ABSTRACT
The free energy is a functional of the function r(z): for every profile r(z) the free energy can be calculated using the above formula. We are now interested in finding that profile r(z) that minimizes the above free energy functional under the constraint that the total volume is constant. This is achieved by introducing ∆p as the Lagrange multiplier to fix the volume and add to the above free energy a term
(24)
Taking the functional derivative of F−∆pV yields the following equation to determine r(z)
(25)
The expression above comprises two terms, the first involving the complete profile r(z) and the second, which is multiplied by the delta function δ(z), involving only the profile and its derivative, at the substrate. In order for the above expression to be zero, both terms in the curly brackets have to be zero separately. Two conditions therefore result, the first of which can be written as
(26)
where we have used the explicit expressions for the radii of curvature in terms of the function r(z)
(27)
The Lagrange multiplier ∆p is determined by relating it to the pressure difference at the apex of the droplet (z=h); see Eq. (15). Eq. (26) then becomes
(28)
Using Eqs. (14) and (15), we now see that the above equation is in fact the Laplace equation as given in Eq. (12), the only difference coming from the fact that a different convention for the location of the origin has been used.
