Article pubs.acs.org/JPCC
Interaction between Mechanical Wave and Nanoporous Energy Absorption System Hailong Liu and Guoxin Cao* HEDPS, Center for Applied Physics and Technology, College of Engineering, Peking University, Beijing 100871, China ABSTRACT: The interaction between mechanical wave initiated by impact load and nanoporous energy absorption system (NEAS) is investigated using molecular dynamics (MD) simulations, which includes the forward procedure (stage I) and the reflected procedure (stage II) of mechanical wave. The current NEAS is made of one-end closed single-walled carbon nanotubes (CNTs) and water. The effects of impact loading rate and tube size on the energy absorption of NEAS are also considered. After the bulk water is pushed into CNTs by the impact load, a new water interface will be created. The water molecules on interface have a much higher potential energy than their bulk counterpart, and this energy change can be defined as liquid interfacial energy, which gives the main contribution to the energy absorption of NEAS. As compared to stage I, the liquid interfacial energy density at stage II is significantly increased, especially for smaller tubes with a higher loading rate, and thus stage II can provide the significant contribution to the system energy absorption and must be considered in the energy absorption procedure. In addition, when water transports through CNTs, the solid−liquid friction based on the ven der Waals (VDW) interaction between CNTs and water is very small because the VDW repulse force can be partially canceled by the VDW attraction force, and the net force is very small; thus, the contribution of the solid−liquid friction to the energy absorption of NEAS can be neglected.
1. INTRODUCTION To meet the requirement for the lightweight, small-volume, and high-efficiency energy absorption system, nanoporous energy absorption system (NEAS) has attracted considerable research interest.1−7 NEAS typically includes nanoporous particles and nonwetting liquid, which are confined in a closed environment. Because of the nonwetting characteristic of the system, it is energetically favorable for the liquid to stay outside of the nanopores under the normal state (i.e., the potential energy of the liquid inside nanopore is higher than that outside). However, under an external load (impact or blast wave), the potential energy of the liquid outside nanopore increases with the pressure, and when the liquid potential energy outside nanopores is higher than that inside, the liquid will be pushed into nanopores to reduce the system energy. In this way, the work done by external load can be transferred into the solid− liquid interfacial energy (proportional to the specific surface area A and the solid−liquid interfacial energy γ), the liquid potential energy difference between inside nanopores and outside, and the heat created by the solid−liquid interfacial friction.6 Because nanoporous materials typically have a very high specific surface area A (∼1000 m2/g)3, NEAS can be a good candidate for the new high-efficiency energy absorption/ damping system. Under the effect of mechanical wave, the liquid infiltrates into nanopores, transports through nanopores, and fills the nanopores finally. The energy damping efficiency of NEAS mainly depends on the interaction between the mechanical wave and NEAS, which includes two stages: (I) the forward © 2013 American Chemical Society
procedure of mechanical wave, transporting from the nanopores’ opening to their end; and (II) the reflecting procedure of mechanical wave, reflecting from the pore end back to the opening. Currently, most of the studies about NEAS are based on quasistatic loading conditions, and very few studies investigate the energy damping behavior under dynamic loading condition (impact/blast),6,7 which unfortunately is the main loading mode for NEAS to protect against under the realistic condition. The interaction between the mechanical wave and NEAS is still not very clear, especially for the reflecting procedure of wave, which has not been studied yet (to our best knowledge). Because of the small length scale, and neutral and smooth surface, the nanofluid behavior (infiltration/transporting) is mainly investigated based on carbon nanotubes (CNTs) and especially for numerical simulations.6−14 On the basis of the CNTs plus water system, our recent work6 studied the fundamental working mechanism of NEAS on the basis of the forward procedure (stage I) (before the mechanical wave reaches the end of CNTs). We found that the potential energy increase of the water molecules entering into CNTs as well as the solid−liquid friction are the main contributors to the system energy absorption. In the present work, the entire energy damping procedure of NEAS is investigated, which includes both the forward stage (stage I) and the reflecting Received: October 10, 2012 Revised: February 5, 2013 Published: February 6, 2013 4245
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Figure 1. The interaction between mechanical wave initiated by the impact load with NEAS modeled using MD simulation: (a) the initial state of NEAS; (b) the forward procedure of mechanical wave; (c) the reflecting procedure of mechanical wave; and (d) the end state of energy absorption (the mechanical wave reaches the opening of CNT).
