Abstract
The dynamic impact response of giant buckyball C_{720} is investigated by using molecular dynamics simulations. The nonrecoverable deformation of C_{720} makes it an ideal candidate for highperformance energy absorption. Firstly, mechanical behaviors under dynamic impact and lowspeed crushing are simulated and modeled, which clarifies the bucklingrelated energy absorption mechanism. Onedimensional C_{720} arrays (both vertical and horizontal alignments) are studied at various impact speeds, which show that the energy absorption ability is dominated by the impact energy per buckyball and less sensitive to the number and arrangement direction of buckyballs. Threedimensional stacking of buckyballs in simple cubic, bodycentered cubic, hexagonal, and facecentered cubic forms are investigated. Stacking form with higher occupation density yields higher energy absorption. The present study may shed lights on employing C_{720} assembly as an advanced energy absorption system against lowspeed impacts.
Keywords:
Impact; Energy absorption; Buckyball; BucklingBackground
Absorption of external impact energy has long been a research topic with the pressing need from civil [1,2] to military needs [3,4]. In particular, effective absorption of mechanical energy at lowimpact speed, i.e., below 100 m/s is of great interest [5,6]. As one of the major branches of fullerene family, the carbon nanotube (CNT) has demonstrated an outstanding mechanical energy dissipation ability through waterfilled CNT [7], CNT forest and bundle [7], CNT/epoxy nanocomposites [8], CNT immersed in nonaqueous liquid [9], intercalating vertical alignment with aligned existing layered compounds [10], and spongelike material containing selfassembled interconnected CNT skeletons [11], among others. The advantage lies within the CNTs’ intriguing mechanical properties, i.e., ultrastrong (Young’s modulus of 0.9 to 5.5 TPa [1214] and tensile strength of 60 GPa [12]) and ultralight, as well as the tube structure which buckles upon external loadings [15]. Both theoretical modeling [1618] and experiments [1921] have proposed that the energy dissipation density of CNTs could be on the order of 200 J/cm^{3}, about 12 order of magnitudes over traditional engineering material [1].
Naturally, another branch of fullerene family with a spherical shape, i.e., the buckyball, also possesses excellent mechanical properties similar to CNTs. Man et al. [22] examined a C_{60} in collision with a graphite surface and found that the C_{60} would first deform into a disklike structure and then recover to its original shape. It is also known that C_{60} has a decent damping ability by transferring impact energy to internal energy [23,24]. This large deformation ability under compressive strain of C_{60} was also verified by Kaur et al. [25]. For higher impact energy, Zhang [26] employed C_{60}/C_{320} to collide with mono/double layer graphene, and the penetration of graphene and the dissociation of buckyball were observed. Furthermore, Wang and Lee [27] observed a novel phenomenon of heat wave propagation driven by impact loading between C_{60} and graphene which was responsible for the mechanical deformation of the buckyball. Meanwhile, giant buckyballs, such as C_{720}, have smaller system rigidity as well as nonrecoverable morphology upon impact, and thus they are expected to have higher capabilities for energy dissipation [28]. However, to the best knowledge of the authors, currently, only few studies about the mechanical behavior of giant buckyball are available [2931].
To understand the mechanical behavior of C_{720} and investigate its energy absorption potential in this paper, the dynamic response of C_{720} is studied at various impact speeds below 100 m/s by employing molecular dynamics (MD) simulations. Firstly, the buckling behaviors under both lowspeed crushing and impact are discussed and described using classical thin shell models. Next, 1D alignment of C_{720} system is investigated to identify the influence of packing of the buckyball on unit energy absorption. Finally, 3D stacking of C_{720} system is considered, where four types of packing forms are introduced and the relationship between unit energy absorption and stacking density are elucidated by an empirical model.
Methods
Computational model and method
The C_{720} is a spherical buckyball with diameter of 2.708 nm (where the van der Waals equilibrium distance is considered), volume of 7.35 nm^{3}, and mass of 1.45 × 10^{−20} g. C_{720} with varying numbers and packing directions (both vertical and horizontal) are selected in this study. Computational cells from single buckyball to 3D buckyball stacking system are illustrated in selected examples in Figure 1. In the scenario of the impact, the buckyball system subjects to the impact of a top rigid plate with incident energy E_{impactor}, and the initial impact speed is below 100 m/s; in the scenario of crushing, the top rigid plate compresses the buckyball system at a constant speed below 100 m/s. The bottom plate, which is rigid and fixed, serves as a receiver, and the force history it experiences could indicate the energy mitigation capability of the protective buckyball system. The buckyball is not allowed to slip with respect to the impactor and receiver plates. Both the impactor and receiver plates are composed of carbon atoms. The masses of the atoms are varied in the following simulation to set various loading conditions (varying impactor mass), while the interactions between the plates and buckyballs remain as carboncarbon interaction.
