On analytical integration of interaction potentials between cylindrical and rectangular bodies with a focus on van der Waals attraction

TL;DR

The paper tackles the challenge of analytically integrating inverse-power interaction potentials over bodies with circular and rectangular cross sections, with a focus on van der Waals attraction. It introduces a set of reduction rules (RIIPI, IUI, RIHBI) to transform complex 3D integrals into tractable lower-dimensional forms and derives extensive exact and approximate laws for disk-disk, disk-plate, disk-cylinder, and rectangle-rectangle interactions, including $m$-dependent expressions and special vdW cases. The main contributions are new exact vdW laws (e.g., disk-disk for $m=6$ and rectangle-rectangle in-plane for $m eq4$), plus computationally efficient approximations (D-D_app, P-C_app, D-C_app) that balance accuracy and speed for coarse-grained fiber simulations. The results enable accurate, fast pre-integration suitable for simulating slender deformable bodies and have practical impact for multiscale modeling of intermolecular forces in fiber and plate-like systems, as demonstrated by a Lennard-Jones fiber example.

Abstract

The paper deals with the analytical integration of interaction potentials between specific geometries such as disks, cylinders, rectangles, and rectangular prisms. Interaction potentials are modeled as inverse-power laws with respect to the point-pair distance, and the complete body-body potential is obtained by pairwise summation (integration). Several exact new interaction laws are obtained, such as disk-plate and (in-plane) rectangle-rectangle for an arbitrary exponent, and disk-disk and rectangle-rectangle for van der Waals attraction. To balance efficiency and accuracy, additional approximate laws are proposed for disk-disk, point-cylinder, and disk-cylinder interactions. A brief numerical example illustrates the application of the pre-integrated Lennard-Jones disk-disk interaction potential for the interaction between elastic fibers.

Figures (20)

• Figure 1: Various interaction pairs considered in this research. Due to parallel orientation and symmetry, the distance between all interaction bodies is defined by a gap (horizontal double arrow) and, for disk-disk and rectangle-rectangle, by an offset (vertical double arrow).
• Figure 2: Interaction of two bodies with constant cross sections. a) Front view. b) and c) Top view in case of circular cross sections (disks). Two relative polar coordinate systems are employed (RPCS1 and RPCS2). d) Top view in case of rectangular cross sections.
• Figure 3: a) Illustration of IUI: The interaction potential between a point $X$ and an infinite plane $Y_\infty$ is $\Pi_{\operatorname{P-PN_{\infty}}}^m$. A plane line and a plane figure are defined in a plane passing through $X$ and parallel to $Y_\infty$. The interaction potential of the plane line or the plane figure with the plane $Y_\infty$ is obtained by multiplying $\Pi_{\operatorname{P-PN_{\infty}}}^m$ with the line's length or the figure's area, respectively. b) Application of RIIPI and IUI: Interaction potential between finite- and infinite-length cylinders in parallel orientation, $\Pi_{\operatorname{C-C_{\infty}}}^m$, equals an interaction potential $\Pi_{\operatorname{D-D_{IP}}}^{m-1}$ between two in-plane disks multiplied by the factor $f_m$ and by the cylinder's length.
• Figure 4: Scaling of the vdW disk-disk laws for different ratios $c=q_1/q_2$ ($R_x=R_y=1$). a) Exact D-D law from Eq. \ref{['DDlaw']}. b) Approximate $\operatorname{D-D_{app}}$ law from Eq. \ref{['appDD2']}.
• Figure 5: Error of the ISSIP and $\operatorname{D-D_{app}}$ laws for $m=6$ and different ratios $c=q_1/q_2$ ($R_x=R_y=1$): a) relative error, b) absolute error. $\operatorname{D-D_{app}}$ is much more accurate than ISSIP, especially for large separations, but it still shows errors at large separations compared to the exact D-D law.

Theorems & Definitions (6)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6