ARTICLE pubs.acs.org/Langmuir
Universal Curves for the van der Waals Interaction between Single-Walled Carbon Nanotubes Evgeny G. Pogorelov,† Alexander I. Zhbanov,*,‡ Yia-Chung Chang,† and Sung Yang§ †
Research Center for Applied Sciences, Academia Sinica, 128, Section 2, Academia Road, Nankang, Taipei 115, Taiwan Graduate Program of Medical System Engineering, Gwangju Institute of Science and Technology (GIST), 261 Oryong-dong, Buk-gu, Gwangju, Republic of Korea 500-712 § Graduate Program of Medical System Engineering, School of Information and Mechatronics, Department of Nanobio Materials and Electronics, Gwangju Institute of Science and Technology (GIST), Gwangju, Republic of Korea ‡
ABSTRACT: We report very simple and accurate algebraic expressions for the van der Waals (VDW) potentials and the forces between two parallel and crossed carbon nanotubes. The Lennard-Jones potential for two carbon atoms and the method of the smeared-out approximation suggested by Girifalco were used. It is found that the interaction between parallel and crossed tubes is described by two universal curves for parallel and crossed configurations that do not depend on the van der Waals constants, the angle between tubes, and the surface density of atoms and their nature but only on the dimensionless distance. The explicit functions for equilibrium VDW distances, well depths, and maximal attractive forces have been given. These results may be used as a guide for the analysis of experimental data to investigate the interaction between nanotubes of various natures.
’ INTRODUCTION The van der Waals (VDW) interaction between graphitic structures is very important for application in nanoelectromechanical systems (NEMS). There are a number of publications devoted to the estimation of VDW potentials for graphite layers,14 two fullerenes,58 a fullerene and a surface,9,10 a carbon nanotube (CNT) and a surface,4,1113 and fullerenes inside and outside of nanotubes.1420 The interactions between the inner and the outer parallel tubes such as single-walled nanotubes (SWNTs),14,2123 double-walled nanotubes,24,25 and multiwalled nanotubes (MWNTs)2628 have also been well studied. The potential between two crossed CNTs was considered in our recent work.29 The physics of a random network of carbon nanotubes has attracted much attention lately because of the diverse range of their potential technological applications, from electronic and acoustic devices to nanocomposite materials with excellent mechanical properties.3032 Usually the raw SWNT synthesized samples are continuously entangled network structures. The mechanisms of network formations and CNT behavior in polymeric and colloidal media are the focus of interest of many researchers.33,34 The achievement of uniform alignment, in turn, is an essential condition in various applications proposed to exploit the unique properties of highly oriented CNTs.34,35 Thus, investigating the VDW interaction for CNTs at various orientations is incredibly important. The continuum Lennard-Jones (LJ) model suggested by Girifalco5 is usually used to evaluate the potential between two r 2011 American Chemical Society
graphitic structures. The LJ potential for two carbon atoms in the graphenegraphene structure is jCC ðrÞ ¼
A B þ 12 6 r r
ð1Þ
where r is a distance and A and B are attractive and repulsive constants. According to Girifalco, the potential between two SWNTs is approximated by the integration of the LJ potential Z ð2Þ j ¼ ν2 jCC ðrÞ dΣ1 dΣ2 where dΣ1 and dΣ2 are the surface elements for each tube. In the case of CNTCNT interaction, the mean surface density of carbon atoms is ν = 4/3(3)1/2a2, where a = 1.42 Å is the lattice constant for the graphene hexagonal structure.24 The VDW force is a complex interaction that involves geometrical as well as many-body effects. The approach proposed by Girifalco is based on the pairwise potential when the relative positions of the C atoms are not taken into account. The limitations of the smeared two-body approximation are discussed in refs 3 and 4. Despite its simplicity, the continuum LJ model was very successful in the description of many complicated technological problems such as carbon-nanotube-based NEMS devices,36 Received: September 27, 2011 Revised: December 1, 2011 Published: December 01, 2011 1276
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the buckling analysis of CNTs,37 the interaction potential for CNTs of arbitrary length and orientation,30,38 and the bulk elastic properties of SWNT bundles.39 The numerical integrations of the potential j were exploited for CNTs both in parallel21,23,27 and in crossed configurations.30,38 Apparently, the integration of potential j in eq 2 in an analytical form for the parallel orientation is impossible. Attempts to take integral eq 2 exactly for crossed CNTs lead to very complicated expressions.29 Thus, the use of simple analytical expressions is very promising for the description of greater random networks and bundles of CNTs. It was found that VDW potential j for C60C60, C60 SWNT, C 60 graphene, graphenegraphene, and parallel SWNTSWNT or MWNTMWNT at different distances d can be described by the universal curve.14,21,27 The universal curve for two tubes means that a plot of j̅ = j/|j0| against d = d/d0 gives the same curve for all radii of tubes, where j0 is the minimum energy and d0 is the equilibrium spacing for the two interacting surfaces. We also have plotted the same universal curve29 for crossed nanotubes and have obtained a deviation from the universal curve suggested by Girifalco et al.14 Unfortunately, both complex analytical expressions and numerical calculations are very difficult to apply to the study of physical mechanisms and dependences. In this work, we report very simple and analytically precise expressions of universal curves for the VDW potentials and forces between parallel and crossed SWNTs in the framework of the smeared-out approximation suggested by Girifalco. These curves depend only on the dimensionless distance and are independent of the attractive and repulsive constants, the diameters of nanotubes, and the angle between their axes. Obtained universal curves without any modification can be directly applied to find the VDW interaction between tubes made of another material such as boron nitrite, silicon carbide, or others nanotubes. Also we give explicit functions for equilibrium VDW distances, potential wells, and maximal attractive forces.
