Simulation of charged systems in heterogeneous dielectric media via a true energy functional

Conduction in heterogeneous systems in the low-frequency regime: variational principles and boundary integral equations

The response of a homogeneous material to the presence of an external low-frequency oscillating electric field can be described by means of an effective complex conductivity. Low frequencies are characterized by negligible magnetic and radiative effects. The properties of heterogeneous systems, composed of multiple homogeneous regions, can be determined from those of the individual components and their geometric arrangement. Examples of such heterogeneous systems include soft materials such as colloidal suspensions, electrolyte systems, and biological tissues. The difference in the intrinsic conductivities between the homogeneous regions leads to the creation of an oscillating charge density localized at the interfaces between these regions. We show how to express key properties of these systems using this dynamic charge as a fundamental variable. We derive a boundary integral equation for the charges and reconstruct potentials and fields from its solution. We present a variational principle that recovers the fundamental equations for the system in terms of the oscillating charge and show that, in some formulations, the associated functional can be interpreted in terms of the power dissipated in the system. The boundary integral equations are numerically solved using a finite element method for a few illustrative cases.

Broken symmetries in quasi-2D charged systems via negative dielectric confinement

We report spontaneous symmetry breaking (SSB) phenomena in symmetrically charged binary particle systems under planar nanoconfinement with negative dielectric constants. The SSB is triggered solely via the dielectric confinement effect, without any external fields. The mechanism of SSB is found to be caused by the strong polarization field enhanced by nanoconfinement, giving rise to charge/field oscillations in the transverse directions. Interestingly, dielectric contrast can even determine the degree of SSB in transverse and longitudinal dimensions, forming charge-separated interfacial liquids and clusters on square lattices. Furthermore, we analytically show that the formed lattice constant is determined by the dielectric mismatch and the length scale of confinement, which is validated via molecular dynamics simulations. The novel broken symmetry mechanism may provide new insights into the study of quasi-2D systems and the design of future nanodevices.

Surface polarization enhances ionic transport and correlations in electrolyte solutions nanoconfined by conductors

Layered materials that perform mixed electron and ion transport are promising for energy harvesting, water desalination, and bioinspired functionalities. These functionalities depend on the interaction between ionic and electronic charges on the surface of materials. Here we investigate ion transport by an external electric field in an electrolyte solution confined in slit-like channels formed by two surfaces separated by distances that fit only a few water layers. We study different electrolyte solutions containing monovalent, divalent, and trivalent cations, and we consider walls made of non-polarizable surfaces and conductors. We show that considering the surface polarization of the confining surfaces can result in a significant increase in ionic conduction. The ionic conductivity is increased because the conductors' screening of electrostatic interactions enhances ionic correlations, leading to faster collective transport within the slit. While important, the change in water's dielectric constant in confinement is not enough to explain the enhancement of ion transport in polarizable slit-like channels.

Rapid Precipitation of Ionomers for Stabilization of Polymeric Colloids

Polymeric colloids have shown potential as "building blocks" in applications ranging from formulations of Pickering emulsions and drug delivery systems to advanced materials, including colloidal crystals and composites. However, for applications requiring tunable properties of charged colloids, obstacles in fabrication can arise through limitations in process scalability and chemical versatility. In this work, the capabilities of flash nanoprecipitation (FNP), a scalable nanoparticle (NP) fabrication technology, are expanded to produce charged polystyrene colloids using sulfonated polystyrene ionomers as a new class of NP stabilizers. Through experimental exploration of formulation parameters, increases in the ionomer content are shown to reduce the particle size, mitigating a significant trade-off between the final particle size and inlet concentration; thus, expanding the processable material throughput of FNP. Further, the degree of sulfonation is found to impact stabilization with optimal performance achieved by selecting ionomers with intermediate (2.45-5.2 mol %) sulfonation. Simulations of single ionomer chains and their arrangement in multicomponent NPs provide molecular insights into the assembly and structure of NPs wherein the partitioning of ionomers to the particle surface depends on the polymer molecular weight and degree of sulfonation. By combining the insights from simulations with diffusion-limited growth kinetics and parametric fits to experimental data, a simple design formulation relation is proposed and validated. This work highlights the potential of ionomer-based stabilizers for controllably producing charged NP dispersions in a scalable manner.

