DLVO Interactions of Carbon Nanotubes with Isotropic Planar Surfaces

Article pubs.acs.org/Langmuir

DLVO Interactions of Carbon Nanotubes with Isotropic Planar Surfaces Lei Wu,† Bin Gao,*,† Yuan Tian,† Rafael Muñoz-Carpena,*,† and Kirk J. Zigler‡ †

Department of Agricultural and Biological Engineering and ‡Department of Chemical Engineering, University of Florida, Gainesville, Florida 32611, United States ABSTRACT: Knowledge of the interaction between carbon nanotubes (CNTs) and planar surfaces is essential to optimizing CNT applications as well as reducing their environmental impact. In this work, the surface element integration (SEI) technique was coupled with the DLVO theory to determine the orientation-dependent interaction energy between a single-walled carbon nanotube (SWNT) and an infinite isotropic planar surface. For the first time, an analytical formula was developed to describe accurately the interaction between not only pristine but also surface-charged CNTs and planar surfaces with arbitrary rotational angles. Compared to other methods, the new analytical formulas were either more convenient or more accurate in describing the interaction between CNTs and planar surfaces, especially with respect to arbitrary angles. The results revealed the complex dependences of both force and torque between SWNTs and planar surfaces on the separation distances and rotational angles. With minor modifications, the analytical formulas derived for SWNTs can also be applied to multiwalled carbon nanotubes (MWNTs). The new analytical expressions presented in this work can be used as a robust tool to describe the DLVO interaction between CNTs and planar surfaces under various conditions and thus to assist in the design and application of CNT-based products.

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INTRODUCTION Carbon nanotubes (CNTs) are cylinder-shaped nanoparticles with an extremely high length-to-diameter ratio.1 Single-walled carbon nanotubes (SWNTs) possess the simplest geometry among the CNTs and have diameters ranging from 0.4 to 3 nm. Multiwalled carbon nanotubes (MWNTs) are composed of a concentric arrangement of many SWNTs, which can reach diameters of up to 100 nm. Their novel properties, such as exceptional mechanical strength and superior electrical and thermal conductivity, prompt their applications in quantum wires,2 high-resolution scanning probes,3 transistors,4 electronfield-emission sources,5 chemical and biological sensors,6,7 reinforced composite materials,8 nanomedicine,9 and many other areas. Some of these applications require assembling or depositing individual CNTs on surfaces of bulk materials with desired parameters, including the location, orientation, geometry, and density.10−12 As a result, a good understanding of the interaction forces between CNTs and the host surfaces is essential to the creation and optimization of CNT-based products. Furthermore, this knowledge may also be used to determine the development of effective strategies to reduce the environmental impact of the CNTs because surface interaction is one of most important factors governing the fate and transport of manufactured materials in soil and aquatic systems.13−16 A theory/model that can accurately describe the interaction between a CNT and a planar surface is therefore critically needed. © 2013 American Chemical Society

Pristine CNTs are crystalline graphitic rods and are often considered to have no surface charge. Their interaction with a surface therefore is mainly controlled by van der Waals forces.17 In the literature, the van der Waals interaction between CNTs and substrate surfaces is determined either by the continuum Lennard-Jones (LJ) model (nanoscopic) with considerations of all pairs of interacting atoms18−22 or by Lifshitz theory (microscopic) in terms of the Hamaker coefficients.17,23 The shape and range of the attractive van der Waals interaction (potential) varies with different dimensions (nanoscopic and microscopic). The LJ model is successful in describing the short-range van der Waals interaction potential in CNT systems. The latter has recently been adapted in a form to describe the long-range van der Waals interaction between pristine SWNTs and anisotropic surfaces with good accuracy.17 However, the application of this method requires ab initio optical properties, which are not only barely documented in the literature but also difficult to measure. Because of strong intertube attractions, pristine CNTs tend to aggregate and form “bundles” or “ropes”. To optimize their use, considerable research has been conducted to disperse CNTs in aqueous or organic media.24−26 Surface modification methods, such as chemical functionalization (e.g., acid oxidization) and polymeric coatings (e.g., surfactants, dissolved Received: December 6, 2012 Revised: February 26, 2013 Published: February 26, 2013 3976

dx.doi.org/10.1021/la3048328 | Langmuir 2013, 29, 3976−3988

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application in describing the interface interactions. Nevertheless, little research effort has been expended to apply the SEI in quantifying the interaction of CNTs with planar surfaces, particularly with respect to obtaining analytical expressions that are more accurate and simpler. Therefore, an accurate, efficient analytical calculation of the interaction energy between CNTs and planar surfaces is of great scientific and practical significance. The overarching objective of this work was to develop analytical formulas that can precisely describe the orientationdependent interaction energy/forces between a CNT and an isotropic planar surface. It was hypothesized that the interaction of CNTs with planar surfaces is mainly controlled by the van der Waals and electrical double layer (EDL) forces, which are the same as the classical DLVO forces. The SEI method was thus integrated into the DLVO theory framework to obtain the analytical expressions of the orientation-dependent interaction energy between an SWNT and an isotropic planar surface. The interaction energy was evaluated for two different situations: (1) a pristine SWNT and an isotropic planar surface and (2) a surface-charged SWNT and an isotropic planar surface. After validations, the analytical expressions were also extended to determine the interaction between an MWNT and the planar surface.

