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     

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     

Beyond Poisson-Boltzmann: Fluctuation effects and correlation functions

We formulate the exact non-linear field theory for a fluctuating counter-ion distribution in the presence of a fixed, arbitrary charge distribution. The Poisson-Boltzmann equation is obtained as the saddle-point of the field-theoretic action, and the effects of counter-ion fluctuations are included by a loop-wise expansion around this saddle point. The Poisson equation is obeyed at each order in this loop expansion. We explicitly give the expansion of the Gibbs potential up to two loops. We then apply our field-theoretic formalism to the case of a single impenetrable wall with counter ions only (in the absence of salt ions). We obtain the fluctuation corrections to the electrostatic potential and the counter-ion density to one-loop order without further approximations. The relative importance of fluctuation corrections is controlled by a single parameter, which is proportional to the cube of the counter-ion valency and to the surface charge density. The effective interactions and correlation functions between charged particles close to the charged wall are obtained on the one-loop level. Received 8 February 1999 and Received in final form 15 May 1999     

Effect of colloidal charge discretization in the primitive model

The effect of fixed discrete colloidal charges in the primitive model is investigated for spherical macroions. Instead of considering a central bare charge, as it is traditionally done, we distribute discrete charges randomly on the sphere. We use molecular dynamics simulations to study this effect on various properties such as overcharging, counterion distribution and diffusion. In the vicinity of the colloid surface the electrostatic potential may considerably differ from the one obtained with a central charge. In the strong Coulomb coupling, we showed that the colloidal charge discretization qualitatively influences the counterion distribution and leads to a strong colloidal charge-counterion pair association. However, we found that charge inversion still persists even if strong pair association is observed. Received 30 June 2000 and Received in final form 28 November 2000     

Monte Carlo simulations of star-branched polyelectrolyte micelles

The concentration profiles of monomers and counterions in star-branched polyelectrolyte micelles are calculated through Monte Carlo simulations, using the freely jointed chain model. We have investigated the onset of different regimes corresponding to the spherical and Manning condensation of counterions as a function of the strength of the Coulomb coupling. The Monte Carlo results are in fair agreement with the predictions of Self-Consistent-Field analytical models. We have simulated a real system of diblock copolymer micelles of (sodium-polystyrene-sulfonate)(NaPSS)-(polyethylene-propylene)(PEP) with f = 54 hydrophilic branches of N = 251 monomers at room temperature in salt-free solution. The calculated form factor compares nicely with our neutron scattering data. Received 18 July 2002 and Received in final form 11 October 2002 RID="a" ID="a"e-mail: roger@drecam.saclay.cea.fr     

Phase behavior of three-component ionic fluids

We study the phase behavior of solutions consisting of positive and negative ions of valence z to which a third ionic species of valence Z>z is added. Using a discretized Debye-Hückel theory, we analyze the phase behavior of such systems for different values of the ratio . We find, for , a three-phase coexistence region and, for , a closed (reentrant) coexistence loop at high temperatures. We characterize the behavior of these ternary ionic mixtures as function of charge asymmetry and temperature, and show the complete phase diagrams for the experimentally relevant cases of and , corresponding to addition of divalent and trivalent ions to monovalent ionic fluids, respectively. Received 6 April 2000 and Received in final form 20 July 2000     

Structure and properties of confined sodium nitrite

We study the lateral and transverse diffusion of amphiphiles in two-component bilayer membranes, using a coarse-grained model for amphiphilic molecules and combined Monte Carlo-Molecular Dynamics simulations. Membrane structural properties, such as the mean thickness, are also measured. The dependence of such properties on membrane composition, inter-molecular interactions, and amphiphile stiffness is determined. In particular, we show that addition of shorter amphiphiles drives the model membrane towards a more fluid state, with increased amphiphile lateral diffusion rates. These results can be understood in the framework of a simple free-volume model. Furthermore, we observe an increase in the trans-membrane diffusion when the interaction energy of amphiphiles with their neighboring molecules is decreased. Received: 6 December 2002 / Accepted: 17 April 2003 / Published online: 27 May 2003 RID="a" ID="a"e-mail: imparato@mpikg-golm.mpg.de RID="b" ID="b"e-mail: shillcock@mpikg-golm.mpg.de RID="c" ID="c"e-mail: lipowsky@mpikg-golm.mpg.de     

Flexible linear polyelectrolytes in multivalent salt solutions: Solubility conditions

