Dust evolution in the circumstellar disc of the unclassified B[e] star HD 50138

We studied the disc of the unclassified B[e] star HD 50138, in order to explore its structure, and to find indications for the evolutionary status of this system, whether it is a young Herbig Be or a post-main-sequence star. Using high spatial resolution interferometric measurements from MIDI instrument (N-band) on the Very Large Telescope Interferometer, we analysed the disc size, the time-variability of the disc's thermal emission, and the spectral shape of the $10~\mu$m silicate feature. By fitting simple disc models, we determined the inclination and the mid-infrared size of the disc, confirming earlier results based on a lower number of observations. We searched for mid-infrared temporal variability of different regions of the disc, and concluded that its morphology is not experiencing significant changes over the observed epochs. We characterized the mid-infrared silicate feature by determining the feature amplitude and the $11.3/9.8~\mu$m flux ratio. The latter parameter is a good indicator of the grain size. The shape of the feature suggests the presence of crystalline silicate grains in the disc. The interferometric data revealed a strong radial trend in the mineralogy: while the disc's innermost region seems to be dominated by forsterite grains, at intermediate radii both forsterite and enstatite may be present. The outer disc may predominantly contain amorphous silicate particles. A comparison of the observed spectral shape with that of a sample of intermediate-mass stars (supergiants, Herbig Ae/Be stars, unclassified B[e] stars) implied that the evolutionary state of HD 50138 cannot be unambiguously decided from mid-IR spectroscopy.