Model of Two Floating Spherical Particles

Kralchevsky et al. [14] also proved from the energy method that Eqn. (3.49) is valid for calculating the horizontal force between the two floating spherical particles. Since r is always positive, the sign of capillary charge Q. depends solely on y/.. If both particles have the same sign of y/., the lateral capillary force between the two floating spheres is attractive otherwise, the force is repulsive. 

Since the contact line moves on the curved surface of the sphere, either r, or y/ is not constant. In order to calculate the capillary charge Q, in Eqn. (3.49), one needs to use the vertical force balance. Because the inclination of the contact line is not significant, the contact line can be assumed horizontal to obtain geometric relations. The volume of the lower part of the sphere immersed in liquid is 

According to the vertical force balance, the vertical component of capillary force at the contact line is coimterbalanced by the gravitational 

Due to extremely small weight (note that the particle weight is proportional to the cube of radius), the interface deformation with small particles is very small. Kralchevsky and Nagayama [26] estimated that particles with radius less than 10 pm cannot deform the interface and, as a result, do not generate any horizontal force, i.e., y/. is small and 0.. Eqn. (3.59) finally becomes 

Combining Eqn. (3.48) with Eqn. (3.60), the horizontal capillary force vs. distance between particles can be calculated. Experimental measurements agree well with this calculation [18,30]. 

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