What you'll need to do is edit the rim at the mesh level. The idea is to separate the parts of the rim you want to apply different materials to. This will leave you with two objects, like rim-inner and rim-outter. They you can apply black to one and white to the other.
Select the rim, goto the modify panel, select the Editable Mesh modifier, select verticie or polygon mode, select the part of the rim you want to disconnect, and hit detach button on that same modify panel (right). Give it a new object name and apply your materials.