We develop a computational method for modeling electrostatic interactions of arbitrarily-shaped, polarizable objects on colloidal length scales, including colloids/nanoparticles, polymers, and surfactants, dispersed in explicit ion electrolytes and nonionic solvents. Our method computes the nonuniform polarization charge distribution induced in a colloidal particle by both externally applied electric fields and local electric fields arising from other charged objects in the dispersion. This leads to expressions for electrostatic energies, forces, and torques that enable efficient molecular dynamics and Brownian dynamics simulations of colloidal dispersions in electrolytes, which can be harnessed to accurately predict structural and transport properties. We describe an implementation in which colloidal particles are modeled as rigid composites of small spherical beads that tessellate the surface of the particle. The electrostatics calculations are accelerated using a spectrally-accurate particle-mesh-Ewald technique implemented on a graphics processing unit (GPU) and regularized such that the electrostatic calculations are well-defined even for overlapping bodies. We demonstrate the effectiveness of this approach through a series of calculations, including the induced dipole moments and forces for one, two, and lattices of spherical colloids in an electric field; the induced dipole moment and torque for anisotropic particles in an electric field; the equilibrium distribution of ions in the double layer around charged colloids; the dynamics of charged colloids; and ions in the double layer around a polarizable colloid exposed to an electric field.
A theory is presented which allows us to accurately calculate the density profile of monovalent and multivalent counterions in suspensions of polarizable colloids or nano-particles. In the case of monovalent ions, we derive a weak-coupling theory that explicitly accounts for the ion-image interaction, leading to a modified Poisson-Boltzmann equation. For suspensions with multivalent counterions, a strong-coupling theory is used to calculate the density profile near the colloidal surface and a Poisson-Boltzmann equation with a renormalized boundary condition to account for the counterion distri…
In the present work, we have extended a weak coupling theory [A. Bakhshandeh, A. P. dos Santos and Y. Levin Phys. Rev. Lett 107, 107801 (2011)] for systems with added 1:1 electrolyte. To study the accuracy of the developed theory, we compare its numerical predictions with Monte Carlo simulation data and a recent theory which accounts for the surface polarization. A very good agreement is found for the case of monovalent electrolytes, up to very high salt concentrations and different colloidal charges
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.
We study the effective interaction between differently charged polyelectrolyte-colloid complexes in electrolyte solutions via Monte Carlo simulations. These complexes are formed when short and flexible polyelectrolyte chains adsorb onto oppositely charged colloidal spheres, dispersed in an electrolyte solution. In our simulations the bending energy between adjacent monomers is small compared to the electrostatic energy, and the chains, once adsorbed, do not exchange with the solution, although they rearrange on the particles surface to accomodate further adsorbing chains or due to the electros…
We present a theory that enables us to calculate the effective surface charge of colloidal particles and to efficiently obtain titration curves for different salt concentrations. The theory accounts for the shift of pH of solution due to the presence of 1:1 electrolyte. It also accounts self-consistently for the electrostatic potential produced by the deprotonated surface groups. To examine the accuracy of the theory we have performed extensive reactive Monte Carlo simulations, which show excellent agreement between theory and simulations without any adjustable parameters.
