Ion condensation in salt‐free dilute polyelectrolyte solutions

Dilute polyelectrolyte solutions: recent progress and open questions

Polyelectrolytes are a class of polymers possessing ionic groups on their repeating units. Since counterions can dissociate from the polymer backbone, polyelectrolyte chains are strongly influenced by electrostatic interactions. As a result, the physical properties of polyelectrolyte solutions are significantly different from those of electrically neutral polymers. The aim of this article is to highlight key results and some outstanding questions in the polyelectrolyte research from recent literature. We focus on the influence of electrostatics on conformational and hydrodynamic properties of polyelectrolyte chains. A compilation of experimental results from the literature reveals significant disparities with theoretical predictions. We also discuss a new class of polyelectrolytes called poly(ionic liquid)s that exhibit unique physical properties in comparison to ordinary polyelectrolytes. We conclude this review by listing some key research challenges in order to fully understand the conformation and dynamics of polyelectrolytes in solutions.

Effects of electrostatic coupling and surface polarization on polyelectrolyte brush structure

In this work, we perform molecular dynamics simulations to study a spherical polyelectrolyte brush. We explore the effects of surface polarization and electrostatic coupling on brush size and distribution of counterions. The method of image charges is considered to take into account surface polarization, considering a metallic, an unpolarizable, and a dielectric nano-core. It is observed that, for all cases, a moderate shrinking-swelling effect appears with an increase in the electrostatic coupling parameter. This effect occurs under high Manning ratios. The curves relating the average size of polyelectrolyte brush as a function of coupling show a minimum. The results show that the grafting density of polyelectrolytes on the nano-core surface plays an important role in the polarization effect. We consider a modified Poisson-Boltzmann theory to describe the counterion profiles around the brush in the case of unpolarizable nano-cores and weak electrostatic coupling.

Explicit characterization of counterion dynamics around a flexible polyelectrolyte

The article presents a comprehensive study of counterion dynamics around a generic linear polyelectrolyte chain with the help of coarse-grained computer simulations. The ion-chain coupling is discussed in the form of binding time, mean-square displacement (MSD) relative to the chain, local ion transport coefficient, and spatiotemporal correlations in the effective charge. We have shown that a counterion exhibits subdiffusive behavior 〈δR^{2}〉∼t^{δ}, δ≈0.9 w.r.t. chain's center of mass. The MSD of ions perpendicularly outward from the chain segment exhibits a smaller subdiffusive exponent compared to the one along the chain backbone. The effective diffusivity of ion is the lowest in chain's close proximity, extending up to the length-scale of radius of gyration R_{g}. Beyond R_{g} at larger distances, they attain diffusivity of free ion with a smooth cross-over from the adsorbed regime to the free ion regime. We have shown that the effective diffusivity drastically decreases for the multivalent ions, while the crossover length scale remains the same. Conversely, with increasing salt concentration the coupling-length scale reduces, while the diffusivity remains unaltered. The effective diffusivity of adsorbed-ion reveals an exponential reduction with electrostatic interaction strength. We further corroborate this from the binding time of ions on the chain, which also grows exponentially with the coupling strength of the ion-polymer duo. Moreover, the binding time of ions exhibits a weak dependence with salt concentration for the monovalent salt, while for multivalent salts the binding time decreases dramatically with concentration. Our work also elucidates fluctuations in the effective charge per site, where it exhibits strong negative correlations at short length-scales.

