The Contact Angle Variation of Floating Particles Makes It Difficult to Use the Neumann Condition To Quantify the Air-Water Interface Deformation in Three-Dimensional Space.

Capillary force is critical to the floatability of particles at the air-water interface. Quantification of the capillary force requires solving the Young-Laplace equation using suitable boundary conditions (BCs) at the triple contact line. For axisymmetric (two-dimensional, 2D) systems, such as single spheres floating at an initially flat air-water surface, both the Dirichlet (constant contact depth) and Neumann (constant contact angle) BCs can be applied. For three-dimensional (3D) systems, Neumann BCs (NBCs) have been successfully used. In this paper, we have challenged the use of NBCs for the 3D deformation of the air-water surface induced by floating particles, which always exhibit intrinsic contact angle (CA) hysteresis that is significantly amplified in 3D systems. Specifically, we designed and conducted the experiments using single prismatic particles, which allowed for the determination of two characteristic CAs at the two diagonal axes with a high degree of certainty. We calibrated the numerical solution to the 3D Young-Laplace equation using the deformed air-water interface profiles at the two diagonal axes and then validated the numerical solution for the capillary force on the floating particles with the measured force. We obtained reliable data for the CA along the three-phase contact line (TPCL), which displayed a significant distribution. We also discussed the findings that were significant to floating spheres in asymmetric systems, such as pairs of floating spheres. This paper provides experimental and theoretical evidence that the CA is not constant along the contact line in a 3D geometry, which invalidates the use of NBCs for 3D systems of floating particles. This study highlights the significance of the CA variation known as CA hysteresis, which should be considered when predicting the floatability of particles at the air-water interface.