Context. Barium stars are s-process enriched giants. They owe their chemical peculiarities to a past mass transfer phase. During this phase they were polluted by their binary companion, which at the time was an asymptotic giant branch (AGB) star, but is now an extinct white dwarf. Barium stars are thus ideal targets for understanding and constraining the s-process in low- and intermediate-mass AGB stars. Aims: We derive the abundances of a large number of heavy elements in order to shed light on the conditions of operation of the neutron source responsible for the production of s-elements in the former companions of the barium stars. Methods: Adopting a recently used methodology, we analyse a sample of eighteen highly enriched barium stars observed with the high-resolution HERMES spectrograph mounted on the Mercator telescope (La Palma). We determine the stellar parameters and abundances using MARCS model atmospheres. In particular, we derive the Nb-Zr ratio which was previously shown to be a sensitive thermometer for the s-process nucleosynthesis. Indeed, in barium stars, 93Zr has fully decayed into mono-isotopic 93Nb, so Nb/Zr is a measure of the temperature-sensitive 93Zr/Zr isotopic ratio. Results: HD 28159, previously classified as K5III and initially selected to serve as a reference cool K star for our abundance analysis, turns out to be enriched in s-process elements, and as such is a new barium star. Four stars are characterised by high nitrogen abundances, and among those three have high [Nb/Zr] and [hs/ls] ratios. The derived Zr and Nb abundances provide more accurate constraints on the s-process neutron source, identified to be 13C(α, n)16O for barium stars. The comparison with stellar evolution and nucleosynthesis models shows that the investigated barium stars were polluted by a low-mass (M ˜ 2 - 3 M⊙) AGB star. HD 100503 is potentially identified as a high metallicity analogue of carbon-enhanced metal-poor star enriched in both r- and s-process elements (CEMP-rs).