Charge regulation mechanism in end-tethered weak polyampholytes

Weak polyampholytes, containing oppositely charged dissociable groups, are expected to be responsive to changes in ionic conditions. Here, we determine structural and thermodynamic properties, including the charged groups' degrees of dissociation, of end-tethered weak polyampholyte layers as a function of salt concentration, pH, and the solvent quality. For diblock weak polyampholytes grafted by their acidic blocks, we find that the acidic monomers increase their charge while the basic monomers decrease their charge with decreasing salt concentration for pH values less than the pKa value of both monomers and vice versa when the pH > pKa. This complex charge regulation occurs because the electrostatic attraction between oppositely charged blocks is stronger than the repulsion between monomers with the same charge in both good and poor solvents when the screening by salt ions is weak. This is evidenced by the retraction of the top block into the bottom layer. In the case of poor solvent conditions to the basic block (the top block), we find lateral segregation of basic monomers into micelles, forming a two-dimensional hexagonal pattern on the surface at intermediate and high pH values for monovalent salt concentrations from 0.01 to 0.1 M. When the solvent is poor to both blocks, we find lateral segregation of the grafted acidic block into lamellae with longitudinal undulations of low and high acidic monomer density. By exploiting weak block polyampholytes, our work expands the parameter space for creating responsive surfaces stable over a wide range of pH and salt concentration.

Theoretical Modeling of Chemical Equilibrium in Weak Polyelectrolyte Layers on Curved Nanosystems

Surface functionalization with end-tethered weak polyelectrolytes (PE) is a versatile way to modify and control surface properties, given their ability to alter their degree of charge depending on external cues like pH and salt concentration. Weak PEs find usage in a wide range of applications, from colloidal stabilization, lubrication, adhesion, wetting to biomedical applications such as drug delivery and theranostics applications. They are also ubiquitous in many biological systems. Here, we present an overview of some of the main theoretical methods that we consider key in the field of weak PE at interfaces. Several applications involving engineered nanoparticles, synthetic and biological nanopores, as well as biological macromolecules are discussed to illustrate the salient features of systems involving weak PE near an interface or under (nano)confinement. The key feature is that by confining weak PEs near an interface the degree of charge is different from what would be expected in solution. This is the result of the strong coupling between structural organization of weak PE and its chemical state. The responsiveness of engineered and biological nanomaterials comprising weak PE combined with an adequate level of modeling can provide the keys to a rational design of smart nanosystems.

Assembly and Stability of Simian Virus 40 Polymorphs

Understanding viral assembly pathways is of critical importance to biology, medicine, and nanotechology. Here, we study the assembly path of a system with various structures, the simian vacuolating virus 40 (SV40) polymorphs. We simulate the templated assembly process of VP1 pentamers, which are the constituents of SV40, into icosahedal shells made of N = 12 pentamers (T = 1). The simulations include connections formed between pentamers by C-terminal flexible lateral units, termed here "C-terminal ligands", which are shown to control assembly behavior and shell dynamics. The model also incorporates electrostatic attractions between the N-terminal peptide strands (ligands) and the negatively charged cargo, allowing for agreement with experiments of RNA templated assembly at various pH and ionic conditions. During viral assembly, pentamers bound to any template increase its effective size due to the length and flexibility of the C-terminal ligands, which can connect to other VP1 pentamers and recruit them to a partially completed capsid. All closed shells formed other than the T = 1 feature the ability to dynamically rearrange and are thus termed "pseudo-closed". The N = 13 shell can even spontaneously "self-correct" by losing a pentamer and become a T = 1 capsid when the template size fluctuates. Bound pentamers recruiting additional pentamers to dynamically rearranging capsids allow closed shells to continue growing via the pseudo-closed growth mechanism, for which experimental evidence already exists. Overall, we show that the C-terminal ligands control the dynamic assembly paths of SV40 polymorphs.

Recent progress in the science of complex coacervation

Complex coacervation is an associative, liquid-liquid phase separation that can occur in solutions of oppositely-charged macromolecular species, such as proteins, polymers, and colloids. This process results in a coacervate phase, which is a dense mix of the oppositely-charged components, and a supernatant phase, which is primarily devoid of these same species. First observed almost a century ago, coacervates have since found relevance in a wide range of applications; they are used in personal care and food products, cutting edge biotechnology, and as a motif for materials design and self-assembly. There has recently been a renaissance in our understanding of this important class of material phenomena, bringing the science of coacervation to the forefront of polymer and colloid science, biophysics, and industrial materials design. In this review, we describe the emergence of a number of these new research directions, specifically in the context of polymer-polymer complex coacervates, which are inspired by a number of key physical and chemical insights and driven by a diverse range of experimental, theoretical, and computational approaches.

