The phase-field (PF) method for solidification phenomena is an open formulation based on a free-energy functional. Two common choices for the PF potential, here referred to briefly as the Caginalp and Kobayashi models, are compared with respect to their numeric results within the classical sharp-interface limit. Both qualitative and quantitative behavior are addressed, and an assessment of the computational effort required to approximate a sharp-interface problem is made. It is shown that the specific form of the free-energy potential does have a strong influence on the convergence of the PF results to their sharp-interface limit. Compliance of the PF solutions with the linear kinetic model for the interface temperature is also investigated. A simple one-dimensional solidification problem in the presence of kinetic undercooling is solved by the PF model and also by a deforming grid method. Our results support the view that, if care is exercised in formulating the phase-temperature coupling, there is a high degree of confidence in using the PF method for the numerical modeling of general solidification phenomena.