We propose a novel actuation mechanism for colloids, which is based on the Seebeck effect of the electrolyte solution: Laser heating of a nonionic particle accumulates in its vicinity a net charge Q, which is proportional to the excess temperature at the particle surface. The corresponding long-range thermoelectric field provides a tool for controlled interactions with nearby beads or with additional molecular solutes. An external field Eext drags the thermocharged particle at a velocity that depends on its size and absorption properties; the latter point could be particularly relevant for sep…
The liquid-gas transition of an electroneutral mixture of oppositely charged colloids, studied by Monte Carlo simulations, is found in the low temperature -- low density region. The critical temperature shows a non-monotonous behavior as a function of the interaction range, $κ^{-1}$, with a maximum at $κσ\approx 10$, implying an island of coexistence in the $κ$-$ρ$ plane. The system is arranged in such a way that each particle is surrounded by shells of particles with alternating charge. In contrast with the electrolyte primitive model, both neutral and charged clusters are obtained in the vap…
The electrostatic interaction between colloidal particles trapped at the interface between two immiscible electrolyte solutions is studied in the limit of small inter-particle distances. Within an appropriate model exact analytic expressions for the electrostatic potential as well as for the surface and line interaction energies are obtained. They demonstrate that the widely used superposition approximation, which is commonly applied to large distances between the colloidal particles, fails qualitatively at small distances and is quantitatively unreliable even at large distances. Our results c…
Extensive Monte Carlo simulations and scaling arguments are used to study the colloidal charge reversal. The critical colloidal surface charge density $σ_c$ at which the reversal first appears is found to depend strongly on the ionic size. We find that $σ_c$ has an inflection point as a function of the electrolyte concentration. The width of the plateau region in the vicinity of the inflection point depends on the temperature and the ionic radius $a$. In agreement with the theoretical predictions it is found that the critical colloidal charge above which the electrophoretic mobility becomes re…
We show that describing the screened electrostatic interactions in a periodic stack of rigid parallel identical charged plate-like colloids within a local density functional theory approach, generically leads to a swelling behaviour. Within the same framework, we find that the effective potential between a pair of such plates immersed in an electrolyte is repulsive at all distances. This general result is in contradiction with a theory put forward by Sogami, Shinohara and Smalley, that we criticize and thoroughly reconsider to show how the two approaches can be reconciled.
Dynamic processes in dispersions of charged spherical particles are of importance both in fundamental science, and in technical and bio-medical applications. There exists a large variety of charged-particles systems, ranging from nanometer-sized electrolyte ions to micron-sized charge-stabilized colloids. We review recent advances in theoretical methods for the calculation of linear transport coefficients in concentrated particulate systems, with the focus on hydrodynamic interactions and electrokinetic effects. Considered transport properties are the dispersion viscosity, self- and collective…
The equilibrium phase diagram of a 1:1 symmetrical mixture composed of oppositely charged colloids is calculated using Monte Carlo simulations. We model the system by the DLVO effective interaction potential. The phase diagram is similar to that of its atomic analog (the ionic fluid), where a liquid-gas first order transition emerges in the low $T-ρ$ regions being stable with respect to crystallization. As in the ionic fluids, we have found two different crystals: at high $T$ the fluid crystallizes in a FCC lattice, whereas at low $T$, the liquid coexists with a BCC crystal. The region of gas-…
We perform computational investigations of electrolyte-mediated interactions of charged colloidal particles confined within nanochannels. We investigate the role of discrete ion effects, valence, and electrolyte strength on colloid-wall interactions. We find for some of the multivalent charge regimes that the like-charged colloids and walls can have attractive interactions. We study in detail these interactions and the free energy profile for the colloid-wall separation. We find there are energy barriers and energy minima giving preferred colloid locations in the channel near the center and at…
We report on the low temperature behaviour of the colloidal electrolyte by means of Molecular Dynamics simulations, where the electrostatic interactions were modeled using effective screened interactions. As in previous works, we have found a region of gas-liquid coexistence located in the low $T$- low $ρ$ region. At temperatures much lower than the critical one, the system cannot reach equilibrium, that is, the gas-liquid transition is arrested. Two different mechanisms have been identified to cause arrest: gelation in the lowest $T$ and crowding at intermediate $T$ values, the latter associa…
