Interaction between two inhomogeneously charged parallel surfaces in the strong coupling regime

Electrostatics of patchy surfaces

In the study of colloidal, biological and electrochemical systems, it is customary to treat surfaces, macromolecules and electrodes as homogeneously charged. This simplified approach is proven successful in most cases, but fails to describe a wide range of heterogeneously charged surfaces commonly used in experiments. For example, recent experiments have revealed a long-range attraction between overall neutral surfaces, locally charged in a mosaic-like structure of positively and negatively charged domains ("patches"). Here, we review experimental and theoretical studies addressing the stability of heterogeneously charged surfaces, their effect on ionic profiles in solution, and the interaction between two such surfaces. We focus on electrostatics, and highlight the important new physical parameters appearing in the heterogeneous case, such as the largest patch size and inter-surface charge correlations.

Bundling in semiflexible polymers: A theoretical overview

Supramolecular assemblies of polymers are key modules to sustain the structure of cells and their function. The main elements of these assemblies are charged semiflexible polymers (polyelectrolytes) generally interacting via a long(er)-range repulsion and a short(er)-range attraction. The most common supramolecular structure formed by these polymers is the bundle. In the present paper, we critically review some recent theoretical and computational advances on the problem of bundle formation, and point a few promising directions for future work.

Strong-coupling electrostatic theory of polymer counterions close to planar charges

Strong-coupling phenomena, such as like-charge macroion attraction, opposite-charged macroion repulsion, charge renormalization, and charge inversion, are known to be mediated by multivalent counterions. Most theories treat the counterions as point charges and describe the system by a single coupling parameter that measures the strength of the Coulomb interactions. In many biological systems, the counterions are highly charged and have finite sizes and can be well-described by polyelectrolytes. The shapes and orientations of these polymer counterions play a major role in the thermodynamics of these systems. In this work we apply a field-theoretic description in the strong-coupling regime to the polymer counterions in the presence of a fixed charge distribution. We work out the special cases of rodlike polymer counterions confined by one, and two charged walls, respectively. The effects of the geometry of the rodlike counterions and the excluded volume of the walls on the density, pressure, and free energy of the rodlike counterions are discussed.

Ion-mediated interactions between net-neutral slabs: Weak and strong disorder effects

We investigate the effective interaction between two randomly charged but otherwise net-neutral, planar dielectric slabs immersed in an asymmetric Coulomb fluid containing a mixture of mobile monovalent and multivalent ions. The presence of charge disorder on the apposed bounding surfaces of the slabs leads to substantial qualitative changes in the way they interact, as compared with the standard picture provided by the van der Waals and image-induced, ion-depletion interactions. While, the latter predict purely attractive interactions between strictly neutral slabs, we show that the combined effects from surface charge disorder, image depletion, Debye (or salt) screening, and also, in particular, their coupling with multivalent ions, give rise to a more diverse behavior for the effective interaction between net-neutral slabs at nano-scale separations. Disorder effects show large variation depending on the properly quantified strength of disorder, leading either to non-monotonic effective interaction with both repulsive and attractive branches when the surface charges are weakly disordered (small disorder variance) or to a dominating attractive interaction that is larger both in its range and magnitude than what is predicted from the van der Waals and image-induced, ion-depletion interactions, when the surfaces are strongly disordered (large disorder variance).

Electrostatic interaction between nonuniformly charged colloids: experimental and numerical study

The influence of the surface charge distribution on the interaction between nanosized particles in water is reported. The distribution of charges at the surface of initially neutral microemulsion droplets has been modulated by additions of various oligomeric cationic surfactants. The osmotic compressibility of the doped microemulsions was measured by light and small-angle neutrons scattering and reveals that the overall effective interaction induced by the ionic groups is repulsive. However, particular charge distributions decrease the osmotic compressibility much less than others. Independent measurements of the activity of the bromide counterions with specific electrodes evidence a significant decrease in the effective charge, which, however, cannot account for the osmotic compressibility in the framework of the primitive model. The q dependence of the structure factor reveals an attractive contribution over a short distance. Numerical studies assign this attractive contribution to the overlap of hydration shells that are extended as a result of the charge localization.

Repulsion between oppositely charged planar macroions

The repulsive interaction between oppositely charged macroions is investigated using Grand Canonical Monte Carlo simulations of an unrestricted primitive model, including the effect of inhomogeneous surface charge and its density, the depth of surface charge, the cation size, and the dielectric permittivity of solvent and macroions, and their contrast. The origin of the repulsion is a combination of osmotic pressure and ionic screening resulting from excess salt between the macroions. The excess charge over-reduces the electrostatic attraction between macroions and raises the entropic repulsion. The magnitude of the repulsion increases when the dielectric constant of the solvent is lowered (below that of water) and/or the surface charge density is increased, in good agreement with experiment. Smaller size of surface charge and the cation, their discreteness and mobility are other factors that enhance the repulsion and charge inversion phenomenons.

