Effective macroion-macroion potentials in asymmetric electrolytes

Ion correlation-driven like-charge attraction in multivalent salt solutions

The electrostatic double layer force is key to determining the stability and self-assembly of charged colloids and many other soft matter systems. Fully understanding the attractive force between two like-charged surfaces remains a great challenge. Here, we apply the modified Gaussian renormalized fluctuation theory to study ion correlation-driven like-charge attraction in multivalent salt solutions. The effects of spatially varying ion correlations on the structure of overlapping double layers and their free energy are self-consistently accounted for. In the presence of multivalent salts, increasing surface charge or counterion valency leads to a short-range attraction. We demonstrate that although both overcharging and like-charge attraction are outcomes of ion correlation, there is no causal relationship between them. Our theory also captures the non-monotonic dependence of like-charge attraction on multivalent salt concentration. The reduction of attraction at high salt concentrations could be a contributing factor toward the reentrant stability of charged colloidal suspensions. Our theoretical predictions are consistent with the observations reported in experiments and simulations.

Assessment of the Wolf method using the Stillinger-Lovett sum rules: From strong electrolytes to weakly charged colloidal dispersions

The Ewald method has been the cornerstone in molecular simulations for modeling electrostatic interactions of charge-stabilized many-body systems. In the late 1990s, Wolf and collaborators developed an alternative route to describe the long-range nature of electrostatic interactions; from a computational perspective, this method provides a more efficient and straightforward way to implement long-range electrostatic interactions than the Ewald method. Despite these advantages, the validity of the Wolf potential to account for the electrostatic contribution in charged fluids remains controversial. To alleviate this situation, in this contribution, we implement the Wolf summation method to both electrolyte solutions and charged colloids with moderate size and charge asymmetries in order to assess the accuracy and validity of the method. To this end, we verify that the proper selection of parameters within the Wolf method leads to results that are in good agreement with those obtained through the standard Ewald method and the theory of integral equations of simple liquids within the so-called hypernetted chain approximation. Furthermore, we show that the results obtained with the original Wolf method do satisfy the moment conditions described by the Stillinger-Lovett sum rules, which are directly related to the local electroneutrality condition and the electrostatic screening in the Debye-Hückel regime. Hence, the fact that the solution provided by the Wolf method satisfies the first and second moments of Stillinger-Lovett proves, for the first time, the reliability of the method to correctly incorporate the electrostatic contribution in charge-stabilized fluids. This makes the Wolf method a powerful alternative compared to more demanding computational approaches.

Clustering and assembly dynamics of a one-dimensional microphase former

Both ordered and disordered microphases ubiquitously form in suspensions of particles that interact through competing short-range attraction and long-range repulsion (SALR). While ordered microphases are more appealing materials targets, understanding the rich structural and dynamical properties of their disordered counterparts is essential to controlling their mesoscale assembly. Here, we study the disordered regime of a one-dimensional (1D) SALR model, whose simplicity enables detailed analysis by transfer matrices and Monte Carlo simulations. We first characterize the signature of the clustering process on macroscopic observables, and then assess the equilibration dynamics of various simulation algorithms. We notably find that cluster moves markedly accelerate the mixing time, but that event chains are of limited help in the clustering regime. These insights will inspire further study of three-dimensional microphase formers.

Multiscale Modelling of Bionano Interface

We present a framework for coarse-grained modelling of the interface between foreign nanoparticles (NP) and biological fluids and membranes. Our model includes united-atom presentations of membrane lipids and globular proteins in implicit solvent, which are based on all-atom structures of the corresponding molecules and parameterised using experimental data or atomistic simulation results. The NPs are modelled by homogeneous spheres that interact with the beads of biomolecules via a central force that depends on the NP size. The proposed methodology is used to predict the adsorption energies for human blood plasma proteins on NPs of different sizes as well as the preferred orientation of the molecules upon adsorption. Our approach allows one to rank the proteins by their binding affinity to the NP, which can be used for predicting the composition of the NP-protein corona for the corresponding material. We also show how the model can be used for studying NP interaction with a lipid bilayer membrane and thus can provide a mechanistic insight for modelling NP toxicity.