Computer simulations are used to investigate the response of a charged colloid and its surrounding microion cloud to an external electric field. Both static fields (DC) and alternating fields (AC) are considered. A mesoscopic simulation method is implemented to account in full for hydrodynamic and electrostatic interactions. The response of the system can be characterized by two quantities: the mobility and the polarizability. Due to the interplay of the electrostatic attraction and hydrodynamic drag, the response of the microions close to the colloid surface is different from that of the micr…
From a multiple scale analysis, we find an analytic solution of spherical and cylindrical Poisson-Boltzmann theory for both a 1:2 (monovalent co-ions, divalent counter-ions) and a 2:1 (reversed situation) electrolyte. Our approach consists in an expansion in powers of rescaled curvature $1/(κa)$, where $a$ is the colloidal radius and $1/κ$ the Debye length of the electrolytic solution. A systematic comparison with the full numerical solution of the problem shows that for cylinders and spheres, our results are accurate as soon as $κa>1$. We also report an unusual overshooting effect where the c…
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 $κa $, 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…
The effective electrostatic interaction between a pair of colloids, both of them located close to each other at an electrolyte interface, is studied by employing the full, nonlinear Poisson-Boltzmann (PB) theory within classical density functional theory. Using a simplified yet appropriate model, all contributions to the effective interaction are obtained exactly, albeit numerically. The comparison between our results and those obtained within linearized PB theory reveals that the latter overestimates these contributions significantly at short inter-particle separations. Whereas the surface co…
This paper builds on two previous works, Lindgren et al. J. Comp. Phys. 371, 712-731 (2018) and Quan et al. arXiv:1807.05384 (2018), to devise a new method to solve the problem of calculating electrostatic interactions in a system composed by many dielectric particles, embedded in a homogeneous dielectric medium, which in turn can also be permeated by charge carriers. The system is defined by the charge, size, position and dielectric constant of each particle, as well as the dielectric constant and Debye length of the medium. The effects of taking into account the dielectric nature of the part…
The interplay between shape anisotropy and directed long-range interactions enables the self-assembly of complex colloidal structures. As a recent highlight, ellipsoidal particles polarized in an external electric field were observed to associate into well-defined tubular structures. In this study, we investigate systematically such directed self-assembly using Monte Carlo simulations of a two-point-charge model of polarizable prolate ellipsoids. In spite of its simplicity and computational efficiency, we demonstrate that the model is capable of capturing the complex structures observed in exp…
Using a field-theoretic approach, we derive the first few coefficients of the exact low-density (``virial'') expansion of a binary mixture of positively and negatively charged hard spheres (two-component hard-core plasma, TCPHC). Our calculations are nonperturbative with respect to the diameters $d_+$ and $d_-$ and charge valences $q_+$ and $q_-$ of positive and negative ions. Consequently, our closed-form expressions for the coefficients of the free energy and activity can be used to treat dilute salt solutions, where typically $d_+ \sim d_-$ and $q_+ \sim q_-$, as well as colloidal suspensio…
The physical behaviour of anisotropic charged colloids is determined by their material dielectric anisotropy, affecting colloidal self-assembly, biological function and even out-of-equilibrium behaviour. However, little is known about anisotropic electrostatic screening, which underlies all electrostatic effective interactions in such soft or biological materials. In this work, we demonstrate anisotropic electrostatic screening for charged colloidal particles in a nematic electrolyte. We show that material anisotropy behaves markedly different from particle anisotropy: The electrostatic potent…
We use a reactive Monte Carlo simulation method and primitive model of electrolyte to study acid-base equilibrium that controls charge regulation in colloidal systems. The simulations are performed in a semi-grand canonical ensemble in which colloidal suspension is in contact with a reservoir of salt and strong acid. The interior of colloidal particles is modeled as a low dielectric medium, different from the surrounding water. The effective colloidal charge is calculated for different number of surface acidic groups, pH, salt concentrations, and types of electrolyte. In the case of potassium …
