  The interplay between the diffusion-controlled dynamics of a solidification
front and the trajectory of a grain boundary groove at the solid-liquid
interface is studied by means of thin-sample directional solidification
experiments of a transparent alloy, and by numerical simulations with the
phase-field method in two dimensions. We find that low-angle grain boundaries
(subboundaries) with an anisotropic interfacial free energy grow tilted at an
angle $\theta_t$ with respect to the temperature gradient axis. $\theta_t$
remains essentially equal to its value imposed at equilibrium as long as the
solidification velocity $V$ remains low. When $V$ increases and approaches the
cellular instability threshold, $\theta_t$ decreases, and eventually vanishes
when a steady-state cellular morphology forms. The absence of mobility of the
subboundary in the solid is key to this transition. These findings are in good
agreement with a recent linear-stability analysis of the problem.
