A problem that was encountered with the response to the Gulf oil spill was the gooey masses which washed up on the beaches. This was likely due to the fact that petroleum (crude oil) has a varied composition. The denser components are known as bitumen and tend to be rather sticky. So what probably happened is that the lighter components were dispersed leaving the gooey mess behind on the surface. One should point out that there are already dispersants in the water since it is a component of the greywater which ends up in the Mississippi.
Another problem is that diffusion is not a very efficient process. The fundamental law of diffusion is known as Fick's law which defines the diffusion coefficient. From Brownian dynamics and Einstein's dispersion relation (1908) we know that the root mean square (rms) dispersion of a number of particles is proportional to the square root of the time, t, and inversely proportional to the square root of the particle mass, M,
So diffusion would be more efficient for smaller particles of oil and molecules with lower molecular mass if γ, a drag coefficient, is relatively constant. A similar dependence of the rate on mass is known as Graham's Law*. One should also note that there are other physical processes such as agitation of the water and shear stresses which tend to disperse the particles.
*edit: Graham studied both effusion and diffusion. He showed that the square of the times of diffusion was proportional to the densities.