Charge inversion on membranes induced by multivalent-counterion fluctuations

Based on the two state model, we study the condensation of counterions on oppositely charged membranes in the presence of monovalent salts. Using the Gaussian approximation, we evaluate the contribution of two-dimensional charge fluctuation to the free energy, from which the number of condensed counterions is determined self-consistently. It is shown that charge inversion can occur upon the addition of divalent ions of millimolar range, or of trivalent ions of micromolar range. The driving force for the overcharging mechanism is the charge correlation induced by thermal fluctuation, in which the multivalency of the counterions plays a crucial role. (Some figures in this article are in colour only in the electronic version) Counterions in watery solution dominantly distribute near an oppositely charged surface of macroions, forming a diffuse electric double layer. The structure of a double layer of counterions, which has been usually described by the mean-field Poisson–Boltzmann theory, is of great importance in interaction and phase behaviours of macroions [1, 2]. When the macroions are subject to an external electric field, they instantaneously reach a stationary velocity due to the viscous friction from the ambient solution. This is called electrophoresis, and it provides an efficient way to probe the double layer structure and thus to measure the surface potential of charged colloidal systems [3, 4]. The counterions strongly bound to the macroion play an important role in this transport property by renormalizing the electrokinetic charges of the surface [5]. According to the Poisson–Boltzmann theory, the macroion charge is only partially compensated (neutralized) by the counterions so that the sign of the net charge is retained even in the presence of the strongly associated counterions. It has been however observed that the counterions can further condense on the surface beyond compensation of the macroion charge, leading to reversal of the drift direction in electrophoretic experiments [6]. This phenomenon is called charge inversion, and has been 0953-8984/05/312943+07$30.00 © 2005 IOP Publishing Ltd Printed in the UK S2943 S2944 Y W Kim and W Sung Figure 1. Snapshot for the distribution of multivalent cations near a negatively charged planar surface. For clarification, the monovalent electrolytes that screen the Coulomb interaction among the counterions are not shown. According to the two state model, the counterions are assumed to be either in the bound state (dark sphere), or in the free state (less dark sphere). appealing for many practical applications such as gene delivery through a similarly charged cell membrane. Therefore, significant theoretical effort has been devoted to study this phenomenon over the last few years [7]. Nguyen et al [8] have recently explained the charge inversion with strong structural correlations based on the Wigner crystal picture [9], which is legitimate for a zero temperature limit. The charge inversion, along with the attraction between similarly charged molecules [10, 11], constitutes the apparent evidences for the important role of charge correlations in biological systems. In this work we investigate the charge inversion from a different point of view from the previous study [8]: based on the Gaussian approximation, thermally induced charge fluctuations are explicitly taken into account, which would be relevant for a finite temperature regime. For this purpose, we consider a charged planar surface or membrane with negative surface charge density −σ0, when both multivalent ions and monovalent salts such as NaCl are present in solution. After integrating out the degrees of freedom for all monovalent salts in the bulk, the electrostatic interactions between the remaining (bound) charges are screened. On further integrating out the surface bound charges including the condensed counterions as slow degrees of freedom, we obtain the free energy arising from two-dimensional charge fluctuations, beyond the mean-field theory. The number of condensed counterions is determined self-consistently by considering the phase equilibria between counterions on the surface and in the bulk. It is shown that the effects of correlation and thermal fluctuation can lead to charge inversion, for which the presence of multivalent ions is essential, in consistency with the previous study [8]. Here we employ the two state model (figure 1) in which counterions are divided into two classes, namely, condensed and non-condensed (free) ions [12]: some of counterions can bind to an oppositely charged surface, in spite of entropic penalty, due to the electrostatic attraction being greater than the thermal energy. On the other hand, counterions tend to remain in the bulk to gain entropy. Considering that the ions in solution are either monovalent or multivalent with valency Z , the effective surface charge density of the membrane is reduced due to counterions condensed within the diffusive layer of a finite thickness as