Na-montmorillonite is a natural clay mineral and is available in abundance in nature. The aqueous dispersions of charged and anisotropic platelets of this mineral exhibit non-ergodic kinetically arrested states ranging from soft glassy phases dominated by interparticle repulsions to colloidal gels stabilized by salt induced attractive interactions. When the salt concentration in the dispersing medium is varied systematically, viscoelasticity and yield stress of the dispersion show non-monotonic behavior at a critical salt concentration, thus signifying a morphological change in the dispersion …
We discuss the peculiarities of the Seebeck effect in stabilized electrolytes containing the colloidal particles. Its unusual feature is the two-stage character, with the linear increase of differential thermopower as the function of colloidal particles concentration $n_{\odot}$ during the first stage and dramatic drop of it at small $n_{\odot}$ during the second one (steady state). We show that the properties of the initial state are governed by the thermo-diffusion flows of the mobile ions of the stabilizing electrolyte medium itself and how the colloidal particles participate in formation o…
Colloidal diffusiophoresis in crossed electrolyte gradients: experimental demonstration of an 'action at a distance' effect predicted by the Nernst-Planck equations
The hypernetted chain/mean spherical approximation (HNC/MSA) integral equation is obtained and solved numerically for a totally asymmetric primitive model electrolyte around a spherical macroparticle. The ensuing radial distribution functions show a very good agreement when compared to our Monte Carlo and molecular dynamics simulations for spherical geometry and with respect to previous anisotropic reference HNC calculations in the planar limit. We report an analysis of the potential vs charge relationship, radial distribution functions, mean electrostatic potential and cumulative reduced char…
We study the Soret effect of charged polystyrene particles as a function of temperature and electrolyte composition. As a main result we find that the Soret coefficient is determined by charge effects, and that non-ionic contributions are small. In view of the well-kown electric-double layer interactions, our thermal field-flow fractionation data lead us to the conclusion that the Soret effect originates to a large extent from diffusiophoresis in the salt gradient and from the electrolyte Seebeck effect, both of which show strong specific-ion effects. Moreover, we find that thermophoresis of p…
When an electric field is applied to an electrolyte-saturated polymer gel embedded with charged colloidal particles, the force that must be exerted by the hydrogel on each particle reflects a delicate balance of electrical, hydrodynamic and elastic stresses. This paper examines the displacement of a single charged spherical inclusion embedded in an uncharged hydrogel. We present numerically exact solutions of coupled electrokinetic transport and elastic-deformation equations, where the gel is treated as an incompressible, elastic Brinkman medium. This model problem demonstrates how the displac…
Because micro-ions accumulate around highly charged colloidal particles in electrolyte solutions, the relevant parameter to compute their interactions is not the bare charge, but an effective (or renormalized) quantity, whose value is sensitive to the geometry of the colloid, the temperature or the presence of added-salt. This non-linear screening effect is a central feature in the field of colloidal suspensions or polyelectrolyte solutions. We propose a simple method to predict effective charges of highly charged macro-ions, that is reliable for mono-valent electrolytes (and counter-ions) in …
The forces between colloidal particles at a decane-water interface, in the presence of low concentrations of a monovalent salt (NaCl) and of the surfactant sodium dodecylsulfate (SDS) in the aqueous subphase, have been studied using laser tweezers. In the absence of electrolyte and surfactant, particle interactions exhibit a long-range repulsion, yet the variation of the interaction for different particle pairs is found to be considerable. Averaging over several particle pairs was hence found to be necessary to obtain reliable assessment of the effects of salt and surfactant. It has previously…
A numerical method is presented for first-principle simulations of charged colloidal dispersions in electrolyte solutions. Utilizing a smoothed profile for colloid-solvent boundaries, efficient mesoscopic simulations are enabled for modeling dispersions of many colloidal particles exhibiting many-body electrostatic interactions. The validity of the method was examined for simple colloid geometries, and the efficiency was demonstrated by calculating stable structures of two-dimensional dispersions, which resulted in the formation of colloidal crystals.
The pair-correlation functions for fluid ionic mixtures in arbitrary spatial dimensions are computed in hypernetted chain (HNC) approximation. In the primitive model, all ions are approximated as non-overlapping hyperspheres with Coulomb interactions. Our spectral HNC solver is based on a Fourier-Bessel transform introduced by Talman [J. Comput. Phys., 29, 35 (1978)], with logarithmically spaced computational grids. Numeric efficiency for arbitrary spatial dimensions is a commonly exploited virtue of this transform method. Here, we highlight another advantage of logarithmic grids, consisting i…