Hydrotrope induced structural modifications in CTAB/butanol/water/isooctane reverse micellar systems

SAXS and NMR studies to gain insight of structural alterations in reverse micellar nanotemplates in presence of hydrotropes.

Microsecond simulations of DNA and ion transport in nanopores with novel ion-ion and ion-nucleotides effective potentials

We developed a novel scheme based on the grand-canonical Monte Carlo/Brownian dynamics simulations and have extended it to studies of ion currents across three nanopores with the potential for single-stranded DNA (ssDNA) sequencing: solid-state nanopore Si₃N₄, α-hemolysin, and E111N/M113Y/K147N mutant. To describe nucleotide-specific ion dynamics compatible with ssDNA coarse-grained model, we used the inverse Monte Carlo protocol, which maps the relevant ion-nucleotide distribution functions from all-atom molecular dynamics (MD) simulations. Combined with the previously developed simulation platform for Brownian dynamics simulations of ion transport, it allows for microsecond- and millisecond-long simulations of ssDNA dynamics in the nanopore with a conductance computation accuracy that equals or exceeds that of all-atom MD simulations. In spite of the simplifications, the protocol produces results that agree with the results of previous studies on ion conductance across open channels and provide direct correlations with experimentally measured blockade currents and ion conductances that have been estimated from all-atom MD simulations.

MagiC: Software Package for Multiscale Modeling

We present software package MagiC, which is designed to perform systematic structure-based coarse graining of molecular models. The effective pairwise potentials between coarse-grained sites of low-resolution molecular models are constructed to reproduce structural distribution functions obtained from the modeling of the system in a high resolution (atomistic) description. The software supports coarse-grained tabulated intramolecular bond and angle interactions, as well as tabulated nonbonded interactions between different site types in the coarse-grained system, with the treatment of long-range electrostatic forces by the Ewald summation. Two methods of effective potential refinement are implemented: iterative Boltzmann inversion and inverse Monte Carlo, the latter accounting for cross-correlations between pair interactions. MagiC uses its own Metropolis Monte Carlo sampling engine, allowing parallel simulation of many copies of the system with subsequent averaging of the properties, which provides fast convergence of the method with nearly linear scaling at parallel execution.

Coarse-graining intermolecular interactions in dispersions of highly charged colloids

Effective pair potentials between charged colloids, obtained from Monte Carlo simulations of two single colloids in a closed cell at the primitive model level, are shown to reproduce accurately the structure of aqueous salt-free colloidal dispersions, as determined from full primitive model simulations by Linse et al. (Linse, P.; Lobaskin, V. Electrostatic Attraction and Phase Separation in Solutions of Like-Charged Colloidal Particles. Phys. Rev. Lett.1999, 83, 4208). Excellent agreement is obtained even when ion-ion correlations are important and is in principle not limited to spherical particles, providing a potential route to coarse-grained colloidal interactions in more complex systems.

Renormalized jellium mean-field approximation for binary mixtures of charged colloids

In this work the renormalized jellium model of colloidal suspensions, originally proposed by Trizac and Levin [Phys. Rev. E 69, 031403 (2004)], is extended to study mechanisms of charge renormalization in binary mixtures of charged colloids. We here apply our recent reformulation that introduces the requirement of self-consistency directly into the Poisson-Boltzmann equation, i.e., the background charge is explicitly replaced by the effective one, thus facilitating the whole charge renormalization scheme. We briefly discuss the reformulated model for monodisperse charged suspensions composed of either spheres or rods. In particular, we put emphasis on the effects of the surface charge variation, mixture composition, and particle size on the charge regulation of charge-stabilized colloidal suspensions.