Predictive Molecular Models for Charged Materials Systems: From Energy Materials to Biomacromolecules

Electrostatic interactions play a dominant role in charged materials systems. Understanding the complex correlation between macroscopic properties with microscopic structures is of critical importance to develop rational design strategies for advanced materials. But the complexity of this challenging task is augmented by interfaces present in the charged materials systems, such as electrode-electrolyte interfaces or biological membranes. Over the last decades, predictive molecular simulations that are founded in fundamental physics and optimized for charged interfacial systems have proven their value in providing molecular understanding of physicochemical properties and functional mechanisms for diverse materials. Novel design strategies utilizing predictive models have been suggested as promising route for the rational design of materials with tailored properties. Here, an overview of recent advances in the understanding of charged interfacial systems aided by predictive molecular simulations is presented. Focusing on three types of charged interfaces found in energy materials and biomacromolecules, how the molecular models characterize ion structure, charge transport, morphology relation to the environment, and the thermodynamics/kinetics of molecular binding at the interfaces is discussed. The critical analysis brings two prominent field of energy materials and biological science under common perspective, to stimulate crossover in both research field that have been largely separated.

Pimples reduce and dimples enhance flat dielectric surface image repulsion

In solid-liquid, or liquid-liquid, interfaces with dielectric contrast, charged particles interact with the induced polarization charge of the interface. These interactions contribute to an effective self-energy of the bulk ions and mediate ion-ion interactions. For flat interfaces, the self-energy and the mediated interactions are neatly constructed by the image charge method. For other geometries, explicit results are scarce and the problem must be treated via approximations or direct computation. The case of interfaces with roughness is of great practical importance. This article provides analytical results, valid to first-order in perturbation theory, for the self-energy of particles near rough substrates. Explicit formulas are provided for the case of a sinusoidal deformation of a flat surface. Generic deformations can be treated by superposition. In addition to results for the self-energy, the surface polarization charge is presented as a quadrature. The interaction between an ion and the deformed surface is modified by the change in relative distance as well as by the local curvature of the surface. Solid walls, with a lower dielectric constant than the liquid, repel all ions. We show that the repulsion is reduced by local convexity and enhanced by concavity; dimples are more repulsive than pimples.

From predictive modelling to machine learning and reverse engineering of colloidal self-assembly

An overwhelming diversity of colloidal building blocks with distinct sizes, materials and tunable interaction potentials are now available for colloidal self-assembly. The application space for materials composed of these building blocks is vast. To make progress in the rational design of new self-assembled materials, it is desirable to guide the experimental synthesis efforts by computational modelling. Here, we discuss computer simulation methods and strategies used for the design of soft materials created through bottom-up self-assembly of colloids and nanoparticles. We describe simulation techniques for investigating the self-assembly behaviour of colloidal suspensions, including crystal structure prediction methods, phase diagram calculations and enhanced sampling techniques, as well as their limitations. We also discuss the recent surge of interest in machine learning and reverse-engineering methods. Although their implementation in the colloidal realm is still in its infancy, we anticipate that these data-science tools offer new paradigms in understanding, predicting and (inverse) design of novel colloidal materials.

Electric-field-based Poisson-Boltzmann theory: Treating mobile charge as polarization

Mobile charge in an electrolytic solution can in principle be represented as the divergence of ionic polarization. After adding explicit solvent polarization a finite volume of an electrolyte can then be treated as a composite nonuniform dielectric body. Writing the electrostatic interactions as an integral over electric-field energy density we show that the Poisson-Boltzmann functional in this formulation is convex and can be used to derive the equilibrium equations for electric potential and ion concentration by a variational procedure developed by Ericksen for dielectric continua [J. L. Ericksen, Arch. Rational Mech. Anal. 183, 299 (2007)AVRMAW0003-952710.1007/s00205-006-0042-4]. The Maxwell field equations are enforced by extending the set of variational parameters by a vector potential representing the dielectric displacement which is fully transverse in a dielectric system without embedded external charge. The electric-field energy density in this representation is a function of the vector potential and the sum of ionic and solvent polarization making the mutual screening explicit. Transverse polarization is accounted for by construction, lifting the restriction to longitudinal polarization inherent in the electrostatic potential based formulation of Poisson-Boltzmann mean field theory.