Manipulation of Confined Polyelectrolyte Conformations through Dielectric Mismatch

We demonstrate that a highly charged polyelectrolyte confined in a spherical cavity undergoes reversible transformations between amorphous conformations and a four-fold symmetry morphology as a function of dielectric mismatch between the media inside and outside the cavity. Surface polarization due to dielectric mismatch exhibits an extra "confinement" effect, which is most pronounced within a certain range of the cavity radius and the electrostatic strength between the monomers and counterions and multivalent counterions. For cavities with a charged surface, surface polarization leads to an increased amount of counterions adsorbed in the outer side, further compressing the confined polyelectrolyte into a four-fold symmetry morphology. The equilibrium conformation of the chain is dependent upon several key factors including the relative permittivities of the media inside and outside the cavity, multivalent counterion concentration, cavity radius relative to the chain length, and interface charge density. Our findings offer insights into the effects of dielectric mismatch in packaging and delivery of polyelectrolytes across media with different relative permittivities. Moreover, the reversible transformation of the polyelectrolyte conformations in response to environmental permittivity allows for potential applications in biosensing and medical monitoring.

Ion Transport and the True Transference Number in Nonaqueous Polyelectrolyte Solutions for Lithium Ion Batteries

Nonaqueous polyelectrolyte solutions have been recently proposed as high Li+ transference number electrolytes for lithium ion batteries. However, the atomistic phenomena governing ion diffusion and migration in polyelectrolytes are poorly understood, particularly in nonaqueous solvents. Here, the structural and transport properties of a model polyelectrolyte solution, poly(allyl glycidyl ether-lithium sulfonate) in dimethyl sulfoxide, are studied using all-atom molecular dynamics simulations. We find that the static structural analysis of Li+ ion pairing is insufficient to fully explain the overall conductivity trend, necessitating a dynamic analysis of the diffusion mechanism, in which we observe a shift from largely vehicular transport to more structural diffusion as the Li+ concentration increases. Furthermore, we demonstrate that despite the significantly higher diffusion coefficient of the lithium ion, the negatively charged polyion is responsible for the majority of the solution conductivity at all concentrations, corresponding to Li+ transference numbers much lower than previously estimated experimentally. We quantify the ion-ion correlations unique to polyelectrolyte systems that are responsible for this surprising behavior. These results highlight the need to reconsider the approximations typically made for transport in polyelectrolyte solutions.

Electrostatic Origin of Element Selectivity during Rare Earth Adsorption

Rare earths, which are fundamental components of modern technologies, are often extracted from aqueous solutions using surfactants at oil-water interfaces. Heavier lanthanides are more easily extracted, even though all lanthanides are chemically very similar. Using x-ray fluorescence measurements and theoretical arguments, we show that there is a sharp bulk-concentration-dependent transition in the interfacial adsorption of cations from aqueous solutions containing Er^{3+} or Nd^{3+} in contact with a floating monolayer. The threshold bulk concentration of erbium (Z=68) is an order of magnitude lower than that of neodymium (Z=60), and erbium is preferentially adsorbed when the solution contains both ions. This implies that elemental selectivity during separation originates at the surfactant interface. Electrostatic effects arising from the interface dielectric mismatch, ionic correlations, and sizes of the ions explain the sharp adsorption curve and selectivity.

