Effects of gravity on equilibrium crystal shapes

The effects of gravity on the two-dimensional equilibrium shapes (ES) of crystals and menisci are investigated for different geometries (positions) of the substrate.

In the gravity-free case, the equilibrium crystal shape (ECS) is characterized by a scale invariance. The presence of gravity breaks the scale invariance and the resulting ECS changes as the volume of the crystal V is changed. Moreover, the presence of gravity breaks the translational invariance along the direction it acts. Physically realized by the necessity of a support, this is manifested by the existence of an inhomogeneous effective pressure P_eff, which divides the space into two regions, with P_eff either negative or positive. The ECS changes as the crystal passes from one region to another, being concave where P_eff < 0, and convex where P_eff > 0.

In all cases it was possible to express the corresponding ECS in terms of the gravity-free one.

For the hung crystal, i.e., a crystal pinned to a vertical wall at the top, it is shown that some orientations are missing from the ECS that otherwise will be present in the gravity-free ECS, adsorbed on the same substrate. Thus, facets could disappear from the crystal shape as the volume V or the gravitational acceleration g is increased. A critical volume V_c is found, so that if the crystal volume V exceeds V_c, the crystal cannot be pinned. The ECS can exhibit both concave and convex portions.

For a crystal, pinned to a vertical wall at its lower end, we find that it will never develop a concave part. On the other hand, new orientations, absent from the gravity-free crystal, will be present on its ECS.

The ES of a free and pinned crystal meniscus is also solved and an expression for the excess (depleted) volume AV is derived. The solution for the crystal meniscus between two walls is also presented.

For the pendant crystal, i.e., a crystal hanging from a horizontal support, we find that it can exhibit both concave and convex portions on its ECS. When it develops a concave part, new orientations will appear, compared to the gravity-free case. An intuitive stability criterion is introduced, according to which only crystals wetting the substrate can develop a concave portion before they break.

The treatment of a crystal on an inclined substrate shows the complications that arise in determining the ES for a general position of the support as a result of the conflict between the directions associated with gravity and support.

An expression for the facet length in the presence of gravity is obtained that is valid for all types of support. For crystal shapes that display a concave portion it offers a very convenient way to experimentally measure step free energies. Thus, by breaking scale invariance, the presence of gravity allows absolute measures of surface energy in contrast to the gravity-free case, where the facet length is proportional to the step free energy by an unknown scale.