Electrostatistics of counter-ions at and between planar charged walls: From Poisson-Boltzmann to the strong-coupling theory

The Poisson-Boltzmann (PB) approach gives asymptotically exact counter-ion density profiles around macroscopic charged objects and forces between macroscopic charged objects in the weak-coupling limit of low counter-ion valency, low surface-charge density, and high temperature. In this paper we derive, using field-theoretic methods, a theory which becomes exact in the opposite limit of strong coupling (SC). Formally, it corresponds to a standard virial expansion. Long-range divergences render the virial expansion intractable for homogeneous bulk systems, giving rise to non-analyticities in the low-density expansion of the free-energy density of electrolyte solutions. We demonstrate that for the case of inhomogeneous density distribution functions at macroscopic charged bodies these divergences are renormalizable by a systematic expansion in powers of the fugacity. For a single planar charged wall, we obtain the counter-ion density profile in the SC limit, which decays exponentially, in contrast to the PB result, which predicts algebraic decay, and in agreement with previously published numerical results. Similarly and highly charged plates in the presence of multivalent counter-ions attract each other in the SC limit and form electrostatically bound states, in contrast to the PB limit, where the interaction is always repulsive. By considering next-leading corrections to both the PB and SC theories, we estimate the range of validity for both theories.     

Electrostatic image effects for counterions between charged planar walls

We study the effect of dielectric inhomogeneities on the interaction between two planparallel charged surfaces with oppositely charged mobile charges in between. The dielectric constant between the surfaces is assumed to be different from the dielectric constant of the two semiinfinite regions bounded by the surfaces, giving rise to electrostatic image interactions. We show that on the weak-coupling level the image charge effects are generally small, making their mark only in the second-order fluctuation term. However, in the strong-coupling limit, the image effects are large and fundamental. They modify the interactions between the two surfaces in an essential way. Our calculations are particularly useful in the regime of parameters where computer simulations would be difficult and extremely time consuming due to the complicated nature of the long-range image potentials.     

Counter-ions at charged walls: Two-dimensional systems

We study equilibrium statistical mechanics of classical point counter-ions, formulated on 2D Euclidean space with logarithmic Coulomb interactions (infinite number of particles) or on the cylinder surface (finite particle numbers), in the vicinity of a single uniformly charged line (one single double layer), or between two such lines (interacting double layers). The weak-coupling Poisson-Boltzmann theory, which applies when the coupling constant G \Gamma is small, is briefly recapitulated (the coupling constant is defined as G \Gamma o \equiv b \beta e ² , where b \beta is the inverse temperature, and e the counter-ion charge). The opposite limit ( G \Gamma ? \rightarrow ∞ is treated by using a recent method based on an exact expansion around the ground-state Wigner crystal of counter-ions. These two limiting results are compared at intermediary values of the coupling constant G \Gamma = 2g \gamma (g \gamma = 1, 2, 3) , to exact results derived within a 1D lattice representation of 2D Coulomb systems in terms of anti-commuting field variables. The models (density profile, pressure) are solved exactly for any particles numbers N at G \Gamma = 2 and up to relatively large finite N at G \Gamma = 4 and 6. For the one-line geometry, the decay of the density profile at asymptotic distance from the line undergoes a fundamental change with respect to the mean-field behavior at G \Gamma = 6 . The like-charge attraction regime, possible for large G \Gamma but precluded at mean-field level, survives for G \Gamma = 4 and 6, but disappears at G \Gamma = 2 .     

Oppositely charged polyelectrolytes grafted onto planar surface: Mean-field lattice theory

Equilibrium structures of planar polyelectrolyte brushes formed by grafted chains carrying charges of opposite sign are examined by employing mean-field lattice theory. Two brushes of different architecture are considered: one formed by grafted diblock copolymers with oppositely charged blocks and the other being a mixed brush composed of oppositely charged homopolymers. The systems display nontrivial intrinsically inhomogeneous brush structures originating from the chain connectivity and the electrostatic interaction among the segments. In addition, a coexistence of stretched and coiled chains inside the brush is observed. The influence of the charges of the blocks, the relative length of the oppositely charged blocks, and the ionic strength of the solution on the brush inhomogeneity and structural differences between the two types of brushes are discussed. Received 14 March 2001 and Received in final form 18 June 2001     

Electrofriction and dynamic stern layers at planar charged surfaces

Using dynamic simulations, the electrophoretic mobility of counterions at a substrate with fixed or mobile surface charges under the action of a lateral electric field is studied. The lateral charge inhomogeneity and corrugation of the substrate is taken into account. Because of the pronounced electrofriction between counterions and surface ions, a large fraction of counterions is practically immobilized for highly charged substrates. This explains the experimentally observed saturation of the electrophoretic mobility of charged particles in the limit of high surface charge density.     

The strong-coupling spectrum of the Seiberg-Witten theory

We carefully study the global structure of the solution of the N = 2 supersymmetric pure Yang-Mills theory with gauge group SU(2) obtained by Seiberg and Witten. We exploit itsZsymmetry and describe the curve in moduli space where BPS states can become unstable, separating the strong-coupling from the weak-coupling region. This allows us to obtain the spectrum of stable BPS states in the strong-coupling region: we prove that only the two particles responsible for the singularities of the solution (the magnetic monopole and the dyon of unit electric charge) are present in this region. Our method also permits us to very easily obtain the weak-coupling spectrum, without using semiclassical methods. We discuss how the BPS states disintegrate when crossing the border from the weak- to the strong-coupling region.     

