Steady solidification along a heat removal surface (either a free-surface or a solid wall) was studied with the goal of characterizing the behavior in the vicinity of the triple junction between the solid, liquid, and surface. By performing a local analysis of the Stefan problem around this point, it was shown that if the free-surface or wall heat flux is continuous, there is no solution where the solid grows along the surface with a nonzero wedge angle, i.e. the angle at which the solidification front intersects the external surface. This result was verified by unsteady numerical simulations of horizontal ribbon growth that showed that the wedge angle decreased to zero in the transient simulations. When the surface heat flux is discontinuous at the triple junction (such as would occur with a solid and liquid having different radiative emissivity), the jump in heat flux leads to a finite wedge angle which is proportional to the jump in heat flux divided by the latent heat release rate. The analytical prediction of the wedge angle agreed with numerical results to within 1%. It was also shown that solidification with a nonzero "growth angle" behaves similarly to the case with a discontinuous surface flux. These results contradict the prediction of simplified energy analyses that the wedge angle is proportional to the magnitude of the heat flux from the external surface. (C) 2015 Elsevier B.V. All rights reserved.