We present models to predict high frequency counts of extragalactic radio sources using physically grounded recipes to describe the complex spectral behaviour of blazars, that dominate the mm-wave counts at bright flux densities. We show that simple power-law spectra are ruled out by high-frequency (nu>100 GHz) data. These data also strongly constrain models featuring the spectral breaks predicted by classical physical models for the synchrotron emission produced in jets of blazars (Blandford & Konigl 1979; Konigl 1981). A model dealing with blazars as a single population is, at best, only marginally consistent with data coming from current surveys at high radio frequencies. Our most successful model assumes different distributions of break frequencies, nu_M, for BL Lacs and Flat-Spectrum Radio Quasars (FSRQs). The former objects have substantially higher values of nu_M, implying that the synchrotron emission comes from more compact regions; therefore, a substantial increase of the BL Lac fraction at high radio frequencies and at bright flux densities is predicted. Remarkably, our best model is able to give a very good fit to all the observed data on number counts and on distributions of spectral indices of extragalactic radio sources at frequencies above 5 and up to 220 GHz. Predictions for the forthcoming sub-mm blazar counts from Planck, at the highest HFI frequencies, and from Herschel surveys are also presented.
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