Contuum modelg of van der Waals teractions between carbon nanotube walls W.B. Lu 1, B. Liu 1a), J. Wu 1, J. Xiao, K.C. Hwang 1, S. Y. Fu 3,4 a), Y. Huang 1 FML, Department of Engeerg Mechanics, Tsghua University, Beijg 184, People's Republic of Cha, Department of Mechanical Engeerg, Northwestern University, Evanston, Illois 6181, USA, 3 Technical Inst. of Physics & Chemistry, Chese Academy of Sciences, Beijg 119, People's Republic of Cha, 4 Department of Civil & Environmental Eng., Northwestern University, Evanston, Illois 6181, USA, ABSTRACT Prior contuum models of van der Waals force between carbon nanotube walls assume that the pressures be either the same on the walls, or versely proportional to wall radius. A new contuum model is obtaed analytically from the Lennard-Jones potential for van der Waals force, with the above assumptions. Bucklg of a double-wall carbon nanotube under external pressure is studied, and the critical bucklg pressure is much smaller than those models volvg the above assumptions. a) Authors to whom correspondence should be addressed. Electrical addresses: y-huang@northwestern.edu and liub@tsghua.edu.cn 1
In order to overcome limitations of atomistic simulations, contuum models have been developed for carbon nanotubes (CNT), such as the lear elastic shell theory 1,, fite-deformation membrane theory 3 and shell theory 4 based on teratomic potentials. For multi-wall CNTs, the teraction between walls is governed by the van der Waals force, which is characterized by the Lennard-Jones 6-1 potential, V ( d) = 4ε ( σ d) ( σ d) the distance between a pair of atoms, and derivative gives the force between two atoms, F V ( d) 1 6, where d is σ =.3415nm and ε =.39ev for carbon 5. Its = (positive for attractive force). There are two types of contuum models to represent the van der Waals force between CNT walls. One assumes the pressure to be the same, p = p, between two adjacent CNT walls 6, where p and p are pressures on the ner and er walls. The other assumes the pressure to be versely proportional to the wall radius 7,8, i.e., pr = pr, where R and R are the ner and er wall radii (Fig. 1). The purpose of this letter is to avoid the above assumptions and obta the pressures on ner and er walls analytically from the Lennard-Jones 6-1 potential. Such analytical expressions will be useful the contuum modelg of multi-wall CNTs and their composite materials. Analytical expressions are also obtaed for the effective pressures on the ner and er walls of electrically charged double-wall CNTs. Cyldrical coordates ( R, θ, z ) are used for the double-wall CNT with the ner and er radii R and R shown Fig. 1. With losg generality, the distance between an atom on the ner wall ( R,,) and an atom on the er wall ( R, θ, z ) is
d = R + R R R cosθ + z. The forces between two atoms are equal and opposite, and are obtaed from the Lennard-Jones potential as F V ( d) the normal direction give the normal forces as ( cosθ ) ( cosθ ) R = (Fig. 1). Their projections along F = F R R d and R F = F R R d, which are not equal anymore (Fig. 1). Figure 1. A schematic diagram of double-wall carbon nanotube The carbon atoms on each wall are represented by their area density ρ c = 4 ( 3 3l ) such that the number of carbon atoms is ρ da over the area da, where l c is the equilibrium bond length (of graphene). This accurately represents the average behavior of van der Waals teractions between atoms from adjacent CNT walls 9. For an fitesimal area da on the ner wall, the net force the normal direction is ( cos ) π 1 c R c c = A c, which gives the pressure on the ρ da F ρ da ρ da dz F R θ R d R dθ ner wall as = π ( cos ) (positive for compression). p ρ R dz F R θ R d dθ 1 c Similarly, the pressure on the er wall is = π ( cos ). p ρ R dz F R R θ d dθ 1 c For the Lennard-Jones 6-1 potential, the above pressures are obtaed analytically as 3
p 11 σ R R 31 E 13 E 11 3πεσρcR R + R R + R = 5, (1) 3R σ R R 16 E 7 E 5 R + R R + R and p π where m = ( 1 s θ ) 11 σ R R 31 E 13 + E 11 3πεσρcR R + R R + R = 5, () 3R σ R R 16 E 7 + E 5 R + R R + R m θ, and k 4R ( ) R R R E k d relation ( )( 1 ) ( 3)( ) ( 4) 1 m k E = m k E m E, and π ( ) ( 1 s θ ) 1 E = K k = k dθ m m m 4 and 1 ( ) elliptic tegrals of 1 st and nd kd, respectively. = + ; E m satisfies the recurrg E = E k = π 1 k s d θ θ are the complete Figure. The normalized pressure, and pressure multiplied by the wall radius R, on the ner and er walls of a double-wall CNT versus the normalized sum of ner and er wall radii. Results of Ref. 8 are also shown for comparison. Figure shows the normalized pressure, ( 8 c ) p πεσρ, versus normalized radius, ( ) R + R σ, for the ter-wall spacg R R = σ, which is the equilibrium spacg due to 4