loading rates are used (including 100, 200, 400, and 1000 m/s) similar to our previous work. The computational cell includes a single-walled CNT segment and 3570 water molecules confined in a “reservoir” shown in Figure 1a. Typically, the nanoporous particles or the nanotube forest (the size is in micrometer level) are used in energy absorption/damping system, and the nanopores are embedded into nanoporous particles. The wall of each pore is constrained by its environmental materials and has a very low flexibility. In addition, Joseph and Aluru15 have reported that the flexibility of the flexible wall only has a weak effect on the mean flow rate of water inside CNTs. Therefore, the rigid CNTs are used in the present work, and the effect of the flexibility of the pore wall (including the vibration behavior) on the energy absorption is neglected. To more accurately show the effect of liquid interface on the energy absorption, we select much longer CNT segments (∼10 nm) than our previous work,6 and the left end of the CNT is closed by a rigid carbon
stage (stage II) of the mechanical wave. The present study can provide us with a further understanding of the mechanism of NEAS.
2. COMPUTATION METHODS Because of their simple structures, CNTs and water are selected as the basic model of nanopores and nonwetting liquids in NEAS to simulate the interaction between mechanical wave and NEAS. Because CNTs are hydrophobic, water cannot wet the internal surface of CNTs; that is, it is energetically favorable for water molecules to stay outside nanotubes than inside nanotubes under the normal condition. Therefore, the empty CNTs are used to simulate the hydrophobic nanopores. Similar to our recent work,6 the impact load is applied by a piston (modeled by a rigid carbon atom plane) with a constant speed. Three CNTs are selected (including (10, 10), (15, 15), and (20, 20) tubes with the radii of 0.67, 1.00, and 1.34 nm) to investigate the pore size effect. For each tube, four different 4246
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water molecules under external loading are monitored in the whole energy absorption procedure.
atom plane to simulate the reflection procedure of the mechanical wave. The right-open end of the CNT is connected with the reservoir, which is bounded by two parallel rigid carbon atom planes. The size of the reservoir is 4.65 × 4.65 × 4.95 nm, and the density of the reservoir is set to about 0.997 g/cm3. The right plane is movable to mimic a piston, and the left one is fixed. Periodic boundary conditions are applied on the lateral directions of the computation cell to remove the boundary effect. The MD simulations are carried out using LAMMPS,16 which is a classical molecular dynamics software from Sandia National Laboratories. Water molecules are simulated by the TIP3P model (transferable intermolecular potential 3P).17 Although the TIP3P model gives poor water bulk properties, it is still widely used in describing the behavior of water confined inside CNTs.18−22 Falk et al.22 compared two different water models (TIP3P and SPC/E) in their study of water transport behavior in CNT membranes. Their results show that those two models can predict a very close value of slip length for water on graphene; thus the transport behavior of water confined inside CNTs is essentially not very sensitive to the water model used. The nonbond interactions between water molecules are modeled by the Lennard-Jones (LJ) potential and the Coulombic potential. Typically, the nonbond interactions between water and C atoms are mainly described by the LJ potential. Zgarbova et al.23 recently showed that the repulse term of the commonly used 6-12 LJ potential might be not accurate in the case without the Coulombic interaction and should be replaced by an exponential function on the basis of comparing the results from AMBER forcefield and the results from density functional theory (DFT). However, the VDW interaction between C atoms and water is very weak because the very high flow rate of water through CNT has been widely reported.15,21,24 For the present work, if the solid−liquid interface friction (created by the VDW interaction) will not have a large contribution to the overall energy absorption of NEAS, the slight variation of the repulse term of LJ potential will not essentially affect the results of the energy absorption of NEAS. The COMPASS class2 forcefield (in LAMMPS) is used to describe the VDW interaction between water and C atoms. The VDW interaction between water and C atoms is simplified as the