An explicit relationship between the dielectric anisotropy and the average electric field gradient in nematic solvents

Nowadays, the interaction between the electric quadrupole moment of a probe-solute and the so-called average electric field gradient (EFG) of the nematic medium is commonly suggested as an important long-range mechanism contributing to the ordering of small molecules dissolved in nematic solvents. Anyway, an explicit relationship between the solvent's EFG and some macroscopic property of the medium has never been established explicitly. In this work, a derivation is carried out leading to a simple formula that shows that the EFG of a nematic solvent is directly related to the dielectric permittivity of the medium (in particular, to its dielectric anisotropy, Delta epsilon) and to the quadrupole moment of the solute. The obtained expression (a) allows for immediate considerations about the sign and the value of EFG (and could be very useful in designing proper nematic mixtures with null EFG) and (b) reconciles two existing conceptual ways of describing long-range orientational interactions, revealing they are only seemingly different.