Effect of charge inhomogeneity and mobility on colloid aggregation

The aggregation of inhomogeneously charged colloids with the same average charge is analyzed using Monte Carlo simulations. We find aggregation of colloids for sizes in the range 10-200 nm, which is similar to the range in which aggregation is observed in several experiments. The attraction arises from the strongly correlated electrostatic interactions associated with the increase in the counterion density in the region between the particles; this effect is enhanced by the discreteness and mobility of the surface charges. Larger colloids attract more strongly when their surface charges are discrete. We study the aggregation as functions of the surface charge density, counterion valence, and volume fraction.

Distribution of counterions and interaction between two similarly charged dielectric slabs: roles of charge discreteness and dielectric inhomogeneity

The distribution of counterions and the electrostatic interaction between two similarly charged dielectric slabs is studied in the strong coupling limit. Dielectric inhomogeneities and discreteness of charge on the slabs have been taken into account. It is found that the amount of dielectric constant difference between the slabs and the environment, and the discreteness of charge on the slabs have opposing effects on the equilibrium distribution of the counterions. At small interslab separations, increasing the amount of dielectric constant difference increases the tendency of the counterions toward the middle of the intersurface space between the slabs and the discreteness of charge pushes them to the surfaces of the slabs. In the limit of point charges, independent of the strength of dielectric inhomogeneity, counterions distribute near the surfaces of the slabs. The interaction between the slabs is attractive at low temperatures and its strength increases with the dielectric constant difference. At room temperature, the slabs may completely attract each other, reach to an equilibrium separation, or have two equilibrium separations with a barrier in between, depending on the system parameters.

Interactions between charged surfaces with ionizable sites

A key factor controlling the interactions between surfaces in aqueous solutions is the surface charge density. Surfaces typically become charged though a titration process where surface groups can become ionized based on their dissociation constant and the pH of the solution. In this work, we use a Monte Carlo method to treat this process in a system with two planar surfaces with explicitly described ionizable sites in a salt solution. We focus on a system with a surface density of ionizable sites set to 4.8 nm(-2), corresponding to silica. We find that the surface charge density changes as the surfaces come close to contact due to interactions between the ionizable groups on each surface. In addition, we observe an attraction between the surfaces above a threshold surface charge, in good agreement with previous theoretical predictions based on uniformly charged surfaces. However, close to contact we find the force is significantly different than for the uniformly charged case.

Monte carlo simulations of tau proteins: effect of phosphorylation

We perform Monte Carlo simulations of tau proteins bound to a cylinder that mimics a microtubule (MT), and then study them in solution. Tau protein binds to a highly anionic MT surface to stabilize the cylindrical structure of MT. The negatively charged tail domain floats away from the anionic MT surface while positively charged tau segments localize near the MT surface. Monte Carlo simulations demonstrate that, in 3RS tau isoform (which has three imperfect repeats (R) short (S) isoform), amino acids are more condensed near a highly charged interface compared to 4RL isoform (which has four imperfect repeats (R) long (L) isoform). In 4RL isoform, amino acids in tail domain stay mostly apart from the MT surface. In the bulk solution, dephosphorylated taus are separated due to Coulomb repulsion between similarly charged isoforms. Moderate phosphorylation of 3RS isoform decreases average intermolecular distance between dephosphorylated and phosphorylated taus and lead to their overlap. Further phosphorylation does not change noticeably the intermolecular distances.

Insights from Monte Carlo simulations on charge inversion of planar electric double layers in mixtures of asymmetric electrolytes

Monte Carlo simulations of a planar negatively charged dielectric interface in contact with a mixture of 1:1 and 3:1 electrolytes are carried out using the unrestricted primitive model under more realistic hydrated ion sizes. Two typical surface charge densities are chosen to represent the systems from the weak to strong coupling regimes. Our goal is to determine the dependence of the degree of charge inversion on increasing concentration of both mono- and trivalent salts and to provide a systematic study on this peculiar effect between short-range and electrostatic correlations. The numerical results show that addition of monovalent salt diminishes the condensation of trivalent counterions due to either the favorable solvation energy or the available space constraints. As the concentration of trivalent salt increases, on the other hand, the inclusion of the ionic size and size asymmetry results in a damped oscillatory charge inversion at low enough surface charge and another counterintuitive surface charge amplification. It is proposed that both of the anomalous events in the weak coupling regime are thought to be entropic in origin which is completely different from the electrostatic driven charge inversion in the strong coupling regime. In addition, the electrostatic images arising from the dielectric mismatch lead to a decaying depletion effect on the structure of double layer with growing salt concentration in the case of low charged interface but have no effect at high surface charge values. The microscopic information obtained here points to the need for a more quantitative theoretical treatment in describing the charge inversion phenomenon of real colloidal systems.