Figure 1. Various alignments of buckyball system as a protector.
MD simulation is performed based on largescale atomic/molecular massively parallel simulator platform with the microcanonical ensembles (NVE) [32] after equilibration. A pairwise LennardJones (LJ) potential term is added to the buckyball potential to account for the steric and van der Waals carboncarbon interaction
where ɛ_{CC} is the depth of the potential well between carboncarbon atoms, σ_{CC} is the finite distance where the carboncarbon potential is zero, r_{ij} is the distance between the two carbon atoms. Here, LJ parameters for the carbon atoms of the buckyball and ε_{CC} = 0.27647 kJ/mol as used in the original parametrization of Girifalco [33] and van der Waals interaction govern in the platebuckyball interaction. A time integration step of 1 fs is used, and periodical boundary conditions are applied in the xy plane to eliminated the boundary effect.
Single buckyball mechanical behavior
Atomistic simulation result
The distinctive mechanical behavior of a single buckyball should underpin the overall energy absorption ability of a buckyball assembly. The force F and displacement W are normalized as FR/Eh^{3} and W/D, respectively, where R, h, D, and E are the radius, effective thickness, diameter, and effective Young’s modulus of the buckyball, respectively. Considering that bending is involved during the buckyball compression, h = 0.66 nm and E = 5 TPa [34,35]. Here a crushing speed at 0.01 m/s is employed to mimic quasistatic loading, because the normalized forcedisplacement curves are verified to be the same at various loading rates from 0.1 to 0.001 m/s in trial simulations. The forcedisplacement response under both quasistatic and a representative dynamic impact loading (with impact speed of 50 m/s and energy of 1.83 eV) are studied, as shown in Figure 2. Two obvious forcedrops could be observed in lowspeed crushing, while only one prominent forcedrop exists in dynamic loading which is related to the lessevident snapthrough deformation shape.
Figure 2. Normalized force displacement curves at both lowspeed crushing and impact loading. The entire process from the beginning of loading to the bowlforming morphology can be divided into four phases. Morphologies of C_{720} are shown at the corresponding normalized displacements.
The entire compression process could be divided into four phases according to the FR/Eh^{3} ~ W/D curve, i.e., buckling (W/D < 10%), postbuckling (10% ≤ W/D < 30%), densification (30% ≤ W/D < 40%), and invertedcapforming phase (W/D > 40%). Upon the ricochet of the plate, the deformation remains as a bowl shape with great volume shrinkage. The stabilization of such a buckled morphology is owing to a lower system potential energy in the buckled configuration due to van der Waals interaction; similar energy dissipation mechanism in CNT network is also revealed by [36].
The derivative of curve undergoes a sudden change at the same W/D value but in two completely different loading rates, suggesting that the sudden forcedrop points are highly dependent on the buckyball deformation rather than the loading rate. And theoretical insights may be obtained from the fourphase deformation.
Phenomenological mechanical models
Note that due to the property of FR/Eh^{3} ~ W/D curve, among the phases of compression process, those with significant reduction of force (Figure 2) are relatively unimportant for energy absorption and not included in the modeling effort. A threephase model for lowspeed crushing and a twophase model for impact loading are proposed separately in the following sections.
Threephase model for lowspeed crushing (quasistatic loading)
(1) Phase I. Buckling phase
In the range of small deformation in the beginning of compression, the model describing thinshell deformation under a point force is applicable [37,38]. Considering a buckyball with wall thickness h = 0.066 nm compressed by F with deformation of W (with the subscript number denoting the phase number sketched in Figure 3), the forcedeflection relation should be expressed as [39]
where the bending stiffness G = Ehc^{2}; the reduced wall thickness and ν is the Poisson’s ratio. The linear deformation behavior continues until it reaches the critical normalized strain W_{b1}. Experimental results for bulk thin spherical shell show that the transition from the flattened to the buckled configuration occurs at a deformation close to twice the thickness of the shell [40]; while W_{b1} here is about 4 h, indicating a larger buckling strain in nanoscale structure.