C−O interaction because the C−H interaction is very weak. The LJ potential parameters between C and O atoms are ε = 0.126 kcal/mol and σ = 3.835 Å. The cutoff distance for the van der Waals (VDW) interaction is set to be 10 Å, which is typically used in the MD simulations of water.15,21 The longrange Coulombic potential is calculated using the particle− particle particle-mesh (PPPM) method.25 A time integration step is set to 1 fs. Initially, water molecules cannot enter into CNTs (the right end of CNT is covered), and the water molecules in the reservoir are equilibrated for about 100 ps using the NVT ensemble. The Nose−Hoover thermostat is used to keep the temperature of 298 K. After the whole system is equilibrated, the cover on the CNT opening is removed (water can freely enter into CNTs), and then the piston is moved to push water with a constant loading rate. The NVE ensemble is used to study the impact-induced infiltration behavior. For all loading rates, the final piston displacement is set to 2 nm. The simulations are stopped after the reflecting procedure is finished (the reflected mechanical wave reaches the CNT opening). The location, velocity, and acceleration of
3. RESULTS AND DISCUSSION The impact energy absorption procedure of NEAS can be simplified as two stages. At stage I, the water molecules are pushed into CNTs by the forward mechanical wave (created by the piston), and then transport through CNTs to the tube end (shown as Figure 1b); at stage II, the mechanical wave is reflected by the tube end and transports back to the tube opening (shown as Figure 1c). The above two stages can be simply separated by the water infiltration length Lin: (I) Lin < L (L is the tube length) and (II) Lin = L. In our recent work,6 the impact-induced infiltration behavior of water at stage I has been studied, and the working mechanism of NEAS has been illustrated on the basis of it. The impact energy absorbed by NEAS can be transferred to the potential energy increase of water molecules after they enter into CNTs (ΔEH2O), the CNT−water interaction energy (Eint), and the heat created by the internal friction (Hf), among which ΔEH2O provides the main contribution. In this section, the interaction between the mechanical wave and NEAS in the whole energy absorption procedure (including stages I and II) is investigated, and the energy damping mechanism of NEAS based on the whole procedure is reevaluated. In the present work, the subscripts of I and II are used to represent the quantities at stages I and II, respectively. 3.1. Liquid Interfacial Energy Density. The water molecules inside CNTs include both surface molecules (directly contact with the CNT wall) and internal molecules. The structure of surface molecules is significantly different from its bulk counterpart due to less hydrogen bonds and the CNT− water interactions. The structure difference between internal water molecules and bulk water caused by the CNT−water interactions rapidly reduces with the increase of the distance from interface to center. Because of the structure difference, the water molecules inside CNTs have a higher potential energy than their bulk counterpart, which is mainly caused by the new created water interface. Therefore, the absorbed impact energy can be converted into the interfacial energy of water. It should be noted that the interfacial energy is not the solid−liquid interaction energy but mainly from the energy change of water inside CNTs. The interfacial energy density (ϕ) of water is defined as: ϕ = n ins(Δes + Δe in)
nsin
(1)
where is the interface molecular density = Nsin/ s (2πReLin), where Nin is the number of infiltrated water molecules on the interface), Δes is defined as the mean energy difference between interface water molecule and its bulk counterpart, and Δein is the equivalent interfacial energy created by all internal molecules, which is calculated by the energy difference between all internal molecules and their bulk counterpart over the number of interface water molecules Nsin. Δein < Δes and with the increase of R, Δein rapidly reduces to zero. Because it is very difficult to accurately determine both nsin and Δes in MD simulations, the following equation is used to calculate ϕ in the present work: ϕ= 4247
(nsin
ECNT + H 2O − ECNT − Nine HB 2O 2πR eL in
(2)
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Figure 2. The relationship between the number of infiltrated water molecules (Nin) and the interfacial energy density of water (ϕ) for (a) (10,10) tube, (b) (15,15) tube, and (c) (20,20) tube.
