  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.
  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.
  We identify two issues. Not all models can reproduce the average dust
emission per unit extinction: there are differences of up to a factor $\sim2$
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.
  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 $\sim20\%$ ($40-50\%$ in extreme cases), while ISRF intensity varies
by $\sim30\%$ ($\sim60\%$ in extreme cases). This must be accounted for in
future modelling.
