Columbia Nanomechanics Research Center, Department of Earth and Environmental Engineering, Columbia University, New York, NY 10027, USA

State Key Laboratory of Automotive Safety and Energy, Department of Automotive Engineering, Tsinghua University, Beijing, 100084, People’s Republic of China

State Key Lab of Electronic Thin Films and Integrated Devices, School of Energy Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, People’s Republic of China

Department of Civil and Environmental Engineering, Hanyang University, Seoul, 133-791, South Korea

International Center for Applied Mechanics, SV Lab, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

Abstract

The dynamic impact response of giant buckyball C_{720} is investigated by using molecular dynamics simulations. The non-recoverable deformation of C_{720} makes it an ideal candidate for high-performance energy absorption. Firstly, mechanical behaviors under dynamic impact and low-speed crushing are simulated and modeled, which clarifies the buckling-related energy absorption mechanism. One-dimensional 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. Three-dimensional stacking of buckyballs in simple cubic, body-centered cubic, hexagonal, and face-centered 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 low-speed impacts.

Background

Absorption of external impact energy has long been a research topic with the pressing need from civil
^{3}, about 1-2 order of magnitudes over traditional engineering material

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.
_{60} in collision with a graphite surface and found that the C_{60} would first deform into a disk-like 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
_{60} was also verified by Kaur et al.
_{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
_{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 non-recoverable morphology upon impact, and thus they are expected to have higher capabilities for energy dissipation

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 low-speed crushing and impact are discussed and described using classical thin shell models. Next, 1-D alignment of C_{720} system is investigated to identify the influence of packing of the buckyball on unit energy absorption. Finally, 3-D 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 3-D buckyball stacking system are illustrated in selected examples in Figure
_{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 carbon-carbon interaction.

Various alignments of buckyball system as a protector

**Various alignments of buckyball system as a protector.**

MD simulation is performed based on large-scale atomic/molecular massively parallel simulator platform with the micro-canonical ensembles (NVE)

where _{CC} is the depth of the potential well between carbon-carbon atoms, _{CC} is the finite distance where the carbon-carbon potential is zero, _{
ij
} is the distance between the two carbon atoms. Here, L-J parameters for the carbon atoms of the buckyball
_{CC} = 0.27647 kJ/mol as used in the original parametrization of Girifalco

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 ^{3} and

Normalized force displacement curves at both low-speed crushing and impact loading

**Normalized force displacement curves at both low-speed crushing and impact loading.** The entire process from the beginning of loading to the bowl-forming 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 ^{3} ~

The derivative of curve undergoes a sudden change at the same

Phenomenological mechanical models

Note that due to the property of ^{3} ~

Three-phase model for low-speed crushing (quasi-static loading)

(1) Phase I. Buckling phase

In the range of small deformation in the beginning of compression, the model describing thin-shell deformation under a point force is applicable

where the bending stiffness ^{2}; the reduced wall thickness
_{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
_{b1} here is about 4

Illustration of deformation phases during compression for C_{720}

**Illustration of deformation phases during compression for C**_{720}**.** Dynamic loading and low-speed 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 low-speed 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 ^{*} ≈ 2.95 should be expanded to Equation 2 as

It is reminded that this equation is only valid for C_{720} under low-speed (or quasi-static) crushing.

(2) Phase II. Post-buckling phase

As the compression continues, buckyball continues to deform. Once the compressive strain reaches _{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

where
_{b2}. The value of _{b2} could be fitted from the simulation data for C_{720} where _{b2} ≈ 11

The first force-drop phenomenon is obvious once the buckling occurs where the loading drops to nearly zero. Therefore, by applying the boundary condition of _{2}(_{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 force-displacement relation becomes nonlinear

where

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 force-displacement model which may be used to describe the mechanical behavior of the buckyball under quasi-static loading condition from small to large deformation.

Figure

Comparison between computational results and analytical model at low-speed crushing of 0.01 m/s

**Comparison between computational results and analytical model at low-speed crushing of 0.01 m/s.**

Two-phase model for impact

The mechanical behaviors of buckyball during the first phase at both low-speed crushing and impact loadings are similar. Thus, Equation 2 is still valid in phase I with a different ^{*} ≈ 4.30. The characteristic buckling time, the time it takes from contact to buckle, is on the order of ^{− 1} ~ 10^{0} ns ~ _{1} ≈ 5.71 × 10^{− 5}ns, where _{720} and
^{*} should be caused by the inertia effect

As indicated before, the buckyball behaves differently during the post-buckling 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

where the bending rigidity
^{’} is the ‘enlarged’ thickness, the result of smaller snap-through phenomenon. Here, ^{’} ≈ 1.40_{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 **c** = _{40}, _{50}, _{60}, _{70}, _{80}, _{90} = [0.83, 1.00, 1.12, 1.14, 1.17]. Figure

Comparison between computational results and analytical model

**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-D alignment buckyball system

The _{
720
} can be arranged both vertically and horizontally in a 1-D chain-like alignment. Figure

Characteristic normalized force-displacement curve of 1-D system with vertically lined C_{720 } buckyballs

**Characteristic normalized force-displacement curve of 1-D system with vertically lined C**_{720}**buckyballs.** The characteristic normalized force-displacement curve of 1-D system with five vertically lined C_{720}s at impact speed of 50 m/s.

Another 1-D 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

Characteristic normalized force-displacement curve of 1-D buckyball system with various numbers of horizontally lined C_{720 } buckyballs

**Characteristic normalized force-displacement curve of 1-D buckyball system with various numbers of horizontally lined C**_{720}**buckyballs.** The characteristic normalized force-displacement curve of 1-D 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

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.** 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

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 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 five-buckyball systems.

3-D stacking buckyball system

The packing density of a 3-D stacking system can be different than that of the 1-D system, and thus the performance is expected to vary. Four types of 3-D stacking forms are investigated, i.e., simple cubic (SC), body-centered cubic (BCC), face-centered cubic (FCC) (a basic crystal structure of buckyball
_{
SC
} =

Figure

Normalized force-displacement curves for SC, BCC, FCC and HCP packing of C_{720}

**Normalized force-displacement curves for SC, BCC, FCC and HCP packing of C**_{720}**.** Typical normalized force-displacement 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

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.** 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 _{m} = UME/(_{impactor}/_{
v
} = UVE/(_{impactor}/_{volume}) where _{volume} is the volume of the buckyball and impact speed as

where

where _{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 non-recovery 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 quasi-static (low-speed 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 1-D 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 low-speed impact. In addition, different 3-D 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 energy-related 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 co-authored more than 40 articles, with an

Acknowledgments

The work is supported by National Natural Science Foundation of China (11172231 and 11102099), DARPA (W91CRB-11-C-0112), National Science Foundation (CMMI-0643726), International joint research project sponsored by Tsinghua University (20121080050), Individual-research founding State Key Laboratory of Automotive Safety and Energy, Tsinghua University (ZZ2011-112), and World Class University program through the National Research Foundation of Korea (R32-2008-000-20042-0).