where ECNT is the potential of CNT, the potential per molecule of bulk water eBH2O ≈ −7.07 × 10−20 J,6 ECNT6+H2O is the total potential energy of CNT and water inside, Nin is the number of infiltrated water molecules, and Re is the effective radius occupied by the water molecules in CNTs. Re is calculated from the radius of the cylindrical space containing 95% of the infiltrated water molecules. For the small tubes, Re is far less than the tube radius R, and (R − Re)/R rapidly decreases with the increase of R. In the present work, ϕ is used to analyze the energy absorption performance of NEAS instead of the water potential energy change (ΔeH2O) and the CNT−water interaction energy (eint) used in our previous work6 because ϕ is a macroscale quantity with more physical meaning and might be measured experimentally. The relationships between ϕ and Nin with the different loading rates v for the whole energy absorption procedure (including stages I and II) are shown in Figure 2. In stage I (ϕ = ϕI), ϕ is essentially not sensitive to Nin (with a large Nin). ϕ increases with v and slightly increases with the tube radius R in
the present study range. These results are mainly caused by the water structure change and quite consistent with our previous findings.6 When Nin is large, a stable water structure can be generated inside CNT, especially for a stable boundary layer (with a fixed ϕ). This structure will not be changed by adding more water molecules but depends on v, which can be proved by the radial distribution functions (RDFs) of the water molecules inside CNTs as shown in Figure 3. Actually, when Nin is enough large, RDFs will not be changed with increasing Nin. For a given R, although nsin decreases with the increase of v (as shown by the peak value of RDFs in Figure 3), both Δes and Δein sharply increase with v because the water molecules can easily deviate from their equilibrium positions with a high v, and thus ϕ increases with v. The peaks of RDFs increase with the decrease of v. For a given v, both nsin and Δes are essentially not sensitive to R, but Δein increases with R becaause there are more internal molecules affected by the CNT−water interactions in a larger tube, especially for a high v, which leads to a slightly increasing ϕ with R. The number of internal 4248
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passed by the reflected wave is changed into structure B (i.e., ϕII = wAϕA + wBϕB), and finally the whole water structure inside CNTs is changed into structure B (wA = 0, wB = 1, and ϕII = ϕB) when the reflected wave reaches the tube opening. After the reflected mechanical wave passed by, the water structure is sharply changed due to the effect of impact pressure, which can be shown by the RDFs (Figure 3). The RDF peak of structure B is much higher than that of structure A, and the peak position of structure B is slightly closer to the tube wall than that of structure A. These results show that for a given tube radius R, structure B has a larger nsin than structure A, and both Δes and Δein will also increase due to the higher molecular density of structure B, and thus the effective surface energy ϕ of structure B is higher than that of structure A (ϕB > ϕA) . For a small tube with a high v, nsin can be significantly increased when the reflected wave passed the structure because structure A has a very low value of nsin, which leads to a significant increase of ϕ from stage I to stage II. However, for a large tube with a low v, nsin is quite high, and there is a stronger CNT−water interaction at stage I, and thus it is very difficult to further compress the interface molecules to create more space to hold new molecules on the interface. Most of the new entered molecules become the internal molecules; thus ϕ slightly increases from stage I to stage II. The ϕ−R relationships under different v are summarized in Figure 4. At stage I, ϕI increases with R in the present study
Figure 3. The radial distribution function (RDF) of the infiltrated water molecules at stages I (displaced as dashed lines) and II (displaced as solid lines) of the energy absorption procedure for (a) (10,10) tube and (b) (20,20) tube. The peak value of RDFs increases with the decrease of the impact velocity (v) for both stages I and II in the figures. Figure 4. The relationship between the water interfacial energy density (ϕ) and the CNT radius (R).