’ CROSSED CONFIGURATION OF TWO SINGLEWALLED CARBON NANOTUBES van der Waals Potential. Figure 1 illustrates the interaction between two SWNTs. In this figure, t1 and t2 are the radii, R is the distance between axes, and d is the gap between the surfaces of tubes. If angle γ = 0, then the tubes are parallel. Let us consider first the case of crossed tubes. We have found that the LJ potential from eq 2 can be integrated exactly for two crossed SWNTs.29 The analytical equation is expressed in terms of elementary functions and elliptic integrals. Introducing the dimensionless variables
b1 ¼
R R , b2 ¼ t1 t2
we obtained for two SWNTs that the VDW potential is ν2 AgA ðb1 , b2 Þ BgB ðb1 , b2 Þ jðt1 , t2 , RÞ ¼ þ R2 R8 sin γ
ð3Þ
ð4Þ
where gA and gB are complicated expressions in terms of elliptical functions.29 It is worth noting that the VDW potential for nanotubes in a crossed configuration is limited and measured in energy units. The VDW potential between two crossed SWNTs is inversely
Figure 1. Schematic drawing of the interaction between two SWNTs.
proportional to the sine of the angle γ. This expression becomes singular for a parallel orientation, γ = 0. The energy of the VDW interaction between two infinite parallel tubes tends toward infinity. Thus, the VDW potential for parallel tubes is defined per unit of length. Therefore, the full VDW energy for parallel nanotubes is infinite and minimum. Approximation of the VDW Potential. Equation 4 can be essentially simplified when d/t1 , 1 and d/t2 , 1. Introducing α = 1/t1 + 1/t2 and using the small-parameter expansion, we obtain the approximating equations pffiffiffiffiffiffiffi π2 t1 t2 1 αd þ gA ≈ ð5Þ 3 48 d3 pffiffiffiffiffiffiffi 2π2 t1 t2 gB ≈ 45d9
ð6Þ
Using the approximations of eqs 5 and 6, we may modify eq 4 to pffiffiffiffiffiffiffi ν2 π 2 t 1 t 2 1 αd 2B ð7Þ þ jðt1 , t2 , dÞ ¼ 3 A þ 3 48 45d6 d sin γ This approximation allows us to explain the existence of universal curves for crossed SWNTs. Also, it allows us to obtain simple expressions of equilibrium VDW distance d0, minimum VDW potential j0, universal curve j̅ = j/|j0|, and total force F for the interaction between two crossed SWNTs. Equilibrium Distance. From eq 7 we find ∂j/∂d = 0 and write the recurrent equation for equilibrium distance d0: d0 ¼
1=6 1=6 B 48 A 5αd0 þ 120
ð8Þ
One can solve eq 8, passing a few steps of this recursive equation with very fast convergence. If αd0 , 1, that is, t1 and t2 are large enough in comparison to d0, then we have the first approximation of equilibrium spacing 1=6 2B ð1Þ ð9Þ d0 ¼ 5A It is remarkable that for SWNTs with large radii the equilibrium distance depends only on attractive and repulsive constants A and B. The independence of the equilibrium distance on the angle between nanotube axes was also observed in ref 38. By measuring the gap d0, it is possible to find the ratio between A and B. We find that eq 9 works well if t1, t2 > 6 Å and gives a small error for tubes with the smallest radii. The second approximation 1277
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provides high accuracy even for t1, t2 ≈ 2 Å: ð2Þ d0 ðαÞ
