of the first "liquid" phase. Simplification to planar geometry, Young's relation derived from variational computation Consider the interface as a curve for where is a free parameter. The free energy to be minimized is with the constraints which we can write as and fixed volume . The modified Lagrangian, taking into account the constraints is therefore where are Lagrange multipliers. By definition, the momentum and the Hamiltonian which is computed to be: Now, we recall that the boundary is free in the direction and is a free parameter. Therefore, we must have: At the boundary and , therefore we recover the Young equation. Non-ideal smooth surfaces and the Young contact angle The Young equation assumes a perfectly flat and rigid surface often referred to as an ideal surface. In many cases, surfaces are far from this ideal situation, and two are considered here: the case of rough surfaces and the case of smooth surfaces that are still real (finitely rigid). Even in a perfectly smooth surface, a drop will assume a wide spectrum of contact angles ranging from the so-called advancing contact angle, , to the so-called receding contact angle, . The equilibrium contact angle () can be calculated from and as was shown by Tadmor as, where The Young–Dupré equation and spreading coefficient The Young–Dupré equation (Thomas Young 1805; Anthanase Dupré and Paul Dupré 1869) dictates that neither γSG nor γSL can be larger than the sum of the other two surface energies. The consequence of this restriction is the prediction of complete wetting when γSG > γSL + γLG and zero wetting when γSL > γSG + γLG. The lack of a solution to the Young–Dupré equation is an indicator that there is no equilibrium configuration with a contact angle between 0 and 180° for those situations. A useful parameter for gauging wetting is the spreading parameter S, When S > 0, the liquid wets the surface completely (complete wetting). When S < 0, partial wetting occurs. Combining the spreading parameter definition with the Young relation yields the Young–Dupré equation: which only has physical solutions for θ when S < 0. A Generalized Model for the Contact Angle of Droplets on Flat and curved surfaces With improvements in measuring techniques such as AFM, confocal microscopy and SEM, researchers were able to produce and image droplets at ever smaller scales. With the reduction in droplet size came new experimental