Poisson-Boltzmann theory of charged colloids: limits of the cell model for salty suspensions

Thermodynamic properties of charge-stabilized colloidal suspensions and polyelectrolyte solutions are commonly modelled by implementing the mean-field Poisson-Boltzmann (PB) theory within a cell model. This approach models a bulk system by a single macroion, together with counterions and salt ions, confined to a symmetrically shaped, electroneutral cell. While easing numerical solution of the nonlinear PB equation, the cell model neglects microion-induced interactions and correlations between macroions, precluding modelling of macroion ordering phenomena. An alternative approach, which avoids the artificial constraints of cell geometry, exploits the mapping of a macroion-microion mixture onto a one-component model of pseudo-macroions governed by effective interparticle interactions. In practice, effective-interaction models are usually based on linear-screening approximations, which can accurately describe strong nonlinear screening only by incorporating an effective (renormalized) macroion charge. Combining charge renormalization and linearized PB theories, in both the cell model and an effective-interaction (cell-free) model, we compute osmotic pressures of highly charged colloids and monovalent microions, in Donnan equilibrium with a salt reservoir, over a range of concentrations. By comparing predictions with primitive model simulation data for salt-free suspensions, and with predictions from nonlinear PB theory for salty suspensions, we chart the limits of both the cell model and linear-screening approximations in modelling bulk thermodynamic properties. Up to moderately strong electrostatic couplings, the cell model proves accurate for predicting osmotic pressures of deionized (counterion-dominated) suspensions. With increasing salt concentration, however, the relative contribution of macroion interactions to the osmotic pressure grows, leading predictions from the cell and effective-interaction models to deviate. No evidence is found for a liquid-vapour phase instability driven by monovalent microions. These results may guide applications of PB theory to colloidal suspensions and other soft materials.

Density dependent interactions and structure of charged colloidal dispersions in the weak screening regime

We determine the structure of charge-stabilized colloidal suspensions at low ionic strength over an extended range of particle volume fractions using a combination of light and small angle neutron scattering experiments. The variation of the structure factor with concentration is analyzed within a one-component model of a colloidal suspension. We show that the observed structural behavior corresponds to a nonmonotonic density dependence of the colloid effective charge and the mean interparticle interaction energy. Our findings are corroborated by similar observations from primitive model computer simulations of salt-free colloidal suspensions.

Electroneutrality and phase behavior of colloidal suspensions

Several statistical mechanical theories predict that colloidal suspensions of highly charged macroions and monovalent microions can exhibit unusual thermodynamic phase behavior when strongly deionized. Density-functional, extended Debye-Hückel, and response theories, within mean-field and linearization approximations, predict a spinodal phase instability of charged colloids below a critical salt concentration. Poisson-Boltzmann cell model studies of suspensions in Donnan equilibrium with a salt reservoir demonstrate that effective interactions and osmotic pressures predicted by such theories can be sensitive to the choice of reference system, e.g., whether the microion density profiles are expanded about the average potential of the suspension or about the reservoir potential. By unifying Poisson-Boltzmann and response theories within a common perturbative framework, it is shown here that the choice of reference system is dictated by the constraint of global electroneutrality. On this basis, bulk suspensions are best modeled by density-dependent effective interactions derived from a closed reference system in which the counterions are confined to the same volume as the macroions. Lower-dimensional systems (e.g., monolayers, clusters), depending on the strength of macroion-counterion correlations, may be governed instead by density-independent effective interactions tied to an open reference system with counterions dispersed throughout the reservoir, possibly explaining the observed structural crossover in colloidal monolayers and anomalous metastability of colloidal crystallites.

Interactions from diffraction data: historical and comprehensive overview of simulation assisted methods

A large part of statistical mechanics is concerned with the determination of condensed matter structure on the basis of known microscopic interactions. An increasing emphasis has been put on the opposite situation in the last decades as well, where structural data, e.g. pair-distance statistics, are known from diffraction experiments, and one looks for the corresponding interaction functions. The solution of this inverse problem was searched for within the integral equation theories of condensed matter in the early investigations, but before long computer simulation assisted methods were suggested. The interest in this field showed an increasing trend after some attempts appeared in the late 1980s. Several methods were published in the 1990s, and one-two methods appear annually nowadays. In this paper a comprehensive and historical overview is given on the solution of the inverse problem with simulation assisted methods. Emphasis is put on the theoretical grounds of the methods, on the choice of possible input structural functions, on the numerically local or global schemes of the potential modifications, on some advantages and limits of the different methods and on the scientific impact of the methods.