Single- and double-stranded DNA and many biological and synthetic polyelectrolytes undergo two structural transitions upon increasing the concentration of multivalent salt or molecules. First, the expanded-stretched chains in low monovalent salt solutions collapse into nearly neutral compact structures when the density of multivalent salt approaches that of the monomers. With further addition of multivalent salt the chains redissolve acquiring expanded-coiled conformations. We study the redissolution transition using a two-state model (F.J. Solis, M. Olvera de la Cruz, J. Chem. Phys. 112, 2030 (2000)). The redissolution occurs when there is a high degree of screening of the electrostatic interactions between monomers, thus reducing the energy of the expanded state. The transition is determined by the chemical potential of the multivalent ions in the solution, μ and the inverse screening length, κ. The transition point also depends on the charge distribution along the chain but is nearly independent of the molecular weight and degree of flexibility of the polyelectrolytes. We generate a diagram of μversusκ² where we find two regions of expanded conformations, one with charged chains and the other with overcharged (inverted charge) chains, separated by a collapsed nearly neutral conformation region. The collapse and redissolution transitions occur when the trajectory of the properties of the salt crosses the boundaries between these regions. We find that in most cases the redissolution occurs within the same expanded branch from which the chain precipitates. Received 15 May 2000 and Received in final form 28 June 2000     

Salty solutions near a charged modulated interface

Charged monolayers at a liquid-vapor interface may be found in a crystalline state, resulting in a surface density of charge that displays periodic modulations. In this paper we discuss how these modulations affect different thermodynamical and mechanical properties (compared with the equivalent uniform charge density) of a system consisting of the charged monolayer and a bulk solution including a finite concentration of counter-ions and co-ions. It is shown that very accurate results for low and moderate salt concentrations are possible within an expansion in the Fourier modes of the modulations, the Weak Amplitude Perturbation (WAP), if the finite size of the ions are included as a Stern layer. We conclude by discussing the implications and the relevance of these results for both theoretical studies and experiments.     

Simple model for attraction between like-charged polyions

We present a simple model for the possible mechanism of appearance of attraction between like charged polyions inside a polyelectrolyte solution. The attraction is found to be short ranged, and exists only in the presence of multivalent counterions. It is produced by the correlations in layers of condensed counterions surrounding each polyion and is only weakly temperature dependent. We find the attraction to be maximum at zero temperature and dimish as the temperature is raised. The attraction is only possible if the number of condensed counterions exceeds the threshold, , where is the valence of counterions and Z is the polyion charge. Received 10 March 1999 and Received in final form 20 April 1999     

Collapse of polyelectrolyte brushes: Scaling theory and simulations

We investigate polyelectrolyte brushes using both scaling arguments and molecular dynamics simulations. As a main result, we find a novel collapsed brush phase. In this phase, the height of the brush results from a competition between steric repulsion between ions and monomers and an attractive force due to electrostatic correlations. As a result, the monomer density inside the brush is independent of the grafting density and the polymerization index. For small ionic and monomer radii (or for large Bjerrum length) the brush undergoes a first-order phase transition from the osmotic into the collapsed state. Received 26 September 2000     

Structure of colloidal complexes obtained from neutral/poly- electrolyte copolymers and oppositely charged surfactants

We report on the phase behavior and scattering properties of colloidal complexes made from block copolymers and surfactants. The copolymer is poly(sodium acrylate)-b-poly(acrylamide), hereafter abbreviated as PANa-PAM, with molecular weight 5000 g/mol for the first block and 30000 g/mol for the second. In aqueous solutions and neutral pH, poly(sodium acrylate) is a weak polyelectrolyte, whereas poly(acrylamide) is neutral and in good-solvent conditions. The surfactant is dodecyltrimethylammonium bromide (DTAB) and is of opposite charge with respect to the polyelectrolyte block. Combining dynamical light scattering and small-angle neutron scattering, we show that in aqueous solutions PANa-PAM diblocks and DTAB associate into colloidal complexes. For surfactant-to-polymer charge ratios Z lower than a threshold (Z(C) approximately 0.3), the complexes are single surfactant micelles decorated by few copolymers. Above the threshold, the colloidal complexes reveal an original core-shell microstructure. We have found that the core of typical radius 100-200 A is constituted from densely packed surfactant micelles connected by the polyelectrolyte blocks. The outer part of the colloidal complex is a corona and is made from the neutral poly(acrylamide) chains. Typical hydrodynamic sizes for the whole aggregate are around 1000 A. The aggregation numbers expressed in terms of numbers of micelles and copolymers per complex are determined and found to be comprised between 100-400, depending on the charge ratio Z and on the total concentration. We have also shown that the sizes of the complexes depend on the exact procedure of the sample preparation. We propose that the driving mechanism for the complex formation is similar to that involved in the phase separation of homopolyelectrolyte/surfactant systems. With copolymers, the presence of the neutral blocks prevents the macroscopic phase separation from occurring.     

