I thought Chi gave a good post. for piling up charges, let's take an absurd example, and assume that we have an aluminum sphere in the vacuum of space and with no nearby electrical fields. Imagine the charge fairy could pluck electrons from atoms far removed and gently deposit them on the aluminums surface. By what mechanism do you imagine that the aluminum would heat. I have no point of view, just wondering what yours might be?
OTOH, movement of electrons in a confined material of arbitrary volume implies heat. They bang around to such an extent that we have both microscopic and macroscopic versions of Joule heating. Bad if you're tying to make a supercomputer the size of a bread box, good if you're arc welding.
To try to formalize problem, consider a situation where to get from point a to point b involves traversing a thin patch of aluminum. You might even call this an electrical fuse. Its not as simple as I suspect the following notions would point to:
For aluminum you have the following properties to consider--at what temp does it melt, what resistance does it offer to electrical passage, and how nuch heat energy is required to heat a gram a single degree. As current moves, it heats in relation I^2*R where I is in amps and R is resitance. Now like pipes for water, a very thin strand of aluminum will have the greatest R. Since it has little mass, it takes very little heat to gain temperature fast.
Etc, etc, Point is is you want to know how much current is required to melt or even blow a hole thru aluminum as in vaporize, depends on a lot of variables.