[Paper Review] Van der Waals interactions of parallel and concentric nanotubes

Elsebeth Schröder, Per Hyldgaard|arXiv (Cornell University)|Jul 1, 2003

Graphene research and applications11 references14 citations

TL;DR

This paper presents a first-principles density functional theory (DFT)-based approach to accurately describe van der Waals (vdW) interactions between parallel and concentric carbon nanotubes by incorporating non-local electron response and screening effects. The method uses a frequency-integrated, geometry-dependent interaction model that reduces the 6+1 dimensional integral to manageable 2D or 1D numerical integrals, yielding quantitative binding energies consistent with macroscopic d⁻⁵ scaling in the asymptotic limit.

ABSTRACT

For sparse materials like graphitic systems and carbon nanotubes the standard density functional theory (DFT) faces significant problems because it cannot accurately describe the van der Waals interactions that are essential to the carbon-nanostructure materials behavior. While standard implementations of DFT can describe the strong chemical binding within an isolated, single-walled carbon nanotube, a new and extended DFT implementation is needed to describe the binding between nanotubes. We here provide the first steps to such an extension for parallel and concentric nanotubes through an electron-density based description of the materials coupling to the electrodynamical field. We thus find a consistent description of the (fully screened) van der Waals interactions that bind the nanotubes across the low-electron-density voids between the nanotubes, in bundles and as multiwalled tubes.

Motivation & Objective

• Address the failure of standard DFT to describe long-range van der Waals (vdW) interactions in sparse, low-electron-density materials like carbon nanotubes.
• Develop a consistent, first-principle method to compute inter-tube vdW binding by coupling electron density response to electrodynamic fields.
• Account for local-field screening and anisotropic dielectric response in nanotubes, which are critical for accurate binding energy predictions.
• Provide a framework applicable to both parallel nanotube bundles and concentric multiwalled nanotubes, enabling quantitative comparison with experimental systems.

Proposed method

• Use first-principle DFT to compute the radial electron density and static susceptibility of single-walled nanotubes.
• Model the local dynamic electron response using a plasmon-pole approximation with a frequency-dependent susceptibility χ₀(n(r), u, u₀).
• Derive an effective susceptibility tensor χ_eff that includes local-field screening effects via the relation χ_eff(u)E_applied = −χ₀(u)∇φ(s, u).
• Apply the thin-wall approximation to represent the nanotube as a radial delta function, reducing the problem to 1D radial response functions.
• Express the vdW interaction energy per unit length as a product of a frequency integral J_tot(R₁, R₂, u₀⁽¹⁾, u₀⁽²⁾) and a spatial integral over angular coordinates.
• Leverage cylindrical symmetry to reduce the 2D angular integrals to a single hypergeometric function integral for identical nanotubes, enabling efficient numerical evaluation.

Experimental results

Research questions

• RQ1 How can standard DFT be extended to accurately describe non-local van der Waals interactions between sparse carbon nanostructures?
• RQ2 What is the role of local-field screening and anisotropic dielectric response in determining the strength and distance dependence of inter-tube binding?
• RQ3 How do the geometric factors (radius, separation, concentricity) influence the vdW interaction energy in nanotube systems?
• RQ4 Can the interaction energy be expressed in a form that recovers known macroscopic limits (e.g., d⁻⁵ scaling) while remaining fully first-principles?
• RQ5 To what extent can symmetry and analytical reduction simplify the otherwise high-dimensional integral for inter-tube vdW energy?

Key findings

• The method successfully computes the vdW interaction energy per unit length for parallel and concentric nanotubes using only first-principle DFT inputs and a consistent electrodynamic model.
• For identical parallel nanotubes, the interaction energy is expressed as a single integral involving a hypergeometric function, significantly improving computational efficiency.
• The asymptotic limit of the interaction energy follows a d⁻⁵ dependence, consistent with macroscopic London theory, but with higher-order corrections due to the hollow cylindrical geometry.
• For concentric nanotubes, the spatial integral is analytically solvable in terms of complete elliptic integrals K(k) and E(k), yielding a closed-form expression for the geometry factor.
• The computed binding energy per unit length shows that larger nanotubes (lower curvature) exhibit stronger attraction at the same center-to-center separation due to enhanced contact area.
• The model improves upon traditional Hamaker estimates by including screening and anisotropic response, reducing errors from approximations in bulk dielectric models.