van der Waals force between graphenes 9. It is observed that p and p have different signs, and therefore p p. The result of He et al. 8, π ρc p dz F R dθ =, also shown Fig., is ( absolute value) much larger than Eq. (1) because the force F was not projected to the normal direction, and therefore misses the factor ( cos ) R θ R d side the tegration. For small ter-wall spacg R R = σ (not shown Fig. ), all pressures are positive (i.e., repulsive force), while they become negative (i.e., attractive force) for large ter-wall spacg R R = σ. Figure also shows the normalized product of pressure and radius, ( 8 c ) pr πεσ ρ, which has different signs for the ner and er walls, and therefore pr pr. There exist other potentials that are more accurate than the Lennard-Jones 6-1 potential to represent the C-C repulsive term, such as a registry dependent potential to describe the corrugation terlayer teractions graphene nanostructures (Kolmogorov et al. 1 ), or a modified Morse potential based on the ab itio LDA calculations (Wang et al. 11 ) that has been verified for graphite subject to high pressure 1. We use the Morse-type potential 11 to calculate the pressures p and p of a double-wall CNT with an ner wall radius.35nm and the ter-wall spacg.3nm. The resultg pressures are p =35. and p =6.7GPa, which give p p and p R p R. These conclusions also hold, general, for other potentials. Equations (1) and () are used to study bucklg of a double-wall CNT subjected to external pressure p ext. The deformation is uniform prior to bucklg, but its rate becomes 5
non-uniform at the onset of bucklg. The analysis is similar to that for a sgle-wall CNT 13 except that (i) the er wall is subjected to a net external pressure p ext -p ; (ii) the ner wall is subjected to an external pressure p ; and (iii) p and p are related to the ner and er wall radii R and R via Eqs. (1) and (). Figure 3 shows the critical external pressure at the onset of bucklg, (p ext ) cr, versus L/R for a (8,8)@(13,13) double-wall CNT, where L and R are the length and er radius; (p ext ) cr is much larger than that for the er wall [(13,13) sgle-wall CNT] 13, almost 4 times for long CNTs. This reflects the resistance of vdw force between walls agast bucklg. Results based on prior assumptions overestimate the critical external pressure for bucklg; the assumption p = p overestimates by 75% for long CNTs, and pr = pr overestimates by 6%. Figure 3. The critical external pressure at the onset of bucklg, (p ext ) cr, versus the length to er radius ratio, L/R, for a (8,8)@(13,13) double-wall CNT. Results for the (13,13) sgle-wall CNT, and those based on prior assumptions are also shown. Equations (1) and () can be expressed the Taylor series of the small parameter ( R R ) ( R R ) ξ = + as 6
p 5 11 σ 1 σ 7 1+ ξ 1+ ξ R R 8 R R, (3) 3 σ 6 σ 3 ± ξ + O( ξ ) 4 R R 5 R R = 8πεσρc 5 11 where + terms are for p and terms for p. Similarly, p R and p R are given by pr 4 1 σ 1 5 σ 1 11 ξ ξ R R ξ 16 R R ξ 4, (4) 1 σ 1 σ ± + + O( ξ ) 8 R R R R = 8πεσ ρc 4 1 where + terms are for pr and terms for p R. For the ter-wall spacg R R = σ as Fig., p, p.6πεσρc ( ξ 3ξ ),.3 c = ± + (and therefore p p ), and ( ) p R p R = πεσ ρ ± + ξ (and therefore p R p R ). Figure 3 also shows the critical external pressure at the onset of bucklg, (p ext ) cr, based on the Taylor series expansion (3) and (4), which agrees very well with the exact solution based on Eqs. (1) and (). The last example to illustrate p p and p R p R is for two concentric tubes 1 with electrical charges, which are characterized by the electrostatic potential V ( d) ~ d and F = V d ~ d, where force ( ) d = R + R R R cosθ + z is the distance between electrical charges. The electrical charges on the ner and er tubes are represented by their area densities (number of charges per unit area) ρ and ρ. The pressures on the ner and π er walls are then given by = ( cos ) and ρ ρ θ θ 1 p R dz F R R d d ( cos ) π 1 p = R dz F R R d d. For F V ( d) ~ d ρ ρ θ θ =, it can be shown 7
analytically that the pressure on the ner wall is identically zero, p, while the pressure on the er wall is not, p, such that p p and p R p R. In summary, contuum models for van der Waals force or electrostatic force between CNT walls are obtaed analytically. They bypass assumptions prior studies, and have been applied to determe the critical external pressure for bucklg of double-wall CNTs. Y.H. acknowledges the supports from the NSF (Grant No. CMMI 8417) and ONR Composites for Mare Structures Program (Grant No. N14-1-1-5, Program Manager Dr. Y. D. S. Rajapakse). B. Liu acknowledges the supports from the NSFC (Grant Nos. 1734, 1735, 98166). K.C. Hwang also acknowledges the support from NSFC (Grant No. 17789) and National Basic Research Program of Cha (973 Program, Grant No. 7CB93683). S.Y. Fu acknowledges the support from CAS (Grant Nos. 5--1) and NSFC (Grant Nos. 167161). 8
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