Counterions at charge-modulated substrates

– We consider counterions in the presence of a single planar surface with a spatially inhomogeneous charge distribution using Monte-Carlo simulations and strong-coupling theory. For high surface charges, multivalent counterions, or pronounced substrate charge modulation the counterions are laterally correlated with the surface charges and their density profile deviates strongly from the limit of a smeared-out substrate charge distribution, in particular exhibiting a much increased laterally averaged density at the surface. The recently revived interest in charged soft-condensed matter systems reflects that there are still many open questions, despite the enormous amount of work available in this field. One example of high experimental relevance is the discrete nature of charged surface groups, or, more generally, the inhomogeneity of substrate charge distributions, and how it affects various thermodynamic properties in an aqueous environment such as forces between charged particles or the counterion distribution. The importance of the discreteness of charged surface groups has been experimentally established in colloidal flocculation[1] and deposition studies[2] and was recently reviewed[3]. On a much larger length scale, chemically micropatterned substrates with charged and neutral patches can be used for controlled colloidal deposition[4] and DNA immobilization[5]. Theoretically, charge-modulated surfaces have been studied by various mean-field approximations[6, 7, 8, 9, 10], liquid state theory[11, 12], as well as computer simulations[13, 14, 15]. In these studies, the importance of inhomogeneous surface charge distributions has been recognized, and for a number of different charge-distribution models and parameters the ionic distribution functions as well as forces between charged surfaces have been calculated. In this paper we use a simple model for a charge-modulated surface which depends on a single geometric parameter and includes as limiting cases both smeared-out and deltapeaked charge distributions. We study the distribution of counterions without added salt at a single charged surface for all different values of the electrostatic coupling parameter (depending on temperature and counterion valence) and for different degrees of surface charge modulation, both using Monte-Carlo (MC) simulation methods and the recently introduced strong-coupling Typeset using EURO-TEX 2 EUROPHYSICS LETTERS (SC) theory[16, 17, 18]. Our study therefore encompasses different experimental situations such as mono/multivalent ions at surfaces with discrete charged chemical groups, as well as charged colloids at microscopically charge-modulated surfaces. As we will demonstrate, the assumption of smeared-out charges dramatically breaks down when the charge-modulation at the surface is pronounced, but also for moderate substratecharge modulation when the electrostatic coupling is large (i.e. for highly charged surfaces, low temperatures, or multivalent counterions): in both cases the counterions become highly correlated with the surface charges and tend to form a two-dimensional, laterally ordered layer close to the surface. A convenient measure for the effects of substrate charge modulation is the counterion contact density, i.e., the laterally averaged counter-ion density at the substrate surface, since it is known exactly in the smeared-out case and can be easily determined from simulations[16]. We find that the contact density for charge-modulated substrates can be much larger than for the smeared-out case, which is in agreement with recent experimental measurements on highly charged surfactant monolayers[19]. Quite surprisingly, substrate charge modulation tends to have a more drastic effect on the counter-ion distribution than fluctuations and correlations, which have been the subject of numerous recent studies (see [16] and references therein). As our numerical results show, the SC theory describes the counterion distributions quantitatively in the SC limit and in particular in the limit of pronounced surfacecharge modulation. We also consider a dielectric-constant jump at the substrate, as relevant for charged biological and colloidal surfaces[20, 21, 22, 23]. In contrast to the mean-field Poisson-Boltzmann (PB) approach, which severely fails even in the smeared-out case, our SC approach compares well with numerical results and helps to understand the intricate interplay of dielectric-jump and charge-modulation effects. The substrate charge distribution is modeled, for simplicity, by point charges distributed on a square lattice with lattice constant a. The point-like oppositely charged counterions are confined to the positive half-space with z > 0, the minimal distance between surface charges and counterions is given by D, as shown in Fig. 1a. The dielectric constant of the positive half-space ε> is allowed to be different from ε<. The system is globally neutral, and only the fixed surface charges and counterions are present, i.e., no salt is added. In principle, one could interpret D as the sum of the surface-ion and counterion hard-core radii. In our model, however, we use the ratio D/a more generally to control the degree of substrate-charge modulation, which allows to describe a variety of experimental situations and systems by a single parameter. The limit D/a → ∞ is equivalent to a smeared-out surface-charge distribution, while D/a → 0 corresponds to a delta-peaked distribution. The typical height of the counterion layer in the smeared-out limit defines the GouyChapman length μ = 1/(2πqlBσs), where σs = Q/a 2 is the number charge density at the wall, q and Q are the valences of counterions and surface ions, and lB = e /4πε>ε0kBT is the Bjerrum length (the distance at which two elementary charges interact with thermal energy, kBT ). In the following, we will rescale all lengths by μ according to r̃ ≡ r/μ. For the simple double layer (smeared out charges at the surface and no dielectric jump) the coupling parameter Ξ = qlB/μ = 2πq lBσs is the only parameter in the problem. In the limit Ξ → 0 (weakly charged surfaces, low-valence counterions, or high temperatures) the PB theory is asymptotically exact, while in the opposite limit Ξ → ∞ (highly charged surfaces, high-valence counterions, or low temperatures) the SC theory is exact[16]. The present model is in addition characterized by D/a, the dielectric-constant ratio ∆ = (ε> − ε<)/(ε> + ε<) and the valence ratio q/Q between counterions and surface charges. The Coulomb interaction in the presence of dielectric discontinuities is given by the solution of the Poisson equation with the appropriate boundary conditions[24]. For a single dielectric jump located at z = 0 as shown in Fig.1a, the electrostatic energy reads A.G. MOREIRA ET AL COUNTERIONS AT NANO-STRUCTURED SUBSTRATES 3