Line-tension effect on the equilibrium and stability of a spherical floating particle at a fluid–fluid interface

A recent matched-asymptotic framework for a spherical particle floating at a fluid–fluid interface reduced the coupled Young equation and vertical force balance to a single nonlinear equation but assumed a constant macroscopic Young angle. We extend the framework to include line tension. A variational derivation places the dimensionless line tension κ=k/(r0σ), the ratio of the line tension k to the product of the sphere radius r0 and the surface tension σ, in the modified Young equation. Because the line-tension energy depends on the filling angle alone and that angle is set by the modified Young equation, the line-tension term drops out of the vertical force balance identically. The coupled system reduces to a single substitution of the constant contact angle by a filling-angle dependent contact angle. Closed-form first-order corrections agree with the numerical solution of the nonlinear asymptotic equilibrium equation with O(κ2) remainders away from the floatability boundary, and the line-tension response of the framework is confirmed independently by a direct numerical solution of the Young–Laplace equation that shares no asymptotic ingredient. The hydrophilic regime is about an order of magnitude more sensitive to κ than the hydrophobic one across the full range of contact angles. The modified Young equation rewrites as a linear regression that recovers k from multi-sphere datasets, and its regression form is shown to be consistent with the original droplet dataset of Klauser and coworkers.