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.     

Simulations of counterions at charged plates

Using Monte Carlo simulations, we study the counterion distribution close to planar charged walls in two geometries: i) when only one charged wall is present and the counterions are confined to one half-space, and ii) when the counterions are confined between two equally charged walls. In both cases the surface charge is smeared out and the dielectric constant is the same everywhere. We obtain the counterion density profile and compare it with both the Poisson-Boltzmann theory (asymptotically exact in the limit of weak coupling, i.e. low surface charge, high temperature and low counterion valence) and the strong-coupling theory (valid in the opposite limit of high surface charge, low temperature and high counterion valence) and with previously calculated correction terms to both theories for different values of the coupling parameter, thereby establishing the domain of validity of the asymptotic limits. Gaussian corrections to the leading Poisson-Boltzmann behavior (obtained via a systematic loop expansion) in general perform quite poorly: At coupling strengths low enough so that the Gaussian (or one-loop) correction does describe the numerical deviations from the Poisson-Boltzmann result correctly, the leading Poisson-Boltzmann term by itself matches the data within high accuracy. This reflects the slow convergence of the loop expansion. For a single charged plane, the counterion pair correlation function indicates a behavioral change from a three-dimensional, weakly correlated counterion distribution (at low coupling) to a two-dimensional, strongly correlated counterion distribution (at high coupling), which is paralleled by the specific-heat capacity which displays a rounded hump at intermediate coupling strengths. For the case of counterions confined between two equally charged walls, we analyze the inter-wall pressure and establish the complete phase diagram, featuring attraction between the walls for large enough coupling strength and at intermediate wall separation. Depending on the thermodynamic ensemble, the phase diagram exhibits a discontinuous transition where the inter-wall distance jumps to infinity (in the absence of a chemical potential coupling to the inter-wall distance, as for charged lamellae in excess solvent) or a critical point where two coexisting states with different inter-wall distance become indistinguishable (in the presence of a chemical potential, as for charged lamellae with a finite fixed solvent fraction). The attractive pressure decays with the inter-wall distance as an inverse cube, similar to analytic predictions, although the amplitude differs by an order of magnitude from previous theoretical results. Finally, we discuss in detail our simulation methods and compare the finite-size scaling behavior of different boundary conditions (periodic, minimal image and open). Received 6 November 2001     

Enskog-Landau kinetic equation for multicomponent mixture. Analytical calculation of transport coefficients

The Enskog-Landau kinetic equation is considered to describe non-equilibrium processes of a mixture of charged hard spheres. This equation has been obtained in our previous papers by means of the non-equilibrium statistical operator method. The normal solution of this kinetic equation found in the first approximation using the standard Chapman-Enskog method is given. On the basis of the found solution the flows and transport coefficients have been calculated. All transport coefficients for multicomponent mixture of spherical Coulomb particles are presented analytically for the first time. Numerical calculations of thermal conductivity and thermal diffusion coefficient are performed for some specific mixtures of noble gases of high density. We compare the calculations with those ones for point-like neutral and charged particles. Received 10 June 1999 and Received in final form 15 October 1999     

Corrections to the wave function and the hyperfine structure in exotic atoms

We present an analytical evaluation of radiative corrections in exotic atoms induced by the one-loop electronic vacuum polarization. We evaluate corrections to the energy levels, to the wave function (at the origin) and to the hyperfine structure. We treat all corrections analytically within a non-relativistic approximation. Agreement is found with a few available numerical results. The analytical treatment allows to determine the asymptotic forms of the corrections in the limit of a small atomic radius, which for the atomic systems considered corresponds to a large mass of the constituent particle as compared to the electron mass. The asymptotics can be verified using the effective charge approach. Received: 28 January 1998 / Accepted: 13 March 1998     

An efficient renormalization group improved implementation of the MSSM effective potential

In the context of MSSM, a novel improving procedure based on the renormalization group equation is applied to the effective potential in the Higgs sector. We focus on the one-loop radiative corrections computed in Landau gauge by using the mass independent renormalization scheme . Thanks to the decoupling theorem, the well-known multimass scale problem is circumvented by switching to a new effective field theory every time a new particle threshold is encountered. We find that, for any field configuration, there is a convenient renormalization scale at which the loop expansion respects the perturbation series hierarchy and the theory retains the vital property of stability. Received: 31 August 1999 / Revised version: 22 March 2000 / Published online: 23 October 2000     