Figure 3. Illustration of deformation phases during compression for C_{720}. Dynamic loading and lowspeed crushing share the same morphologies in phase I while they are different in phase II. Analytical models are based on the phases indicated above and below the dash line for lowspeed crushing and impact loading, respectively.
The nanostructure has higher resistance to buckle than its continuum counterpart and based on the fitting of MD simulation, a coefficient f^{*} ≈ 2.95 should be expanded to Equation 2 as
It is reminded that this equation is only valid for C_{720} under lowspeed (or quasistatic) crushing.
(2) Phase II. Postbuckling phase
As the compression continues, buckyball continues to deform. Once the compressive strain reaches W_{b1}, the flattened area becomes unstable and within a small region, the buckyball snaps through to a new configuration in order to minimize the strain energy of the deformation, shown in Figure 3. The ratio between the diameter and thickness of buckyball is quite large, i.e., D/h ≈ 36.5, and only a small portion of volume is involved thus the stretching energy is of secondary order contribution to the total strain energy. Hubbard and Stronge [41] developed a model to describe the postbuckling behavior of a thin spherical shell under compression based on Steele’s [42] model
where . This nonlinear deformation behavior extends until it reaches the densification critical normalized strain W_{b2}. The value of W_{b2} could be fitted from the simulation data for C_{720} where W_{b2} ≈ 11h.
The first forcedrop phenomenon is obvious once the buckling occurs where the loading drops to nearly zero. Therefore, by applying the boundary condition of F_{2}(W_{2}) ≈ 0, Equation 4 maybe further modified as
(3) Phase III. Densification phase
When the compression goes further, the crushing displacement eventually becomes much larger than the thickness and thus the forcedisplacement relation becomes nonlinear [42]. The buckled buckyball is densified during this process. A phenomenological nonlinear springlike behavior could be fitted as
where γ is a coefficient and n is fitted as n ≈ 1.16. Considering the relationship [41,42]
and
we may come to the equation
Thus, by considering the continuity of two curves in adjacent phases, we may rewrite Equation 9 as
Therefore, Equations 3, 5, and 10 together serve as the normalized forcedisplacement model which may be used to describe the mechanical behavior of the buckyball under quasistatic loading condition from small to large deformation.
Figure 4 shows the simulation data at lowspeed crushing compared with the model calculation. A good agreement between two results is observed which validates the effectiveness of the model.
Figure 4. Comparison between computational results and analytical model at lowspeed crushing of 0.01 m/s.
Twophase model for impact
The mechanical behaviors of buckyball during the first phase at both lowspeed crushing and impact loadings are similar. Thus, Equation 2 is still valid in phase I with a different f^{*} ≈ 4.30. The characteristic buckling time, the time it takes from contact to buckle, is on the order of τ ≈ 10^{− 1} ~ 10^{0} ns ~ T ≈ 2.5R/c_{1} ≈ 5.71 × 10^{− 5}ns, where ρ is the density of C_{720} and . It is much longer than the wave traveling time; thus, the enhancement of f^{*} should be caused by the inertia effect [43].
As indicated before, the buckyball behaves differently during the postbuckling phase if it is loaded dynamically, i.e., no obvious snap through would be observed at the buckling point such that the thin spherical structure is able to sustain load by bending its wall. Therefore, a simple shell bending model is employed here to describe its behavior as shown in Figure 3; the top and bottom flattened wall with length of L experiences little stretching strain, whereas the side wall bends with finite deformation, governing the total system strain energy
where the bending rigidity and M is the bending moment. A denotes the integration area. The h^{’} is the ‘enlarged’ thickness, the result of smaller snapthrough phenomenon. Here, h^{’} ≈ 1.40h via data fitting. Substituting geometrical constraints and taking the derivative, the forcedisplacement relation becomes (for C_{720} under 100 m/s impact)
Therefore, Equations 3 and 12 together provide a model to describe the mechanical behavior of the buckyball under dynamic loadings.
When the impact speed is varied, the corresponding force is modified by a factor α owing to strain rate effect [4446]. With the subscript representing the impact speed (in units of m/s), the correction factor c = α_{40}, α_{50}, α_{60}, α_{70}, α_{80}, α_{90} = [0.83, 1.00, 1.12, 1.14, 1.17]. Figure 5 illustrates the comparison between atomistic simulation and model (for impact speeds of 40 to 90 m/s), with good agreements.
