Two-Particle Lateral Capillary Force from Matched Asymptotic Expansions: Sphere and Oblate Spheroid

The lateral capillary force between two floating particles at a deformable fluid interface is derived analytically from a second-order matched asymptotic expansion (MAE) framework. Superposition of two outer monopoles yields the interaction energyEint(d)=-2πσc1c2r0,1r0,2K0(d/lc) and the lateral force F(d)=2πσc1c2r0,1r0,2K1(d/lc)/lc, with the capillary charge ci obtained from the nonlinear single-particle equilibrium rather than its linearized Nicolson form; over the asymptotic range ε=r0/lc≤0.4, the linearized approximation overestimates the force by a few percent. Extension to oblate spheroidal particles of aspect ratio k = cpol/a ≤ 1 follows from the single substitution βeff(θe) = α - atan2(k sin θe,cos θe), giving c ∝ k and Foblate/Fsphere ≈ k2 at fixed equatorial radius; the maximum floatable density ratio scales as Dmax ∝ k-1 in the strongly hydrophobic regime. All predictions are validated against the experimental data of Vassileva et al. [Langmuir 2005, 21, 11190] and against direct numerical integration of the Young-Laplace equation.