organic matter, and ligands) are often used to improve the stability of CNT suspensions by introducing repulsive electrostatics forces.26−28 For example, Mamedov et al.29 used nitric acid oxidization to introduce negative charges onto SWNTs, which enable the assembly of a stable SWNT composite film. For surface-modified CNTs, the dispersion of CNTs is mainly controlled by electrostatic and steric interactions. Unfortunately, there are only a limited number of studies that have been attempted to determine theoretically the interactions, particularly electrostatic interactions, of surface-modified CNTs with charged surfaces. Chapot et al.30 and Lowen31 proposed frameworks allowing one to compute the interactions between charged rodlike colloidal particles; however, it is unclear whether these frameworks can be applied to determine the electrostatic interactions between charged CNTs and planar interfaces. The formation of an electric double layer on charged CNT surfaces is similar to the phenomenon observed with colloidal particles.32 In addition, previous studies have indicated that although CNTs are molecular objects with two dimensions in the nanometric range, their dispersion, deposition, and aggregation behaviors follow the principles of the classical colloidal system (especially for MWNTs).33,11,13,34 The Derjaguin−Landau−Verwey−Overbeek (DLVO) theory, which was originally developed for spherical colloids, thus has been used to describe semiquantitatively the stability of surfacemodified CNTs and their interaction with planar surfaces.13,15,16,35−37 Because an oversimplified assumption that tubular CNTs have “equivalent” spherical diameters was used, the DLVO theory often failed to provide an accurate estimation of the interaction forces under various conditions.16 Furthermore, the interaction of CNTs and planar surfaces is orientation-dependent, which gives rise to a torque orienting the CNTs in an energetically favorable configuration to approach/leave the planar surfaces. Such dynamic behavior cannot be explained merely on the basis of spherically symmetric interaction potentials of classical DLVO theory. Several techniques have been developed to calculate the interaction force/energy between curved surfaces/bodies, including the Derjaguin approximation (DA) and surface element integration (SEI).38 The DA method estimates the interaction energy between two finite-sized bodies by relating it to that between two infinite parallel flat plates. It can be applied only to surfaces that are separated by a small distance and to circumstances when the interaction range is substantially smaller than the radii of curvature of the surfaces. For very small nonspherical particles such as SWNTs, the DA method may lead to inaccuracies in calculating their interaction with planar surfaces.39 The SEI technique takes into account curvature effects over the whole object by integrating the interaction energy between a surface element of the object and the plane surface using the exact surface geometry of the object. It can precisely determine the interaction forces between a planar surface and a curved body with any defined shape, including CNTs.16 For instance, Stolarczyk et al.39 successfully applied the SEI method to evaluate numerically the interaction forces between functionalized MWNTs and ligand-stabilized gold nanoparticles (modeled as a cylinder−sphere system). However, the calculation is quite complex and time-consuming. Thus, although the SEI has made a remarkable breakthrough in the accurate calculation of the interaction energy between a curved body and a planar surface or two curved bodies, the difficulty in numerical implementation restricts its wide

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THEORY The general expression representing the interaction between an SWNT and a planar surface is presented in Figure 1. For convenience, two coordinate systems were used: a set of bodyfixed coordinates (x, y, z) to account for the internal geometric properties of the tubular SWNT and a space-fixed coordinate system (X, Y, Z) to account for the orientation of the tubular SWNT relative to the planar surface. The SWNT was modeled as a hollow cylinder (e.g., SWNTs can be converted from nearly

Figure 1. Schematic illustration of interaction of an SWNT with an infinite isotropic planar surface. Body-fixed coordinates (xyz) and space-fixed coordinates (XYZ) both originate at O. The body-fixed coordinates are rotated at an arbitrary angle ϕ relative to the spacefixed coordinates. The two required geometrical properties for the SEI technique are the local unit normal, n̂, to the surface of the SWNT (at point P), and the local distance h of the point P from the infinite planar surface. H is the closest distance between the SWNT surface and the planar surface. D is the distance between the center of the SWNT and the planar surface. The relation between H and D can be written as D = H + L cos φ + a sin φ. 3977


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where D is the separation distance of an infinite planar surface from the center of the SWNT and L is the semiaxis of an SWNT directed along the z axis. The interaction between the surface element and the planar surface was assumed to be mainly controlled by the DLVO forces,32,41,42 thus E(hi) can be written as

endless, highly tangled ropes into short, open-ended pipes after acid oxidation).40 The top, side, and bottom surfaces of the SWNT were defined as S1, S2, and S3, respectively. It is worth noting that the presence of hemispherical caps at the ends of the SWMTs is also important and is currently a topic of ongoing investigation but is beyond the scope of this study. The SEI method23,25,26 was used in this work to determine the total interaction energy between an SWNT of finite length and an infinite planar surface. The governing equation of the SEI method can be written as Φ(D , φ) =

∫S nŜ ·kÊ (hi) dS i

E(hi) = Evdw (hi) + Eedl(hi)

where Evdw(hi) and Eedl(hi) are the van der Waals and EDL interaction energies per unit area between two infinite flat plates, respectively. For the van der Waals interaction, instead of the Lifshitz approach that is obtained from complex optical properties, the Hamaker approach was used in this study and Evdw(hi) can be written as43

(1)

i

where Φ(D, ϕ) is the total interaction energy when the center of the SWNT with arbitrary angle (φ) is located at a distance D from the planar surface, n̂Si is the outer unit normal vector to the SWNT surface element dS, (i = 1, 2, 3, and represents the top, side, and bottom surfaces of the SWNT, respectively), k̂ is a unit vector normal to the planar surface, and E(hi) represents the unit interaction energy between the surface element and the planar surface at a distance of hi . It is worth noting that the sign of n̂Si·k̂ governs the magnitude of the total interaction energy. When the unit normal to the surface at a given point makes an obtuse angle with k,̂ then the sign of n̂Si·k̂ is negative and hence the energy contribution of the corresponding surface element is also negative. Thus, we need to subtract the interaction energy of those regions of the surface that “face away” from the planar plate. The differential area of the surface element dS of the SWNT then can be written as dS = a dz dθ for outer S2

(2.1.1)

dS = (a − R ) dz dθ for inner S2

(2.1.2)

dS = ρ dρ dθ for S1 and S3

Evdw (hi) = −

⎛ kT ⎞2 Eedl(hi) = 32ε0εrγ1γ2κ ⎜ ⎟ exp( −κhi) ⎝ νe ⎠

nŜ 1·k ̂ = cos φ for S1

(3.2)

nŜ 3 ·k ̂ = −cos φ for S3

(3.3)

where φ is the orientation angle of the tubular SWNT. The distance between the surface element and the planar surface, hi, can be written as h1 = D − L cos φ − ρ sin φ sin θ for S1

h2 = D − z cos φ − a sin φ sin θ for outer S2

(6.2)

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RESULTS AND DISCUSSION DLVO Interactions between a Pristine SWNT and an Isotropic Planar Surface. The interaction between a pristine SWNT (without charge) and an isotropic planar surface is mainly controlled by the attractive van der Waals forces. For this case, mathematical analysis of the interaction energy between a pristine SWNT and an isotropic planar surface with arbitrary angle position yields the analytical solution