Figure 5. Comparison between computational results and analytical model. Comparison between computational results and analytical model at various impact speeds from 40 to 90 m/s.
Results and discussion
Buckyball assembly
In practice, buckyballs need to be assembled (shown in Figure 1) so as to protect materials/devices. Various stacking arrays are investigated as follows.
1D alignment buckyball system
The C_{720} can be arranged both vertically and horizontally in a 1D chainlike alignment. Figure 6 shows the mechanical behavior of a fivebuckyball array subjecting to a rigid plate impact with impact energy and speed of 9.16 eV and 50 m/s respectively. Progressive buckling and bowlshape forming behavior takes the full advantage of single buckyball energy absorption ability one by one and controls the force on the receiver within a relatively low value during first section of deformation (within W/D < 1.5) which provides cushion protections.
Figure 6. Characteristic normalized forcedisplacement curve of 1D system with vertically lined C_{720}buckyballs. The characteristic normalized forcedisplacement curve of 1D system with five vertically lined C_{720}s at impact speed of 50 m/s.
Another 1D arrangement direction is normal to a plate impact. Unlike the progressive buckling behavior in the vertical system, all buckyballs buckle simultaneously in the horizontal array. Figure 7 shows the scenario with impact energy of 1.83 eV per buckyball and impact speed of 50 m/s, where the total reaction force scales with the number of buckyballs. Systems with different buckyball numbers show almost uniform deformation characteristics of individual buckyballs.
Figure 7. Characteristic normalized forcedisplacement curve of 1D buckyball system with various numbers of horizontally lined C_{720}buckyballs. The characteristic normalized forcedisplacement curve of 1D buckyball system with various numbers of horizontally lined C_{720}s at impact speed of 50 m/s.
The energy absorption per unit mass (UME, J/g) and unit volume (UVE, J/cm^{3}) are given in Figure 8, which shows that the UME and UVE are almost invariant regardless of buckyball number or arrangement. In Figure 8 the impact energy per buckyball is fixed as 1.83 eV; if the impact energy or speed changes, the value of UME or UVE alters; however, the result is still insensitive to buckyball number or arrangement. The major responsible reason is that the energy absorption ability of the system stems from the nonrecoverable deformation of individual buckyball which is almost uniform.
Figure 8. UME and UVE values of both vertical and horizontal buckyball systems with various buckyball numbers. UME and UVE values of both vertical and horizontal buckyball systems with various buckyball numbers at impact speed of 50 m/s.
By fixing either the impact speed or mass and varying the other parameter, the impact energy per buckyball can be varied. It imposes a nonlinear influence on the UME and the maximum force on the receiver, as shown in Figure 9 for the vertical alignment of fivebuckyball system. No matter how the impact speed or mass varies, it is the impact energy per buckyball that dominates the values of UME and maximum transmitted force.
Figure 9. UME and maximum contact force at constant impact speed (50 m/s) with various impact masses. UME and maximum contact force at constant impact speed (50 m/s) with various impact mass (from 8.7 × 10^{−19} to 7.1 × 10^{−17} g), and constant impact mass (2.8 × 10^{−18} g) with various impact speeds (from 10 to 90 m/s), for fivebuckyball systems.
3D stacking buckyball system
The packing density of a 3D stacking system can be different than that of the 1D system, and thus the performance is expected to vary. Four types of 3D stacking forms are investigated, i.e., simple cubic (SC), bodycentered cubic (BCC), facecentered cubic (FCC) (a basic crystal structure of buckyball [47]), and hexagonalclosed packing (HCP). The occupation density η _{SC} = π/6 ≈ 0.52, , [48] for SC, BCC, FCC, and HCP, respectively. Convergence study indicates that the profiles of forcedisplacement curves as well as the energy absorption rate at increasing buckyball numbers at one computational cell keep the same. In this case, a fundamental unit, such as containing 2 × 2 × 3 buckyballs for SC arrangement is shown in Figure 1c.
Figure 10 illustrates the normalized forcedisplacement curves for SC, BCC, FCC, and HCP units under the same impact energy per buckyball (1.83 eV). As expected, the mechanical behaviors of FCC and HCP are similar, while the BCC and SC units (with lower η) have more space for system to comply and hence the impact force is smaller yet the displacement is larger. Consequently, FCC and HCP have the same energy absorption ability and that of BCC and SC are inferior.
