Accurate stellar parameters of individual objects in binary systems are essential to constrain the effects of binarity on stellar evolution. These parameters serve as a prerequisite to probing existing and future theoretical evolutionary models. We aim to derive the atmospheric parameters of the 31 SB2s in the TMBM sample. This sample, composed of detached, semi-detached and contact systems with at least one of the components classified as an O star, is an excellent test-bed to study how binarity can impact our knowledge of the evolution of massive stars. 32 epochs of FLAMES/GIRAFFE spectra are analysed using spectral disentangling to construct the individual spectra of 62 components. We apply the CMFGEN atmosphere code to determine their stellar parameters and their He, C and N surface abundances. From these properties, we show that the effects of tides on chemical mixing are limited. Components on longer-period orbits show higher nitrogen enrichment at their surface than those on shorter-period orbits, in contrast to expectations of rotational or tidal mixing, implying that other mechanisms play a role in this process. Components filling their Roche lobe are mass donors. They exhibit higher nitrogen content at their surface and rotate more slowly than their companions. By accreting new material, their companions spin faster and are rejuvenated. Their locations in the N-vsini diagram tend to show that binary products are good candidates to populate the two groups of stars (slowly rotating, nitrogen-enriched and rapidly rotating non-enriched) that cannot be reproduced through single-star population synthesis. This sample is the largest sample of binaries to be studied in such a homogeneous way. The study of these objects gives us strong observational constraints to test theoretical binary evolutionary tracks.
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