(4.1) (4.2.1)

h2 = D − z cos φ − (a − R )sin φ sin θ for inner S2 (4.2.2)

h3 = D + L cos φ − ρ sin φ sin θ for S3

(6.1)

where ε0 is the permittivity of vacuum, εr is the relative permittivity of the solution, γ = tanh(veψ/4kT), ψ is the surface potential, κ is the Debye−Huckel parameter, k is the Boltzmann constant, T is the temperature, v is the valence of the electrolyte, and e is the electron charge. Ideally, the surface potential must be used in eq 6.2. In recent studies, ψ is often approximated by the zeta potential (ζ), the potential located at the electrokinetic plane of shear.45−50 It is worth noting that there is a debate over charge distribution on the surface of CNTs: Paillet et al.51 provided the experimental confirmation that charges are distributed uniformly along the nanotubes whereas open ends and defect sites were reported to be preferentially charged areas.34,52,53 By using a theoretical atomic charge-dipole model and experimental electrostatic force microscopy, Wang et al.54 demonstrate that the charge enhancement at the end is insignificant for an SWNT with a length of around 30 nm and will become negligible for micrometer-long CNTs. Hence, in this study, we assume that the charge on the SWNT surface is uniformly distributed on the surface.

where a represents semiaxes of the SWNT directed along the x and y axes, ρ and θ are radial and angular coordinates in a cylindrical coordinate system, respectively, and R is the diameter of a carbon atom (1.54 × 10−10 m). The expressions for n̂Si·k̂ in eq 1 can be written as (3.1)

A 12π (hi)2

where A is the effective Hamaker constant. For the EDL interaction, the linear superposition approach is regarded as the most accurate physical description of the EDL interaction for CNTs because it gives intermediate values between those for the constant potential (mobile charges that keep the potential between the two surfaces constant) and constant charge (assuming immobile charges) cases, and Eedl(hi) is given by Gregory44

(2.2)

nŜ 2 ·k ̂ = sin θ sin φ for S2

(5)

(4.3)

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Figure 2. van der Waals interaction energy between a pristine SWNT and an isotropic planar surface: (A) Predictions of (−Φvdw/kT) with the closest separation distance (H) and an arbitrary angle (ϕ). (B) Predictions of (−Φvdw/kT) with the closest separation distance (H) for four angles: 0, π /6, π/4, and π/2, respectively. (C) Comparison of predictions of (−Φvdw/kT) between a tube and a cylinder. (D) Comparison of predictions of (−Φvdw/kT) among the SEI approach, DA approach, and Rajter approach (A = 9.81 × 10−21 J *, a = 0.7 nm, L = 0.2 μm, and T = 298 K). The effective Hamaker constant between the CNTs and the quartz surface was obtained from the literature values62−64 based on the combining law A132 = (A11A22)1/2 − (A11A33)1/2 − (A33A22)1/2 + A33, where A11, A22, and A33 are the Hamaker constants of SWNTs, sand, and water, respectively. A recent study suggested that the Hamaker constant of type-purified SWNTs could be much larger than the value used in this study because of the influence of purification by the electronic type used in that study.10

⎧ ⎪ ⎪ ⎪ ⎪− ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Φvdw (D , φ) = ⎨ ⎪− ⎪ ⎪ ⎪ ⎪ ⎪ ⎪− ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

⎡ a sin φ a sin φ (D − L cos φ)arc tan M (D + L cos φ)arc tan M ⎢ 1 2 ⎢ − M1 M2 A sec φ ⎢ ⎢ 3π ⎢ (a − R )sin φ (a − R )sin φ (D − L cos φ)arc tan (D + L cos φ)arc tan M3 M4 ⎢− + ⎢⎣ M3 M4 ⎛ 1 ⎛ 1 A cos φ csc φ2 ⎡ 1 ⎞ 1 ⎞⎤ ⎢(D − L cos φ)⎜ − − − ⎟ + (D + L cos φ)⎜ ⎟⎥, ⎢⎣ 6 M3 ⎠ M4 ⎠⎥⎦ ⎝ M1 ⎝ M2

(

)

(

)

(

(

⎤ A(2aR − R2) ⎡ 1 1 + , ⎢ 2 2⎥ 12 ⎣ (D − L) (D + L) ⎦ ⎡ ⎛ 2aAL⎢ D2 − a2 + a arc tan⎜ ⎝ ⎣

+

a D2 − a 2

⎞⎤ ⎟ ⎠⎥⎦

3π(D2 − a2)3/2 ⎡ ⎛ 2(a − R )AL⎢ D2 − (a − R )2 + (a − R )arc tan⎜ ⎝ ⎣ 3π(D2 − (a − R )2 )3/2

where M1 = ((D − L cos φ)2 − (a sin φ)2)1/2, M2 = ((D + L cos φ)2 − (a sin φ)2)1/2, M3 = ((D − L cos φ)2 − ((a − R) sin φ)2)1/2, and M4 = ((D + L cos φ)2 − ((a − R) sin φ)2)1/2.

(a − R ) D2 − (a − R )2

⎞⎤ ⎟⎥ ⎠⎦

,

)

)

⎫ ⎤ ⎪ ⎥ ⎪ ⎥ ⎪ ⎥ 0 < < /2 φ π ⎪ ⎥ ⎪ ⎥ ⎪ ⎥ ⎪ ⎥⎦ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ φ=0 ⎪ ⎪ ⎪ ⎪ ⎪ φ = π /2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(7)

Detailed mathematical derivations of eq 7 can be found in the Appendix. 3979


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Equation 7 was used to determine the interaction energy profiles (scaled to kT) between a pristine SWNT and a planar quartz surface in water for arbitrary angle approaching patterns. The interaction energy between the pristine SWNT and the flat surface was attractive, suggesting that pristine SWNTs intend to attach to surfaces because of the attractive van der Waals forces. Overall, the magnitude of the van der Waals interaction energy depends on the orientation of the SWNT with respect to the planar surface (Figure 2A). The attractive interaction energy increases when the arbitrary angle increases from φ = 0 (“endon”, attached by the tube end) to φ = π/2 (“side-on”, attached to the tube side). The attractive energy of the side-on pattern was much higher (at least two orders of magnitude) than that of the end-on pattern, indicating that it is more favorable for pristine SWNTs to attach to planar surfaces through the sideon approach pattern (Figure 2B). For a randomly positioned pristine SWNT in the system, the difference in attractive energy between the two approaching patterns may generate a torque to drive the SWNT to attach to the planar surface through the side-on pattern. Because of their nanosized diameter, SWNTs are often modeled as solid cylinders (without open ends) instead of tubes for convenience.39 This practice, however, may generate errors when used to determine the interaction energy between SWNTs and flat surfaces using the SEI method. When a pristine SWNT of a relatively small diameter (e.g., 0.2 nm) was modeled as a solid cylinder, the energy profile matched that of the tubular SWNT closely (Figure 2C). When a pristine SWNT of a relatively large diameter (e.g., 1.5 nm) was modeled as a solid cylinder, however, the results showed large deviations between the two energy profiles. These results suggest that SWNTs, especially with large diameters, should be modeled as tubes instead of solid cylinders to describe their interaction with planar surfaces accurately. Recently, Rajter et al.17 developed a model to calculate the van der Waals interaction energy between optically anisotropic SWNTs and planar surfaces based on the optical properties of the SWNTs and Lifshitz theory. This model was applied in this work to determine the energy profile between the pristine SWNT and the planar surface for the side-on pattern. The interaction energy profile obtained from the Rajter model was compared to that in this work and in the widely used DA method (Figure 2D). The results demonstrated excellent agreement between the new model and the Rajter model, indicating that the SEI approach can be integrated into the DLVO theory to describe the interaction between SWNTs and planar surfaces accurately. In contrast, the DA method overestimated the interaction between an SWNT and a planar surface up to three orders of magnitude, confirming that the method is not suitable for nanosized nonspherical particles.39 DLVO Interactions between a Surface-Modified SWNT and a Charged Isotropic Planar Surface. Both van der Waals and EDL forces are important to the interaction between a surface-modified SWNT and a charged isotropic planar surface. Whereas the attractive van der Waals interaction energy is the same as discussed previously (eq 7), the analytical expression of the EDL interaction energy between the surfacecharged SWNT and the planar surface with an arbitrary angle can be written as

