  We calculate the conductance variation of several metallic carbon nanotubes
as their end is being dipped into a liquid metal electrode, where experiments
have shown an achievable conductance close to 1 quantum of conductance. The
calculated conductance for a (40,40) nanotube indicates that the current flows
almost entirely through the pi mode, and not the pi* mode. The calculation also
predicts that for narrower nanotubes (~1 nm in diameter), in a 'weak coupling'
regime, the saturation of both pi and pi* modes should be observable. An
experiment is proposed to verify this point.
