Study of the heating effect contribution to the nonlinear dielectric response of a supercooled liquid

We present a detailed study of the heating effects in dielectric measurements carried out on a liquid. Such effects come from the dissipation of the electric power in the liquid and give contribution to the nonlinear third harmonics susceptibility χ(3), which depends on the frequency and temperature. This study is used to evaluate a possible "spurious" contribution to the recently measured nonlinear susceptibility of an archetypical glassforming liquid (glycerol). Those measurements have been shown to give a direct evaluation of the number of dynamically correlated molecules temperature dependence close to the glass transition temperature T(g) ≈ 190 K [Crauste-Thibierge et al., Phys. Rev. Lett. 104, 165703 (2010)]. We show that the heating contribution is totally negligible (i) below 204 K at any frequency; (ii) for any temperature at the frequency where the third harmonics response χ(3) is maximum. Besides, this heating contribution does not scale as a function of f/f(α), with f(α)(T) the relaxation frequency of the liquid. In the high frequency range, when f/f(α) ≥ 1, we find that the heating contribution is damped because the dipoles cannot follow instantaneously the temperature modulation due to the heating phenomenon. An estimate of the magnitude of this damping is given.