Since the early observation of nematic phases of disc-like clay colloids by Langmuir in 1938, the phase behaviour of such systems has resisted theoretical understanding. The main reason is that there is no satisfactory generalization for charged discs of the isotropic DLVO potential describing the effective interactions between a pair of spherical colloids in an electrolyte. In this contribution, we show how to construct such a pair potential, incorporating approximately both the non-linear effects of counter-ion condensation (charge renormalization) and the anisotropy of the charged platelets…
Recently reported colloidal aggregation data obtained for different monovalent salts (NaCl, NaNO$_3$, and NaSCN) and at high electrolyte concentrations are matched with the stochastic solutions of the master equation to obtain bond average lifetimes and bond formation probabilities. This was done for a cationic and an anionic system of similar particle size and absolute charge. Following the series Cl$^-$, NO$_3^-$, SCN$^-$, the parameters obtained from the fitting procedure to the kinetic data suggest: i) The existence of a potential of mean force (PMF) barrier and an increasing trend for it …
A theory is presented which allow us to accurately calculate the critical coagulation concentration (CCC) of hydrophobic colloidal suspensions. For positively charged particles the CCC's follow the Hofmeister (lyotropic) series. For negatively charged particles the series is reversed. We find that strongly polarizable chaotropic anions are driven towards the colloidal surface by electrostatic and hydrophobic forces. Within approximately one ionic radius from the surface, the chaotropic anions loose part of their hydration sheath and become strongly adsorbed. The kosmotropic anions, on the othe…
A method for measuring the pair interaction potential between colloidal particles by extrapolation measurement of collective structure to infinite dilution is presented and explored using simulation and experiment. The method is particularly well suited to systems in which the colloid is fluorescent and refractive index matched with the solvent. The method involves characterizing the potential of mean force between colloidal particles in suspension by measurement of the radial distribution function using 3D direct visualization. The potentials of mean force are extrapolated to infinite dilutio…
We study the thermal diffusion coefficient DT of a charged colloid in a temperature gradient, and find that it is to a large extent determined by the thermoelectric response of the electrolyte solution. The thermally induced salinity gradient leads in general to a strong increase with temperature. The difference of the heat of transport of coions and counterions gives rise to a thermoelectric field that drives the colloid to the cold or to the warm, depending on the sign of its charge. Our results provide an explanation for recent experimental findings on thermophoresis in colloidal suspension…
We study, using Monte Carlo simulations, the interaction between charged colloidal particles confined to the air-water interface. The dependence of force on ionic strength and counterion valence is explored. For 1:1 electrolyte, we find that the electrostatic interaction at the interface is very close to the one observed in the bulk. On the other hand, for salts with multivalent counterions, an interface produces an enhanced attraction between like charged colloids. Finally, we explore the effect of induced surface charge at the air-water interface on the interaction between colloidal particle…
Dielectrophoresis (DEP), the motion of polarizable particles in non-uniform electric fields, has become an important tool for the transport, separation, and characterization of microparticles in biomedical and nanoelectronics research. In this article we present, to our knowledge, the first molecular dynamics simulations of DEP of nanometer-sized colloidal particles. We introduce a simplified model for polarizable nanoparticles, consisting of a large charged macroion and oppositely charged microions, in an explicit solvent. The model is then used to study DEP motion of the particle at differen…
We theoretically study the electrolyte Seebeck effect in the vicinity of a heated metal nanostructure, such as the cap of an active Janus colloid in an electrolyte, or gold- coated interfaces in optofluidic devices. The thermocharge accumulated at the surface varies with the local temperature, thus modulating the diffuse part of the electric double layer. On a conducting surface with non-uniform temperature, the isopotential condition imposes a significant polarization charge within the metal. Surprisingly, this does not affect the slip velocity, which takes the same value on insulating and co…
An approach is proposed which allows to self-consistently calculate the structural and thermodynamic properties of highly charged aqueous colloidal suspensions. The method is based on the renormalized Jellium model with the background charge distribution related to the colloid-colloid correlation function. The theory is used to calculate the correlation functions and the effective colloidal charges for suspension containing additional monovalent electrolyte. The predictions of the theory are in excellent agreement with the Monte Carlo simulations.
