The self-assembly of ink molecules deposited using dip-pen nanolithography (DPN) is modeled as a two-dimensional diffusion with a source (tip). A random walk simulation and simple analytic theory are used to study how the diffusion dynamics affects patterns generated in DPN. For a tip generating a constant flux of ink molecules, circles, lines, and letters are studied by varying the deposition rate of ink molecules and the tip scan speed. Even under the most favorable condition studied here, peripheries of patterns fluctuate from perfect circles or lines, due to the random, diffusional nature of self-assembly. The degree of fluctuation is quantified for circles and lines. Circles generated by fixing the tip position do not depend on the deposition rate if the same amount of ink is deposited. For a moving tip, patterns change drastically depending on tip speed and deposition rate. Overall, fast scan or slow deposition relative to the diffusion time scale makes lines narrower. When the tip deposits ink too slowly or scans too fast, patterns become incoherent, making molecules in patterns separated from each other. Therefore, there seems to be an optimal choice of the deposition rate and tip speed that gives both narrow and coherent patterns. We also explore the consequences of varying the relative rates of diffusion of ink molecules on bare surface and on previously deposited molecules.