¼
1=6
48B A
!1=6 2B 1=6 5α þ 120 5A
Table 1. Minimal and Maximal Equilibrium Distances and Potential Energy for Interaction between Two Identical Armchair SWNTs Crossed at Right Anglesa
ð10Þ
If A = 15.2 eV Å6, B = 24 100 eV Å12 as in ref 14, and t1 and t2 are changing from 3.40 to 20.35 Å, then d0(α) is in the 2.914 2.927 Å range. If t1 and t2 tend toward infinity, then d0 = d(1) 0 = 2.931 Å. Using potential eq 4, we can calculate the exact value of equilibrium distance dex 0 (t1, t2). If t1, t2 > 3.4 Å, then the maximal (1) difference between dex 0 (t1, t2) and approximation d0 does not ex (2) exceed 0.6% and that between d0 (t1, t2) and d0 does not exceed 0.1%. Potential Depth. We can rewrite eq 7 for the potential energy in the form pffiffiffiffiffiffiffi ν2 t1 þ t2 ð11Þ C1 t1 t2 þ C2 pffiffiffiffiffiffiffi jðt1 , t2 , dÞ ¼ t1 t2 sin γ where C1 ¼ π
2
A 2B Aπ2 3 þ ¼ , C 2 3d 45d9 48d2
ð12Þ
are the parameters that do not depend on the radii of tubes. Substitution of equilibrium VDW gap d0 into eq 7 gives us the potential well ν2 0 pffiffiffiffiffiffiffi 0 t1 þ t2 ð13Þ C t1 t2 þ C2 pffiffiffiffiffiffiffi j0 ðt1 , t2 Þ ¼ t1 t2 sin γ 1 Let us note here that a similar function has been obtained numerically in our work.29 In limiting case γ = 0, the potential depth of the parallel configuration is infinite. If we use the first approximation d(1) 0 for equilibrium spacing in eq 9, then pffiffiffiffiffi A3=2 10π2 0 A4=3 22=3 51=3 π2 pffiffiffi , C2 ¼ C01 ¼ ð14Þ 96B1=3 9 B We used exact potential j0(t1, t2) based on exact analytical expression eq 4 and have found that the potential may be very well approximated by eq 13. Our numerically fitted parameters29 are Cnum = 1.3232 eV Å3 and Cnum = 0.4012 eV Å4. These 1 2 values are very close to C01 = 1.3238 eV Å3 and C02 = 0.3638 eV Å4 from eq 14. In deriving our expressions for the equilibrium distance and the potential depth, first we used the smeared-out approximation suggested by Girifalco and second we assumed that the nanotubes are rigid enough to maintain a cylindrical geometry. Undoubtedly, both presumptions are valid in the limit of large tube separations. At first glance, the soundness of eqs 10 and 13 can result in serious uncertainty at a small gap between tubes. We used molecular dynamics (MD) simulations to estimate the limits of applicability for our model. It was shown by Ruoff et al.40 that VDW forces between adjacent nanotubes can deform them, destroying the cylindrical symmetry. Using molecular mechanics, Hertel et al.41 calculated the radial deformations of carbon nanotubes adsorbed on surfaces. The radial compressions of adsorbed SWNTs with respect to the undistorted free tubes do not exceed 2% for tube diameters of 13.5 Å and smaller. When the number of
equilibrium
potential
distance (Å)
energy (eV)
MD
MD
simulation
simulation
tube-type radius (Å)
a
min
max
eq 10
min
max
eq 13
(3, 3) 2.034
2.778
2.953
2.876
0.529
0.466
0.501
(5, 5) 3.39 (10, 10) 6.78
2.859 2.904
2.929 2.928
2.897 2.914
0.770 1.428
0.727 1.407
0.761 1.415
(15, 15) 10.17
2.917
2.928
2.920
2.084
2.065
2.069
MD simulations and analytical model.