ΦEDL(D , φ) ⎫ ⎧ ⎛ kT ⎞2 ⎪ 32πε0εrγ1γ2⎜ ⎟ e−κD{2a tan φN1N4 0 < φ < π /2 ⎪ ⎝ ⎠ ve ⎪ ⎪ ⎪ + κ cos φN5[a 2 Ν2 − (a − R )2 Ν3]}, ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ 32π (2aR − R2)ε ε γ γ κ ⎛⎜ kT ⎞⎟ ⎪ φ=0 0 r12 ⎬ =⎨ ⎝ ve ⎠ ⎪ ⎪ −κ(D − L) −κ(D + L) e e [ ], + ⎪ ⎪ ⎪ ⎪ 2 ⎛ ⎞ kT ⎪128πaLε ε γ γ κ ⎜ ⎟ exp(−κD) ⎪ φ = π /2 0 r12 ⎝ ve ⎠ ⎪ ⎪ ⎪ L (κa), ⎪ ⎩ ⎭ −1

(8)

where N1 = L−1(κa sin φ); N2 = I0(κa sin φ) − I2(ka sin φ); N3 = I0(κ(a − R)sin φ) − I2(κ(a − R)sin φ); N4 = (eκL cos φ − e−κL cos φ), N5 = (eκL cos φ + e−κL cos φ); I0(x), I0(x), and I2(x) are modified Bessel functions of zeroth, first, and second order, respectively; and L−1(x) is a modified Struve function of order −1. Detailed mathematical derivations of eq 8 can be found in the Appendix. Here, the electrostatic interaction was considered as the only force introduced by the surface modification, which is reasonable for surface-charged CNTs. Other interaction forces, such as steric repulsion and hydrophobic interactions, however, may be trigged by other surface modification methods (e.g., coating with surfactants or polymers),20,22 which are beyond the scope of this study. Thus, further investigations are necessary to include these non-DLVO interaction forces to describe the extended DLVO interaction of surfactant- or polymer-modified SWNTs and planar surfaces in the future. Equation 8 was used to determine the scaled EDL energy profiles between a charged SWNT (i.e., humic acid-coated SWNT36) and a planar quartz surface in water for arbitrary angle approaching patterns (Figure 3A). The ξ potential values of the SWNT and the glass beads under two ionic strength conditions were obtained from reported values in the literature.35,36 It should be mentioned that the reported ξ values were often determined by assuming that the SWNTs are spherical, which might overestimate the actual numbers by up to 20% under certain conditions.28 This potential error could slightly affect the magnitude of the EDL energy profiles by less than 5% but does not change the shape of and the trend in the overall DLVO energy profiles as discussed in the examples in following sections. Although much research effort has been expended to develop new methods to determine ξ values of higher-aspect-ratio structures (i.e., cylindrical particles),55−57 it is still unclear whether these methods could provide more accurate measurements of the ξ-potential values of tubular SWNTs. The EDL interaction energy between the charged SWNTs and the quartz surface was repulsive, and the repulsive energy was 4 orders of magnitude stronger for the side-on pattern than for the end-on pattern under both ionic strength conditions (Figure 3B). Furthermore, the predicted EDL energy was very sensitive to ionic strength and decreased dramatically for an arbitrary angle pattern when the solution ionic strength increased from 0.001 to 0.1 M. This is consistent with the DLVO theory in which increases in ionic strength can compress the double layer to reduce the repulsive electrostatic forces between two charged surfaces in electrolyte solutions.32,42 On the basis of eqs 7 and 8, the total interaction energy between the surface-modified SWNT and the charged planar surface for arbitrary angle attachment thus can be written as 3980


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arbitrary angle patterns, the EDL interaction became shortranged in the presence of electrolyte (i.e., 0.1 M). As a result, shallow secondary minimum energy wells were identified in the energy profiles for all approaching patterns. For example, the depth of the secondary minimum for side-on pattern is 0.87kT, indicating that the SWNT could also attach to the planar surface through a side-on pattern in the secondary minima under high-energy-barrier conditions although the deposition in the secondary minimum is temporary because the depth is close to the average kinetic energy of a particle (1.5kT). No secondary minimum wells were found in the energy profiles for the low-ionic-strength conditions over the entire range of separation distance shown (0−100 nm) for both patterns. Figure 4F−I shows the coupled effects of radii and orientation (approaching angle) on the total interaction energy profile of SWNTs and planar surfaces. For easy comparison, the zeta potential and length of SWNTs of different radii were assumed to be the same under the tested conditions. Overall, both of the interaction energy barriers and depths of the secondary minimum increase when the radii of the SWNTs increase for all of the approaching patterns. When the SWNT radii are fixed, the heights of the interaction energy barrier and the depths of the secondary minimum increased as the rotational angles increased from 0 to π/2, which is consistent with the discussion above. When the orientation angle is fixed, the separation distance at the secondary minimum well is independent of SWNT radius. For example, the secondary minimum locations (separation distances) of three tested SWNTs with different radii were at 10.1, 7.4, 7.3, and 9.5 nm when the approaching angles were 0, π/6, π/4, and π/2, respectively (dashed lines in insets). These results are in agreement with findings from previous studies using the LJ potential approach.19 On the basis of the results shown in Figure 4F−I, there may exist a critical rotational angle at which the secondary minimum well separation distance reaches the minimum value. Volkov and Zhigilei21 also reported that the equilibrium distance between two CNTs of finite length depends on their rotational angles. DLVO Forces and Torques of SWNTs with Planar Surfaces. The analytical expressions of the interaction energy of SWNTs with planar surfaces (eqs 7−9) enable a straightforward analysis of force F(D, φ) and torque τ(D, φ), which can be written as the derivatives of the energy potential:

Figure 3. Electrostatic double layer interaction energy (ΦEDL/kT) between a surface-modified SWNT and a charged isotropic planar surface: (A) predictions of (ΦEDL/kT) with the closest separation distance (H) and arbitrary angle (ϕ) and (B) predictions of (ΦEDL/ kT) with the closest separation distance (H) under different ionic strength conditions for four angles: 0, π/6, π/4, and π/2, respectively. (a = 0.7 nm, L = 0.2 μm, IS = 0.1, 0.001 M, ζ(SWNT,high) = −34.50 mV, ζ(SWNT,low) = −44.50 mV, ζ(GB,high) = −38.81 mV, ζ(GB,low) = −52.07 mV, and T = 298 K).

Φtotal (D , φ) = Φvdw (D , φ) + ΦEDL(D , φ)

(9)

Equation 9 was used to determine the scaled DLVO energy profiles between a surface-modified SWNT and a planar quartz surface (using the physicochemical and surface properties mentioned above) in water for arbitrary angle approaching patterns (Figure 4A). The coupled effects of orientation and ionic strength on the total interaction energy profile of SWNTs and planar surfaces were also investigated (Figure 4B−E). Both the height of the energy barrier and the depth of the secondary minimum increased with the increase in the approaching angle for the two tested ionic strength conditions. The total interaction energy of the side-on pattern was several orders of magnitude higher than that of the end-on pattern over the entire range of separation distances. The energy barrier of the side-on pattern was (0.001 M, 173.52kT; 0.1 M, 87.64kT) more than 300 times higher than that of the end-on pattern (0.001 M, 0.55kT; 0.1 M, 0.47kT). These results indicated that it might be much easier for the SWNT to overcome the energy barrier to attach to the planar surface through the end-on pattern than through the side-on pattern when repulsive EDL forces are present. For all

F (D , φ ) = −

∂Φtotal (D , φ) ∂D

(10.1)

τ (D , φ ) = −

∂Φtotal (D , φ) ∂φ

(10.2)

Thus, the DLVO force and torque between a charged SWNT and a planar quartz surface (with the same properties as mentioned above) in water for arbitrary angle approaching patterns can be determined using the two equations (Figure 5A,B, respectively). Figures 5C shows the DLVO force profiles of SWNTs with different radii interacting with the planar surface under a side-on approaching pattern. The results indicated that the distance where the attractive force reaches its maximum is a constant (9.7 nm) for all of the tested conditions, which is inconsistent with the fact that the corresponding secondary energy minimum wells are located at around 9.5 nm (Figure 5D) as discussed above. The inset in Figure 5C indicates that the surfaces jumped to a primary minimum from 3981


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Figure 4. Total interaction energy (Φtotal/kT) between a surface-modified SWNT and a charged isotropic planar surface: (A) predictions of (Φtotal/ kT) with the closest separation distance (H) and arbitrary angle (ϕ). (B−E) Coupled effects of ionic strength and rotational angle on the total interaction energy (Φtotal/kT) between a surface-modified SWNT (a = 0.7 nm) and a charged isotropic planar surface. (F−I) Coupled effects of the SWNT radius and rotational angle on the total interaction energy (Φtotal/kT) between a surface-modified SWNT and a charged isotropic planar surface under high ionic strength conditions. Insets are plotted on a smaller y-axis scale to highlight the secondary minimum depth. The depths of secondary energy minima for an SWNT with a = 3 nm under four approaching angles are −0.0018kT, −0.05kT, −0.077kT, and −1.12kT, respectively. The corresponding secondary minimum locations (separation distances) were at 10.1, 7.4, 7.3, and 9.5 nm, respectively. (A = 9.81 × 10−21 J, aSWNT = 0.4, 1, and 3 nm, respectively; L = 0.2 μm, IS = 0.1, 0.001 M, ζ(SWNT,high) = −34.50 mV, ζ(SWNT,low) = −44.50 mV, ζ(GB,high) = −38.81 mV, ζ(GB,low) = −52.07 mV, and T = 298 K.)

a constant (i.e., 3.39 nm).58 For the EDL interaction of modified MWNTs, previous studies have shown that all of the surface charge may be uniformly distributed on their outer layer surfaces,59 and thus the EDL expression (i.e., eq 8) can be applied directly. Figure 6 compares normalized interaction energy profiles (Φtotal/|Φsec| vs (H/(Hsec)), where Φsec is the secondary minimum and Hsec is the corresponding separation distance) of a modified SWNT and a modified MWNT (with the same surface charge density and length) interacting with the charged planar surface at three different rotational angles. When the rotational angle is fixed, the normalized interaction energy profiles of SWNTs and MWNTs are almost identical regardless of the differences in their radii. This result is consistent with the findings from previous studies based on the continuum LJ potential model18,19,60,61 and confirms that the analytical expressions are applicable to the MWNT systems. Because the normalized energy profile is independent of the Hamaker constant, surface charge density, and CNT properties, further research is needed to explore the mathematical connections among the maximum energy barrier, the secondary energy minimum, and their corresponding separation distances

H = 0.87 nm. Figure 5D shows the dependences of torque on the rotational angle φ of the SWNT approaching the surface. The results suggested that that torque direction is depended on the separation distance. For close separation distance (e.g., 2 nm) between the SWNT and the surface, where repulsion dominates, the torque (negative) acts to misalign the SWNT to be end-on (perpendicular) to the planar surface. For intermediate separation distances, in the vicinity of the equilibrium position, the equilibrium angle (when the torque is zero) may not necessarily be 0 or π/2 (e.g., when H = 10 nm, the equilibrium angle is π/10). For a relatively long separation distance, where attraction may dominate, the torque (positive) acts to align the CNTs to be side-on (parallel) to the planar surface. DLVO Interactions of MWNTs and Planar Surfaces. The analytical expressions can be extended to describe the interaction of MWNTs and planar surfaces. Because MWNTs are made of several layers of SWNTs, it is reasonable to assume that each layer interacts with the planar surface as one SWNT. Thus, the total van der Waals interaction energy between an MWNT and a planar surface can be obtained by a summation over all layers with the assumption that the interlayer spacing is 3982