Explicit Ion Effects on the Charge and Conformation of Weak Polyelectrolytes

The titration behavior of weak polyelectrolytes is of high importance, due to their uses in new technologies including nanofiltration and drug delivery applications. A comprehensive picture of polyelectrolyte titration under relevant conditions is currently lacking, due to the complexity of systems involved in the process. One must contend with the inherent structural and solvation properties of the polymer, the presence of counterions, and local chemical equilibria enforced by background salt concentration and solution acidity. Moreover, for these cases, the systems of interest have locally high concentrations of monomers, induced by polymer connectivity or confinement, and thus deviate from ideal titration behavior. This work furthers knowledge in this limit utilizing hybrid Monte Carlo⁻Molecular Dynamics simulations to investigate the influence of salt concentration, pK a , pH, and counterion valence in determining the coil-to-globule transition of poorly solvated weak polyelectrolytes. We characterize this transition at a range of experimentally relevant salt concentrations and explicitly examine the role multivalent salts play in determining polyelectrolyte ionization behavior and conformations. These simulations serve as an essential starting point in understanding the complexation between weak polyelectrolytes and ion rejection of self-assembled copolymer membranes.

Role of Associative Charging in the Entropy-Energy Balance of Polyelectrolyte Complexes

Polyelectrolytes may be classified into two primary categories (strong and weak) depending on how their charge state responds to the local environment. Both of these find use in many applications, including drug delivery, gene therapy, layer-by-layer films, and fabrication of ion filtration membranes. The mechanism of polyelectrolyte complexation is, however, still not completely understood, though experimental investigations suggest that entropy gain due to release of counterions is the key driving force for strong polyelectrolyte complexation. Here we perform a comprehensive thermodynamic investigation through coarse-grained molecular simulations permitting us to calculate the free energy of complex formation. Importantly, our expanded-ensemble methods permit the explicit separation of energetic and entropic contributions to the free energy. Our investigations indicate that entropic contributions indeed dominate the free energy of complex formation for strong polyelectrolytes, but are less important than energetic contributions when weak electrostatic coupling or weak polyelectrolytes are present. Our results provide a new view of the free energy of polyelectrolyte complex formation driven by polymer association, which should also arise in systems with large charge spacings or bulky counterions, both of which act to weaken ion-polymer binding.

Competitive calcium ion binding to end-tethered weak polyelectrolytes

We have developed a molecular model to describe the structural changes and potential collapse of weak polyelectrolyte layers end-tethered to planar surfaces and spherical nanoparticles as a function of pH and divalent ion concentration. In particular, we describe the structural changes of polymer-coated nanoparticles end-tethered to copolymers of poly(acrylic acid) (pAA) and poly arcrylamido-2-methylpropane sulfonate (pAMPS) in the presence of Ca²⁺ ions. We find that end-grafted poly(acrylic acid) layers will collapse in aqueous solutions containing sufficient amounts of Ca²⁺ ions, while polymers and copolymers with sufficient AMPS monomers will not collapse. The collapse of end-tethered pAA is due to the formation of calcium bridges between two acrylic acid monomers and one calcium ion. On the other hand pAMPS layers do not collapse due to the lack of calcium bridges. The collapse of pAA layers is strongly dependent on the pH as well as divalent and monovalent salt concentrations of the environment. The collapse is also strongly influenced by the curvature of the tethering surface.

Defining the Structure of a Protein-Spherical Nucleic Acid Conjugate and Its Counterionic Cloud

