by the looks of your screen shot you may be able to make a circle and create an emitter for it that emits from the vertices or control points (depends on what you're emitting from)

and with your balls you can just make them bounce towards the circle (maybe use the circle as a goal) and when they get there they vanish

so then the circle emitter then starts to emit spheres

so you have two seperate emitters....

it could work, although i dont know about the pattern you have drawn on there...

i bet there's another way... some way to like make your logo as a texture and have the particles collide with the surface and the texture starts to appear...

although, i currently have no idea how this would be done