The Planck-HFI all-sky survey from 353 to 857GHz combined with the 100 microns IRAS show that the dust properties vary in the diffuse ISM at high Galactic latitude (1e19<NH<2.5e20 H/cm2). Our aim is to explain these variations with changes in the ISM properties and grain evolution. Our starting point is the latest core-mantle dust model. It consists of small aromatic-rich carbon grains, larger amorphous carbon grains with aliphatic-rich cores and aromatic-rich mantles, and amorphous silicates with Fe/FeS nano-inclusions covered by aromatic-rich carbon mantles. We explore whether variations in the radiation field or in the gas density distribution in the diffuse ISM could explain the observations. The dust properties are also varied in terms of mantle thickness, Fe/FeS inclusions, carbon abundance, and size distribution. Variations in the radiation field intensity and gas density distribution cannot explain the observed variations but radiation fields harder than the standard ISRF may participate in creating part of them. We further show that variations in the grain mantle thickness coupled with changes in the grain size distribution can reproduce most of the observations. We put a limit on the mantle thickness of the silicates (~10-15nm), and find that aromatic-rich mantles are needed for the carbon grains (at least 5-7.5nm thick). We also find that changes in the carbon abundance in the grains could explain part of the observed variations. Finally, we show that varying the composition of Fe/FeS inclusions in the silicates cannot account for the variations. With small variations in the dust properties, we are able to explain most of the variations in the dust emission observed by Planck-HFI in the diffuse ISM. We also find that the small realistic changes in the dust properties that we consider almost perfectly match the anti-correlation and scatter in the observed beta-T relation.
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