  A long-standing goal of materials science is to understand, predict and
control the evolution of microstructures in crystalline materials. Most
microstructure evolution is controlled by interface motion; hence, the
establishment of rigorous interface equations of motion is a universal goal of
materials science. We present a new model for the motion of arbitrarily curved
interfaces that respects the underlying crystallography of the two
phases/domains meeting at the interface and is consistent with microscopic
mechanisms of interface motion; i.e., disconnection migration (line defects in
the interface with step and dislocation character). We derive the equation of
motion for interface migration under the influence of a wide range of driving
forces. In Part II of this paper [Salvalaglio, Han and Srolovitz, 2021], we
implement the interface model and the equation of motion proposed in this paper
in a diffuse interface simulation approach for complex morphology and
microstructure evolution.
