Aims. We compare the performance of several dust models in reproducing the dust spectral energy distribution (SED) per unit extinction in the diffuse interstellar medium (ISM). We use our results to constrain the variability of the optical properties of big grains in the diffuse ISM, as published by the Planck collaboration. Methods. We use two different techniques to compare the predictions of dust models to data from the Planck HFI, IRAS, and SDSS surveys. First, we fit the far-infrared emission spectrum to recover the dust extinction and the intensity of the interstellar radiation field (ISRF). Second, we infer the ISRF intensity from the total power emitted by dust per unit extinction, and then predict the emission spectrum. In both cases, we test the ability of the models to reproduce dust emission and extinction at the same time. Results. We identify two issues. Not all models can reproduce the average dust emission per unit extinction: there are differences of up to a factor ~2 between models, and the best accord between model and observation is obtained with the more emissive grains derived from recent laboratory data on silicates and amorphous carbons. All models fail to reproduce the variations in the emission per unit extinction if the only variable parameter is the ISRF intensity: this confirms that the optical properties of dust are indeed variable in the diffuse ISM. Conclusions. Diffuse ISM observations are consistent with a scenario where both ISRF intensity and dust optical properties vary. The ratio of the far-infrared opacity to the V band extinction cross-section presents variations of the order of ~20% (40?50% in extreme cases), while ISRF intensity varies by ~30% (~60% in extreme cases). This must be accounted for in future modelling.
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