inner shells is greater than eight, the compressions are less than 1% for 54.3 Å MWNTs. Experimental observations of a bundle of two SWCNTs show that the deformation does not exceed 4% for 21 Å and smaller SWNTs.42,43 Therefore, we can apply cylindrical geometry to SWNTs with small radii and MWNTs consisting of many inner shells. We recognize that our model will fail for SWNTs with large radii because of its essential deformation, but this model can be used for MWNTs with large radii. We applied the MD simulation to determine the equilibrium distance and the minimum in the potential energy for two SWNTs at various angles γ of orientation at different rotations of nanotubes about their axes and different shifts of carbon atom structures along these axes. It is clear that averaging over rotations and shifts is exactly the smeared-out approximation used in our model. We found that for larger radii of SWNTs and smaller angles γ the MD simulation gives a smaller deviation from our model. We obtained the largest variation in the equilibrium distance and the VDW potential when two tubes crossed at right angles. We demonstrate in Table 1 the maximum and minimum in the potential energy and the corresponding equilibrium distances over rotations and shifts for several identical armchair SWNTs crossed at right angles. According to Table 1, we conclude that our analytical model is quite adequate when the radii of tubes are 3.39 Å or higher. A comparison between the analytical magnitude of the well depth (eq 4) and the approximate value (eq 13) for armchair SWNTs of different sizes at right angles is shown in Table 2. The maximal error is 3%. Universal Curve for the VDW Potential. Let us consider the dimensionless potential j̅ = j/j0 as function of the dimensionless distance d = d/d0, the radii of tubes t1 and t2, and the material parameters A and B (sin γ will be reduced), where j0 and d0 are the minimum VDW potential and the equilibrium distance, respectively. According to the Buckingham π theorem, this function can be written as t1 B jðd, t1 , t2 , A, BÞ ð15Þ j̅ d̅ , , 6 ¼ j0 ðt1 , t2 , A, BÞ t2 Ad0 Assuming that t1/t2 is some arbitrary constant and using eqs 9, 13, and 14, we analytically derived the dimensionless potential as 1278
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Table 2. Comparison of Analytical Potential Depth |u0| (eV) from Equation 4 (Upper-Right Side)29 and Approximation from Equation 13 (Lower-Left Side) for Two SWNTs Crossed at Right Angles tube-type radius (Å)
(5, 5) 3.40
(10, 10) 6.79
(15, 15) 10.18
(5, 5) 3.40
0.761\0.785
1.060
1.277
1.463
1.627
1.777
(10,10) 6.79 (15, 15) 10.18
1.039 1.256
1.415\1.434 1.711
1.728 2.069\2.081
1.979 2.384
2.202 2.653
2.403 2.895
(20, 20) 13.57
1.442
1.963
2.373
2.722\2.731
3.039
3.317
(25, 25) 16.96
1.606
2.186
2.643
3.031
3.376\3.382
3.691
(30, 30) 20.35
1.755
2.388
2.887
3.312
3.688
4.029\4.031
Figure 2. Universal curves for potentials between two SWNTs.
ð16Þ where ð1Þ
d0 ðA, B0 Þ B ¼B d0 ðt1 , t2 , A, BÞ
!6 , d0 ¼ d
(25, 25) 16.96
(30, 30) 20.35
Figure 3. Universal curves for forces between two SWNTs.
a function of the dimensionless distance only ! d t1 B j ̅ , , d0 ðt1 , t2 , A, BÞ t2 Ad06 ðt1 , t2 , A, BÞ t1 , t2 f ∞ t1 =t2 ¼ const ! d0 t1 B0 1 3̅d6 , , ¼j ̅ ð̅dÞ ¼ j̅ ¼ ð1Þ ð1Þ 6 2̅d9 d0 ðA, B0 Þ t2 A½d0 ðA, B0 Þ
0
(20, 20) 13.57
ð1Þ
d0 ðA, B0 Þ , and d0 ðt1 , t2 , A, BÞ
where j0 and d0 are exact functions of their parameters. It is enough to know the values of j0 and d0 to recover the full potential j(d). We emphasize that this general proof is valid for the universal potential and force curves for crossed and parallel SWNTs. The universal curve j̅ (d) is shown in Figure 2 as a solid blue line. The plots for SWNTs of different radii fall on the universal curve with an accuracy of the line thickness. The splitting between all plots does not exceed 0.013 at d = 2. van der Waals Force. The VDW force for two crossed SWNTs can be obtained by simple differentiation ∂j/∂d of the VDW potential from eq 11 pffiffiffiffiffiffiffi ν2 t1 þ t2 ð18Þ D1 t1 t2 þ D2 pffiffiffiffiffiffiffi Fðt1 , t2 , dÞ ¼ t1 t2 sin γ where
d0
d d̅ ¼ ð1Þ ¼ 0 d ðt , t 0 1 2 , A, BÞ d0 ðA, B Þ
D1 ¼ π
Therefore, it is convenient to define this function as the universal curve for VDW interaction between crossed SWNTs. It is also worth highlighting that the universal curve does not depend on material constants A and B, radii of tubes t1 and t2, the angle between tubes, and the density of atoms and their nature and that it is an exact, analytically derived expression. That is the exact VDW potential in eq 4, which can be written as d ̅ jðd, t1 , t2 , A, BÞ ¼ j0 ðt1 , t2 , A, BÞ 3 j d0 ðt1 , t2 , A, BÞ
2
A 2B Aπ2 ¼ , D 2 d4 5d10 24d3
ð19Þ
Using eq 18, we can find the distance dmax where the attractive VDW force reaches its maximum. If t1 and t2 tend to infinity in the first approximation, then we have 1=6 B ð1Þ ð20Þ dmax ¼ A The second approximation of higher accuracy is
ð2Þ dmax ðαÞ
ð17Þ 1279
!1=6 1=6 B α B 1=6 ¼ þ 1 A 32 A
ð21Þ
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the expansion is taken into account: ϕðt1 , t2 , dÞ ¼ ν2
Figure 4. Schematic drawing for the VDW interaction between two parallel SWNTs.