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Figure 5. DLVO force and torque acting on a surface-modified SWNT and a charged isotropic planar surface: (A) the DLVO force (Ftotal), (B) the DLVO torque (τtotal), (C) the DLVO force of SWNTs of different radii interacting with the planar surface under side-on approaching pattern, and (D) the DLVO torque of the SWNT (a = 3 nm) interacting with the planar surface with different rotational angles at various separation distances. (A = 9.81 × 10−21 J; aSWNT = 0.4, 1, and 3 nm, respectively; ζ(SWNT) = −34.50 mV; LSWNT = 0.2 μm; IS = 0.1 M; ζ(GB,high) = −38.81 mV; and T = 298 K).

by integrating the SEI method into the DLVO theory. For the first time, exact analytical solutions of the DLVO interaction energy were developed not only for pristine SWNTs and planar surfaces but also for surface-modified SWNTs and charged planar surfaces with arbitrary orientation angles. Simplified formulas were also given for the case of “end-on” and “side-on” approaching patterns. Compared to the results of other methods, the new solutions were either convenient or more accurate than existing approaches at describing the interaction of SWNTs with isotropic surfaces. The analytical formulas derived for SWNTs can also be applied to MWNTs with minor modifications. The analysis of the DLVO force and torque showed that in the region close to the planar surface the repulsive interaction creates a preferential alignment of CNTs perpendicular to the planar surface; without this interaction, parallel alignment is favored. The new model presented in this work provides a clear picture of the interaction energy/forces/ torques between CNTs and planar surfaces with arbitrary orientation and sheds light on the approaching patterns of CNTs with respect to the planar surfaces under various conditions. It can be used as an effective tool by end users to predict and optimize the interaction between CNTs and planar surfaces for a wide variety of fields of interests (e.g., biodevices, biomedicine, etc.). Although in this work the interaction energy is developed for CNTs, it is not limited to CNTs and can be readily applied to various types of nanostructures and microtubular structures for the analysis of their interaction with planar surfaces.

Figure 6. Normalized total interaction energy between a surfacemodified SWNT or MWNT and a charged isotropic planar surface at selected rotational angles. (A = 9.81 × 10−21 J, aSWNT= 3 nm, aMWNT= 40 nm, LSWNT = 0.2 μm, LMWNT = 0.2 μm, IS = 0.1 M, ζ(SWNT,high) = ζ(MWNT,high) = −34.50 mV, ζ(GB,high) = −38.81 mV, and T = 298 K).

(equilibrium distance). This will allow the development of new mathematical tools to determine the interaction of CNTs and planar surfaces.

■

CONCLUSIONS The interaction between a tubular SWNT and an isotropic planar surface with arbitrary orientation angles was quantified 3983


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APPENDIX

The right-hand side of eq A-5 becomes

Derivation of van der Waals Interaction Energy (Equation 7)

−

∫S nŜ ·kÊ (h2) ds + ∫S nŜ ·kÊ (h1) ds

Φvdw (D) =

2

1

2

+

1

∫S nŜ ·kÊ (h3) ds 3

3

(A-1)

By using eqs 2.1, 3.1, 4.2, and 6.1 in the main text of the Article, for outer S2 we obtain

L

∫−L ∫0

π

∫ ∫

∫S nŜ ·kÊ (h2) ds = − Aa12sinπ φ 2

2

⎡ ⎤ 1 ⎢ ⎥ 2 π (D − z cos φ − a sin φ sin θ) ⎥ dθ dz sin θ ⎢⎢ ⎥ 0 1 ⎢+ 2⎥ ⎣ (D − z cos φ + a sin φ sin θ) ⎦

L

⎡ 1 sin θ ⎢ ⎣ (α − β sin θ )2 ⎤ 1 + ⎥ dθ dz (α + β sin θ )2 ⎦ Aa sin φ ∂ =− 12π ∂β L π⎛ 1 ⎜ −L 0 ⎝ α − β sin θ ⎞ 1 − ⎟ dθ dz α + β sin θ ⎠ Aa sin φ 12π

∫−L ∫

=−

L

Aa sin φ ∂ 3π ∂β

∫−L

⎛ arctan⎜ ⎝

β α2 − β2

α2 − β2

⎞ ⎟ ⎠

dz (A-9)

(A-2)

Now, let α/β = t, D − L cos φ = γ1, and D + L cos φ = γ2.

Similarly, from eqs 2.2, 3.2, 4.1, and 6.1 and eqs 2.2, 3.3, 4.3, and 6.1 in the main text of the Article, we obtain eqs A-3 and A-4, respectively,

Then dz = −(β/(cos φ)) dt, and eq A-9 becomes

∫S nŜ ·kÊ (h1) ds = − A 12cosπ φ 1

1

a

∫a−R ∫0

2π

−

ρ dθ dρ (D − L cos φ − ρ sin φ sin θ )2 (A-3)

Aa sin φ 3π

L

∫−L

Aa sin φ = 3π cos φ

∫S nŜ ·kÊ (h3) ds = − A 12cosπ φ

⎛ arctan⎜ ⎝

β 2

α −β

2

⎞ ⎟ ⎠

2

α − β2 γ1/ β

(

arctan

∫γ /β 2

dz

1 t2 − 1

t2 − 1

) dt (A-10)

3

3

a

∫a−R ∫0

2π

By differentiating eq A-10 with respect to β and appealing to

ρ dθ dρ (D + L cos φ − ρ sin φ sin θ )2

eq A-7, we obtain (A-4)

For eq A-2, to simplify the notation, we let D − z cos φ = α and a sin φ = β. Then eq A-2 can be written as

∫S

2

2

2

L

π

⎡ ⎤ 1 1 dθ dz + sin θ ⎢ 2 2⎥ ⎣ (α − β sin θ) (α + β sin θ) ⎦ (A-5)

To evaluate eq A-5, we appeal to the following formulas: b

d dt

∫a

d dx

∫f (x)

f (x , t ) d x =

∫a

b

∂f (x , t ) dx ∂t

(A-6)

Similarly, for inner S2, we obtain

g (x)

∫S

F(t ) dt = F(g (x)) g ′(x) − F(f (x)) f ′(x)

Together with the following integral identity π

=

⎛ ⎞ 1 1 − ⎜ ⎟ dθ α + β sin θ ⎠ ⎝ α − β sin θ ⎛ ⎞ β ⎟ arctan⎜⎜ 2 2 ⎟ − α2 − β2 α β ⎝ ⎠