Universal Reduction in Dielectric Response of Confined Fluids

Dielectric permittivity is central to many biological and physiochemical systems, as it affects the long-range electrostatic interactions. Similar to many fluid properties, confinement greatly alters the dielectric response of polar liquids. Many studies have focused on the reduction of the dielectric response of water under confinement. Here, using molecular dynamics simulations, statistical-mechanical theories, and multiscale methods, we study the out-of-plane (z-axis) dielectric response of protic and aprotic fluids confined inside slit-like graphene channels. We show that the reduction in perpendicular permittivity is universal for all the fluids and exhibits a Langevin-like behavior as a function of channel width. We show that this reduction is due to the favorable in-plane (x-y plane) dipole-dipole electrostatic interactions of the interfacial fluid layer. Furthermore, we observe an anomalously low dielectric response under an extreme confinement.

Surface polarization effects in confined polyelectrolyte solutions

Understanding nanoscale interactions at the interface between two media with different dielectric constants is crucial for controlling many environmental and biological processes, and for improving the efficiency of energy storage devices. In this contributed paper, we show that polarization effects due to such dielectric mismatch remarkably influence the double-layer structure of a polyelectrolyte solution confined between two charged surfaces. Surprisingly, the electrostatic potential across the adsorbed polyelectrolyte double layer at the confining surface is found to decrease with increasing surface charge density, indicative of a negative differential capacitance. Furthermore, in the presence of polarization effects, the electrostatic energy stored in the double-layer structure is enhanced with an increase in the charge amplification, which is the absorption of ions on a like-charged surface. We also find that all of the important double-layer properties, such as charge amplification, energy storage, and differential capacitance, strongly depend on the polyelectrolyte backbone flexibility and the solvent quality. These interesting behaviors are attributed to the interplay between the conformational entropy of the confined polyelectrolytes, the Coulombic interaction between the charged species, and the repulsion from the surfaces with lower dielectric constant.

Simulations of Coulomb systems confined by polarizable surfaces using periodic Green functions

We present an efficient approach for simulating Coulomb systems confined by planar polarizable surfaces. The method is based on the solution of the Poisson equation using periodic Green functions. It is shown that the electrostatic energy arising from the surface polarization can be decoupled from the energy due to the direct Coulomb interaction between the ions. This allows us to combine an efficient Ewald summation method, or any other fast method for summing over the replicas, with the polarization contribution calculated using Green function techniques. We apply the method to calculate density profiles of ions confined between the charged dielectric and metal surfaces.

Surface Polarization Effects on Ion-Containing Emulsions

Surface polarization in ion-containing heterogeneous dielectric media such as cell media and emulsions is determined by and determines the positions of the ions. We compute the surface polarization self-consistently as the ions move and analyze their effects on the interactions between electro-neutral, ion-containing droplets using coarse-grained molecular dynamics simulations based on the true energy functional. For water droplets immersed in oil, the interdroplet interaction is attractive, and the surface polarization makes the major contribution. By contrast, for oil droplets in water, the ion-surface induced charge interaction is repulsive and counteracts the attraction between the ions, leading to a small attractive interaction between the droplets. This research improves our understanding of self-assembly in mixed phases such as metal extraction for recovering rare earth elements and nuclear waste as well as water purification.

Ion Structure Near a Core-Shell Dielectric Nanoparticle

A generalized image charge formulation is proposed for the Green's function of a core-shell dielectric nanoparticle for which theoretical and simulation investigations are rarely reported due to the difficulty of resolving the dielectric heterogeneity. Based on the formulation, an efficient and accurate algorithm is developed for calculating electrostatic polarization charges of mobile ions, allowing us to study related physical systems using the Monte Carlo algorithm. The computer simulations show that a fine-tuning of the shell thickness or the ion-interface correlation strength can greatly alter electric double-layer structures and capacitances, owing to the complicated interplay between dielectric boundary effects and ion-interface correlations.

Hybrid Monte Carlo and continuum modeling of electrolytes with concentration-induced dielectric variations

The distribution of ions near a charged surface is an important quantity in many biological and material processes, and has been therefore investigated intensively. However, few theoretical and simulation approaches have included the influence of concentration-induced variations in the local dielectric permittivity of an underlying electrolyte solution. Such local variations have long been observed and known to affect the properties of ionic solution in the bulk and around the charged surface. We propose a hybrid computational model that combines Monte Carlo simulations with continuum electrostatic modeling to investigate such properties. A key component in our hybrid model is a semianalytical formula for the ion-ion interaction energy in a dielectrically inhomogeneous environment. This formula is obtained by solving for the Green's function Poisson's equation with ionic-concentration-dependent dielectric permittivity using a harmonic interpolation method and spherical harmonic series. We also construct a self-consistent continuum model of electrostatics to describe the effect of ionic-concentration-dependent dielectric permittivity and the resulting self-energy contribution. With extensive numerical simulations, we verify the convergence of our hybrid simulation scheme, show the qualitatively different structures of ionic distribution due to the concentration-induced dielectric variations, and compare our simulation results with the self-consistent continuum model. In particular, we study the differences between weakly and strongly charged surfaces and multivalencies of counterions. Our hybrid simulations conform particularly the depletion of ionic concentrations near a charged surface and also capture the charge inversion. We discuss several issues and possible further improvement of our approach for simulations of large charged systems.