water molecules affected by the CNT−water interactions will not continually increase when R is larger than the VDW interaction range (∼1 nm); thus the effect of Δein on ϕ rapidly decreases with the increase of R. In stage II, the mechanical wave is reflected back from the tube end to the opening, and water molecules continually enter into CNTs at the same time. The water structure in CNTs at stage II is more complicated than that at stage I, which can be considered as a weighted average of the following two structures: (A) the untouched water structure (including new entered water molecules) by the reflected wave; (B) the water structure affected by the reflected wave. Thus, ϕ = ϕII = wAϕA + wBϕB, and wA + wB = 1, where wA and wB are the weighting parameters. Structure A is similar to the water structure at stage I because it does not feel the impact pressure. At the beginning of stage II, the whole water structure inside CNTs is the same as structure A (i.e., wA = 1, wB = 0, and ϕII = ϕA = ϕI), and then, with the propagation of the reflected wave, the water structure
range, while at stage II, (15,15) tube (R = 1 nm) has the larger value of ϕII than both (10,10) tube (R = 0.67 nm) and (20,20) tube (R = 1.34 nm). The difference between ϕII and ϕI represents the effect of the reflected wave on the water structure. It is clearly shown that the value of ϕ is significantly increased in the reflecting procedure, especially for a small tube under a high speed impact (e.g., ϕII/ϕI ≈ 3.8 for R = 0.67 nm and v = 1000 m/s), which decreases with the increase of R or the decrease of v (e.g., ϕII/ϕI ≈ 1.6 for R = 1.33 nm and v = 100 m/s). In addition, the trend of ϕ with R at stage II is also changed: ϕ reaches the maximum value round R = 1 nm, and then ϕ decreases with the further increase of R. Thus, the reflecting procedure is highly important for the energy absorption. 4249
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3.2. Interfacial Friction of Water Molecules. It has been reported that water transports through CNTs as a plug flow; that is, the flow front is quite flat, and the boundary layer has an axial velocity similar to that of the center part.15,21,22,24 When water flows through CNTs, there might be a friction between solid CNT wall and liquid due to the solid−liquid interaction (VDW interaction). Our previous work21 and Falk et al.22 investigated the water− CNT friction behavior of the stead flow in CNTs by checking the deceleration of the water flow inside CNTs, and found that the water−CNT shear stress (τ) increases with the tube size R or the flow rate (vt). In addition, on the basis of the VDW repulse force between CNT and water molecules, our recent finding6 showed that the friction at the stage I of the impact induced infiltration procedure slightly increases with R with v = 100 m/s. In this section, we will discuss the solid−liquid friction in the whole energy absorption procedure (including stages I and II) with different R and v. Similar to our previous work,6 the solid−liquid friction is calculated by the summation of the van der Waals (VDW) force between the carbon atoms and infiltrated water molecules. In our previous work, only the repulse force of the VDW interaction is used to calculate the friction because the repulse force is much larger than the attraction force, while the whole VDW interaction (including both repulse and attraction) is considered in the present work, and we also select much longer CNT segments (L ≈ 10 nm) than those used in our previous work, which should provide the more accurate friction results. To remove the tube end effect, the tube ends (about 1 nm long of both ends) will not be considered in the friction calculations. The work done by the water−CNTs friction force at each time step can be calculated as:
