Simulation of deposit growth on a two-dimensional substrate was studied based on a new model that tracks individual cubic particles as they form a deposit structure. The present model is an extension of the classical ballistic deposition model. Effects of three different parameters were studied. These include an attraction parameter that is a measure of the particle to particle attractions, an interaction length within which the particles are assumed to influence and be influenced by surrounding particles, and allowed sticking positions (face-face, edge-edge and corner-corner) that favor particular growth directions. Structures with widely varying properties were obtained using this model. The three parameters were found to have considerable effect on the structure including indications of morphological phase transformations. A new property of the system (saturated roughness/deposit growth rate) was identified that can classify the different types of growth into a single type.