If we use the first approximation, eq 20, for dmax in eqs 18 and 19, then we have the maximal attractive force pffiffiffiffiffiffiffi ν2 max t1 þ t2 p ffiffiffiffiffiffi ffi ð22Þ t þ D t Fmax ðt1 , t2 Þ ¼ Dmax 1 2 2 t1 t2 sin γ 1 where Dmax ¼ 1
3A5=3 π2 max A3=2 π2 , D2 ¼ pffiffiffi 2=3 5B 24 B
ð23Þ
Universal Curve for the VDW Force. If we define the universal curve for the VDW force as a plot of F = F/|Fmax| against df = d/|dmax|, then we have
Fð̅ ̅ df Þ ¼
5̅d6f 2 3̅d10 f
ð24Þ
The universal curve for the VDW force between two crossed SWNTs is shown in Figure 3 as a solid blue line. We obtained that at small intertube gaps, below the equilibrium distance, the relative error between approximation eq 24 and the force calculated according to precise eq 4 tends to zero. For example, for two tubes of 3.4 Å radii at distance d = 2.8 Å the relative error is 4%.
’ PARALLEL CONFIGURATION OF TWO SINGLEWALLED CARBON NANOTUBES van der Waals Potential. The VDW interaction between two parallel SWNTs is illustrated in Figure 4. For the integral of the LJ potential between the point and line, we have Z 3πA 63πB ð25Þ jCC ðrÞ dl ¼ 5 þ 8r 256r 11
þ ðt1 sin θ1 t2 sin θ2 Þ2 g1=2
ð26Þ
Thus, the VDW potential is ZZ 3A 63B 2 ϕðt1 , t2 , dÞ ¼ πν t1 t2 5 þ 11 dθ1 dθ2 8r12 256r12 ð27Þ
where ϕ is expressed in units of energy per unit of length. Approximation of the VDW Potential. We assume that if d/ t1 , 1 and d/t2 , 1 then the surfaces of tubes in the range of small angles θ1 and θ2 give the main contribution in the VDW interaction. Using the expansion in small parameters θ1 and θ2, we obtain an approximating formula where only the main term of
ð28Þ
By analogy to a previous part of this work, we can calculate the equilibrium VDW distance, the potential well, and the maximal attractive force for interaction between two parallel SWNTs. Equilibrium Distance and Potential Depth. The equilibrium distance is 1 92 378 1=6 B 1=6 ð29Þ d0 ¼ 4 35 A If we use lattice constant a = 2.49 Å, attractive constant A = 15.1636 eV Å6, and repulsive constant B = 24 052 eV Å12 as in ref 21, then the equilibrium gap is d0 = 3.174 Å. The equilibrium VDW potential per unit of length is rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 240π2 35 7=12 A19=12 2t1 t2 ϕ0 ðt1 , t2 Þ ¼ ν2 19 92378 B7=12 t1 þ t2 ð30Þ Following ref 21, we calculated the well depth for SWNTs of different radii from 2 to 22 Å. A comparison between the numerically calculated magnitude of the well depth21 and approximating value eq 30 is shown in Table 3. We can conclude that the maximal difference is 6.9%. The independence of the equilibrium gap on the tubes’ radii or a very weak dependence was observed in several studies.14,21,29,38 On the basis of this fact and by applying dimensional analysis of the problem, it was possible to assume that the equilibrium distance mainly depends on the sixth root of the ratio B/A. Above we gave a rigorous proof of this relationship both for crossed and parallel configurations. Using eq 9 together with eq 29, it is possible to find not only the ratio B/A but also the ratio between equilibrium distances for crossed and parallel tubes. Absolute values of the attractive and repulsive constants can be determined from experimental observations after additional measurements of the force or potential. Universal Curve for the VDW Potential. The equation for the universal curve is ϕ̅ ð̅dÞ ¼
