A mechanical model of open filament shape and-growth driven by phase ordering is formulated. For a given phase-ordering driving force, the model output is the filament shape evolution and the filament end-point kinematics. The linearized model for the slope of the filament is the Cahn-Hilliard model of spinodal decomposition, where the buckling corresponds to concentration fluctuations. Two modes are predicted: (i) sequential growth and buckling and (ii) simultaneous buckling and growth. The relation among the maximum buckling rate, filament tension, and matrix viscosity is given. These results contribute to ongoing work in smectic A filament buckling.