Analytical sphere–thin rod interaction potential

A compact analytical form is derived through an integration approach for the interaction between a sphere and a thin rod of finite and infinite lengths, with each object treated as a continuous medium of material points interacting by the Lennard-Jones 12-6 potential and the total interaction potential as a summation of the pairwise potential between material points on the two objects. Expressions for the resultant force and torque are obtained. Various asymptotic limits of the analytical sphere-rod potential are discussed. The integrated potential is applied to investigate the adhesion between a sphere and a thin rod. When the rod is sufficiently long and the sphere sufficiently large, the equilibrium separation between the two (defined as the distance from the center of the sphere to the axis of the rod) is found to be well approximated as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mn>0.787</mml:mn> <mml:mi>σ</mml:mi></mml:mrow> </mml:math> , where a is the radius of the sphere and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>σ</mml:mi></mml:math> is the unit of length of the Lennard-Jones potential. Furthermore, the adhesion between the two is found to scale with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msqrt><mml:mi>a</mml:mi></mml:msqrt> </mml:math> .