Ion distribution in aqueous electrolytes near the interface plays critical roles in electrochemical, biological and colloidal systems and is expected to be particularly significant inside nanoconfined regions. Electroneutrality of the total charge inside nanoconfined regions is commonly assumed a priori in solving ion distribution of aqueous electrolytes nanoconfined by uncharged hydrophobic surfaces with no direct experimental validation. Here, we use a quantitative nuclear magnetic resonance approach to investigate the properties of aqueous electrolytes nanoconfined in graphitic-like nanopor…
Within a modified Poisson-Boltzmann theory we study the effect of Bjerrum pairs on the typical length scale $1/\barκ$ over which electric fields are screened in electrolyte solutions, taking into account a simple association-dissociation equilibrium between free ions and Bjerrum pairs. At low densities of Bjerrum pairs, this length scale is well approximated by the Debye length $1/κ\propto 1/\sqrt{ρ_\mathrm{s}}$, with $ρ_\mathrm{s}$ the free ion density. At high densities of Bjerrum pairs, however, we find $1/\barκ\propto \sqrt{ρ_\mathrm{s}}$ which is significantly larger than $1/κ$ due to the…
Dielectric particles in weakly conducting fluids rotate spontaneously when subject to strong electric fields. Such Quincke rotation near a plane electrode leads to particle translation that enables physical models of active matter. Here, we show that Quincke rollers can also exhibit oscillatory dynamics, whereby particles move back and forth about a fixed location. We explain how oscillations arise for micron-scale particles commensurate with the thickness of a field-induced boundary layer in the nonpolar electrolyte. This work enables the design of colloidal oscillators.
From microscopic fluid clusters to macroscopic droplets, the structure of fluids is governed by the Van der Waals force, a force that acts between polarizable objects. In this Letter, we derive a general theory that describes the non-equilibrium counterpart to the Van der Waals force, which emerges in spatially coherently fluctuating electromagnetic fields. We describe the formation of a novel and complex hierarchy of self-organized morphologies in magnetic and dielectric colloid systems. Most striking among these morphologies are dipolar foams - colloidal superstructures that swell against gr…
We study the combined effects of electrostatic and hydrodynamic interactions (HI) on the short-time dynamics of charge-stabilized colloidal spheres. For this purpose, we calculate the translational and the rotational self-diffusion coefficients, $D^t_s$ and $D^r_s$, as function of volume fraction $ϕ$ for various values of the effective particle charge $Z$ and various concentrations $n_s$ of added 1--1 electrolyte. Our results show that the self-diffusion coefficients in deionized suspensions are less affected by HI than in suspensions with added electrolyte. For very large $n_s$, we recover th…
We use a hydrodynamic reciprocal approach to phoretic motion to derive general expressions for the electrophoretic and thermophoretic mobility of weakly charged colloids in aqueous electrolyte solutions. Our approach shows that phoretic motion can be understood in terms of the interfacial transport of thermodynamic excess quantities that arises when a colloid is kept stationary inside a bulk fluid flow. The obtained expressions for the mobilities are extensions of previously known results as they can account for different hydrodynamic boundary conditions at the colloidal surface, irrespective …
Charge-stabilized colloidal spheres dispersed in weak 1:1 electrolytes are supposed to repel each other. Consequently, experimental evidence for anomalous long-ranged like-charged attractions induced by geometric confinement inspired a burst of activity. This has largely subsided because of nagging doubts regarding the experiments' reliability and interpretation. We describe a new class of thermodynamically self-consistent colloidal interaction measurements that confirm the appearance of pairwise attractions among colloidal spheres confined by one or two bounding walls. In addition to supporti…
We calculate phase diagrams of charged colloidal spheres (valency $Z$ and radius $a$) in a 1:1 electrolyte from multi-centered nonlinear Poisson-Boltzmann theory. Our theory takes into account charge-renormalization of the colloidal interactions and volume terms due to many-body effects. For valencies as small as Z=1 and as large as $10^4$ we find a gas--liquid spinodal instability in the colloid-salt phase diagram provided $Z\lB/a\gtrsim24\pm1$, where $\lB$ is the Bjerrum length.
The interactions between charged colloidal particles in an electrolyte may be described by usual Debye-Hückel theory provided the source of the electric field is suitably renormalized. For spherical colloids, we reconsider and simplify the treatment of the popular proposal put forward by Alexander {\it et al.} [J. Chem. Phys. {\bf 80}, 5776 (1984)], which has proven efficient in predicting renormalized quantities (charge and salt content). We give explicit formulae for the effective charge and describe the most efficient way to apply Alexander's et al. renormalization prescription in practice.…
Within the framework of a Modified Poisson-Boltzmann theory accounting for steric effects of microions, we prove analytically that the effective pair interactions between like-charge colloids immersed in a confined electrolyte are repulsive. Our approach encompasses and extends previously known results to the case of complete confinement, and further incorporates the finite size of the microions which is absent in the standard Poisson-Boltzmann theory.