2

nŜ 2 ·kÊ (h2) ds =

−A 3π cos φ

⎞ ⎛ ⎛ ⎞ (a − R )sin φ ⎜ arctan⎜ ⎟(D − L cos φ) ⎟ 2 2 ⎝ (D − L cos φ) − ((a − R)sin φ) ⎠ ⎟ ⎜ ⎟ ⎜ 2 2 (D − L cos φ) − ((a − R )sin φ) ⎟ ⎜ ⎟ ⎜ ⎞ (a − R )sin φ ⎜ arctan⎛⎜ ⎟(D + L cos φ)⎟ 2 2 ⎟ ⎜ ⎝ (D + L cos φ) − ((a − R)sin φ) ⎠ ⎟⎟ ⎜⎜− 2 2 (D + L cos φ) − ((a − R )sin φ) ⎠ ⎝ (A-11-2)

(A-7)

∫0

−A 3π cos φ

⎞ ⎛ ⎛ ⎞ a sin φ ⎜ arctan⎜ ⎟(D − L cos φ) ⎟ 2 2 D L a − φ − φ ( cos ) ( sin ) ⎝ ⎠ ⎟ ⎜ ⎟ ⎜ 2 2 ( cos ) ( sin ) D L a − φ − φ ⎟ ⎜ ⎟ ⎜ ⎛ ⎞ a sin φ ⎟ ⎜ arctan⎜ ( cos ) D L + φ ⎟ ⎟ ⎜ ⎝ (D + L cos φ)2 − (a sin φ)2 ⎠ ⎟⎟ ⎜⎜− 2 2 (D + L cos φ) − (a sin φ) ⎠ ⎝ (A-11-1)

∫S nŜ ·kÊ (h2) ds = − Aa12sinπ φ ∫−L ∫0

nŜ 2 ·kÊ (h 2) ds =

4

(A-8) 3984


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For eq A-3, to simplify the notation, we let α1 = D − L cos φ and β1 = ρ cos φ. Then eq A-3 can be written as

⎛

3

a

∫a−R ∫0

1

1

∫a−R ∫0

2π

ρ dθ dρ (α1 + β1 sin θ )2

1 dθ = α1 + β1 sin θ

2π

L

α12 − β12

π

∫−L ∫0

(A-13)

∫S nŜ ·kÊ (h1) ds = − A(D −6Lsincos2 φφ)cos φ ⎤ ⎡ 1 ⎥ ⎢ 2 2 ⎥ ⎢ (D − L cos φ) − (a sin φ) ⎥ ⎢ 1 ⎥ ⎢− 2 2 2 ⎥ ⎢ φ φ D L a R ( − cos ) − ( − ) sin ⎦ ⎣ (A-14)

∫−L ∫0

=

e−κD (e κL cos φ − e−κL cos φ) κ cos φ π

sin θ(e α sin θ + e−α sin θ) dθ

function of zeroth order L0(z): L 0 (z ) =

2

f (α ) = =4

∫0

∫0

π

(e α sin θ − e−α sin θ) dθ

π /2

sinh(α sin θ ) dθ = 4

∫0

π /2

sinh(α cos θ) dθ

By differentiating f(α) with respect to α, we obtain (A-16)

∫0

⎛ kT ⎞2 nŜ 2 ·kÊ (h2) ds = 32aε0εrγ1γ2κ ⎜ ⎟ sin φ ⎝ ve ⎠ S2

∫

π

π

sin θe−κ(D − z cos φ + a sin φ sin θ) dz dθ )

L−1(α) + L1(α) = 2L0′(α) − (A-17)

(A-23)

2 , π

L1(α) = L0′(α) −

2 π

(A-24)

We obtain ⎛ kT ⎞2 e−κD nŜ 2 ·kÊ (h2) ds = 64πa sin φε0εrγ1γ2⎜ ⎟ ⎝ ve ⎠ cos φ S2

⎛ kT ⎞2 nŜ 1·kÊ (h1) ds = 32ε0εrγ1γ2κ ⎜ ⎟ cos φ ⎝ ve ⎠ S1

∫

∫

ρe−κ(D − L cos φ − ρ sin φ sin θ) dθ dρ

sin θ(e α sin θ + e−α sin θ) dθ = 2πL0′(α)

functions, we know that

Similarly, from eqs 2.2, 3.2, 4.1, and 6.2 and eqs 2.2, 3.3, 4.3, and 6.2 in the main text of the Article, we obtain eqs A-18 and A-19, respectively,

2π

π

On the basis of the recurrence relations of modified Struve

sin θe−κ(D − zcos φ − a sin φ sin θ) dz dθ

∫−L ∫0

a

(A-21)

(A-22)

By using eqs 2.1, 3.1, 4.2, and 6.2 in the main Article context, we obtain

∫a−R ∫0

sinh(z cos θ ) dθ

1

3

L

π /2

1

3

+

∫0

∫S nŜ ·kÊ (h2) ds + ∫S nŜ ·kÊ (h1) ds

∫S nŜ ·kÊ (h3) ds

L

2 π

We rewrite f(α) in terms of L0(z):

Derivation of Electrostatic Double Layer Interaction Energy (Equation 8)

∫−L ∫0

(A-20)

recall the integral representation of the modified Struve

∫

(

sin θe−κ(D − z cos φ + a sin φ sin θ) dz dθ

obtained by differentiation f(α) with respect to α. Now we

A(D + L cos φ)cos φ nŜ 3· kÊ (h3) ds = − S3 6 sin 2 φ ⎤ ⎡ 1 ⎥ ⎢ 2 2 ⎥ ⎢ (D + L cos φ) − (a sin φ) ⎥ ⎢ 1 ⎥ ⎢− 2 2 2 ⎥ ⎢ D L φ a R φ ( + cos ) − ( − ) sin ⎦ ⎣ (A-15)

2

(A-19)

Let f(α) = ∫ π0(eα sin θ − e−α sin θ) dθ and eq A-20 can be

Similarly, for eq A-4, we obtain

+

π

+

∫0

1

ΦEDL (D) =

ρe−κ(D + L cos φ − ρ sin φ sin θ) dθ dρ

sin θe−κ(D − z cos φ − a sin φ sin θ) dz dθ

L

Equation A-12 can be obtained by differentiating eq A-13 with respect to α1 and integrating with respect to ρ, and after a routine algebraic operation, we obtain 1

2π

A-19 can be written as

We recall a well-known integral identity 2π

⎟

After letting α = κa sin φ and integrating with respect to z, eq

(A-12)