Protein-spherical nucleic acid conjugates (Pro-SNAs) are an emerging class of bioconjugates that have properties defined by their protein cores and dense shell of oligonucleotides. They have been used as building blocks in DNA-driven crystal engineering strategies and show promise as agents that can cross cell membranes and affect both protein and DNA-mediated processes inside cells. However, ionic environments surrounding proteins can influence their activity and conformational stability, and functionalizing proteins with DNA substantively changes the surrounding ionic environment in a nonuniform manner. Techniques typically used to determine protein structure fail to capture such irregular ionic distributions. Here, we determine the counterion radial distribution profile surrounding Pro-SNAs dispersed in RbCl with 1 nm resolution through in situ anomalous small-angle X-ray scattering (ASAXS) and classical density functional theory (DFT). SAXS analysis also reveals the radial extension of the DNA and the linker used to covalently attach the DNA to the protein surface. At the experimental salt concentration of 50 mM RbCl, Rb⁺ cations compensate ∼90% of the negative charge due to the DNA and linker. Above 75 mM, DFT calculations predict overcompensation of the DNA charge by Rb⁺. This study suggests a method for exploring Pro-SNA structure and function in different environments through predictions of ionic cloud densities as a function of salt concentration, DNA grafting density, and length. Overall, our study demonstrates that solution X-ray scattering combined with DFT can discern counterionic distribution and submolecular features of highly charged, complex nanoparticle constructs such as Pro-SNAs and related nucleic acid conjugate materials.

The Influence of Polymer and Ion Solvation on the Conformational Properties of Flexible Polyelectrolytes

The study of the coupling between the conformational properties of a polyelectrolyte chain and the distribution of counter-ions surrounding the chain is important in developing predictive theories for more complex polymer materials, such as polyelectrolyte gels. We investigated the influence of solvent affinity to counter-ions and the polyelectrolyte backbone on the conformational properties of highly charged flexible polymer chains using molecular dynamics simulations that include both ions and an explicit solvent. We find that the solvation of the polyelectrolyte backbone can be achieved by either increasing the solvent affinity for the polyelectrolyte segments or by increasing the solvent affinity for the counter-ions. However, these two mechanisms influence the conformational properties of the polyelectrolyte chain in rather different ways, suggesting the inadequacy of polyelectrolyte solution models that treat the solvent as a continuum medium.

Assembly of Triblock Amphiphilic Peptides into One-Dimensional Aggregates and Network Formation

Peptide assembly plays a key role in both neurological diseases and development of novel biomaterials with well-defined nanostructures. Synthetic model peptides provide a unique platform to explore the role of intermolecular interactions in the assembly process. A triblock peptide architecture designed by the Hartgerink group is a versatile system which relies on Coulomb interactions, hydrogen bonding, and hydrophobicity to guide these peptides' assembly at three different length scales: β-sheets, double-wall ribbon-like aggregates, and finally a highly porous network structure which can support gels with ≤1% by weight peptide concentration. In this study, by using molecular dynamics simulations of a structure based implicit solvent coarse grained model, we analyzed this hierarchical assembly process. Parametrization of our CG model is based on multiple-state points from atomistic simulations, which enables this model to represent the conformational adaptability of the triblock peptide molecule based on the surrounding medium. Our results indicate that emergence of the double-wall β-sheet packing mechanism, proposed in light of the experimental evidence, strongly depends on the subtle balance of the intermolecular forces. We demonstrate that, even though backbone hydrogen bonding dominates the early nucleation stages, depending on the strength of the hydrophobic and Coulomb forces, alternative structures such as zero-dimensional aggregates with two β-sheets oriented orthogonally (which we refer to as a cross-packed structure) and β-sheets with misoriented hydrophobic side chains are also feasible. We discuss the implications of these competing structures for the three different length scales of assembly by systematically investigating the influence of density, counterion valency, and hydrophobicity.

Barrier-induced dielectric counterion relaxation at super-low frequencies in salt-free polyelectrolyte solutions

Based on the thermally activated diffusion of counterions over the barrier of the electrostatic binding potential, we construct a scaling theory for the slow dielectric response in dilute and semi-dilute polyelectrolyte solutions. The theory is based on an analytic evaluation of the mean-escape time of a single counterion from the surface of a polyelectrolyte chain and uses a variational expression for the electrostatic potential of a charged cylinder including counterion condensation. This mean-escape time shows a range of characteristic power-law dependencies on the polyelectrolyte length and the polyelectrolyte monomer concentration. The existence of this novel dielectric mode at super-low frequencies reflects the wide spectrum of experimental findings for the super-low-frequency dielectric relaxation mode and thereby helps to reconcile conflicting interpretations of experimental data in terms of conventional scaling laws. We also devise a scaling theory for the counterion condensation of finite-length polyelectrolyte chains at finite concentration, which allows us to include polyelectrolyte charge renormalization in dilute as well as semi-dilute solutions in a unified theoretical framework.

