Near critical electrolytes: are the charge-charge sum rules obeyed?

In an electrolyte solution the charge-charge structure factor obeys S(ZZ)(k;T,ρ)=0+ξ(Z,1)(2)k(2)-ξ(Z,2)(4)k(4)+···, where ξ(Z, 1) and ξ(Z, 2) are the second- and fourth-moment charge-charge correlation lengths depending on the temperature T and the overall ion density ρ. The vanishing of the leading term, the first Stillinger-Lovett (SL) sum rule, simply reflects bulk electroneutrality. The second SL rule, or second-moment condition, dictates that ξ(Z, 1) = ξ(D), where the Debye screening length ξ(D) is proportional to √(T/ρ). In this paper we present results from grand canonical Monte Carlo simulations of a fully size and charge symmetric 1:1 (finely-discretized) hard-sphere electrolyte, or restricted primitive model. By design, electroneutrality is imposed during the simulations, so satisfying the first sum rule automatically. However, careful finite-size scaling analyses of extensive histogram reweighted data indicate that the second-moment condition is violated at criticality, ξ(Z,1)(c) exceeding ξ(D)(c) by approximately 8%. It is also found that ξ(Z,2)(4) diverges to +∞ as T → T(c) in a manner closely mirroring the density-density fluctuations, S(NN)(0). These findings contradict generalized Debye-Hückel theory and also the exactly soluble charge-symmetric spherical models, both of which support the second-moment condition at criticality and the finiteness of the fourth-moment. Nevertheless, the observed behavior is strikingly similar to that of the charge-asymmetric spherical models.