Surfaces with quenched and annealed disordered charge distributions

We consider surfaces with disordered charge distribution. The disorder can be caused by mobile charges, as for example in mixed lipid bilayers, or by weakly charged surfaces where charge regulation takes place (e.g. carboxyl groups). Using Monte-Carlo simulation methods we find for quenched as well as annealed disordered charge distributions counterion densities close to the surface that are significantly larger than for ordered regularly spaced surface ions. Our field-theoretic results agree well with results obtained from Monte-Carlo simulations of the system. Furthermore, we obtain expressions for the effective interaction between charged colloids and charged rods close to a charged surface and discuss the effect of the surface-ion mobility and polarization charges on the interaction. In general, polarization effects as well as surface-ion mobility lead to a weakening of the effective interaction between charged objects.     

Static van der Waals interactions in electrolytes

Using a field-theoretic formalism, we calculate the static contribution to the van der Waals interaction between two dielectric semi-infinite half-spaces in the presence of mobile salt ions. The ions can be located in the slab, in one, or in both half-spaces. We include an excess polarizability of the salt ions, i.e., each (spherical) ion has a dielectric constant which in general is different from the surrounding medium. This leads to a modification of the effective dielectric constant of the medium hosting the ions. This shift can be large for high salt concentrations and therefore influences the Hamaker constant decisively. Salt ions in the slab screen the static van der Waals interaction, as was shown by Davies and Ninham. The salt-modified van der Waals interaction also contains salt-confinement and salt-correlation effects. This is clearly demonstrated by the fact that the interaction is non-zero even in the case when the dielectric constant is homogeneous throughout the system, in which case salt correlations are solely responsible for the interaction. If the salt ions are in one or both of the two half-spaces (and no ions in the slab), the van der Waals interaction is not screened but the effective Hamaker constant approaches a universal value for large slab thickness which is different from the value in the absence of salt ions and which is independent of the salt concentration and of the effective electrolyte dielectric constant. If both half-spaces contain salt, the asymptotic value of the Hamaker constant for large separation between the half-spaces is the one obtained for the interaction between two metallic half-spaces through an arbitrary dielectric medium, which is given by A≃ - 1.20. As is explicitly demonstrated, ion enrichment or depletion at the interfaces due to image-charge effects is already included on the one-loop level and therefore does not lead to a change of the screened van der Waals interaction as given by Davies and Ninham. For positive and negative ions with different valencies or different excess polarizabilities, we obtain different adsorbed surface excesses of positive and negative ions, leading to a non-vanishing surface potential, which is computed explicitly. All of these results are obtained on the linear one-loop level. For a special case we extend the calculation of the dispersion interaction to the two-loop level. We find the corrections to the one-loop results to be quite large for high salt concentrations or multivalent ions. Received 17 February 2000     

Quantum corrections to the energy density of a homogeneous Bose gas

Quantum corrections to the properties of a homogeneous interacting Bose gas at zero temperature can be calculated as a low-density expansion in powers of , where is the number density and a is the S-wave scattering length. We calculate the ground state energy density to second order in . The coefficient of the correction has a logarithmic term that was calculated in 1959. We present the first calculation of the constant under the logarithm. The constant depends not only on a, but also on an extra parameter that describes the low energy scattering of the bosons. In the case of alkali atoms, we argue that the second order quantum correction is dominated by the logarithmic term, where the argument of the logarithm is ,and is the length scale set by the van der Waals potential. Received 2 February 1999     

Asymmetry of Stark profiles

The theoretical basis is presented that allows to compute the Stark broadened line shapes of atomic ions up to the quadrupole terms in the interaction potential between the radiator and the plasma electric microfields and their gradients. The nature of the corrections due to the plasma polarization effects associated with the electron distribution around ion perturbers are carefully analyzed. The relevant universal plasma functions are evaluated in a cluster expansion or by Monte Carlo simulations, and the line shape is calculated with ion dynamic effects by the Model Microfield Method. The asymmetry of the Lyman line of hydrogenic ions is then studied. Received 21 January 2000 and Received in final form 27 April 2000     

