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Homework Statement
Consider a planar liquid-gas interface and a solid sphere partially immersed in liquid. A fraction of the solid surface area is wet by the liquid, call it ##A_{sl}##. The complement of the solid’s area is ‘wet’ by the gas, say ##A_{sg}##. There is also an area of contact between liquid and gas ##A_{lg}## (make this finite by imagining a large cylindrical control volume). Define system energy ##E## to be the weighted sum of the interfacial areas. Call these weights ##\sigma_{sl}##, ##\sigma_{sg}##, and ##\sigma_{lg}##, respectively. Think of the weights as constants for fixed materials, having units energy/area. Find the equilibrium states -- those at which the energy is stationary, ##\delta E = 0##.
Homework Equations
The Attempt at a Solution
Assume the sphere is small enough that gravity and buoyancy are insignificant. Then an energy balance would be $$E = \cos\theta(\sigma_{sl}A_{sl}-\sigma_{sg}A_{sg})$$ where I define ##\theta## as the angle the sphere makes with the liquid at the point of contact. I feel I am missing more for the total energy balance though. Any ideas?
Thanks!
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