Dynamics and stability of dispersions of polyelectrolyte-filled multilayer microcapsules

The authors report dynamic and coagulation properties of a dispersion of polyelectrolyte multilayer microcapsules filled with solutions of a strong polyelectrolyte. Microcapsules are shown to take a charge of the sign of encapsulated polyions and are characterized by a nonuniform distribution of inner polyions, which indicates a semipermeability of the shell and a leakage of counterions. The capsule self-diffusion coefficient in the vicinity of the similarly charged wall is measured using a particle tracking procedure from confocal images of the dispersion. The diffusion of capsules in the force field suggests that the effective interaction potential contains an electrostatic barrier, so that we deal with the same types of interaction forces as for solid particles. The theoretical estimates of the authors show that when microcapsules are in close proximity, their interaction should even be quantitatively the same as that of colloids with the same surface potential. However, due to the mobility of inner polyions they might repel stronger at large distances. The authors thus conclude that the encapsulation of charged polymers is an important factor in determining the adhesion and interaction properties of multilayer microcapsules.

Smoluchowski equation and the colloidal charge reversal

Smoluchowski equation and the Monte Carlo simulations are used to study the conditions leading to the reversal of the electrophoretic mobility. Zeta (zeta) potential is identified with the diffuse potential at the shear plane which, we argue, must be placed at least one ionic diameter away from the colloidal surface. For sufficiently strongly charged colloids, zeta potential changes sign as a function of the multivalent electrolyte concentration, resulting in a reversal of the electrophoretic mobility. This behavior occurs even for very small ions of 4 A diameter as long as the surface charge density of the colloidal particles is sufficiently large and the concentration of 1:1 electrolyte is sufficiently low.

Measuring inter-DNA potentials in solution

Interactions between short strands of DNA can be tuned from repulsive to attractive by varying solution conditions and have been quantified using small angle x-ray scattering techniques. The effective DNA interaction charge was extracted by fitting the scattering profiles with the generalized one-component method and inter-DNA Yukawa pair potentials. A significant charge is measured at low to moderate monovalent counterion concentrations, resulting in strong inter-DNA repulsion. The charge and repulsion diminish rapidly upon the addition of divalent counterions. An intriguing short range attraction is observed at surprisingly low divalent cation concentrations, approximately 16 mM Mg2+. Quantitative measurements of inter-DNA potentials are essential for improving models of fundamental interactions in biological systems.

Phase separation in charge-stabilized colloidal suspensions: influence of nonlinear screening

The phase behavior of charge-stabilized colloidal suspensions is modeled by a combination of response theory for electrostatic interparticle interactions and variational theory for free energies. Integrating out degrees of freedom of the microions (counterions, salt ions), the macroion-microion mixture is mapped onto a one-component system governed by effective macroion interactions. Linear response of microions to the electrostatic potential of the macroions results in a screened-Coulomb (Yukawa) effective pair potential and a one-body volume energy, while nonlinear response modifies the effective interactions [A. R. Denton, Phys. Rev. E 70, 031404 (2004)]. The volume energy and effective pair potential are taken as input to a variational free energy, based on thermodynamic perturbation theory. For both linear and first-order nonlinear effective interactions, a coexistence analysis applied to aqueous suspensions of highly charged macroions and monovalent microions yields bulk separation of macroion-rich and macroion-poor phases below a critical salt concentration, in qualitative agreement with predictions of related linearized theories [R. van Roij, M. Dijkstra, and J.-P. Hansen, Phys. Rev. E 59, 2010 (1999); P. B. Warren, J. Chem. Phys. 112, 4683 (2000)]. It is concluded that nonlinear screening can modify phase behavior but does not necessarily suppress bulk phase separation of de-ionized suspensions.