Charge distribution on annealed polyelectrolytes

We investigate the equilibrium charge distribution along a single annealed polyelectrolyte chain under different conditions. The coupling between the conformation of the chain and the local charge distribution is described for various solvent qualities and salt concentration. In salt free solution, we find a slight charge depletion in the central part of the chain: the charges accumulate at the ends. The effect is less important if salt is added to the solution since the charge inhomogeneity is localized close to the chain ends over a distance of order of the Debye length. In the case of poor solvent conditions we find a different charge per monomer in the beads and in the strings in the framework of the necklace model. This inhomogeneity leads to a charge instability and a first order transition between spherical globules and elongated chains. Received 19 March 1999 and Received in final form 2 August 1999     

A correlation-hole approach to the electric double layer with counter-ions only

ABSTRACT

We study a classical system of identically charged counter-ions near a planar wall carrying a uniform surface charge density. The equilibrium statistical mechanics of the system depends on a single dimensionless coupling parameter. A new self-consistent theory of the correlation-hole type is proposed which leads to a modified Poisson–Boltzmann integral equation for the density profile, convenient for analytical progress and straightforward to solve numerically. The exact density profiles are recovered in the limits of weak and strong couplings. In contrast to previous theoretical attempts of the test-charge family, the density profiles fulfil the contact-value theorem at all values of the coupling constant and exhibit the mean-field decay at asymptotically large distances from the wall, as expected. We furthermore show that the density corrections at large couplings exhibit the proper dependence on coupling parameter and distance to the charged wall. The numerical results for intermediate values of the coupling provide accurate density profiles which are in good agreement with those obtained by Monte Carlo simulations. The crossover to mean-field behaviour at large distance is studied in detail.     

Variational charge renormalization in charged systems

We apply general variational techniques to the problem of the counterion distribution around highly charged objects where strong condensation of counterions takes place. Within a field-theoretic formulation using a fluctuating electrostatic potential, the concept of surface-charge renormalization is recovered within a simple one-parameter variational procedure. As a test, we reproduce the Poisson-Boltzmann surface potential for a single-charged planar surface both in the weak-charge and strong-charge regime. We then apply our techniques to non-planar geometries where closed-form solutions of the non-linear Poisson-Boltzmann equation are not available. In the cylindrical case, the Manning charge renormalization result is obtained in the limit of vanishing salt concentration. However, for intermediate salt concentrations a slow crossover to the non-charged-renormalized regime (at high salt) is found with a quasi-power-law behavior which helps to understand conflicting experimental and theoretical results for the electrostatic persistence length of polyelectrolytes. In the spherical geometry charge renormalization is only found at intermediate salt concentrations, in agreement with previous numerical results.Received: 16 December 2002, Published online: 22 July 2003PACS: 82.70.-y Disperse systems; complex fluids - 61.20.Ja Computer simulation of liquid structure - 61.20.Qg Structure of associated liquids: electrolytes, molten salts, etc.     

A model for anomalous directed percolation

We introduce a model for the spreading of epidemics by long-range infections and investigate the critical behaviour at the spreading transition. The model generalizes directed bond percolation and is characterized by a probability distribution for long-range infections which decays in d spatial dimensions as . Extensive numerical simulations are performed in order to determine the density exponent and the correlation length exponents and for various values of . We observe that these exponents vary continuously with , in agreement with recent field-theoretic predictions. We also study a model for pairwise annihilation of particles with algebraically distributed long-range interactions. Received: 4 September 1998 / Accepted: 22 September 1998     

Restoring entanglement in atomic collisions: A gedanken experiment

A new type of collision experiments is discussed, where observations of two successive collisions of the same pair of particles would be possible. When such technology is available, a surprising restoring of entanglement, normally considered broken in usual collision experiments, could be observed. As an illustration the collision partners He⁺ and He⁺⁺ in a collision regime where the resonant charge transfer is dominating are considered. In the analysis it is shown that in such experiments, two spatially widely separated ion paths, corresponding in fact to two different charge states, would contribute coherently to the final amplitudes, describing which of the ions emerges as singly charged, i.e. which carries the single electron involved. The double collision experiments are not trivial, since their overall cross-sections are extremely small. Development of relevant experimental techniques will decide if the proposed phenomena remain in the field of gedanken experiments or enter the world of real experimental physics. Received 2 December 1999 and Received in final form 12 May 2000     

