Modeling analysis of the growth of a cubic crystal in a finite space

The applications of semiconductor nanocrystals in optoelectronics are based on the unique characteristic of quantum confinement. There is great interest to tailor the performance of optoelectronic nanodevices and systems through the control of the sizes of nanocrystals. In this work, we develop a general mathematical formulation for the growth of a crystal/particle in a liquid solution, which takes account of the combinational effect of diffusion-limited growth and reaction-limited growth, and formulate the growth equations for the size of a cubic crystal grown under three different scenarios - isothermal and isochoric conditions, isothermal growth with the evaporation and/or extraction of the solvent and isochoric growth with continuous change in temperature. For the growth of a cubic crystal under isothermal and isochoric conditions, there are three growth stages - linear growth, nonlinear growth and plateau, and the growth rate in the stage of linear growth and the final size of the cubic crystal are dependent on the degree of supersaturation. For the growth of multi-crystals with a Gaussian distribution of crystal sizes, the change of the monomer concentration in a liquid solution is dependent on the change rates of average size and the standard deviation of the crystal sizes.