  We investigate the effects of element diffusion on the structure and
evolution of low-mass helium white dwarfs (WD). Attention is focused on the
occurrence of hydrogen shell flashes induced by diffusion processes during
cooling phases. Initial models from 0.406 to 0.161 solar masses are constructed
by applying mass loss rates at different stages of the RGB evolution of a solar
model. The multicomponent flow equations describing gravitational settling, and
chemical and thermal diffusion are solved and the diffusion calculations are
coupled to an evolutionary code. In addition, the same sequences are computed
but neglecting diffusion. We find that element diffusion strongly affects the
structure and cooling history of helium WD. In particular, diffusion induces
the occurrence of hydrogen shell flashes in models with masses ranging from
0.18 to 0.41 solar masses, which is in sharp contrast from the situation when
diffusion is neglected. In connection with the further evolution, these
diffusion-induced flashes lead to much thinner hydrogen envelopes, preventing
stable nuclear burning from being an appreciable energy source at advanced
stages of evolution. This implies much shorter cooling ages than in the case
when diffusion is neglected. These new WD models are discussed in light of
recent observational data of some millisecond pulsar systems with WD
companions. We find that age discrepancies between the predictions of standard
evolutionary models and such observations appear to be the result of ignoring
element diffusion in such models. Indeed, such discrepancies vanish when
account is made of diffusion.
