Long-range correlations of polarization and number densities in dilute electrolytes
In dilute electrolytes, we calculate the pair correlation functions among the solvent polarization p, the solvent density n1, the cation density n2, and the anion density n3. We set up a simple Ginzburg-Landau free energy for these variables, so our results are valid at distances longer than the molecular size σ. In particular, we reproduce the Hoye-Stell result for the polarization correlation ⟨pα(r)pβ(0)⟩ (α, β = x, y, z) [J. S. Hoye and G. Stell, J. Chem. Phys. 68, 4145 (1978)], which is proportional to the second derivative ∂2(e-κr/r)/∂xα∂xβ for r ≫ σ with κ being the Debye wave number. We also show that size asymmetry between the cations and the anions gives rise to similar long-range correlations in ⟨pα(r)δn1(0)⟩ and ⟨δni(r)δn1(0)⟩ (i = 1, 2, 3). Moreover, we calculate the polarization time-correlation function. As a unique feature in dynamics, the longitudinal polarization fluctuations (∝∇ · p) consist of rapidly decaying and slowly decaying components, where the latter relax with the charge density ρ. As a result, the long-range part of the equal-time polarization correlation changes into a different long-ranged and long-lived form after the shorter polarization relaxation.