  Nucleon exchange mechanism is investigated in the central collisions of
${}^{40}$Ca + ${}^{238}$U and ${}^{48}$Ca + ${}^{238}$U systems near the
quasi-fission regime in the framework of the Stochastic Mean-Field (SMF)
approach. Sufficiently below the fusion barrier, di-nuclear structure in the
collisions is maintained to a large extend. Consequently, it is possible to
describe nucleon exchange as a diffusion process familiar from deep-inelastic
collisions. Diffusion coefficients for proton and neutron exchange are
determined from the microscopic basis of the SMF approach in the semi-classical
framework. Calculations show that after a fast charge equilibration the system
drifts toward symmetry over a very long interaction time. Large dispersions of
proton and neutron distributions of the produced fragments indicate that
diffusion mechanism may help to populate heavy trans-uranium elements near the
quasi-fission regime in these collisions.
