Calculate total energy surface tension

[member 428835]

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!

 

[Charles Link]

If I understand your $\theta$ correctly, you are trying to say $dA=R^2 sin(\theta) \, d \theta \, d \phi$ so that, upon integrating, $A_{immersed}=2 \pi R^2 (1-cos(\theta))$ and $A_{above \, surface}=2 \pi R^2(1+cos(\theta))$. To work out energetically how well this object will float (with the surface tension included), you need to also include the immersed volume (the weight of the fluid displaced for the buoyant force=you can neglect gas buoyancy), and you need the downward gravitational force which is $F_g= \delta V g$ where $\delta$ is the density of the object. You can also write the immersed volume as a function of $\theta$, so that $\theta$ can be the parameter that you solve for. The buoyant force and the gravitational force can be written in terms of their energy. For gravity, $E=mgz$ where $z=R cos(\theta)+Constant$. For the buoyant energy, you can integrate $\int F_{buoyant} \, dz$. Anyway, hopefully this was helpful, and steers you in the right direction. (I haven't done the algebra myself to check to see if this will work, but I think it might.)

 

[member 428835]

To work out energetically how well this object will float (with the surface tension included), you need to also include the immersed volume (the weight of the fluid displaced for the buoyant force=you can neglect gas buoyancy), and you need the downward gravitational force which is $F_g= \delta V g$ where $\delta$ is the density of the object. You can also write the immersed volume as a function of $\theta$, so that $\theta$ can be the parameter that you solve for. The buoyant force and the gravitational force can be written in terms of their energy. For gravity, $E=mgz$ where $z=R cos(\theta)+Constant$. For the buoyant energy, you can integrate $\int F_{buoyant} \, dz$. Anyway, hopefully this was helpful, and steers you in the right direction. (I haven't done the algebra myself to check to see if this will work, but I think it might.)

Thanks for replying Charles Link! Actually, this sphere is very small so we are asked (my the professor) to ignore buoyancy and gravity and just consider surface tension. The Bond number is sufficiently small so viscous terms dominate over gravity. You are correct regarding the description of the areas!