Several works have found an increase of the abundances of the s-process neutron-capture elements in the youngest Galactic stellar populations, giving important constraints to stellar and Galactic evolution. We aim to trace the abundance patterns and the time-evolution of five s-process elements in the first peak, Y and Zr, and in the second peak, Ba, La and Ce using the Gaia-ESO idr5 results. From the UVES spectra of cluster member stars, we determined the average composition of clusters with ages >0.1 Gyr. We derived statistical ages and distances of field stars, and we separated them in thin and thick disc populations. We studied the time evolution and dependence on metallicity of abundance ratios using open clusters and field stars. Using our large and homogeneous sample of open clusters, thin and thick disc stars, spanning an age range larger than 10 Gyr, we confirm an increase towards young ages of s-process abundances in the Solar neighbourhood. These trends are well defined for open clusters and stars located nearby the solar position and they may be explained by a late enrichment due to significant contribution to the production of these elements from long-living low-mass stars. At the same time, we found a strong dependence of the s-process abundance ratios with the Galactocentric distance and with the metallicity of the clusters and field stars. Our results, derived from the largest and homogeneous sample of s-process abundances in the literature, confirm the growth with decreasing stellar ages of the s-process abundances in both field and open cluster stars. At the same time, taking advantage of the abundances of open clusters located in a wide Galactocentric range, they open a new view on the dependence of the s-process evolution on the metallicity and star formation history, pointing to different behaviours at various Galactocentric distances.
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