Interaction between macroions mediated by divalent rod-like ions

– Attractive interactions between identical like-charged macroions in aqueous multivalent salt solution arise due to ion-ion correlations. The mean-field level Poisson-Boltzmann (PB) theory does not predict such behavior for point-like structureless ions. Various multivalent ions, such as certain DNA condensing agents or short stiff polyelectrolytes, do have an internal, often rod-like, structure. Applying PB theory to the generic case of divalent rod-like salt ions, we find attraction between like-charged macroions above a critical distance between the two individual charges of the rod-like ions. We calculate this distance analytically within linearized PB theory. Numerical results for the non-linear PB theory indicate strong enhancement of the tendency to mediate attractive interactions. Introduction. – Poisson-Boltzmann (PB) theory is a widely used mean-field level method to calculate interactions between macroions in aqueous (salty) solutions. For monovalent salt, its predictions are generally found to agree well with experimental results and computer simulations. However, the presence of multivalent ions can affect the nature of the interactions between macroions in a way that qualitatively differs from the PB prediction. A remarkable example is the possibility of attraction between two identical, like-charged, macroions that the mean-field approach is unable to predict. This attraction currently receives much interest [1] because it is observed for a number of biologically relevant processes such as condensation of DNA [2], network formation in actin solutions [3], virus aggregation [4] and interactions between lipid membranes that occur during adhesion and fusion. Various theoretical approaches ascribe this attraction to the presence of ion-ion correlations [5,6]. An intuitive understanding of these correlations can be based on the formation of a periodic counterion arrangement in the vicinity of each macroion, similar to a Wigner crystal. The two inter-locked counteriondecorated macroions then experience short-ranged attraction. c © EDP Sciences K. Bohinc et al.: Interaction between macroions 495 Multivalent ions are commonly treated as point charges. However, real ions (and particularly organic ones) often possess an internal structure with the individual charges being located at distinct, well-separated positions. Among others [7], a characteristic example is the rod-like backbone structure of various DNA condensing agents such as the triand tetravalent ions spermine and spermidine, protamine sulfate, or poly-lysine. Clearly, the spatial separation of the individual charges within a multivalent ion is expected to affect the role that correlations play for the energetics of interacting macroions. That is, large separation is expected to reduce the importance of correlations between different multivalent ions (“inter-ionic” correlations) but retains the steric constraints between the individual charges of the multivalent ions (“intra-ionic” correlations). As is well known, PB theory entirely neglects inter-ionic correlations. Yet, intra-ionic correlations can be accounted for within PB theory. Hence, with growing separation between the individual charges of a multivalent ion, PB theory is expected to become increasingly more applicable. In line with this, a recent simulation study [8] on a mixture of monovalent and rigid dumbell-like ions observed a substantial decrease in the critical temperature compared to an ordinary 1 : 2 electrolyte of point-like ions. The questions arise, how to apply PB theory [9] and if it is able to predict attraction (or at least an attractive component) for the interaction of like-charged macroions. We note that PB theory offers a particularly simple tool to model the interaction between macroions of low dielectric constant immersed in aqueous salt solution. Such systems are not captured by the so-called primitive model where —nevertheless— simulations as well as analytical model calculations have provided valuable information on the structure and phase behavior of electrolyte solutions [10], including their role for biological systems [11]. Obtaining attractive interactions between likecharged macroions is also possible on the level of PB theory if additional order parameters are considered. This is the case, for instance, upon including the solvent structure [12] or the presence of polyelectrolytes [13,14] into PB theory. In the present work we formulate PB theory for an electrolyte with an internal structure of the individual ions. For simplicity, we shall focus on the simplest, namely that of a symmetric 2 : 2 electrolyte in which the two charges of each (positive and negative) divalent ion are separated by a distance l. Figure 1 schematically illustrates the rod-like structure of the divalent ions. More involved cases such as asymmetric electrolytes with higher than divalent ions and with any distribution of the individual charges on the rod-like ion can be treated analogously (but are not expected to give rise to qualitatively new behavior). We shall derive a modified PB equation that takes into account the charge distribution of the rod-like divalent Fig. 1 – Schematic illustration of two like-charged macroions of (bare) surface charge density σ0 and overall surface area A, interacting in a symmetric 2 : 2 electrolyte solution enclosed in a volume V . The divalent ions of the electrolyte are rod-like, with a separation l between their individual charges. 496 EUROPHYSICS LETTERS ions up the quadrupolar order. For the generic case of two interacting like-charged planar surfaces we show that the modified PB equation can give rise to an attractive interaction. Single rod-like ion in external field. – Consider first a single rod-like divalent ion, located at fixed position r in the external electric field E(r). The two individual charges of the ion are localized at positions r1,2 = r± (l/2) t, where the unit vector t = {sin θ cosφ, sin θ sinφ, cos θ} describes the instantaneous spatial orientation of the ion. The ion’s orientation-dependent interaction energy, uel(t) = e[Φ(r1) + Φ(r2)] with the electric field E(r) = −∇Φ(r), can be written up to quadratic order in l as uel(t) = 2eΦ(r) + el 4 t · [∇ ◦∇Φ(r)] · t, (1) where e denotes the elementary charge and Φ is the electrostatic potential. Statistical averaging over all possible orientations yields the free energy up to quadratic order in l: fel = − ln [ 1 4π ∫ 2π