I ran into this problem working on differential equations that model pattern formation (reaction-diffusion equations, originally postulated by Turing in the 1950s). The equations are highly nonlinear, but some solutions can be found when solving the problem on a sphere. You get spot solutions that dynamically move essentially to the minimum energy configuration (Fekete points I believe are called). BTW, Neil Sloane, of OEIS fame, has a list of the best packings, up to n=100 I believe [0].
Things get interesting when you also allow the sphere to grow, the spots start to split (and sometimes annihilate), understanding how the spots move on the sphere is itself a very interesting problem.