The distance between the surfaces of tubes in a plane perpendicular to their axes is r12 ¼ f½d þ t1 ð1 cos θ1 Þ þ t2 ð1 cos θ2 Þ2
pffiffiffiffiffiffiffiffiffi π2 2t1 t2 2431B p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5A þ 2048d6 32d7=2 t1 þ t2
7 19̅d6 12̅d19=2
ð31Þ
The universal curve is shown in Figure 2 as a dotted red line. The splitting between all plots for SWNTs of different radii does not exceed 0.015 at d = 2. It is remarkable that the universal curve proposed by Girifalco et al.14 " 4 10 # 1 3:41 3:41 ϕ̅ G ð̅dÞ ¼ 0:4 0:6 3:13̅d þ 0:28 3:13̅d þ 0:28 is in very good agreement with our eq 35. If d > 1, then max|ϕ ̅ (d)| = 0.019 at d = 1.7. The comparison between uni̅ G(d) ϕ versal curves for crossed eq 16 and parallel eq 31 SWNTs and Girifalco’s universal curve (green dotted line) is shown in Figure 2. According to Girifalco et al.,14 the universal curve is the plot of ϕ̅ = ϕ/|ϕ0| against d = (R t1 t2 δ)/(R0 t1 t2 δ), where R is the distance between the centers of graphitic 1280
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Table 3. Comparison of the Numerically Calculated Potential Depth |O0| (meV/Å) (Upper-Right Side)21 and Our Approximation (Lower-Left Side) for Two Parallel SWNTs radius (Å)
2
6
10
2
50.88\48.19
61.82
67.40
70.53
72.58
74.00
6 10
62.31 65.68
88.12\84.74 98.53
95.63 113.77\N/A
102.27 119.66
106.76 126.26
110.01 131.17
14
67.31
104.27
122.88
134.61\131.26
139.62
145.97
18
68.26
107.93
129.00
142.78
152.64\149.47
157.07
22
68.89
110.47
133.40
148.82
160.09
168.74\165.76
structures, R0 is the equilibrium spacing at the minimum energy for the two interacting entities, t1 and t2 are the radii of tubes or fullerenes, and δ is another parameter. For the interaction between two parallel SWNTs, one has δ = 0. In other cases of interaction between graphitic structures (C60C60, C60 SWNT, etc.), the fitting parameter δ is used to adjust a plot to the universal curve. We prefer not to use any fitting parameter at all. Similar to the universal curve for the crossed-tube configuration, we have a direct functional relationship between the dimensionless distance and dimensionless potential, which does not include material parameters A and B and surface density ν. Thus, we have two different universal curves for parallel and crossed SWNTs (Figure 2). van der Waals Force and the Universal Curve. The VDW force per unit of length for two parallel SWNTs is f ðt1 , t2 , dÞ ¼ ν2 π2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2t1 t2 35A 46 189B þ t1 þ t2 64d9=2 131 072d21=2
ð32Þ
The distance where the attractive VDW force reaches its maximum is 46 1891=6 4805=6 B 1=6 ð33Þ dmax ¼ A 960 The maximal attractive force per unit of length is fmax ðt1 , t2 Þ ¼ ν
2 153=4 A7=4 46 1893=4 B3=4
2 80π
2 1=4
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2t1 t2 t1 þ t2
ð34Þ
If we define the universal curve for the VDW force between two parallel SWNTs as a plot of f = f/|fmax| against df = d/dmax, then we have f̅ ð̅df Þ ¼
7̅d6f 3 21=2
4̅df
ð35Þ
This universal curve is shown in Figure 3 as a dotted red line.
