  We report on a detailed characterization of complex dielectric response of
Na-DNA aqueous solutions by means of low-frequency dielectric spectroscopy (40
Hz - 110 MHz). Results reveal two broad relaxation modes of strength
20<\Delta\epsilon_LF<100 and 5<\Delta\epsilon_HF<20, centered at 0.5
kHz<\nu_LF<70 kHz and 0.1 MHz<\nu_HF<15 MHz. The characteristic length scale of
the LF process, 50<L_LF<750nm, scales with DNA concentration as
c_DNA^{-0.29\pm0.04} and is independent of the ionic strength in the low added
salt regime. Conversely, the measured length scale of the LF process does not
vary with DNA concentration but depends on the ionic strength of the added salt
as I_s^{-1} in the high added salt regime. On the other hand, the
characteristic length scale of the HF process, 3<L_HF<50 nm, varyes with DNA
concentration as c_DNA^{-0.5} for intermediate and large DNA concentrations. At
low DNA concentrations and in the low added salt limit the characteristic
length scale of the HF process scales as c_DNA^{-0.33}. We put these results in
perspective regarding the integrity of the double stranded form of DNA at low
salt conditions as well as regarding the role of different types of counterions
in different regimes of dielectric dispersion. We argue that the free DNA
counterions are primarily active in the HF relaxation, while the condensed
counterions play a role only in the LF relaxation. We also suggest theoretical
interpretations for all these length scales in the whole regime of DNA and salt
concentrations and discuss their ramifications and limitations.
