Planck intermediate results. 14. Dust emission at millimetre wavelengths in the Galactic plane

We use Planck HFI data combined with ancillary radio data to study the emissivity index of the interstellar dust emission in the frequency range 10 - 353 GHz, or 3 - 0.8 mm, in the Galactic plane. We analyse the region l=20 degr - 44 degr and |b| $\backslash$leq 4 degr where the free-free emission can be estimated from radio recombination line data. We fit the spectra at each sky pixel with a modified blackbody model and two spectral indices, beta\_mm and beta\_FIR, below and above 353 GHz respectively. We find that beta\_mm is smaller than beta\_FIR and we detect a correlation between this low frequency power-law index and the dust optical depth at 353 GHz, tau\_353. beta\_mm increases from about 1.53 in the more diffuse regions of the Galactic disk, |b| = 3 degr - 4 degr and tau\_353 \~{} 5 x 10\^{}\{-5\}, to about 1.65 in the densest regions with an optical depth of more than one order of magnitude higher. We associate this correlation with an evolution of the dust emissivity related to the fraction of molecular gas along the line of sight. This translates into beta\_mm \~{} 1.53 for a medium which is mostly atomic and beta\_mm \~{} 1.65 when the medium is dominated by molecular gas. We find that both the Two-Level System model and the emission by ferromagnetic particles can explain the results if spatial variations of the component or physical processes responsible for the flattening of the dust emission are allowed. The results improve our understanding of the physics of interstellar dust, and lead towards a complete model of the dust spectrum from far-infrared to millimetre wavelengths.