The evolution of cellular morphology during interfacial instability for liquid-solid transition for pure unary material systems is studied using the maximum entropy production (generation) rate principle (MEPR) for steady-state directional solidification. This approach is dependent on a quantity calledmaximum entropy production rate densitywhich inherently contains key solidification parameters that governs cellular evolution for liquid-solid transformation. Themaximum entropy production rate densityis computationally measured from the solid-liquid interface in diffuse form and considers steady-state solidification at low velocities for both near and far from equilibrium conditions. The results are presented in mathematical expressions for morphological instability that corresponding to the evolution of a cellular morphology which emanates from the solid-liquid interface. The model is formulated to evaluate the solid-liquid interface thickness, the solidification velocity, grain boundary energy, and the size of the cellular morphological at instability. The results are tested with a number of pure single element materials at different temperature gradients which compare well with available experimental data.