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We compare the performance of several dust models in reproducing the dustspectral energy distribution (SED) per unit extinction in the diffuseinterstellar medium (ISM). We use our results to constrain the variability ofthe optical properties of big grains in the diffuse ISM, as published by thePlanck collaboration.  We use two different techniques to compare the predictions of dust models todata from the Planck HFI, IRAS and SDSS surveys. First, we fit the far-infraredemission spectrum to recover the dust extinction and the intensity of theinterstellar radiation field (ISRF). Second, we infer the ISRF intensity fromthe total power emitted by dust per unit extinction, and then predict theemission spectrum. In both cases, we test the ability of the models toreproduce dust emission and extinction at the same time.  We identify two issues. Not all models can reproduce the average dustemission per unit extinction: there are differences of up to a factor $\sim2$between models, and the best accord between model and observation is obtainedwith the more emissive grains derived from recent laboratory data on silicatesand amorphous carbons. All models fail to reproduce the variations in theemission per unit extinction if the only variable parameter is the ISRFintensity: this confirms that the optical properties of dust are indeedvariable in the diffuse ISM.  Diffuse ISM observations are consistent with a scenario where both ISRFintensity and dust optical properties vary. The ratio of the far-infraredopacity to the $V$ band extinction cross-section presents variations of theorder of $\sim20\%$ ($40-50\%$ in extreme cases), while ISRF intensity variesby $\sim30\%$ ($\sim60\%$ in extreme cases). This must be accounted for infuture modelling.

Dust models post-Planck: constraining the far-infrared opacity of dust in the diffuse interstellar medium