Shape of partly dissolved cube Let's say you have a cube of some material, and you put it in a liquid that can dissolve that material. After a while you take out what is left of the cube. What shape will you have?
I imagine it will be like a cube with rounded corners. At the corners, there is more liquid around, so it will dissolve faster there. But what is the exact shape?
I am mostly interested in the mathematics used to define how shapes change this way. The cube is just an example.
 A: The answer depends on the exact nature of the chemistry occurring at the cube's surface, as follows.
For an etchant that acts uniformly, any projections sticking out of the surface will be eaten off first because a projection can be attacked from two sides at once. This will turn the cube into a sphere as time goes by. This etchant is called a leveling etch because it acts to smooth off the surfaces it attacks.
Now as an etch dissolves away a surface, there will be a natural tendency for the etch rate to depend on the availability of fresh etchant molecules which must diffuse up to the surface through the bulk liquid to attack that surface. By putting other molecules into the etch mix which interfere with diffusion, it is possible to magnify the levelling tendency of the etch to the point where the only thing that gets etched are sharp asperities sticking out from the surface that protrude deep into the liquid- and then you have a polishing etch that produces a mirror-like finish.
Alternatively you can imagine an etch which is inhibited by for example the presence of oxygen in solution. If you now have a crevice in the surface, the etch will be accelerated in the crevice because it's difficult for the oxygen to diffuse into the crevice- and you get an etch that preferentially roughens the surface instead of smoothing it. Artful chemistry then allows you to concoct an etch that actively attacks crevices- and you then get what's called an undercutting etch.
Undercutting etches can also be devised which create electrochemical concentration cells in which the depletion of a reactant component in solution causes the depleted zone to  have a different electronegativity relative to the (smooth) region surrounding the crevice, causing a huge magnification of the corrosion rate at the bottom of the crevice.
So- without knowing what the chemistry of the corrosion process is, you can't predict what shape the cube will evolve into as it gets corroded.
