Novel analytical expressions for determining van der Waals interaction between a particle and air–water interface: Unexpected stronger van der Waals force than capillary force

Analytical expressions for calculating Hamaker constant (HC) and van der Waals (VDW) energy/force for interaction of a particle with a solid water interface has been reported for over eighty years. This work further developed novel analytical expressions and numerical approaches for determining HC and VDW interaction energy/force for the particle approaching and penetrating air-water interface (AWI), respectively.The expressions of HC and VDW interaction energy/force before penetrating were developed through analysis of the variation in free energy of the interaction system with bringing the particle from infinity to the vicinity of the AWI. The surface element integration (SEI) technique was modified to calculate VDW energy/force after penetrating.We explain why repulsive VDW energy exists inhibiting the particle from approaching the AWI. We found very significant VDW repulsion for a particle at a concave AWI after penetration, which can even exceed the capillary force and cause strong retention in water films on a solid surface and at air-water-solid interface line. The methods and findings of this work are critical to quantification and understanding of a variety of engineered processes such as particle manipulation (e.g., bubble flotation, Pickering emulsion, and particle laden interfaces).