Analytical and Computational Studies of Lennard-Jones Thin Rods

Abstract

Rod-like particles and structures are ubiquitous in nature and engineered systems. Examples include carbon nanotubes, nanowires, biological filaments, liquid crystal molecules, and colloidal cylinders. Understanding their interactions is crucial for predicting the behavior of these systems at various scales. The Lennard-Jones (LJ) 12-6 potential, one of the most widely used functional forms in molecular simulations, captures both van der Waals attraction and Pauli repulsion between (neutral) atoms. The integrated forms of the LJ potential are frequently used to describe interactions between finite-size objects. While such integrated potentials have been reported for simple geometries such as spheres and planes, compact analytical forms for rod-like particles have remained elusive. This dissertation presents a comprehensive theoretical framework and computational implementation for studying LJ interactions involving thin rod-like particles. Through rigorous mathematical derivations based on Ostrogradsky's integration method, we develop closed form analytical expressions for three fundamental interaction scenarios: (1) rod-rod interaction, (2) sphere-rod interaction, and (3) point particle-rod interaction, all in arbitrary three-dimensional configurations. These analytical forms enable efficient computation of both forces and torques, facilitating their use in molecular dynamics simulations and theoretical analyses. For the rod-rod interaction, we derive an integrated LJ potential for two thin rods of finite or infinite lengths in general skew configurations, including special cases of parallel, crossing, coplanar, and collinear arrangements. Each thin rod is treated as a segment consisting of LJ material points. The expressions encompass the integration of both attractive (1/r^6) and repulsive (1/r^12) components of the LJ potential. We verify these analytical results through comparison with direct numerical integration and explore their physical implications, including the adhesion behavior between rods in various configurations. The sphere-rod interaction potential is derived by integrating the LJ potential between a sphere (treated as a continuous medium) and a thin rod. We obtain compact analytical expressions valid for rods of both finite and infinite lengths. This potential is used to study the adhesion between a sphere and a rod. Interesting scaling relationships are discovered: for sufficiently large spheres and long rods, the equilibrium gap width is approximately constant at 0.787σ, where σ is the LJ length parameter, and the sphere-rod adhesion scales with the square root of the sphere radius. To enable practical applications of these analytical potentials in large-scale molecular dynamics simulations, we implement them as a user package in the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). In this implementation, the motion of each rod is described through the motion of its two ends, with each end treated as a particle in molecular dynamics simulation. We develop efficient algorithms for neighbor list construction, force decomposition, and coordinate transformation between local and global reference frames. The parallel performance of the implementation is analyzed, demonstrating its capability to simulate systems with thousands of rods efficiently.