Water Dielectric Effects in Planar Confinement

We investigate the dielectric profile of water confined between two planar polar walls using atomistic molecular dynamics simulations. For a water slab thickness below 1 nm the dielectric response is highly asymmetric: while the parallel component slightly increases compared to bulk, the perpendicular one decreases drastically due to anticorrelated polarization of neighboring water molecules. We demonstrate the importance of the dielectric contribution due to flexible polar headgroups and derive an effective dielectric tensorial box model suitable for coarse-grained electrostatic modeling.

Computer Simulations of Ion Transport in Polymer Electrolyte Membranes

Understanding the mechanisms and optimizing ion transport in polymer membranes have been the subject of active research for more than three decades. We present an overview of the progress and challenges involved with the modeling and simulation aspects of the ion transport properties of polymer membranes. We are concerned mainly with atomistic and coarser level simulation studies and discuss some salient work in the context of pure binary and single ion conducting polymer electrolytes, polymer nanocomposites, block copolymers, and ionic liquid-based hybrid electrolytes. We conclude with an outlook highlighting future directions.

Charge reversal at a planar boundary between two dielectrics

Despite the ubiquitous character and relevance of the electric double layer in the entire realm of interface and colloid science, very little is known of the effect that surface heterogeneity exerts on the underlying mechanisms of ion adsorption. Herein, computer simulations offer a perspective that, in sharp contrast to the homogeneously charged surface, discrete groups promote multivalent counterion binding, leading to charge reversal but possibly having not a sign change of the electrophoretic mobility. Counterintuitively, the introduction of dielectric images yields a significantly greater accumulation of counterions, which further facilitates the magnitude of charge reversal. The reported results are very sensitive to both the degree of ion hydration and the representation of surface charges. Our findings shed light on the mechanism for charge reversal over a broad range of coupling regimes operating the adsorption of counterions through surface group bridging attraction with their own images and provide opportunities for experimental studies and theoretical development.

Computer simulations of single particles in external electric fields

Applying electric fields is an attractive way to control and manipulate single particles or molecules, e.g., in lab-on-a-chip devices. However, the response of nanosize objects in electrolyte solution to external fields is far from trivial. It is the result of a variety of dynamical processes taking place in the ion cloud surrounding charged particles and in the bulk electrolyte, and it is governed by an intricate interplay of electrostatic and hydrodynamic interactions. Already systems composed of one single particle in electrolyte solution exhibit a complex dynamical behaviour. In this review, we discuss recent coarse-grained simulations that have been performed to obtain a molecular-level understanding of the dynamic and dielectric response of single particles and single macromolecules to external electric fields. We address both the response of charged particles to constant fields (DC fields), which can be characterized by an electrophoretic mobility, and the dielectric response of both uncharged and charged particles to alternating fields (AC fields), which is described by a complex polarizability. Furthermore, we give a brief survey of simulation algorithms and highlight some recent developments.

Importance of varying permittivity on the conductivity of polyelectrolyte solutions

Dissolved ions can alter the local permittivity of water; nevertheless most theories and simulations ignore this fact. We present a novel algorithm for treating spatial and temporal variations in the permittivity and use it to measure the equivalent conductivity of a salt-free polyelectrolyte solution. Our new approach quantitatively reproduces experimental results unlike simulations with a constant permittivity that even qualitatively fail to describe the data. We can relate this success to a change in the ion distribution close to the polymer due to the buildup of a permittivity gradient.

Electrolytes between dielectric charged surfaces: Simulations and theory

We present a simulation method to study electrolyte solutions in a dielectric slab geometry using a modified 3D Ewald summation. The method is fast and easy to implement, allowing us to rapidly resum an infinite series of image charges. In the weak coupling limit, we also develop a mean-field theory which allows us to predict the ionic distribution between the dielectric charged plates. The agreement between both approaches, theoretical and simulational, is very good, validating both methods. Examples of ionic density profiles in the strong electrostatic coupling limit are also presented. Finally, we explore the confinement of charge asymmetric electrolytes between neutral surfaces.

