1 5 M ay 2 00 1 Overcharging by macroions : above all , an entropy effect

Model macroion solutions next to a charged wall show interface true overcharging, charge reversal and inversion, and layering. Macroion layering is present, even if the wall or the macroparticle are uncharged or if the wall and macroions are like-charged. An effective long-range attractive force between the adsorbed macroions is implied. The results are obtained through an integral equation theory and a new extended Poisson-Boltzmann theory, and are in accordance with experiments on confined macroions and polymer layering. PACS: 68.08.-p, 61.20.Qg, 82.70.Dd Typeset using REVTEX 1 The restricted primitive model (RPM) for an electrolyte solution includes the two main forces in this system: the long range Coulombic and the short-range repulsive forces. In RPM the ions are taken to be hard spheres of diameter a and charge ezi (e is the protonic charge and zi is the ionic valence), embedded in a dielectric medium of dielectric constant ε. This model has been shown to be in agreement with Monte Carlo (MC) simulations and experimental results of bulk and confined electrolyte systems [1]. When a divalent electrolyte, at high concentration, is next to a charged wall, the charge of the adsorbed counterions to the wall overcome that on the wall. This effect produces a second layer of ions, where the coions outnumber the counterions. These effects are known as charge reversal (CR), some times (perhaps improperly) referred to as overcharging, and charge inversion (CI), respectively. Although these phenomena have been reported, both theoretically [2] and by computer simulations [3], since 1980, important implications to protein electrophoresis [4] and medicine [5] were later recognized. On the other hand, the long-range attraction between confined like-charged macroparticles [6] and the adsorption of macroions onto oppositely charged [7] or like-charged [8] surfaces have received much attention. The understanding of these phenomena have been recognized as relevant for the colloid science and technology [9], the oil industry, and molecular self-assembly (e.g., DNA encapsulation) and nano-structured films (e.g., polyelectrolyte layering) [5,7,8]. Here, we extend the hypernetted chain/mean spherical approximation (HNC/MSA) integral equation [10] to be applied to model macroion solutions next to a charged wall. The HNC/MSA has been proved to be in good agreement with Monte Carlo data for the electrical double layer (EDL) of closely related models [11,12]. Because of the larger macroion’s size, this theory is expected to be even more reliable than for the simple electrolyte case [2,12]. The macroparticle is taken as a charged, hard sphere of diameter aM , concentration ρM and valence zM , whereas the little ions are modeled by the RPM. The wall has uniform surface charge density σ0. The wall dielectric constant is chosen to be equal to that of the solvent, to avoid image potentials. The ionic distribution, as a function of the distance x from the surface of the wall, gives the structure of the equilibrium 2 EDL, and is expressed in terms of the concentration profiles, ρwi(x) = ρigwi(x). ρι is the bulk concentration, of the ionic species i, and gwi(x) is the species i reduced concentration profile (RCP). The HNC/MSA integral equations for the RCPs are given by [10] gwi(x) ≡ exp[−βWi(x)] = exp [ −β (eziψ(x) + Ji(x))]. Wi(x) is the potential of mean force, i.e., the effective total wall-ion interaction potential. Wi(x) has two contributions: the electrostatic potential part, given by the mean electrostatic potential,ψ(x), plus the short range repulsive potential part, due to the ionic size, given by Ji(x). Both functions are functionals of ρwi(x). β=1/(KBT), where KB is the Boltzmann constant and T is the absolute temperature. The ion-ion and the macroion-ion direct interaction potentials are given by a hard-sphere potential plus the Coulombic potential. In the limiting case of a = 0 HNC/MSA reduces to the integral equation form of a new extended inhomogeneous Poisson-Boltzmann (PB) theory [1,5,9] for point ions plus macroions, next to a charged wall. Since macroions are considered at finite concentration, this approach is an improvement to the classical PB equation for confined macroions, at infinite dilution [5,9], where only two macroions are considered: i.e., in our theory macroion-macroion correlations are included. For a=0 and ρM =0, we recover the integral equation version of the classical Gouy-Chapman (GC) theory for point-ions next to a charged wall [9]. A point-ion model (PIM) for an electrolyte solution is like the RPM, but a=0. We have solved HNC/MSA for several values of ZM , aM , ρM , σ0 and salt parameters: z+ : z−, ρι and a. We calculated gwi(x), ψ(x) and the effective charge density, σ(x) = − ∞∫ x ρel(y)dy [10,13] . The charge profile in the solution is given by ρel(x) ≡ 3 ∑ m=1 ezmρmgm(x), where we have omitted the sub-index w, for notation simplicity. x=a/2 is the distance of closest approach, to the wall, of the small ions. Hence, σ0 = − ∞ ∫ a/2 ρel(y)dy, by the electroneutrality condition, and σ(x) ≡ x ∫ a/2 ρel(y)dy is the charge induced by the wall, on the fluid, between the wall and the distance x to the wall. Hence, σ(x) ( ≡ σ(x) + σ0) is the effective or net charge (wall plus fluid) at the distance x away from the wall. σ(x) measures overcharging, CR or CI at the interface. The effective electrostatic force on an ion