Charge oscillations in Debye-Hückel theory

– The recent generalized Debye-Hückel (GDH) theory is applied to the calculation of the charge-charge correlation function, GZZ(r). The resulting expression satisfies both i) the charge neutrality condition and ii) the Stillinger-Lovett second-moment condition for all T and ρN , the overall ion density, and iii) exhibits charge oscillations for densities above a “Kirkwood line” in the (ρN , T )-plane. This corrects the normally assumed DH charge correlations, and, when combined with the GDH analysis of the density correlations, leaves the GDH theory as the only complete description of ionic correlation functions, as judged by i)–iii), iv) exact low-density (ρN , T ) variation, and v) reasonable behavior near criticality. A complete theory of ionic fluids, and in particular one which describes the ionic critical region, remains an outstanding challenge for statistical physics [1]. Recent progress has been made at the mean-field level by studies based on the Debye-Hückel (DH) theory of the restricted primitive model (RPM) of electrolytes [2], supplemented with ion pairing, free-ion depletion, and the concomitant dipolar-ionic interactions [3]. A satisfactory theory must also include a description of the ionic correlation functions. However, the conventionally assumed DH ion-ion correlation functions, of which there appear to be three varieties, have several shortcomings: the most significant is the absence of a diverging density-density correlation length at the DH critical point, which renders the theory totally uninformative as regards the relevant orderparameter fluctuations. Further, although the predicted charge-charge correlation function, GZZ(r), with the familiar screening decay as e −κDr/r, is exact in the low-density limit, it violates the Stillinger-Lovett second-moment condition [4]. Furthermore, there is much evidence indicating that the charge correlations become oscillatory at moderately high densities [5], [6], a phenomenon also missed by the simple screening form. Similar difficulties arise with the oft-used mean-spherical approximation (MSA), which exhibits, in particular, a complete cancellation of Coulombic effects from the density-density correlation function, GNN (r). In this case some improvement has been found by adding an ad c © Les Editions de Physique 612 EUROPHYSICS LETTERS hoc term to the assumed direct correlation functions and adjusting it to gain consistency with various sum rules; this generalized MSA, or GMSA, yields non-trivial density correlations, including a diverging correlation length at the MSA critical point [7]. However, it fails badly at low densities [8]-[10] and, as explained elsewhere [3], [8], [11], the MSA (and GMSA) appears to be inferior to DH-based theories for describing the critical region. Hence, we have sought to remedy the deficiencies of the DH correlation functions as well and, specifically, to do so in a more natural way. By following the spirit of the DH approximation, we extended the theory to the case of non-uniform, slowly varying ionic densities, ρ±(r), thus enabling derivation of a Helmholtz free-energy functional [8], [10]. Ion correlations may then be obtained by functional differentiation techniques. This generalized Debye-Hückel (GDH) theory was applied to the calculation of GNN (r), and provided not only the expected critical divergence of the second-moment density correlation length, ξ, but also the surprising, universal and exact divergence of ξ in the low-density limit [8], [12] (where the GMSA fails by predicting a finite, non-universal value [9]). This testament to the physical validity of the GDH approach motivated the calculation of the charge-charge correlations via GDH theory that is reported here. We find an expression for GZZ(r) ≡ 〈 [ρ+(r) − ρ−(r)] [ρ+(0) − ρ−(0)] 〉 which in the lowdensity limit approaches the conventional and exact DH result, but which also explicitly satisfies the Stillinger-Lovett second-moment condition. Furthermore, it exhibits charge oscillations for densities above a “Kirkwood line” in the density-temperature plane [7]. More concretely, we find for the RPM (equisized hard-spheres with diameters a, charges ±q0, and solvent dielectric constant D) the closed-form, Fourier transform expression for the charge-charge correlation function ĜZZ(k; ρN , T ) = ρNk /[κD + k 2 + ag0(κDa, ka)] , (1) where ρN = ρ+ + ρ− is the total ion density, while the Debye parameter is given, as usual, by κD = 4πq 2 0ρN/DkBT , and g0(x, q) = x (cos q − 1)− [2 ln(1 + x)− 2x+ x](cos q − sin q/q) . (2) Expansion in powers of k yields ĜZZ(k; ρN , T ) = (DkBT/4πq 2 0) k +O(k), which demonstrates satisfaction of both the Stillinger-Lovett second-moment condition [4], ∫ dr r GZZ(r) = −6ρN/κ 2 D , (3) and the “zeroth-moment” or charge-neutrality condition [2b], [4], ∫ drGZZ(r) = 0 ⇒ ∫ |r|>a drGZZ(r) = −ρN , (4) for all ρN and T . In the low-density limit (1) becomes ĜZZ(k) ≈ ρNk/(κD + k ), the exact, universal limiting behavior. By analyzing the poles of (1) we may obtain the predicted large-distance behavior of GZZ(r; ρN , T ): from that we find that simple exponential decay persists only up to x ≡ κDa = xK; for x > xK the decay is oscillatory. Numerically we obtain the “Kirkwood value”