Counterions at charged cylinders: criticality and universality beyond mean-field theory

The counterion-condensation transition at charged cylinders is studied using Monte Carlo simulations. Employing logarithmically rescaled radial coordinates, large system sizes are tractable and the critical behavior is determined by a combined finite-size and finite-ion-number analysis. Critical counterion localization exponents are introduced and found to be in accord with mean-field theory both in two (2D) and three (3D) dimensions. In 3D, the heat capacity shows a universal jump at the transition, while in 2D, it consists of discrete peaks where single counterions successively condense.     

Revisiting the Poisson-Boltzmann theory: Charge surfaces, multivalent ions and inter-plate forces

The Poisson-Boltzmann (PB) theory is extensively used to gain insight on charged colloids and biological systems as well as to elucidate fundamental properties of intermolecular forces. Many works have been devoted in the past to study PB-related features and to confirm them experimentally. In this work we explore the properties of inter-plate forces in terms of different boundary conditions. We treat the cases of constant surface charge, constant surface potential and mixed boundaries. The interplay between electrostatic interactions, attractive counter-ion release, and repulsive van ’t Hoff contribution is discussed separately for each case. Finally, we discuss how the crossover between attractive and repulsive interactions for constant surface charge case is influenced by the presence of multivalent counter-ions, where it is shown that the range of the attractive interaction grows with the valency.     

Ground-state correlation energy of counterions at a charged planar wall: Gibbs-Bogoliubov lower-bound approach

Recent simulation results imply the lowering of the ground-state correlation energy per counterion at a charged planar wall, compared with that of the 2D and 3D one-component plasma systems. Our aim is to correctly evaluate the ground-state energy of strongly-coupled counterion systems by considering a quasi-2D bound state where bound counterions are confined to a layer of molecular thickness. We use a variational approach based on the Gibbs-Bogoliubov inequality for the lower-bound free energy so that the liquid-state theory can be incorporated into the formulations. The soft mean spherical approximation demonstrates that the lowered ground-state energy can be reproduced by the obtained analytical form of a quasi-2D bound state.     

The quantum mechanical planar propagator: From vector models to planar graphs

In the framework of a systematic investigation of planar field theory, we study the planar two-point Green function. We define and evaluate an expansion which preserves the formal properties of the theory and compute the three lowest lying poles of the propagator in a one-dimensional space-time, finding excellent agreement with known results.     

Influence of surface affinity of planar walls on effective interaction between the wall and colloids

Using density functional theory, we investigate the effective interaction between a big colloid immersed in a sea of small colloids and a wall which has different affinity to the small colloids. Steele 10-4-3 potential is introduced to mimic both short-range repulsive and long-range attractive interactions between the wall and the small colloids. It is found that the surface affinity of the wall has a significant influence on the effective interaction. In the short-range repulsive case, the repulsion greatly enhances the big colloid-wall effective attraction, which sensitively depends on the concentration of small colloids, and is not sensitive to the repulsive strength. In the long-range attractive case, both the concentration of small colloids and the attractive strength have great effect on the effective interaction, and with an increase of the attractive strength, a strong repulsion may be induced when the big colloid is close to the wall. In low density limit of small colloids, the present results agree well with those of the Asakura and Oosawa(AO) approximation.     

Attraction between like-charged walls: Short-ranged simulations using local molecular field theory

Effective attraction between like-charged walls mediated by counterions is studied using local molecular field (LMF) theory. Monte Carlo simulations of the "mimic system" given by LMF theory, with short-ranged "Coulomb core" interactions in an effective single particle potential incorporating a mean-field average of the long-ranged Coulomb interactions, provide a direct test of the theory, and are in excellent agreement with more complex simulations of the full Coulomb system by Moreira and Netz [Eur. Phys. J. E 8, 33 (2002)]. A simple, generally applicable criterion to determine the consistency parameter sigma(min) needed for accurate use of the LMF theory is presented.     

Relation between one-parameter problems of the theory of potential scattering of charged particles

Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 5–8, October, 1989.     

Quantum neuristor: From theory to experiment

Technical Physics - Based on modern advances in electronics and optoelectronics, experimental and theoretical works on verifying the quantum-optical model of a living neuron have been outlined for...     

Conductance of ion channels and nanopores with charged walls: a toy model

We consider ion transport through protein ion channels in lipid membranes and water-filled nanopores in silicon films. It is known that, due to the large ratio of dielectric constants of water and the surrounding material, an ion placed inside the channel faces a large electrostatic self-energy barrier. The barrier leads to an exponentially large resistance of the channel. We study reduction of the electrostatic barrier by immobile charges located on the internal walls of the channel. We show that the barrier practically vanishes already at relatively small concentration of wall charges.     

The comparison between the Mie theory and the Rayleigh approximation to calculate the EM scattering by partially charged sand

The Mie theory and Rayleigh approximation are two basic methods to study the EM scattering of uncharged spherical particle, and when the particle radius is much smaller than the incident wavelength, they are equivalent, but whether the Rayleigh approximation is still equivalent to Mie theory when we use them to calculate the EM scattering of small charged particle, there is still no any report published to discuss this problem. In this paper we make some comparisons between Mie theory and Rayleigh approximation to solve the EM scattering of partially electrification spherical particles. The results showed that the Mie theory would be more suitable to calculate the scattering of charged spherical particles.     

Correlations between two identified charged hadrons at the CERN ISR

We have measured the value of the correlation R(h₁, h₂) between two identified charged hadrons $\text{(}\text{h}\text{= π}{}^{\mathrm{\pm }}\text{,}\text{K}{}^{\mathrm{\pm }}\text{,}\text{p or}\text{p}\text{?}\text{)}$ produced in pp collisions at the CERN ISR. One hardon was produced in the forward direction and the other at a large angle. Quantum number dependent effects have been observed.