  We use the method of free energy minimization based on the first law of
thermodynamics to derive static meniscus shapes for crystal ribbon growth
systems. To account for the possibility of multivalued curves as solutions to
the minimization problem, we choose a parametric representation of the meniscus
geometry. Using Weierstrass' form of the Euler-Lagrange equation we derive
analytical solutions that provide explicit knowledge on the behaviour of the
meniscus shapes. Young's contact angle and Gibbs pinning conditions are also
analyzed and are shown to be a consequence of the energy minimization problem
with variable end-points. For a given ribbon growth configuration, we find that
there can exist multiple static menisci that satisfy the boundary conditions.
The stability of these solutions is analyzed using second order variations and
are found to exhibit saddle node bifurcations. We show that the arc length is a
natural representation of a meniscus geometry and provides the complete
solution space, not accessible through the classical variational formulation.
We provide a range of operating conditions for hydro-statically feasible
menisci and illustrate the transition from a stable to spill-over configuration
using a simple proof of concept experiment.
