  The occurrence and development of phase inhomogeneity in {\rm
Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$} single crystals have been investigated by
measuring their superconducting transition patterns using a high-sensitivity ac
magnetometer. We find that the overdoped {\rm Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$}
single crystals, even if they were single-phased immediately after a
high-temperature annealing process, show progressively pronounced multiple
superconducting transitions with time as they were kept at room temperature in
a dry air atmosphere. The results reveal that the oxygen dopants in the crystal
are still mobile at room temperature. These dopants tend to re-arrange and thus
form different phases, at a characteristic time scale of one or two weeks or
so. Moreover, the diamagnetization loops in the newly segregated phase are very
weak, presumably due to the existence of additional inhomogeneity down to a
nanometer scale. Our results also show that the optimally doped {\rm
Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$} crystals seem capable to remain in a single
phase with time.