Effect of discrete macroion charge distributions in solutions of like-charged macroions

The effect of replacing the conventional uniform macroion surface charge density with discrete macroion charge distributions on structural properties of aqueous solutions of like-charged macroions has been investigated by Monte Carlo simulations. Two discrete charge distributions have been considered: point charges localized on the macroion surface and finite-sized charges protruding into the solution. Both discrete charge distributions have been examined with fixed and mobile macroion charges. Different boundary conditions have been applied to examine various properties. With point charges localized on the macroion surface, counterions become stronger accumulated to the macroion and the effect increases with counterion valence. As a consequence, with mono- and divalent counterions the potential of mean force between two macroions becomes less repulsive and with trivalent counterions more attractive. With protruding charges, the excluded volume effect dominates over the increased correlation ability; hence the counterions are less accumulated near the macroions and the potential of mean force between two macroions becomes more repulsive/less attractive.

Multiscale modeling of lipids and lipid bilayers

A multiscale modeling approach is applied for simulations of lipids and lipid assemblies on mesoscale. First, molecular dynamics simulation of initially disordered system of lipid molecules in water within all-atomic model was carried out. On the next stage, structural data obtained from the molecular dynamics (MD) simulation were used to build a coarse-grained (ten sites) lipid model, with effective interaction potentials computed by the inverse Monte Carlo method. Finally, several simulations of the coarse-grained model on longer length- and time-scale were performed, both within Monte Carlo and molecular dynamics simulations: a periodical sample of lipid molecules ordered in bilayer, a free sheet of such bilayer without periodic boundary conditions, formation of vesicle from a plain membrane, process of self-assembly of lipids randomly dispersed in volume. It was shown that the coarse-grained model, developed exclusively from all-atomic simulation data, reproduces well all the basic features of lipids in water solution.

Nonlinear screening and effective electrostatic interactions in charge-stabilized colloidal suspensions

A nonlinear response theory is developed and applied to electrostatic interactions between spherical macroions, screened by surrounding microions, in charge-stabilized colloidal suspensions. The theory describes leading-order nonlinear response of the microions (counterions, salt ions) to the electrostatic potential of the macroions and predicts microion-induced effective many-body interactions between macroions. A linear response approximation [A.R. Denton, Phys. Rev. E 62, 3855 (2000)] yields an effective pair potential of screened-Coulomb (Yukawa) form, as well as a one-body volume energy, which contributes to the free energy. Nonlinear response generates effective many-body interactions and essential corrections to both the effective pair potential and the volume energy. By adopting a random-phase approximation (RPA) for the response functions, and thus neglecting microion correlations, practical expressions are derived for the effective pair and triplet potentials and for the volume energy. Nonlinear screening is found to weaken repulsive pair interactions, induce attractive triplet interactions, and modify the volume energy. Numerical results for monovalent microions are in good agreement with available ab initio simulation data and demonstrate that nonlinear effects grow with increasing macroion charge and concentration and with decreasing salt concentration. In the dilute limit of zero macroion concentration, leading-order nonlinear corrections vanish. Finally, it is shown that nonlinear response theory, when combined with the RPA, is formally equivalent to the mean-field Poisson-Boltzmann theory and that the linear response approximation corresponds, within integral-equation theory, to a linearized hypernetted-chain closure.

Simulation of electric double layers with multivalent counterions: Ion size effect

In this paper, the structure of the electric double layer in the presence of (mostly) multivalent counterions is investigated through Monte Carlo simulations. Unlike previous similar studies addressing this matter, the difference of this study lies in the use of realistic hydrated ion sizes. Additionally, two different methods for calculating energies in the Metropolis algorithm are applied. The obtained results show that the conclusions of preceding papers must be revised. In particular, our simulations suggest the existence of certain ion layering effects at high surface charge densities, which are not accounted for by integral equation theories in the case of divalent counterions. These layering effects could justify why the overcharging phenomena due to ion size correlations are hardly observable in real colloids with divalent counterions. The existence of charge inversion due to ion size correlations (and without requiring specific counterion adsorption) is probed for trivalent counterions. Moreover, the hypernetted-chain/mean-spherical-approximation is tested under conditions not studied yet.

