  The emission of cold dust grains at long wavelengths will soon be observed by
the Planck and Herschel satellites and provide new constraints on the nature of
interstellar dust. The microwave anomalous emission, proposed to be due to
spinning PAHs, should help to better define these species. Moreover,
understanding the fluctuations of the anomalous emission over the sky is
crucial for CMB studies. We focus on the long wavelength emission of
interstellar PAHs in their rovibrational and rotational transitions. The PAH
emission spectrum from the IR to the microwave range is presented and compared
to anomalous emission observations. To model their long wavelength emission, we
treat PAHs as isolated systems and follow consistently their IR and rotational
emissions. We consider several interstellar phases and discuss how the
anomalous emission may constrain their size distribution. Our model of PAH
emission accounts for the mid-IR spectra of the diffuse interstellar medium and
of the Orion Bar. For lambda<3mm the PAH IR emission does not scale any more
with the radiation field intensity (Go) unlike the mid-IR part of the spectrum.
This emission represents less than 10% of the total dust emission at 100 GHz.
We find the broadband emissivity of spinning PAHs per carbon atom to be rather
constant for Go<100 and nH<100cm-3. Observations of anomalous emission in the
Perseus molecular cloud are explained by our model of spinning PAH emission
with a standard abundance, supporting PAHs as the origin of the anomalous
emission. The behaviour of the spinning dust emissivity with Go provides a
clear test that can be tested against observations of anomalous and dust mid-IR
emissions. Comparison of these emissions will provide constraints on the size
and/or electric dipole moment of interstellar PAHs.
