Brownian dynamics, (BD), simulation has been used to follow the phase separation of Lennard-Jones-type particles quenched from a supercritical state point into the vapor-liquid or vapor-solid co-existence parts of their phase diagrams. Calculations were performed with spherical particles interacting via 12:6, 24:12, and 36:18 interaction laws at subcritical temperatures and low-volume fractions (phi less than or equal to 0.2). Structural properties were followed as the systems evolved using pictures of the configurations, radial distribution function, and the low-angle scattering peak of the structure factor. The time dependence of the interaction energy was also followed. The scaling behavior of these quantities as a function of time was found to be similar to that observed in light scattering experiments during the phase separation of real colloidal systems. The aggregate structure that developed with time was sensitive to the range of the attractive part of the potential and its underlying phase diagram (the 36:18 system does not have a liquid phase). The 12:6 systems soon formed compact structures, whereas the systems generated using the shorter-ranged potentials persisted in a more diffuse, tenuous network for the duration of the simulations. Apart from at very short times for all potential laws, the only convincing evidence for a long-lived fractal structure was for the 36:18 systems at the lowest quench temperatures (k(T)/epsilon= 0.3, where epsilon is the depth of the potential). The local structure in the dense regions of the network was sensitive to the range of the potential, exhibiting in the vapor-solid co-existence part of the phase diagram glassylike features for the 12:6 systems and crystalline local order for the 24:12 and 38:18 states. The 12:6 systems close to the metastable region of the vapor-liquid two-phase part of the phase diagram exhibited latency in the appearance and growth of the small angle scattering peak. The 24:12 and 36:18 systems also displayed latency at the higher temperatures both in the growth of the peak height and its movement to lower scattering vectors.