in both positive region and negative region; it is essentially not sensitive to R but slightly increases with v. The net work of the VDW force on the flow should be the summation of the work of all time steps (ψt = ∑ψ(t)), which is very small for the system of CNTs and water. Thus, the work of the VDW interaction per unit area of CNT surface (φ = ψt/(2πRL)) is very low: for example, φ ≈ 0.0031 J/m2 for the tube with R = 1.33 nm and v = 100 m/s. Because of neglecting the attraction part of the VDW force, our previous finding6 gave a much higher CNT−water friction. Therefore, the VDW attraction part is very important to the solid−liquid friction and cannot be neglected. Our present results are much lower than the water−CNT friction calculated from the steady flow,21,22 which also shows the increasing friction with R. This difference may be caused by the flow rate vt and the water density inside CNTs. In the present work, due to the high impact velocity, the flow rate is higher and the infiltrated water flow density is much lower than those of the stead flow. With the decrease of the loading velocity v, the water flow density increases, which can be clearly shown by the RDF peaks in Figure 3. 3.3. Energy Absorption Efficiency. In the above sections, the liquid interfacial energy density (ϕ) and the effective friction work (ψt) of the whole energy absorption procedure are analyzed. The overall impact work absorbed by NEAS should be converted into the liquid interfacial energy and the effective friction work. As compared to ϕ, the effective friction work density (φ) is about 1−2 order lower, and thus the effect of φ on the energy absorption of NEAS can be neglected. The main purpose to develop NEAS is to meet the requirement for a high energy absorption under a small volume limitation. Thus, the energy absorption efficiency can be effectively represented by the energy absorption density:
Nin
ψ (t ) =
∑ fi si i=1
ξ ≈ A̅ ϕλ (3)
(4)
where A̅ is the specific surface area of CNTs per unit volume (as shown in Table 1), and λ = Re/R, the ratio of the effective
where f i is the VDW force applied on each water molecules from CNTs, and si is the displacement of water molecules. ψ(t) periodically fluctuated around ψ = 0 (as shown in Figure 5), which means that the VDW force can either resist the flow (ψ > 0) or drive the flow (ψ < 0). The amplitude of ψ is quite close
Table 1. System Parameters of NEAS CNT type (10,10) tube radius (Å) specific surface area (m2/cm3) volume fraction of CNT to NEAS (α)
6.71 2982 0.19−0.29
(15,15) 10.03 1993 0.22−0.33
(20,20) 13.37 1496 0.24−0.36
radius of water inside CNTs to the CNT radius, with λ converging to 1 with the increase of R. The interfacial energy density ϕ is calculated from the area of water interface, but the energy absorption density of CNTs should be calculated on the basis of the volume of CNTs. The ξ−R relationships under different v are shown in Figure 6. Because A̅ is inversely propositional to R, ξ essentially increases with the decrease of R. In the present study range, ξ can reach as high as 1.33 kJ/ cm3 for R = 0.67 nm and v = 1000 m/s. If only stage I of the energy absorption procedure is considered, ξ will be significantly underestimated, especially for small tubes with a high speed impact. In addition, the trend of ξ with R is also changed after considering stage II of energy absorption. It should be noted that the energy absorption density shown in Figure 6 is based on isolated CNTs, but it is very difficult for
Figure 5. The variation of the work done by the VDW force with the simulation time. 4250
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fraction of isolated CNTs to the whole NEAS (shown in Table 1). In the present work, we further extend our previous work, including the contribution of stage II to the overall energy absorption and also considering the effect of the water−CNT VDW attraction on the liquid−solid friction. If the water−CNT VDW attraction was neglected, the water−CNT friction was significantly overestimated, and our previous work6 gave a higher value of the energy absorption density of isolated CNTs: ξ = 1.7 kJ/cm3 (for R = 0.67 nm and v = 400 m/s), in which the friction work contributed about 0.95 kJ/cm3, but it can be essentially neglected in the present work. After considering the stage II of energy absorption, the energy absorption density of NEAS is increased by about 60−280%, especially for the small tubes with a high loading velocity. In addition, the response time of NEAS to impact load should be considered as the total time for both stages I and II.