’ CONCLUSIONS We applied the Lennard-Jones potential and the method of the smeared-out approximation suggested by Girifalco to study the interaction between two SWNTs of different diameters. Using an expansion of the small parameter, we obtained very simple and accurate algebraic expressions of the VDW potential and the force for two parallel and crossed carbon nanotubes. It is found that the interaction between parallel and crossed tubes is described by two universal curves. The universal curve is the direct functional relationship between the dimensionless
14
18
22
distance and dimensionless potential. These curves depend only on the dimensionless distance and are independent of the attractive and repulsive constants, the diameters of the nanotubes, and the angle between their axes. That makes it easy to apply this approximation method to estimate the VDW potential between tubes made of another material such as boron nitrite, silicon carbide, or other nanotubes. The application of simple analytical expressions is very promising for the description of greater random networks and bundles of CNTs. Our method of deriving the universal curves can be generalized for the VDW potential between arbitrary nanoobjects with different natures. We found that the equilibrium gap between crossed nanotubes of a fixed radius does not depend on the angle between them, which is what is proven by the mathematically precise equation for our model. This equilibrium gap abruptly increases for the same nanotubes in a parallel configuration. That fact has been already obtained and analyzed by Volkov and Zhigilei.38 We can explain the difference between the equilibrium gaps for similar tubes in crossed and parallel configurations as the following. In the parallel configuration, any cylindrical part of a nanotube is under the same equilibrium conditions as another cylindrical part. Let us mentally remove the ends of one of the tubes and keep only its finite central part. Next we rotate this part by a very small angle. It will be in an equilibrium that is very similar to the parallel configuration. After that, we add back two cylindrical ends of the tube. Because of rotation, these cylindrical ends will be farther from another tube; therefore, they are subjected to an attractive force. This will move the central part, initially being in an equilibrium state, closer to another tube. Unfortunately, we cannot compare the well depth for crossed and parallel tubes because in the crossed case the well depth is measured in eV whereas for parallel tubes the units should be eV per unit of length. We have shown that the equilibrium distances and the distance where the attractive VDW force are maximal mainly depend only on material constants and are practically independent of the diameters of interacting nanotubes. We gave explicit functions for the equilibrium VDW distance, well depth, and maximal attractive force. We plotted universal potential curves and universal force curves for SWNTs. We have demonstrated that by knowing the equilibrium distance d0 and the potential well j0 one can exactly recover the VDW potential or force using the universal curve. The accuracy of this method depends only on the accuracy of j0 and d0. These results may be used as a guide for the analysis of experimental data to investigate the interaction between nanotubes of various natures.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. 1281