Ionic distribution and polymer conformation, near phase separation, in sodium polyacrylate/divalent cations mixtures: small angle X-ray and neutron scattering

Anomalous small angle X-ray scattering experiments show that before demixion in sodium polyacrylate/cobalt and sodium polyacrylate/calcium mixtures all the divalent counterions are in the close vicinity of the polyacrylate chain. The present results are consistent with previous UV/VIS spectroscopy, which have shown that all cobalt ions are chemically associated with acrylate groups. The chemical association dehydrates the acrylate monomers. However, the hydrophobicity of the complexed monomers is not strong enough to induce a collapse of the polymer chain at small spatial scale before the demixion. Indeed, the scattered intensity (X-ray and neutron scattering) decreases with the scattering vector q as q^-x with for q > 0.1 nm ^-1 which indicates that the local conformation of the chain is Gaussian. Received 21 January 1999     

Structural properties of charged diblock copolymer solutions

Aqueous micellar solutions of ionic/neutral block copolymers have been studied by light scattering, small angle neutron scattering and small angle X-ray scattering. We made use of a polymer comprised of a short hydrophobic block (polyethylene-propylene) PEP and of a long polyelectrolytic block (polystyrene-sulfonate) PSSNa which has been shown previously to micellize in water. The apparent polydispersity of these micelles is studied in detail, showing the existence of a few large aggregates coexisting with the population of micelles. Solutions of micelles are found to order above some threshold in polymer concentration. The order is liquid-like, as demonstrated by the evolution with concentration of the peak observed in the structure factor (), and the degree of order is found to be identical over a large range of concentrations (up to 20 wt%). Consistent values of the aggregation number of the micelles are found by independent methods. The effect of salt addition on the order is found to be weak. Received: 19 June 1997 / Received in final form: 4 September 1997 / Accepted: 9 October 1997     

Shapes of nearly cylindrical, axisymmetric bilayer membranes

Shapes of nearly cylindrical sections of axisymmetric phospholipid membranes are studied theoretically. Describing the shape of such sections by their deviation from a reference cylinder, the well-established shape equation for axisymmetric bilayer membranes is expanded in terms of this deviation, and it is then solved analytically. The phase diagram shows the resulting stationary shapes as functions of system parameters and external conditions, i.e., the pressure difference across the membrane, the membrane tension, the difference between the tensions of the two monolayers, and the axial force acting on the vesicle. The accuracy of the approximate analytical solution is demonstrated by comparison with numerical results. The obtained analytical solution allows to extend the analysis to include shapes where numerical methods have failed. Received 27 September 2000 and Received in final form 26 March 2001     

Solubility of highly charged anionic polyelectrolytes in presence of multivalent cations: Specific interaction effect

Studies performed on strong polyelectrolytes and on a weak polyelectrolyte, sodium poly(acrylate), show that their stability in presence of multivalent cations depends on the chemical nature of the charged side groups of the polymer. For sulfonate groups (SO₃ ^-) or sulfate groups (OSO₃ ^-) phase separation generally occurs in presence of inorganic cations of valency 3 (as La³⁺) or larger and a resolubilization takes place at high salt concentration. The interactions of the polyelectrolyte with multivalent cations are of electrostatic origin and the phase diagrams are weakly dependent on the chemical nature of the polymer backbone and on the specificity of the counterions. For acrylate groups, (COO^-), the phase separation was observed with inorganic cations of valency 2 (as Ca²⁺) or larger without resolubilization at high salt concentration. The phase separation is due to a chemical association between cations and acrylate groups of two neighboring monomers of the same chain. This chemical association creates a hydrophobic complex by dehydrating both monomer and cation. With organic trivalent cation, as spermidine ⁺H₃N(CH₂)₄NH₂ ⁺(CH₂)₃NH₃ ⁺, where no chemical association occurs with the charged side groups COO^- or SO₃ ^- of the polyelectrolyte, similar phase diagrams were observed whatever was the polyelectrolyte with a resolubilization at high trivalent cation concentration. Received 3 March 1999 and Received in final form 2 September 1999     

