Forces and the Young Relationship in the Bubble-Spherical Particle-Liquid System

"Flotation is possible when the forces operating between a bubble and the attached particle are properly balanced. There are many forces operating in the bubble-particle aggregate. The magnitude, location and direction of the forces depend on the type of process. The simplest arrangement in flotation is when a spherical bubble is attached to a spherical particle and both are immersed in water. In such a system there are four essential forces which shape the stability of the bubble-particle aggregate. They are excess pressure and interfacial solid/liquid, solid/gas and liquid/gas forces. A proper analysis and balance of the size, location and direction of these four principal forces lead to a general equation delineating the stability of the spherical particle–attached bubble system. Specific cross sections of the system and solution of the general equation lead to different relationships including the Young formula as well as the well-known geometrical relation between contact angle, particle and bubble radiuses. The arrangement of forces in the proposed graphical representation of the Young law is different from that presently used in literature for a bubble attached to an unmovable solid particle. Many real solid/liquid/gas systems are more complex due to the presence of additional forces which change the regular curvature of the liquid/gas interface leading to advancing and receding contact angles.INTRODUCTIONThere are numerous papers (Athukorallage, Aulisa, Iyer & Zhang, 2015; Chau, Bruckard, Koh & Nguyen, 2009; Drzymala, 1994; Lubarda, 2012; Pellicer, Manzanares & Mafe, 1995; Wang, Cui, Zhou, Xu, Sun, & Zhu, 2014; Young, 1804) and books (Davies & Rideal, 1963; Defay & Prigogine, 1966; Lee, 2013; Lyklema, 2000) dealing with the solid/liquid/gas system. Many of them concern the three involved phases having the shape of a sphere or spherical cap and the formed contact angle. The contact angle is usually measured through the liquid phase and can be either smaller or greater than 90o. The essential part of these publications is a graphical representation of the system with the Young equation shown as a result of balance of only three forces that is the excess Gibbs (or Helmholz) free energy per unit area of the three involved interfaces (Davies & Rideal, 1963; Defay & Prigogine, 1966; Israelachvili, 1985) (Figs. 1a-c). The Young equation, when the gravity force/buoyancy is neglected, is:"