4. CONCLUSIONS Using MD simulation, the whole energy absorption procedure of the NEAS made by CNTs and water is investigated, and the effects of both impact loading rate and CNT size are also considered. The whole energy absorption procedure includes the forward procedure of mechanical wave (stage I) and the reflected procedure of mechanical wave (stage II). The impact work can be mainly converted into the liquid interfacial energy, which is calculated as the energy difference between the water molecules inside CNTs and their bulk value. To the best of our knowledge, this is the first work to consider the effect of the interaction between the reflected mechanical wave and NEAS on the energy absorption efficiency of NEAS. Because the water molecules continually enter into CNTs until they meet the reflected wave, NEAS continually absorb the energy at stage II, that is, ϕI < ϕII. The contribution of stage II is more significant for a smaller R and a higher v, and the ratio of ϕI/ϕII = 0.24−0.66. Thus, the interfacial energy density will be significantly underestimated if only stage I is considered, especially for the small tube under the high speed impact v. In addition, the ϕ increases with R or v at stage I, but ϕ reaches the maximum value at the tube radius R ≈ 1 nm at stage II. The trend of ϕ to R has been changed after considering stage II. The VDW interaction between CNTs and water molecules includes both attraction force and repulse force, which can partially cancel with each other, and the net force (the source of the interface friction) is very small. The VDW attraction cannot be neglected; otherwise the solid−liquid friction will be significantly overestimated. As compared to the interfacial energy, the work done by the interface friction can be essentially neglected in the overall energy absorption of NEAS. The energy absorption density (ξ) of isolated CNTs with water is higher for the smaller tube because the specific surface area of isolated CNTs is reversely proportional to R. ξ also increases with v because the interfacial energy density increases with v. On the basis of the present study range, the maximum value of ξ can be as high as 1.33 kJ/cm3 (R = 0.67 nm and v = 1000 m/s). After considering the volume fraction of the isolated CNTs to the whole NEAS, the energy absorption density (ξ) of NEAS can take the maximum value at the tube size (R) around 1 nm, and the value can be estimated as 0.26− 0.4 kJ/cm3, which is still much higher than the conventional energy absorption materials. Therefore, NEAS can be a promising candidate for mechanical energy damping, especially under the small volume limitation condition.
Figure 6. The relationship between the energy absorption density (ξ) and the CNT radius (R) for isolated CNTs.
CNTs to be isolated, and they normally stay as CNT bundles. In addition, CNT bundles need to be sealed inside a nonwetting liquid environment to realize the function of energy absorption, and thus the energy absorption density should be calculated on the basis of the unit volume of NEAS, which can be roughly estimated to be about 2−3 times the volume of CNT bundles sealed inside. In bundles, CNTs are distributed as the centered hexagon pattern, and the center− center distance of CNTs in the bundle is 2R + dVDW, where dVDW = 0.34 nm, the van der Waals distance of carbon atoms. After considering the volume fraction of CNTs to the overall NEAS (α), the energy absorption density (ξ̅ = ξ·α) can take the maximum value at the tube size close to 1 nm (as shown in Figure 7), and the value reduces to 0.26−0.4 kJ/cm3, which is still much higher than that of the conventional energy absorption materials. In Figure 7, the error bar represents the range of ξ̅ calculated from the range of the estimated volume
Figure 7. The relationship between the energy absorption density (ξ̅) and the CNT radius (R) for NEAS. The error bar in the figure represents the range of ξ̅ calculated from the estimated range of the volume fraction of isolated CNTs to NEAS. 4251
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AUTHOR INFORMATION
Corresponding Author
*Tel.: 086-01-62756284. E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We acknowledge the financial support provided by the National Science Foundation of China under grant no. 11172002. REFERENCES
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dx.doi.org/10.1021/jp310028x | J. Phys. Chem. C 2013, 117, 4245−4252
