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L. Samaj

Publications and source records attributed to L. Samaj.

At least 19 recordsLinked to original sources
arXiv2025

A strong-coupling effective-field theory for asymmetrically charged plates with counterions only

We are interested in rationalizing the phenomenon of like-charge attraction between charged bodies, such as a pair of colloids, in the strong coupling regime. The two colloids are modelled as uniformly charged parallel plates, neutralized by mobile counterions. In an earlier work [Palaia et al., J. Phys. Chem. B 126, 3143 (2022)], we developed an effective-field theory for symmetric plates, stemming from the ground-state description that holds at infinite couplings. Here, we generalize the approach to the asymmetric case, where the plates bear charges of the same sign, but of different values. In the symmetric situation, the mobile ions, which are localized in the vicinity of the two plates, share equally between both of them. Here, the sharing is non-trivial, depending both on the coupling parameter and the distance between the plates. We thus introduce a counterion occupation parameter, that is determined variationally to ensure minimum of the free energy. The resulting analytical results for the pressure as a function of the plate-plate distance $d$ agree well with our Monte Carlo data, in a large interval of strong and intermediate coupling constants $\Xi$. We show in particular that within this description, there exists a range of large distances at which the attractive pressure features a $1/d^2$ behavior.

cond-mat.soft↗
arXiv2019

Electric double layers with surface charge modulations: Novel exact Poisson-Boltzmann solutions

Poisson-Boltzmann theory is the cornerstone for soft matter electrostatics. We provide novel exact analytical solutions to this non-linear mean-field approach, for the diffuse layer of ions in the vicinity of a planar or a cylindrical macroion. While previously known solution are for homogeneously charged objects, the cases worked out exhibit a modulated surface charge --or equivalently surface potential-- on the macroion (wall) surface. In addition to asymptotic features at large distances from the wall, attention is paid to the fate of the contact theorem, relating the contact density of ions to the local wall charge density. For salt-free systems (counterions only), we make use of results pertaining to the two-dimensional Liouville equation, supplemented by an inverse approach. When salt is present, we invoke the exact two-soliton solution to the 2D sinh-Gordon equation. This leads to inhomogeneous charge patterns, that are either localized or periodic in space. Without salt, the electrostatic signature of a charge pattern on the macroion fades exponentially with distance for a planar macroion, while it decays as an inverse power-law for a cylindrical macroion. With salt, our study is limited to the planar geometry, and reveals that pattern screening is exponential.

cond-mat.soft↗
arXiv2018

Strong-coupling theory of counterions between symmetrically charged walls: from crystal to fluid phases

We study thermal equilibrium of classical pointlike counterions confined between symmetrically charged walls at distance $d$. At very large couplings when the counterion system is in its crystal phase, a harmonic expansion of particle deviations is made around the bilayer positions, with a free lattice parameter determined from a variational approach. For each of the two walls, the harmonic expansion implies an effective one-body potential at the root of all observables of interest in our Wigner Strong-Coupling expansion. Analytical results for the particle density profile and the pressure are in good agreement with numerical Monte Carlo data, for small as well as intermediate values of $d$ comparable with the Wigner lattice spacing. While the strong-coupling theory is extended to the fluid regime by using the concept of a correlation hole, the Wigner calculations appear trustworthy for all electrostatic couplings investigated. Our results significantly extend the range of accuracy of analytical equations of state for strongly interacting charged planar interfaces.

cond-mat.soft↗
arXiv2018

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

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 fulfill 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 behavior at large distance is studied in detail.

cond-mat.soft↗
arXiv2018

Planar screening by charge polydisperse counterions

We study how a neutralising cloud of counterions screens the electric field of a uniformly charged planar membrane plate, when the counterions are characterised by a distribution of charges (or valence), $n(q)$. We work out analytically the one-plate and two-plate cases, at the level of non-linear Poisson-Boltzmann theory. The (essentially asymptotic) predictions are successfully compared to numerical solutions of the full Poisson-Boltzmann theory, but also to Monte Carlo simulations. The counterions with smallest valence control the long-distance features of interactions, and may qualitatively change the results pertaining to the classic monodisperse case where all counterions have the same charge. Emphasis is put on continuous distributions $n(q)$, for which new power-laws can be evidenced, be it for the ionic density or the pressure, in the one- and two-plates situations respectively. We show that for discrete distributions, more relevant for experiments, these scaling laws persist in an intermediate but yet observable range. Furthermore, it appears that from a practical point of view, hallmarks of the continuous $n(q)$ behaviour is already featured by discrete mixtures with a relatively small number of constituents.