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.

Coulomb energy of uniformly charged spheroidal shell systems

We provide exact expressions for the electrostatic energy of uniformly charged prolate and oblate spheroidal shells. We find that uniformly charged prolate spheroids of eccentricity greater than 0.9 have lower Coulomb energy than a sphere of the same area. For the volume-constrained case, we find that a sphere has the highest Coulomb energy among all spheroidal shells. Further, we derive the change in the Coulomb energy of a uniformly charged shell due to small, area-conserving perturbations on the spherical shape. Our perturbation calculations show that buckling-type deformations on a sphere can lower the Coulomb energy. Finally, we consider the possibility of counterion condensation on the spheroidal shell surface. We employ a Manning-Oosawa two-state model approximation to evaluate the renormalized charge and analyze the behavior of the equilibrium free energy as a function of the shell's aspect ratio for both area-constrained and volume-constrained cases. Counterion condensation is seen to favor the formation of spheroidal structures over a sphere of equal area for high values of shell volume fractions.

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.

Polydispersity-driven topological defects as order-restoring excitations

The engineering of defects in crystalline matter has been extensively exploited to modify the mechanical and electrical properties of many materials. Recent experiments on manipulating extended defects in graphene, for example, show that defects direct the flow of electric charges. The fascinating possibilities offered by defects in two dimensions, known as topological defects, to control material properties provide great motivation to perform fundamental investigations to uncover their role in various systems. Previous studies mostly focus on topological defects in 2D crystals on curved surfaces. On flat geometries, topological defects can be introduced via density inhomogeneities. We investigate here topological defects due to size polydispersity on flat surfaces. Size polydispersity is usually an inevitable feature of a large variety of systems. In this work, simulations show well-organized induced topological defects around an impurity particle of a wrong size. These patterns are not found in systems of identical particles. Our work demonstrates that in polydispersed systems topological defects play the role of restoring order. The simulations show a perfect hexagonal lattice beyond a small defective region around the impurity particle. Elasticity theory has demonstrated an analogy between the elementary topological defects named disclinations to electric charges by associating a charge to a disclination, whose sign depends on the number of its nearest neighbors. Size polydispersity is shown numerically here to be an essential ingredient to understand short-range attractions between like-charge disclinations. Our study suggests that size polydispersity has a promising potential to engineer defects in various systems including nanoparticles and colloidal crystals.

Tunable soft structure in charged fluids confined by dielectric interfaces

Fluids of charged particles act as the supporting medium for chemical reactions and physical, dynamical, and biological processes. The local structure in an electrolytic background is deformed by micro- and nanoscopic polarizable objects. Vice versa, the forces between the objects are regulated by the cohesive properties of the background. We study here the range and strength of these forces and the microscopic origin from which they emerge. We find the forces to be sensitively dependent on the material properties of the charged fluid and the immersed solutes. The induced interactions can be varied over decades, offering high tunability and aided by accurate theory, control in experiments and applications. To distinguish correlational effects from simple ionic screening, we describe electrolyte-induced forces between neutral objects. The interplay of thermal motion, short-range repulsions, and electrostatic forces is responsible for a soft structure in the fluid. This structure changes near polarizable interfaces and causes diverse attractions between confining walls that seem well-exploited by microbiological systems. For parameters that correspond to monovalent electrolytes in biologically and technologically relevant aqueous environments, we find induced forces between nanoscopic areas of the order of piconewtons over a few nanometers.