∫0

⎜

3

∫S nŜ ·kÊ (h1) ds = − A 12cosπ φ a

⎞2

cos φ ∫S nŜ ·kÊ (h3) ds = 32ε0εrγ1γ2κ ⎝ kT ve ⎠

(e κL cos φ − e−κL cos φ)L−1(κa sin φ)

(A-18)

(A-25) 3985


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For eqs A-18 and A-19, letting μ = κ sin φ, we obtain ⎛ kT ⎞ nŜ 3 ·kÊ (h3) ds = 32ε0εrγ1γ2κ ⎜ ⎟ ⎝ ve ⎠ S1 S3 −κD κL cos φ −κL cos φ cos φe (e ) +e

∫

nŜ 1·kÊ (h1) ds + a

∫a−R ∫0

2π

By inserting eqs A-31 and A-32 into eq A-28, we obtain a

2

∫0 ∫0

∫

∫0

∫−a

1

2π

− 2a 2

ρe μρ sin θ dρ dθ = 2a 2

■

a 2 − y 2 e μy dy

1

t

∫−1

2

1−t

2

1

∫−1

1 1−t

2

1 π

1

∫−1

1 − t2

e μat dt

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We gratefully acknowledge partial financial support of this work by the NSF through grant CHE-1213333. We also thank Dr. L.C. Shen of the Department of Mathematics at the University of Florida for his valuable suggestions on derivations of the double integrals.

■

dt (A-29)

e μat

1

1 − t2

(A-30)

dt = J0 ( −iμa)π = I0(μa)π

(A-31)

Differentiating eq A-31 with respect to −iμa, we can write the second integral on the right-hand side in eq A-28 as 1

−

∫−1

t 2e μat 1 − t2

dt = J0 ″( −iμa)π

(A-32)

On the basis of the recurrence relations of the Bessel function, we obtain J0 ″( −iμa) = =

1 [J ( −iμa) − J0 ( −iμa)] 2 2

1 [I2(μa) − I0(μa)] 2

REFERENCES

(1) Zheng, L. X.; O’Connell, M. J.; Doorn, S. K.; Liao, X. Z.; Zhao, Y. H.; Akhadov, E. A.; Hoffbauer, M. A.; Roop, B. J.; Jia, Q. X.; Dye, R. C.; Peterson, D. E.; Huang, S. M.; Liu, J.; Zhu, Y. T. Ultralong singlewall carbon nanotubes. Nat. Mater. 2004, 3, 673−676. (2) Tans, S. J.; Devoret, M. H.; Dai, H. J.; Thess, A.; Smalley, R. E.; Geerligs, L. J.; Dekker, C. Individual single-wall carbon nanotubes as quantum wires. Nature 1997, 386, 474−477. (3) Wong, S. S.; Harper, J. D.; Lansbury, P. T.; Lieber, C. M. Carbon nanotube tips: high-resolution probes for imaging biological systems. J. Am. Chem. Soc. 1998, 120, 603−604. (4) Tans, S. J.; Verschueren, A. R. M.; Dekker, C. Room-temperature transistor based on a single carbon nanotube. Nature 1998, 393, 49− 52. (5) Fan, S. S.; Chapline, M. G.; Franklin, N. R.; Tombler, T. W.; Cassell, A. M.; Dai, H. J. Self-oriented regular arrays of carbon nanotubes and their field emission properties. Science 1999, 283, 512− 514. (6) Kong, J.; Franklin, N. R.; Zhou, C. W.; Chapline, M. G.; Peng, S.; Cho, K. J.; Dai, H. J. Nanotube molecular wires as chemical sensors. Science 2000, 287, 622−625. (7) Collins, P. G.; Bradley, K.; Ishigami, M.; Zettl, A. Extreme oxygen sensitivity of electronic properties of carbon nanotubes. Science 2000, 287, 1801−1804. (8) Haggenmueller, R.; Gommans, H. H.; Rinzler, A. G.; Fischer, J. E.; Winey, K. I. Aligned single-wall carbon nanotubes in composites by melt processing methods. Chem. Phys. Lett. 2000, 330, 219−225.

If we let x = iμa in eq A-29, then the first integral on the right-hand side in eq A-28 can be written as

∫−1

AUTHOR INFORMATION

Notes

(A-28)

I0(x) = J0 (ix)

(A-36)

*E-mail: carpena@ufl.edu and bg55@ufl.edu. Phone: (352) 392-1864, Fax: (352) 392-4092.

e μat dt

e ixt

[I0(κ(a − R )sin φ) + I2(κ(a − R )sin φ)]⎤⎦

Corresponding Author

We recall the integral representations of the Bessel function of zeroth order (J0(x)) and the modified Bessel function of zeroth order (I0(x)): J0 (x) =

3

3

⎣⎡a 2π[I0(κ sin φa) + I2(κ sin φa)] − (a − R )2 π

If we let y = at, then eq A-27 can also be written as a

(A-35)

⎛ kT ⎞2 −κD κL cos φ ⎜ ⎟ cos φe (e + e−κL cos φ) ⎝ ve ⎠

(A-27)

∫0 ∫0

ρe μρ sin θ dρ dθ = (a − R )2

1

We now rewrite the integral in eq A-26 in terms of the Bessel function. To do this, first we convert the integral from polar coordinates to Cartesian coordinates. If we let x = ρ cos θ and y = ρ sin θ, we obtain ρe μρ sin θ dρ dθ = 2

2π

∫S nŜ ·kÊ (h1) ds + ∫S nŜ ·kÊ (h3) ds = 32ε0εrγ1γ2κ

(A-26)

∫0 ∫0

∫0

By inserting eqs A-34 and A-35 into eq A-26, we obtain

∫

a

a−R

π[I0(κ(a − R )sin φ) + I2(κ(a − R )sin φ)]

∫ ∫

2π

(A-34)

Similarly, we obtain

⎛ kT ⎞2 = 32ε0εr γ1γ2κ ⎜ ⎟ cos φe−κD(e κLcos φ + e−κLcos φ) ⎝ ve ⎠ ⎡ a 2π μρ sin θ dρ dθ ρe ⎢⎣ 0 0 2π a−R ⎤ ρe μρ sin θ dρ dθ ⎥ − ⎦ 0 0

a

ρe μρ sin θ dρ dθ

= a 2π[I0(κ sin φa) + I2(κ sin φa)]

ρe μρ sin θ dρ dθ

∫

2π

(A-33) 3986


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DLVO Interactions of Carbon Nanotubes with Isotropic Planar Surfaces

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