  We observationally investigate the relation between the photoelectric heating
efficiency in PDRs and the charge of PAHs, which are considered to play a key
role in photoelectric heating. Using PACS onboard Herschel, we observed six
PDRs spanning a wide range of FUV radiation fields (G_0=100-10^5). To measure
the photoelectric heating efficiency, we obtained the intensities of the main
cooling lines, i.e., the [OI]63um, 145um, and [CII]158um, as well as the FIR
continuum intensity. We used Spitzer/IRS spectroscopic mapping observations to
investigate the MIR PAH features in the same regions. We decomposed the MIR PAH
emission into that of neutral (PAH^0) and positively ionized (PAH^+) species to
derive the fraction of the positively charged PAHs, and compare it to the
photoelectric heating efficiency. The heating efficiency traced by
([OI]63um+[OI]145um+[CII]158um) / TIR, ranges between 0.1% and 0.9% in
different sources, and the fraction of PAH^+ relative to (PAH^0 + PAH^+) spans
from 0(+11)% to 87(+/-10)%. All positions with a high PAH^+ fraction show a low
heating efficiency, and all positions with a high heating efficiency have a low
PAH^+ fraction, supporting the scenario in which a positive grain charge
results in a decreased heating efficiency. Theoretical estimates of the
photoelectric heating efficiency show a stronger dependence on the charging
parameter gamma=G_0 T^{1/2}/n_e than the observed efficiency reported in this
study, and the discrepancy is significant at low gamma. The photoelectric
heating efficiency on PAHs, traced by ([OI]63um+[OI]145um+[CII]158um) /
(PAH+[OI]63um+[OI]145um+[CII]158um), shows a much better match between the
observations and the theoretical estimates. The good agreement of the
photoelectric heating efficiency on PAHs with a theoretical model indicates the
dominant contribution of PAHs to the photoelectric heating. (abridged for
arXiv)