Monte carlo study of coulombic criticality in polyelectrolytes

The role of charges in determining the water solubility of polyelectrolytes, a question of considerable relevance to biology, is currently unresolved. We use computer simulations to study the purely Coulombic phase separation of flexible polyelectrolytes with monovalent counterions in an athermal solvent. In agreement with recent theories we find that the critical temperature for this transition increases with chain length, but that the critical density remains unchanged. We therefore stress that the phase behavior of polyelectrolytes is qualitatively different from uncharged polymers, where the critical density decreases towards zero for long chains.

Generalizations for the Potential of Mean Force between Two Isolated Colloidal Particles from Monte Carlo Simulations

A substantial amount of experimental and numerical evidence has shown that the Derjaguin-Landau-Verwey-Overbeek theory is not suitable for describing those colloidal solutions that contain multivalent counterions. Toward improved understanding of such solutions, the authors report Monte Carlo calculations wherein, following Rouzina and Bloomfield, they postulate that, in the absence of van der Waals forces, the overall force between two isolated charged colloidal particles in electrolyte solutions is determined by a dimensionless parameter Gamma=z(2)l(B)/a, which measures the electrostatic repulsion between counterions adsorbed on the macroion surface, where z = counterion valence, l(B)=Bjerrum length, and a = average separation between counterions on the macroion surface calculated as if the macroion were fully neutralized. The authors find, first, that the maximum repulsion between like-charged macroions occurs at Gamma approximately 0.5 and, second, that onset of attraction occurs at Gamma approximately 1.8, essentially independent of the valence and concentration of the surrounding electrolyte. These observations might provide new understanding of interactions between electrostatic double layers and perhaps offer explanations for some electrostatic phenomena related to interactions between DNA molecules or proteins.

"Ordered" structure in ionic dilute solutions: dendrimers with univalent and bivalent counterions

As an intermediate sample of ionic solutes between colloidal particles (macroions) and simple electrolyte ions, we made small-angle x-ray scattering (SAXS) measurements for aqueous solutions of poly(amido amine) dendrimers of three generations (G4, G7, and G10). The SAXS curves of univalent acid solutions showed a single scattering peak, as observed for synthetic macroions. The peak position was dependent on the dendrimer concentration but independent not only of the acid concentration (degree of protonation) but also of the counterion species. The effective charge density of the dendrimer determined by conductivity measurements was found to be insensitive to the acid concentration and the counterion species. The nearest neighbor interparticle distance 2D(exp) calculated from the peak position of the structure factor of G7 and G10 was obviously smaller, though slightly, than the average interparticle distance 2D(0) calculated from molecular weights and concentrations of dendrimers, implying that acid solutions of dendrimers formed the two-state structures by the attractive force. The ultra-small-angle x-ray scattering curve for the hydrochloric acid solution did not show an upturn, which indicates the existence of large scale structural inhomogeneities such as localized ordered structures, probably due to the weak attraction and hence less clear distinction of the ordered and disordered regions. For sulfuric acid solutions, clear scattering peaks were not observed. The bivalent counterions were more strongly associated with the dendrimer ions than the univalent ones. The resulting low charge number of the dendrimers with the bivalent counterion was confirmed directly by the conductivity measurements. These observations confirm that the counterion-mediated attraction does exist even with the univalent counterions and point out that the frequently advanced claim that the effective potential is essentially repulsive with univalent counterions while attraction appears with bivalent counterions is not necessarily correct. It is noted that the intensity of the counterion-mediated attraction in dendrimer solutions is dictated by both the effective charge density and the effective charge number, in contrast with macroionic solutions or colloidal dispersions in which only the effective charge density appeared to be important.