Application of B-splines finite basis sets to relativistic two-photon decay rates of level in hydrogenic ions

A theoretical study of the one- and two-photon spontaneous emission rates from the 2 s1/2 state of one-electron ions is presented. High-precision values of the relativistic emission rates for ions with nuclear charge Z up to 100 are obtained through the use of finite basis sets for the Dirac equation constructed from B-splines. Furthermore, we analyze the influence of the inclusion of quantum electrodynamics corrections in the initial and final state energies. Received: 6 January 1998 / Accepted: 31 March 1998     

Non-Markovian relaxation in an open quantum system

Non-Markovian dynamics in open quantum systems is characterized by a time-non-locality in the equation of motion valid for the reduced density operator. An expansion of this density matrix equation with respect to Laguerre polynomials is used to tackle the time-non-locality. The applicability and the numerical limitations of the method are discussed in detail. In order to illuminate the characteristics of non-Markovian dynamics the reference example is studied of a single quantum degree of freedom moving in a harmonic potential and being embedded in a heat bath. If interpreted as the photoinduced dynamics of nuclear motion in polyatomic molecules we can suggest two clear signatures of non-Markovian dynamics observable in ultrafast optical experiments, firstly a pronounced and somewhat irregular oscillatory behavior of the vibrational level populations, and secondly a separation of the vibrational wavepacket into a double-structure. Received 12 April 2000 and Received in final form 2 September 2000     

One-component-plasma: Going beyond Debye-Hückel

Using field-theoretic methods, we calculate the internal energy for the One-Component Plasma (OCP). We go beyond the recent calculation by Brilliantov [N. Brilliantov, Contrib. Plasma Phys. 38, 489 (1998)] by including non-Gaussian terms. We show that, for the whole range of the plasma parameter , the effect of the higher-order terms is small and that the final result is not improved relative to the Gaussian theory when compared to simulations. Received 12 April 1999     

Kosterlitz-Thouless vs. Ginzburg-Landau description of 2D superconducting fluctuations

We evaluate the charge and spin susceptibilities of the 2D attractive Hubbard model and we compare our results with Monte Carlo simulations on the same model. We discuss the possibility to include topological Kosterlitz-Thouless superconducting fluctuations in a standard perturbative approach substituting in the fluctuation propagator the Ginzburg-Landau correlation length with the Kosterlitz-Thouless correlation length. Received 30 June 1999     

Energy distribution of ions produced by excimer-laser ablation of solid and molten targets

2 . The measurements reveal components with different charge-to-mass ratio and distinct components with the same charge/mass ratio. The most probable kinetic energy has values of several tens of eV for singly charged ions, and is larger by a factor exceeding 2 for doubly charged ions. The role of target material and state, solid or liquid, laser photon energy and fluence has been investigated and is discussed in comparison to the findings of previous investigations. An estimate of the electrostatic plasma potential in PLA, based on electron loss rate arguments, is presented to account for the high ion energies observed. Received: 9 March 1998 / Accepted: 27 November 1998 / Published online: 24 February 1999     

Interface dynamics in liquid crystals

We have experimentally observed the pattern instabilities of an Ising wall formed in a nematic or cholesteric liquid crystal layer. We have deduced an envelope equation, relevant close to the Fréedericksz transition, from which we derived an equation for the dynamics of the interface in the vicinity of its bifurcation. In the case of the zig-zag instability, this model is characterized by a conservative and variational order parameter whose gradient satisfies a Cahn-Hilliard equation. We have also investigated the influence of slightly broken symmetries on the dynamical behaviour of the system. The disappearance of the interface translational invariance or of the reflection symmetry along the wall axis may induce new interfacial patterns which have been both experimentally and theoretically pointed out. Received 5 August 1999 and Received in final form 13 September 1999