relation, so that it can describe the solitary wave behaviors in 1-D buckyball chain at low temperature very well. The temperature effect is not considered in current nonlinear-spring model which will be included in our future works.
Next, the relation between the solitary wave speed and wave amplitude is discussed. Based on the nonlinear-spring model used in the macroscopic granular chains, the dependence of solitary wave speed on amplitude is u ∝ A (n−1)/2n , where u is wave speed and A is wave amplitude 3 . By substituting the power index of the inter-buckyballs force-displacement relation n = 1.91 into u ∝ A (n−1)/2n , the relation between the normalized solitary wave speed and amplitude can be well depicted as u N = 1.005 × A N 0.238 , as shown in Fig. 4(b), where (n − 1)/2n = 0.238, u N and A N are normalized wave speed and amplitude respectively, i.e. u N = u/u 0 , A N = A/A 0 . The wave speed and wave amplitude is plotted in logarithmic coordinates for the clear view of the power relation. Here, u 0 = 5181 m/s and A 0 = 4.29 nN are the wave speed and amplitude corresponding to the impact speed v N = 0.315. Furthermore, a linear relation between the solitary wave amplitude and the maximum force at the impactor is found, i.e. A impact = 0.887A, as shown in the inset of Fig. 4(b). As the impacting energy is distributed among several granules during the solitary wave propagation, the wave amplitude is smaller than the impact force amplitude. Therefore, the solitary wave speed can be further represented by the impactor force amplitude as u N = 1.034 × (A impact /A 0 ) 0.238 .
In this work, the stress wave behaviors in 1-D buckyball system have been investigated by MD simulation and quantitative modeling. The principal differences between the nanoscale buckyball system and the macroscale granular system lie in the different interaction laws (van der Waals interaction for buckyball system and Hertz law for macroscale granular system) and temperature effect. Due to the discrete nature and strong nonlinear interaction of buckyball chains, the solitary wave propagation is found in the 1-D buckyball chains. Different from the solitary wave in macroscale granular system, the solitary waves in buckyball chains are composed of one symmetrical positive and negative force pulses which is caused by the van der Waals interaction in buckyball system. The temperature effect plays an important role in the solitary wave behaviors of the buckyball system. A phase diagram is proposed in which the solitary wave behaviors are divided into three regions, i.e. the stationary solitary wave region (relatively low initial system temperature and high impacting speed), the disturbed solitary wave region (relatively high initial system temperature and low impacting speed) and the distorted solitary wave region. A nonlinear-spring model is established by accurately describing the van der Waals force-displacement relation of inter-buckyballs with a nonlinear spring, which can describe the MD simulation results at low temperature very well. This work may provide a powerful tool to investigate a series of related problem and open a new area to study the solitary wave propagation at nanoscale.