FIG. 10 Shape of the liquid droplet from the numerical tables of Bashforth and Adams [29] for β=1. The location of the substrate (the broken horizontal line) is such that the liquid-vapor surface meets the substrate with angle θ.

Introduce the arc length s as the distance along the surface from the apex in units of b, and ψ(s) as the angle at the surface defined previously in Fig. 9. The derivative with respect to s of the reduced height Z(s)≡z(s)/b and radial distance R(s)≡r(s)/b are then related to ψ(s) via

R′(s)=cosψ(s), Z′(s)=−sinψ(s), (18)

where the prime indicates a differentiation with respect to the argument s. In terms of the functions ψ(s) and R(s) the radii of curvature, R1 and R2 are given by

(19)

The differential equation in Eq. (16) thus reduces to

(20)

We see that instead of one second-order differential equation, we are now left with a set of coupled first-order differential equations [Eqs. (18) and (20)]. For given value of β these can be solved numerically as a function of s until the angle ψ reaches the contact angle at some s0, ψ(s0)=θ. A schematic version of such a program using the Runge-Kutta method for solving differential equations is given in Table 1. As a function of β, stepsize h and contact angle θ, this program determines the shape (R, Z) parametrized in terms of s as well as the droplet volume

(21)

If instead of β the droplet volume is given, the parameter β needs to be varied iteratively such that the volume is equal to that required.