Dressed counterions: strong electrostatic coupling in the presence of salt

We reformulate the theory of strong electrostatic coupling in order to describe an asymmetric electrolyte solution of monovalent salt ions and polyvalent counterions using field-theoretical techniques and Monte Carlo simulations. The theory is based on an asymmetric treatment of the different components of the electrolyte solution. The weak coupling Debye-Hückel approach is used in order to describe the monovalent salt ions while a strong coupling approach is used to tackle the polyvalent counterions. This combined weak-strong coupling approach effectively leads to dressed interactions between polyvalent counterions and thus directly affects the correlation attraction mediated by polyvalent counterions between like-charged objects. The general theory is specifically applied to a system composed of two uniformly charged plane-parallel surfaces in the presence of salt and polyvalent counterions. In the strong coupling limit for polyvalent counterions, the comparison with Monte Carlo simulations shows good agreement for large enough values of the electrostatic coupling parameter. We delineate two limiting laws that in fact encompass all the Monte Carlo data.

Monte Carlo determination of mixed electrolytes next to a planar dielectric interface with different surface charge distributions

Employing canonical ensemble Monte Carlo simulations, we report a calculation of the distribution of small ions next to a planar negatively charged surface in the presence of mixed electrolytes of monovalent and trivalent salt ions within the framework of the primitive model under more realistic hydrated ion size conditions. The effects of surface charge discreteness and dielectric breakdown on charge inversion are discussed based on increasing concentration of both monovalent and trivalent salt. Moreover, a comparison of the simulation results for different discretization models is made along with the case of uniformly distributed charge in terms of the ionic density profiles as well as the integrated charge distribution function. For finite size charged groups located inside the lower dielectric region, a complete equivalence with the case of uniform distribution is observed if the quantities of interest are exclusively analyzed as a function of the distance to the charged interface. With protruding head groups into the aqueous solution, the excluded volume dominates over the correlation effect, therefore the ions are less accumulated in the vicinity of the charged surface, inducing that the onset position of charge inversion experiences an evident shift toward the aqueous environment. Overall, the effect of repulsive image forces on the diffuse double layer structure can be significant at low surface charge density irrespectively of surface charge distributions.

Lamellar phase coexistence induced by electrostatic interactions

Membranes containing highly charged biomolecules can have a minimal free-energy state at small separations that originates in the strongly correlated electrostatic interactions mediated by counterions. This phenomenon can lead to a condensed, lamellar phase of charged membranes that coexists in thermodynamic equilibrium with a very dilute membrane phase. Although the dilute phase is mostly water, entropy dictates that this phase must contain some membranes and counterions. Thus, electrostatics alone can give rise to the coexistence of a condensed and an unbound lamellar phase. We use numerical simulations to predict the nature of this coexistence when the charge density of the membrane is large, for the case of multivalent counterions and for a membrane charge that is characteristic of biomolecules. We also investigate the effects of counterion size and salt on the two coexisting phases. With increasing salt concentration, we predict that electrostatic screening by salt can destroy the phase separation.

The role of multipoles in counterion-mediated interactions between charged surfaces: strong and weak coupling

We present general arguments for the importance, or lack thereof, of structure in the charge distribution of counterions for counterion-mediated interactions between bounding symmetrically charged surfaces. We show that on the mean field or weak coupling level, the charge quadrupole contributes the lowest order modification to the contact value theorem and thus to the intersurface electrostatic interactions. The image effects are non-existent on the mean field level even with multipoles. On the strong coupling level the quadrupoles and higher order multipoles contribute additional terms to the interaction free energy only in the presence of dielectric inhomogeneities. Without them, the monopole is the only multipole that contributes to the strong coupling electrostatics. We explore the consequences of these statements in all their generality.

Weak- and strong-coupling electrostatic interactions between asymmetrically charged planar surfaces

We compare weak- and strong-coupling theory of counterion-mediated electrostatic interactions between two asymmetrically charged plates with extensive Monte Carlo simulations. Analytical results in both weak- and strong-coupling limits compare excellently with simulations in their respective regimes of validity. The system shows a surprisingly rich structure in terms of interactions between the surfaces as well as fundamental qualitative differences in behavior in the weak- and the strong-coupling limits.

Electrostatic image effects for counterions between charged planar walls

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

Effects of dielectric discontinuities on two charged plates

Counterions in a biological system are charged in water and interact with charged macroions, which are generally made up of hydrocarbons. The dielectric difference between water and the hydrocarbon substrates occurs naturally, and may greatly affect the electrostatic properties of biological systems. Particularly for a slab geometry, bulk counterions that are dissolved in water are driven to the midplane of the slab because of their repulsive interaction with their image charges. The pressure between two charged plates becomes less repulsive since the low dielectric constant of the hydrocarbon substrate creates stronger association between counterions and surface charges as compared to the case of no dielectric discontinuity.