Mechanical and order rigidity of nematic elastomers

We calculate the ratio of moduli for extension parallel and perpendicular to the director of nematic elastomers. Elastomers in practice are not ideal nematics with a jump from a finite to zero-order parameter with increasing temperature. Some classes behave as if under effective external fields. These are really internal in origin and leave the system subcritical (still with jumps) or supercritical. We give expressions for general non-ideality and memory of formation conditions, along with its translation into values of the “fields". For the case of supercritical fields, we find that the modulus ratio deviates more from unity, in accord with experiment, than in the subcritical case. Received 29 September 2000     

Correlation effects,image charge effects and finite size in the macro-ion-electrolyte system: A field-theoretic approach

We consider a model of a macro-ion surrounded by small ions of an electrolyte solution. The finite size of ionic charge distributions of ions, and image charge effects are considered. From such a model it is possible to construct a statistical field theory with a single fluctuating field and derive physical interpretations for both the mean field and two-point correlation function. For point-like charges, at the level of a Gaussian (or saddle point) approximation, we recover the standard Poisson-Boltzmann equation. However, to include ionic correlation effects, as well as image charge effects of individual ions, we must go beyond this. From the field theory considered, it is possible to construct self-consistent approximations. We consider the simplest of these, namely the Hartree approximation. The Hartree equations take the form of two coupled equations. One is a modified Poisson-Boltzmann equation; the other describes both image charge effects on the individual ions, as well as correlations. Such equations are difficult to solve numerically, so we develop an (a WKB-like) approximation for obtaining approximate solutions. This, we apply to a uniformly charged rod in univalent electrolyte solution, for point like ions, as well as for extended spherically symmetric distributions of ionic charge on electrolyte ions. The solutions show how correlation effects and image charge effects modify the Poisson-Boltzmann result. Finite-size charge distributions of the ions reduce both the effects of correlations and image charge effects. For point charges, we test the WKB approximation by calculating a leading-order correction from the exact Hartree result, showing that the WKB-like approximation works reasonably well in describing the full solution to the Hartree equations. From these solutions, we also calculate an effective charge compensation parameter in an analytical formula for the interaction of two charged cylinders. Electronic supplementary material  Supplementary material in the form of a doc file available from the journal web page at and are accessible for authorised users.     

Scaling theory of mixed amphiphilic monolayers

We predict the elastic properties of mixed amphiphilic monolayers in the swollen state within the blob model using scaling arguments. First the elastic moduli and the spontaneous curvature of a bimodal brush are determined as a function of the composition and the relative chain length. We obtain simple and useful scaling functions which interpolate between the elastic moduli of a pure short-chain brush and a pure long-chain brush. By using the analogy between block copolymer interfaces and polymeric brushes, the effect of mixing on self-assembled diblock copolymer monolayers is investigated in the swollen state. We calculate various interfacial properties, such as the equilibrium surface coverage, interface curvature, and the mixing free energy as a function of the composition. In general, we find a nonlinear dependence on the composition, which deviates from the simple linear averaging of the properties of pure components. Our results are used to discuss a recent experiment on the effect of amphiphilic block copolymers on the efficiency of microemulsions. Received 29 December 2000 and Received in final form 19 March 2001     

The elastic anisotropy of nematic elastomers

We examine the robustness of order in nematic elastomers under mechanical strains imposed along and perpendicularly to the director when director rotation is prohibited. In contrast to electric and magnetic fields applied to conventional nematics, mechanical fields are shown theoretically and experimentally to greatly affect the degree of nematic order and related quantities. Unlike in liquid nematics, one can impose fields perpendicular to the director, thereby inducing biaxial order which should be susceptible to experimental detection. Nematic elastomers with unchanging director and degree of order should theoretically have the same elastic moduli for longitudinal and transverse extensions. This is violated when nematic order is permitted to relax in response to strains. Near the transition we predict the longitudinal modulus to be smaller than the transverse modulus; at lower temperatures the converse is true, with a crossover a few degrees below the transition. The differences are ascribed to the different temperature dependence of the stiffness of uniaxial and biaxial order. We synthesised side chain single-crystal nematic polymer networks, performed DSC, X-ray, birefringence, and thermo-mechanical characterisations, and then obtained linear moduli from stress-strain measurements. Received 29 September 2000