Packing of charged chains on toroidal geometries

We study a strongly adsorbed flexible polyelectrolyte chain on tori. In this generalized Thomson problem, the patterns of the adsorbed chain are analyzed in the space of the toroidal coordinates and in terms of the orientation of each chain segment. Various patterns are found, including double spirals, disclination-like structures, Janus tori, and uniform wrappings, arising from the long-range electrostatic interaction and the toroidal geometry. Their broken mirror symmetry is quantitatively characterized by introducing an order parameter, an integral of the torsion. The uniform packing, which breaks the mirror symmetry the least, has the lowest value of the order parameter. In addition, it is found that the electrostatic energy of confined chains on tori conforms to a power law regardless of the screening effect in some typical cases studied. Furthermore, we study random walks on tori that generate chain configurations in the large screening limit or at large thermal fluctuation; some features associated with the toroidal geometry are discussed.

Conformational transitions of flexible hydrophobic polyelectrolytes in solutions of monovalent and multivalent salts and their mixtures

Conformations of cationic polyelectrolytes (PEs), a weak poly(2-vinylpyridine) (P2VP) and a strong poly(N-methyl-2-vinylpyridinium iodide) (qP2VP), adsorbed on mica from saline solutions in the presence of counterions of different valences are studied using in situ atomic force microscopy (AFM). Quantitative characteristics of chain conformations are analyzed using AFM images of the adsorbed molecules. The results of the statistical analysis of the chain contour reveal collapse of the PE coils when ionic strength is in a range from tens to hundreds of millimoles per kilogram and re-expansion of the coils with a further increase of ionic strength up to a region of the saturated saline solutions. The competition between monovalent and multivalent counterions simultaneously present in solutions strongly affects conformations of PE chains even at a very small fraction of multivalent counterions. Shrinkage of PE coils is steeper for multivalent counterions than for monovalent counterions. However, the re-expansion is only incremental in the presence of multivalent counterions. Extended adsorbed coils at low salt concentrations and at very high concentrations of monovalent salt exhibit conformation corresponding to a 2D coil with 0.95 fraction of bound segments (segments in "trains") in the regime of diluted surface concentration of the PE. Shrunken coils in the intermediate range of ionic strength resemble 3D-globules with 0.8 fraction of trains. The incrementally re-expanded PE coils at a high ionic strength remain unchanged at higher multivalent salt concentrations up to the solubility limit of the salt. The formation of a strong PE complex with multivalent counterions at high ionic strength is not well understood yet. A speculative explanation of the observed experimental result is based on possible stabilization of the complex due to hydrophobic interactions of the backbone.

Solubility and transport of cationic and anionic patterned nanoparticles

We analyze bulk diffusion and transport through hydrophobic nanochannels of nanoparticles (NPs) with different hydrophobic-hydrophilic patterns achieved by coating a fraction of the NP sites with positive or negative charges via explicit solvent molecular dynamics simulations. Ten different charge pattern types including Janus charged-hydrophobic NPs are studied. The cationic NPs are more affected by the patterns and have higher diffusion constants and fluxes than their anionic NPs counterparts. The NP-water interaction dependence on surface pattern and field strength explains these observations. The NP-water Coulomb interaction of anionic NPs in the bulk, which are much stronger than the hydrophobic NP-water interactions, are stronger for NPs with higher localized charge, and stronger than in the cationic NPs counterparts. The diffusion and transport of anionic NPs such as proteins and protein charge ladders with the same total charge but different surface charge patterns are slowest for the highest localized charge pattern, which also adsorb strongest onto surfaces. Our model demonstrates the separation (by reverse osmosis, capillary electrophoresis, or chromatography) of cationic NPs, including proteins with equal net charge but different surface charge distributions.