cond-mat.stat-mech↗
arXiv2016

Quasi-universality in mixed counterions systems

The screening of plate-plate interactions by counterions is an age-old problem. We revisit this classic question when counterions exhibit a distribution of charges. While it is expected that the long-distance regime of interactions is universal, the behaviour of the inter-plate pressure at smaller distances should a priori depend rather severely on the nature of the ionic mixture screening the plate charges. We show that is not the case, and that for comparable Coulombic couplings, different systems exhibit a quasi-universal equation of state.

cond-mat.stat-mech↗
arXiv2016

Mean-field beyond mean-field: the single particle view for moderately to strongly coupled charged fluids

In a counter-ion only charged fluid, Coulomb coupling is quantified by a single dimensionless parameter. Yet, the theoretical treatment of moderately to strongly coupled charged fluids is a difficult task, central to the understanding of a wealth of soft matter problems, including biological systems. We show that the corresponding coupling regime can be remarkably well described by a single particle treatment, which, at variance with previous works, takes due account of inter-ionic interactions. To this end, the prototypical problem of a planar charged dielectric interface is worked out. Testing our predictions against Monte Carlo simulation data reveals an excellent agreement.

cond-mat.soft↗
arXiv2016

The rich polymorphic behavior of Wigner bilayers

Self-assembly into target structures is an efficient material design strategy. Combining analytical calculations and computational techniques of evolutionary and Monte Carlo types, we report about a remarkable structural variability of Wigner bilayer ground states, when charges are confined between parallel charged plates. Changing the inter-layer separation, or the plate charge asymmetry, a cascade of ordered patterns emerges. At variance with the symmetric case phenomenology, the competition between commensurability features and charge neutralization leads to long range attraction, appearance of macroscopic charges, exotic phases, and non conventional phase transitions with distinct critical indices, offering the possibility of a subtle, but precise and convenient control over patterns.

cond-mat.soft↗
arXiv2015

Effective charge of cylindrical and spherical colloids immersed in an electrolyte: the quasi-planar limit

We consider the non-linear Poisson-Boltzmann theory for a single cylindrical or spherical macro-ion in symmetric 1:1, together with asymmetric 1:2 and 2:1 electrolytes. We focus on the regime where $\kappa a $, the ratio of the macro-ion radius $a$ over the inverse Debye length in the bulk electrolyte, is large. Analyzing the structure of the analytical expansion emerging from a multiple scale analysis, we uncover a hidden structure for the electrostatic potential. This structure, which appears after a heuristic resummation, suggests a new and convenient expansion scheme that we present and work out in detail. We show that novel exact results can thereby be obtained, in particular pertaining to effective charge properties, in complete agreement with the direct numerical solution to the problem.

cond-mat.soft↗
arXiv2014

Counter-ions between or at asymmetrically charged walls: 2D free-fermion point

This work contributes to the problem of determining effective interaction between asymmetrically (likely or oppositely) charged objects whose total charge is neutralized by mobile pointlike counter-ions of the same charge, the whole system being in thermal equilibrium. The problem is formulated in two spatial dimensions with logarithmic Coulomb interactions. The charged objects correspond to two parallel lines at distance d, with fixed line charge densities. Two versions of the model are considered: the standard "unconstrained" one with particles moving freely between the lines and the "constrained" one with particles confined to the lines. We solve exactly both systems at the free-fermion coupling and compare the results for the pressure (i.e. the force between the lines per unit length of one of the lines) with the mean-field Poisson-Boltzmann solution. For the unconstrained model, the large-d asymptotic behaviour of the free-fermion pressure differs from that predicted by the mean-field theory. For the constrained model, the asymptotic pressure coincides with the attractive van der Waals-Casimir fluctuational force. For both models, there are fundamental differences between the cases of likely-charged and oppositely-charged lines, the latter case corresponding at large distances d to a capacitor.

cond-mat.stat-mech↗
arXiv2013

Ground state structure of a bilayer Wigner crystal with repulsive dielectric images

We study the ground-state structures of identical classical point charges with Coulomb interactions, confined between two symmetric parallel charged walls. For the well understood homogeneous dielectric case with no electrostatic images, the charges evenly condense on the opposite walls, thereby forming a bilayer Wigner crystal; five structures compete upon changing the inter-wall separation. Here, we consider a dielectric jump between the walls and a solvent in which charges are immersed, implying repulsive images. Using recently developed series representations of lattice sums for Coulomb law, we derive the complete phase diagram. In contrast to the homogeneous dielectric case, the particles remain in a hexagonal Wigner monolayer up to a certain distance between the walls. Beyond this distance, a bifurcation occurs to a sequence of Wigner bilayers, each layer having a nonzero spacing from the nearest wall. Another new phenomenon is that the ground-state energy as a function of the wall separation exhibits a global minimum.

