We develop a model of dust evolution in a multiphase , inhomogeneous ISM including dust growth and destruction processes .
The physical conditions for grain evolution are taken from hydrodynamical simulations of giant molecular clouds in a Milky Way-like spiral galaxy .
We improve the treatment of dust growth by accretion in the ISM to investigate the role of the temperature-dependent sticking coefficient and ion-grain interactions .
From detailed observational data on the gas-phase Si abundances { [ Si _ { gas } / H } ] measured in the local Galaxy , we derive a relation between the average { [ Si _ { gas } / H } ] and the local gas density n ( H ) which we use as a critical constraint for the models .
This relation requires a sticking coefficient that decreases with the gas temperature .
The relation predicted by the models reproduces the slope of -0.5 of the observed relation in cold clouds .
It is steeper than that for the warm medium and is explained by the dust growth .
We find that growth occurs in the cold medium for all adopted values of the minimum grain size a _ { \min } from 1 to 5 nm .
For the classical cut-off of a _ { \min } = 5 nm , the Coulomb repulsion results in slower accretion and higher { [ Si _ { gas } / H } ] than the observed values .
For a _ { \min } \lesssim 3 nm , the Coulomb interactions enhance the growth rate , steepen the slope of { [ Si _ { gas } / H } ] – n ( H ) relation and provide a better match to observations .
The rates of dust re-formation in the ISM by far exceed the rates of dust production by stellar sources .
After the initial 140 Myr , the cycle of matter in and out of dust reaches a steady state , in which the dust growth balances the destruction on a similar timescale of 350 Myr .