A large number of extremely low-mass helium white dwarfs ( ELM WDs ) have been discovered in recent years .
The majority of them are found in close binary systems suggesting they are formed either through a common-envelope phase or via stable mass transfer in a low-mass X-ray binary ( LMXB ) or a cataclysmic variable ( CV ) system .
Here , we investigate the formation of these objects through the LMXB channel with emphasis on the proto-WD evolution in environments with different metallicities .
We study for the first time the combined effects of rotational mixing and element diffusion ( e.g .
gravitational settling , thermal and chemical diffusion ) on the evolution of proto-WDs and on the cooling properties of the resulting WDs .
We present state-of-the-art binary stellar evolution models computed with MESA for metallicities of Z = 0.02 , 0.01 , 0.001 and 0.0002 , producing WDs with masses between \sim 0.16 - 0.45 M _ { \odot } .
Our results confirm that element diffusion plays a significant role in the evolution of proto-WDs that experience hydrogen shell flashes .
The occurrence of these flashes produces a clear dichotomy in the cooling timescales of ELM WDs , which has important consequences e.g .
for the age determination of binary millisecond pulsars .
In addition , we confirm that the threshold mass at which this dichotomy occurs depends on metallicity .
Rotational mixing is found to counteract the effect of gravitational settling in the surface layers of young , bloated ELM proto-WDs and therefore plays a key role in determining their surface chemical abundances , i.e .
the observed presence of metals in their atmospheres .
We predict that these proto-WDs have helium-rich envelopes through a significant part of their lifetime .
This is of great importance as helium is a crucial ingredient in the driving of the \kappa - mechanism suggested for the newly observed ELM proto-WD pulsators .
However , we find that the number of hydrogen shell flashes and , as a result , the hydrogen envelope mass at the beginning of the cooling track , are not influenced significantly by rotational mixing .
In addition to being dependent on proto-WD mass and metallicity , the hydrogen envelope mass of the newly formed proto-WDs depends on whether or not the donor star experiences a temporary contraction when the H-burning shell crosses the hydrogen discontinuity left behind by the convective envelope .
The hydrogen envelope at detachment , although small compared to the total mass of the WD , contains enough angular momentum such that the spin frequency of the resulting WD on the cooling track is well above the orbital frequency .