Generic, phenomenological, on-the-fly renormalized repulsion model for self-limited organization of terminal supraparticle assemblies

Self-limited, or terminal, supraparticles have long received great interest because of their abundance in biological systems (DNA bundles and virus capsids) and their potential use in a host of applications ranging from photonics and catalysis to encapsulation for drug delivery. Moreover, soft, uniform colloidal aggregates are a promising candidate for quasicrystal and other hierarchical assemblies. In this work, we present a generic coarse-grained model that captures the formation of self-limited assemblies observed in various soft-matter systems including nanoparticles, colloids, and polyelectrolytes. Using molecular dynamics simulations, we demonstrate that the assembly process is self-limited when the repulsion between the particles is renormalized to balance their attraction during aggregation. The uniform finite-sized aggregates are further shown to be thermodynamically stable and tunable with a single dimensionless parameter. We find large aggregates self-organize internally into a core-shell morphology and exhibit anomalous uniformity when the constituent nanoparticles have a polydisperse size distribution.

Convexity and Stiffness in Energy Functions for Electrostatic Simulations

We study the properties of convex functionals which have been proposed for the simulation of charged molecular systems within the Poisson-Boltzmann approximation. We consider the extent to which the functionals reproduce the true fluctuations of electrolytes and thus the one-loop correction to mean field theory-including the Debye-Hückel correction to the free energy of ionic solutions. We also compare the functionals for use in numerical optimization of a mean field model of a charged polymer and show that different functionals have very different stiffnesses leading to substantial differences in accuracy and speed.

Electrostatics-driven shape transitions in soft shells

Manipulating the shape of nanoscale objects in a controllable fashion is at the heart of designing materials that act as building blocks for self-assembly or serve as targeted drug delivery carriers. Inducing shape deformations by controlling external parameters is also an important way of designing biomimetic membranes. In this paper, we demonstrate that electrostatics can be used as a tool to manipulate the shape of soft, closed membranes by tuning environmental conditions such as the electrolyte concentration in the medium. Using a molecular dynamics-based simulated annealing procedure, we investigate charged elastic shells that do not exchange material with their environment, such as elastic membranes formed in emulsions or synthetic nanocontainers. We find that by decreasing the salt concentration or increasing the total charge on the shell's surface, the spherical symmetry is broken, leading to the formation of ellipsoids, discs, and bowls. Shape changes are accompanied by a significant lowering of the electrostatic energy and a rise in the surface area of the shell. To substantiate our simulation findings, we show analytically that a uniformly charged disc has a lower Coulomb energy than a sphere of the same volume. Further, we test the robustness of our results by including the effects of charge renormalization in the analysis of the shape transitions and find the latter to be feasible for a wide range of shell volume fractions.

Dielectric effects in the self-assembly of binary colloidal aggregates

Electrostatic interactions play an important role in numerous self-assembly phenomena, including colloidal aggregation. Although colloids typically have a dielectric constant that differs from the surrounding solvent, the effective interactions that arise from inhomogeneous polarization charge distributions are generally neglected in theoretical and computational studies. We introduce an efficient technique to resolve polarization charges in dynamical dielectric geometries, and demonstrate that dielectric effects qualitatively alter the predicted self-assembled structures, with surprising colloidal strings arising from many-body effects.

Generating true minima in constrained variational formulations via modified Lagrange multipliers

Variational principles are important in the investigation of large classes of physical systems. They can be used both as analytical methods as well as starting points for the formulation of powerful computational techniques such as dynamical optimization methods. Systems with charged objects in dielectric media and systems with magnetically active particles are important examples. In these examples and other important cases, the variational principles describing the system are required to obey a number of constraints. These constraints are implemented within the variational formulation by means of Lagrange multipliers. Such constrained variational formulations are in general not unique. For the application of efficient simulation methods, one must find specific formulations that satisfy a number of important conditions. An often required condition is that the functional be positive-definite, in other words, its extrema be actual minima. In this article, we present a general approach to attack the problem of finding, among equivalent variational functionals, those that generate true minima. The method is based on the modification of the Lagrange multiplier which allows us to generate large families of effective variational formulations associated with a single original constrained variational principle. We demonstrate its application to different examples and, in particular, to the important cases of Poisson and Poisson-Boltzmann equations. We show how to obtain variational formulations for these systems with extrema that are always minima.