Figure 10. Normalized forcedisplacement curves for SC, BCC, FCC and HCP packing of C_{720}. Typical normalized forcedisplacement curves for SC, BCC, FCC and HCP packing of C_{720} at impact speed of 50 m/s, and the impact energy per buckyball is 1.83 eV.
Energy absorption performances of the three basic units are studied at various impact speeds, i.e., from 10 to 90 m/s while the impact mass is kept a constant, as shown in Figure 11. With the impact speed increases, more mechanical energy is absorbed; but the increasing trend becomes slighter at higher impact speed when the buckyball system reaches its mitigation limit. The improvement is greater in terms of UVE than UME with higher η.
Figure 11. UME and UVE values of SC, BCC, FCC, and HCP packing of C_{720 }at impact speeds. UME and UVE values of SC, BCC, FCC, and HCP packing of C_{720} at impact speeds from 10 m/s to 90 m/s. Fitting surfaces based on the empirical equations are also compared with the simulation. (a) UME values of various packing forms of C_{720} at impact various impact speeds. (b) UVE values of various packing forms of C_{720} at impact various impact speeds.
By normalizing the UME and UVE as Π_{m} = UME/(E_{impactor}/m) and Π_{v} = UVE/(E_{impactor}/V_{volume}) where V_{volume} is the volume of the buckyball and impact speed as where B = 34 GPa [49] is the bulk modulus of graphite. An empirical equation could be fitted as
where A = 5.50, B = −0.25, C = 0.21, and D = 25.0 with fitting correlation coefficient of 0.96 and
where A = 0.46, B = −1.94, C = 0.21, and D = 187.9 with fitting correlation coefficient of 0.96. These equations are valid for lowspeed impact speed (below 100 m/s) on stacked C_{720} buckyballs. When the impact speed is fixed, the unit energy absorption linearly increases with the occupation density; under a particular spatial arrangement, the energy absorption ability increases nonlinearly with the impact speed.
Conclusions
C_{720} as a representative giant buckyball has the distinctive property of nonrecovery deformation after crushing or impact, which makes it capable of absorbing a large amount of energy. The mechanical behaviors of a single C_{720} under quasistatic (lowspeed crushing) and dynamic impact are investigated via MD simulation and analytical modeling. By understanding the mechanism of mechanical behavior of individual C_{720}, the energy absorption ability of a 1D array of buckyball system is studied. It is found that regardless of the direction of alignment and number of buckyballs, the unit energy absorption density is almost the same for lowspeed impact. In addition, different 3D stacking at various impact speeds and stacking forms are investigated. Explicit empirical models are suggested where packing density and impact speed may pose a positive effect on the unit energy absorption. This study may shed lights on the buckyball dynamic mechanical behavior and its application in energy absorption devices and inspire the related experimental work.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
JX carried out the molecular dynamic simulation and drafted the manuscript. YL participated in the design of the study and performed the mechanical analysis. XC and YX conceived of the study and participated in its design and coordination and helped draft the manuscript. All authors read and approved the final manuscript.
Authors’ information
JX is a Ph.D. candidate in Department of Earth and Environmental Engineering at Columbia University, supported by the Presidential Distinguished Fellowship. His research interests are nanomechanics and energyrelated materials. YL is a Professor in Department of Automotive Engineering at Tsinghua University. He has been awarded by the National Science and Technology Advancement Award (second prize) for twice. His major research interests are advanced energy absorption material. YX is a Professor in School of Energy Science and Engineering at University of Electronic Science and Technology of China. His research is focused on combinatorial materials research with emphasis on energy applications, particularly on thin film materials and devices, printed electronics, and power electronics. He has authored and coauthored more than 40 articles, with an hindex of 12. XC is an Associate Professor in Department of Earth and Environmental Engineering at Columbia University. He uses multiscale theoretical, experimental, and numerical approaches to investigate various research frontiers in materials addressing challenges in energy and environment, nanomechanics, and mechanobiology. He has published over 200 journal papers with an hindex over 30.
Acknowledgments
The work is supported by National Natural Science Foundation of China (11172231 and 11102099), DARPA (W91CRB11C0112), National Science Foundation (CMMI0643726), International joint research project sponsored by Tsinghua University (20121080050), Individualresearch founding State Key Laboratory of Automotive Safety and Energy, Tsinghua University (ZZ2011112), and World Class University program through the National Research Foundation of Korea (R322008000200420).
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