  We study the structure and evolution of white dwarf stars with helium-rich
atmospheres (DB) in a self-consistent way with the predictions of
time-dependent element diffusion. Our treatment of diffusion includes
gravitational settling and chemical and thermal diffusion. OPAL radiative
opacities for arbitrary metallicity and carbon-and oxygen-rich compositions are
employed. Emphasis is placed on the evolution of the diffusion-modeled
double-layered chemical structure. This structure, which is characterized by a
pure helium envelope atop an intermediate remnant shell rich in helium, carbon
and oxygen, is expected for pulsating DB white dwarfs, assuming that they are
descendants of hydrogen-deficient PG1159 post-AGB stars. We find that,
depending on the stellar mass, if DB white dwarf progenitors are formed with a
helium content smaller than \approx 10^-3 M_*, a single-layered configuration
is expected to emerge during the DB pulsation instability strip. We also
explore the consequences of diffusively evolving chemical stratifications on
the adiabatic pulsational properties of our DB white dwarf models. In this
context, we find that the evolving shape of the chemical profile translates
into a distinct behaviour of the theoretical period distribution as compared
with the case in which the shape of the profile is assumed to be fixed during
the evolution across the instability strip. Finally, we extend the scope of the
calculations to the domain of the helium-rich carbon-contaminated DQ white
dwarfs. In particular, we speculate that DQ white dwarfs with low detected
carbon abundances would not be descendants of the PG1159 stars.