cond-mat.stat-mech↗
arXiv2012

Like-charge colloidal attraction: a simple argument

By a length scale analysis, we study the equilibrium interactions between two like-charge planes confining neutralising counter-ions. At large Coulombic couplings, approaching the two charged bodies leads to an unbinding of counter-ions, a situation that is amenable to an exact treatment. This phenomenon is the key to attractive effective interactions. A particular effort is made for pedagogy, keeping equations and formalism to a minimum.

cond-mat.soft↗
arXiv2012

Critical phenomena and phase sequence in classical bilayer Wigner crystal at zero temperature

We study the ground-state properties of a system of identical classical Coulombic point particles, evenly distributed between two equivalently charged parallel plates at distance $d$; the system as a whole is electroneutral. It was previously shown that upon increasing d from 0 to infinity, five different structures of the bilayer Wigner crystal become energetically favored, starting from a hexagonal lattice (phase I, d=0) and ending at a staggered hexagonal lattice (phase V, d -> infinity). In this paper, we derive new series representations of the ground-state energy for all five bilayer structures. The derivation is based on a sequence of transformations for lattice sums of Coulomb two-particle potentials plus the neutralizing background, having their origin in the general theory of Jacobi theta functions. The new series provide convenient starting points for both analytical and numerical progress. Its convergence properties are indeed excellent: Truncation at the fourth term determines in general the energy correctly up to 17 decimal digits. The accurate series representations are used to improve the specification of transition points between the phases and to solve a controversy in previous studies. In particular, it is shown both analytically and numerically that the hexagonal phase I is stable only at d=0, and not in a finite interval of small distances between the plates as was anticipated before. The expansions of the structure energies around second-order transition points can be done analytically, which enables us to show that the critical behavior is of the Ginzburg-Landau type, with a mean-field critical index beta=1/2 for the growth of the order parameters.

cond-mat.str-el↗
arXiv2012

Ground state of classical bilayer Wigner crystals

We study the ground state structure of electronic-like bilayers, where different phases compete upon changing the inter-layer separation or particle density. New series representations with exceptional convergence properties are derived for the exact Coulombic energies under scrutiny. The complete phase transition scenario --including critical phenomena-- can subsequently be worked out in detail, thereby unifying a rather scattered or contradictory body of literature, hitherto plagued by the inaccuracies inherent to long range interaction potentials.

cond-mat.str-el↗
arXiv2011

Strong-coupling theory for a polarizable planar colloid

We propose a strong-coupling analysis of a polarizable planar interface, in the spirit of a recently introduced Wigner-Crystal formulation. The system is made up of two moieties: a semi-infinite medium (z<0) with permittivity epsilon' while the other half space in z>0 is occupied by a solution with permittivity epsilon, and mobile counter-ions (no added electrolyte). The interface at z=0 bears a uniform surface charge. The counter-ion density profile is worked out explicitly for both repulsive and attractive dielectric image cases.

cond-mat.soft↗
arXiv2011

Survival probability (heat content) and the lowest eigenvalue of Dirichlet Laplacian

We study the survival probability of a particle diffusing in a two-dimensional domain, bounded by a smooth absorbing boundary. The short-time expansion of this quantity depends on the geometric characteristics of the boundary, whilst its long-time asymptotics is governed by the lowest eigenvalue of the Dirichlet Laplacian defined on the domain. We present a simple algorithm for calculation of the short-time expansion for an arbitrary "star-shaped" domain. The coefficients are expressed in terms of powers of boundary curvature, integrated around the circumference of the domain. Based on this expansion, we look for a Pad\'e interpolation between the short-time and the long-time behavior of the survival probability, i.e. between geometric characteristics of the boundary and the lowest eigenvalue of the Dirichlet Laplacian.

math.SP↗
arXiv2010

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 Gamma is small, is briefly recapitulated (the coupling constant is defined as Gamma = beta e^2 where beta is the inverse temperature, and e the counter-ion charge). The opposite strong-coupling limit (Gamma -> infinity) is treated by using a recent method based on an exact expansion around the ground-state Wigner crystal of counter-ions. The weak- and strong-coupling theories are compared at intermediary values of the coupling constant Gamma=2 gamma (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 Gamma=2 and up to relatively large finite N at 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 Gamma=6. The like-charge attraction regime, possible in the strong coupling limit but precluded at mean-field level, survives for Gamma=4 and 6, but disappears at Gamma=2.

cond-mat.soft↗