arXiv:0806.3716v3 [cond-mat.soft] 20 Jan 2009 Electrostatic interactions of charged bodies from the weak to the strong coupling regime Marius M. Hatlo and Leo Lue ∗ School of Chemical Engineering and Analytical Science The University of Manchester PO Box 88 Sackville Street Manchester M60 1QD United Kingdom (Dated: January 20, 2009) A simple field theory approach is developed to model the properties of charged, dielectric bodies and their associated counterions. This predictive theory is able to accurately describe the properties of systems (as com- pared to computer simulation data) from the weak coupling limit, where the Poisson-Boltzmann theory works well, through to the strong coupling limit. In particular, it is able to quantitatively describe the attraction between like-charged plates and the influence of image charge interactions. Electrostatic interactions play a major role in determining the structure and thermodynamics of many colloidal and bi- ological solutions, which typically contain charged macro- molecular structures with low dielectric interiors, such as DNA, charged micelles, or membranes. These charged struc- tures are always surrounded by neutralizing counterions, and, in many cases, the properties of the system can be mainly at- tributed to properties of the counterions [1]. When the electrostatic interactions are weak, their contri- butions to the system properties are accurately described by the Poisson-Boltzmann (PB) theory. However, as these inter- actions strengthen, the PB theory becomes less and less accu- rate. Perturbation methods, such as the loop expansion can be used to systematically improve the theory; however, the first loop correction offers only a small improvement [1, 2, 3], and higher-order corrections are increasingly complicated to eval- uate. When the electrostatic interactions are extremely strong (e.g., when the surface charge of the macromolecular struc- tures or the valency of the counterions is high), the Poisson- Boltzmann theory can yield qualitatively incorrect predic- tions. For example, in this regime, the counterions can gen- erate attractive forces between similarly charged objects [4]. This phenomenon cannot be explained by the PB theory, but has been observed in Monte Carlo simulations [4] and in ex- periments (e.g., condensation of DNA molecules [5], bundle formation of filamentous actin [6]. In this strong coupling regime, the counterions “collapse” on the neutralizing charged surfaces to form a highly interacting 2D structure that resem- bles a confined one-component plasma (OCP) [7, 8, 9]. In these systems, the average distance a ⊥ between the ions is much larger than the average distance z between the ions and the charged surface (i.e. z ≪ a ⊥ ). Consequently, a single par- ticle theory provides a good description of the system. This leads to the strong coupling (SC) expansion [8, 10], which has been quite successful [11, 12, 13]. However, many systems are in a regime where both the PB theory and SC expansion are inaccurate. The behavior of these systems can be rationalized in terms of a correlation hole [14] — a region of size σ around each counterion where it is unfa- vorable for other counterions to be located. At length scales greater than σ, the counterions are weakly correlated, while at shorter length scales, the counterions are strongly corre- lated but fairly “isolated” [10]. In the weak coupling regime, the counterions form a diffuse 3D layer, and the size of the correlation hole is approximately equal to the Bjerrum length l B = βq 2 /ǫ (where q is the counterion charge), the distance at which two counterions interact with energy k B T . In the SC limit, the size of the correlation hole becomes equal to the average (2D) distance between the ions a ⊥ =2μ  2l B /μ, where μ = (2πβqΣ) −1 is the Gouy-Chapman length (where Σ is the surface charge density), the distance at which the in- teraction between a counterion and the charged surface equals k B T . Based on this observation, Weeks and coworkers [15] and Santangelo [16] developed approaches that split the interac- tion between the ions at short and long range. The long- range interaction is treated within a mean field approximation, and the short-range interactions with a more precise approach (e.g., computer simulation, liquid state theory, etc.). With an appropriate value for σ, these approaches can successfully de- scribe Monte Carlo results for the full range of electrostatic coupling. However, the value of σ is determined empirically. Additionally, these approaches are not capable of describing systems with dielectric inhomogeneities. In this work, we present a self-consistent theory that is in good agreement with Monte Carlo simulations at weak, inter- mediate, and strong coupling. The theory is similar in idea to the work of Weeks and coworkers [15] and Santangelo [16], however, the parameter σ is calculated consistently from the partition function, rather than chosen empirically or adjusted to fit data. In addition, the theory accurately describes the presence of dielectric bodies, even in the SC limit, which has not been demonstrated by any previous theory. For the two plate system in the presence of image charges, the system un- dergoes a transition from from a two peak density profile to a one peak density profile. We limit our attention to systems composed of a fixed charge distribution Σ(r) that is surrounded by a neutralizing cloud of counterions, which are point charges of magnitude