Get a sink full of water with a grease slick on top. Then, apply a tiny dab of detergent on your finger and then just lightly touch the surface of the water with your finger. Almost instantly, the grease in the water dissolves in a very rapidly expanding circle.
However, if you had a drop of red dye on your finger and did the same thing, it would take many minutes for it to diffuse as far.
Is there some mechanism by which detergent can travel very quickly in water? Does the grease slick pull the detergent along as it travels or something?
IIRC, soap reduces the surface tension of water. Surface tension arises because the water molecules are polarized and thus attracted to each other. This attraction is what pulls water into its classic ‘drop’ shape. Since the molecules at the surface are only attracted from below and on the sides, the surface molecules form a weak film. You can think of it like a very very flimsy balloon. When the soap dissolves in the water and the surface tension is released, the “balloon” pops. As the water molecules surrounding the soapy area are pulled away by the remaining surface tension, they carry the detergent with them.
When you release dye into the water, it is dispersed by diffusion and some convection, as the heavier dye falls through the water. This is a slower process and doesn’t have the energy of breaking surface tension bonds to propel it along.
Try putting some dye into soap and seeing how quickly it spreads.
The detergent isn’t diffusing. Its surfactant(s) comprises molecules that strongly prefer to be at a liquid/gas interface, and the first ones there get pushed rapidly outward along that interface by others also trying to join the interface.
Mixing dye and detergent, and then adding that to the water, won’t let you visualize how rapidly the detergent spreads. For the most part, the detergent molecules and the dye molecules will act independantly when they are in the water. This experiment will mostly be showing you how the dye itself behaves; at best you’ll get a little interference between the detergent and dye molecules in areas where they represent a big fraction of the total.