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’ ACKNOWLEDGMENT We gratefully acknowledge support through the National Science Council of Taiwan, Republic of China, under grant NSC 98-2112-M-001-022-MY3, the Asian Office of Aerospace Research & Development (AOARD) under grant no. FA238609-1-4128, and support given by the institute of Medical System Engineering (iMSE) in the GIST, Republic of Korea. ’ REFERENCES (1) Girifalco, L. A.; Lad, R. A. J. Chem. Phys. 1956, 25, 693–697. (2) Allers, W.; Schwarz, A.; Schwarz, U. D.; Wiesendanger, R. Appl. Surf. Sci. 1999, 140, 247–252. (3) Kolmogorov, A. N.; Crespi, V. H. Phys. Rev. B 2005, 71, 235415. (4) Shtogun, Y. V.; Woods, L. M. J. Phys. Chem. Lett. 2010, 1, 1356– 1362. (5) Girifalco, L. A. J. Phys. Chem. 1992, 96, 858–861. (6) Kniaz, K.; Girifalco, L. A.; Fischer, J. E. J. Phys. Chem. 1995, 99, 16804–16806. (7) Guerin, H. J. Phys.: Condens. Matter 1998, 10, L527–L532. (8) Baowan, D.; Thamwattana, N.; Hill, J. M. Eur. Phys. J. D 2007, 44, 117–123. (9) Rey, C.; García-Rodeja, J.; Gallego, L. J.; Alonso, J. A. Phys. Rev. B 1997, 55, 7190–7197. (10) Guo, S.; Nagel, P. M.; Deering, A. L.; Van Lue, S. M.; Kandel, S. A. Surf. Sci. 2007, 601, 994–1000. (11) Dong, L.; Arai, F.; Fukuda, T. IEEE Trans. Mechatron. 2004, 9, 350. (12) Drummond, N. D.; Needs, R. J. Phys. Rev. Lett. 2007, 99, 166401. (13) Sque, S. J.; Jones, R.; Oberg, S.; Briddon, P. R. Phys. Rev. B 2007, 75, 115328. (14) Girifalco, L. A.; Hodak, M.; Lee, R. S. Phys. Rev. B 2000, 62, 13104. (15) Mickelson, W.; Aloni, S.; Han, W.-Q.; Cumings, J.; Zettl, A. Science 2003, 300, 467–469. (16) Baowan, D.; Thamwattana, N.; Hill, J. M. Phys. Rev. B 2007, 76, 155411. (17) Cox, B. J.; Thamwattana, N.; Hill, J. M. Curr. Appl. Phys. 2008, 8, 249–252. (18) Thamwattana, N.; Hill, J. M. J. Nanopart. Res. 2008, 10, 665– 677. (19) Khlobystov, A. N.; Porfyrakis, K.; Britz, D. A.; Kanai, M.; Scipioni, R.; Lyapin, S. G.; Wiltshire, J. G.; Ardavan, A.; Nguyen-Manh, D.; Nicholas, R. J.; Pettifor, D. G.; Dennis, T. J. S.; Briggs, G. A. D. Mater. Sci. Technol. 2004, 20, 969–974. (20) Khlobystov, A. N.; Scipioni, R.; Nguyen-Manh, D.; Britz, D. A.; Pettifor, D. G.; Briggs, G. A. D.; Lyapin, S. G.; Ardavan, A.; Nicholas, R. J. Appl. Phys. Lett. 2004, 84, 792–794. (21) Sun, C.-H.; Yin, L.-C.; Li, F.; Lu, G.-Q.; Cheng, H.-M. Chem. Phys. Lett. 2005, 403, 343–346. (22) Cao, D.; Wang, W. Chem. Eng. Sci. 2007, 62, 6879–6884. (23) Popescu, A.; Woods, L. M.; Bondarev, I. V. Phys. Rev. B 2008, 77, 115443. (24) Saito, R.; Matsuo, R.; Kimura, T.; Dresselhaus, G.; Dresselhaus, M. S. Chem. Phys. Lett. 2001, 348, 187–193. (25) Baowan, D.; Hill, J. M. Z. Angew. Math. Phys. 2007, 2007, 857–875. (26) Xiao, T.; Liao, K. Compos. B, Eng. 2004, 35, 211–217. (27) Sun, C.-H.; Lu, G.-Q.; Cheng, H.-M. Phys. Rev. B 2006, 73, 195414. (28) Zheng, Q.; Liu, J. Z.; Jiang, Q. Phys. Rev. B 2002, 65, 245409. (29) Zhbanov, A. I.; Pogorelov, E. G.; Chang, Y. C. ACS Nano 2010, 4, 5937–5945. (30) Volkov, A. N.; Zhigilei, L. V. ACS Nano 2010, 4, 6187–6195. (31) Cao, Q.; Kim, H. S.; Pimparkar, N.; Kulkarni, J. P.; Wang, C. J.; Shim, M.; Roy, K.; Alam, M. A.; Rogers, J. A. Nature 2008, 454, 495.
ARTICLE
(32) Cao, Q.; Rogers, J. A. Adv. Mater. 2009, 21, 29–53. (33) Xie, X. L.; Mai, Y. W.; Zhou, X. P. Mater. Sci. Eng., R 2005, 49, 89–112. (34) Kyrylyuk, A. V.; van der Schoot, P. Proc. Natl. Acad. Sci. U.S.A. 2008, 105, 8221–8226. (35) Lagerwall, J.; Scalia, G.; Haluska, M.; Dettlaff-Weglikowska, U.; Roth, S.; Giesselmann, F. Adv. Mater. 2007, 19, 359–364. (36) Ke, C. H.; Pugno, N.; Peng, B.; Espinosa, H. D. J. Mech. Phys. Solids 2005, 53, 1314–1333. (37) He, X. Q.; Kitipornchai, S.; Liew, K. M. J. Mech. Phys. Solids 2005, 53, 303–326. (38) Volkov, A. N.; Zhigilei, L. V. J. Phys. Chem. C 2010, 114, 5513–5531. (39) Liu, J. Z.; Zheng, Q. S.; Wang, L. F.; Jiang, Q. J. Mech. Phys. Solids 2005, 53, 123–142. (40) Ruoff, R. S.; Tersoff, J.; Lorents, D. C.; Subramohey, S.; Chan, B. Nature 1993, 364, 514–516. (41) Hertel, T.; Walkup, R. E.; Avouris, P. Phys. Rev. B 1998, 58, 13870–13873. (42) Abrams, Z. R.; Hanein, Y. Carbon 2007, 45, 738–743. (43) Jiang, Y. Y.; Zhou, W.; Kim, T.; Huang, Y.; Zuo, J. M. Phys. Rev. B 2008, 77, 153405.
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