Morphologies of planar polyelectrolyte brushes in a poor solvent: molecular dynamics simulations and scaling analysis

Using molecular dynamics simulations and scaling analysis, we study the effect of the solvent quality for the polymer backbone, the strength of the electrostatic interactions, the chain degree of polymerization, and the brush grafting density on conformations of the planar polyelectrolyte brushes in salt-free solutions. Polyelectrolyte brush forms: (1) vertically oriented cylindrical aggregates (bundles of chains), (2) maze-like aggregate structures, or (3) thin polymeric layer covering a substrate. These different brush morphologies appear as a result of the fine interplay between electrostatic and short-range monomer-monomer interactions. The brush thickness shows nonmonotonic dependence on the value of the Bjerrum length. It first increases with the increasing value of the Bjerrum length, and then it begins to decrease. This behavior is a result of counterion condensation within a brush volume.

Interactions between planar stiff polyelectrolyte brushes

Molecular-dynamics simulations were performed for two opposing flat surfaces sparsely grafted with rigid polyelectrolyte chains whose lengths are smaller than their persistence lengths. The resulting force-distance dependence was analyzed theoretically in terms of two separate physical mechanisms: the pressure arising from osmotically active counterions trapped within the brush and the work required to bend the brush chains under confinement, which can be accurately characterized by a ground-state theory of rigid polymer buckling. These contributions are of the same magnitude and should be distinguishable in experiments of double-stranded DNA brushes.

Asymmetric charge renormalization for nanoparticles in aqueous media

The effective renormalized charge of nanoparticles in an aqueous electrolyte is essential to determine their solubility. By using a molecular model for the supporting aqueous electrolyte, we find that the effective renormalized charge of the nanoparticles is strongly dependent on the sign of the bare charge. Negatively charged nanoparticles have a lower effective renormalized charge than positively charged nanoparticles. The degree of asymmetry is a nonmonotonic function of the bare charge of the nanoparticle. We show that the effect is due to the asymmetric charge distribution of the water molecules, which we model using a simple three-site molecular structure of point charges.

Counterion adsorption on flexible polyelectrolytes: comparison of theories

Counterion adsorption on a flexible polyelectrolyte chain in a spherical cavity is considered by taking a "permuted" charge distribution on the chain so that the "adsorbed" counterions are allowed to move along the backbone. We compute the degree of ionization by using self-consistent field theory (SCFT) and compare with the previously developed variational theory. Analysis of various contributions to the free energy in both theories reveals that the equilibrium degree of ionization is attained mainly as an interplay of the adsorption energy of counterions on the backbone, the translational entropy of the small ions, and their correlated density fluctuations. Degree of ionization computed from SCFT is significantly lower than that from the variational formalism. The difference is entirely due to the density fluctuations of the small ions in the system, which are accounted for in the variational procedure. When these fluctuations are deliberately suppressed in the truncated variational procedure, there emerges a remarkable quantitative agreement in the various contributing factors to the equilibrium degree of ionization, in spite of the fundamental differences in the approximations and computational procedures used in these two schemes. Furthermore, it is found that the total free energies from the truncated variational procedure and the SCFT are in quantitative agreement at low monomer densities and differ from each other at higher monomer densities. The disagreement at higher monomer densities is due to the inability of the variational calculation to accurately compute the solvent entropy at higher concentrations. A comparison of electrostatic energies (which are relatively small) reveals that the Debye-Hückel estimate used in the variational theory is an overestimation of electrostatic energy as compared to the Poisson-Boltzmann estimate. Nevertheless, since the significant effects from density fluctuations of small ions are not captured by the SCFT, and due to the close agreement between SCFT and the other contributing factors in the more transparent variational procedure, the latter is a better computational tool for obtaining the degree of ionization.

Theory of competitive counterion adsorption on flexible polyelectrolytes: divalent salts

The counterion distribution around an isolated flexible polyelectrolyte in the presence of a divalent salt is evaluated using the adsorption model [M. Muthukumar, J. Chem. Phys. 120, 9343 (2004)] that considers the Bjerrum length, salt concentration, and local dielectric heterogeneity as physical variables in the system. Self-consistent calculations of effective charge and size of the polymer show that divalent counterions replace condensed monovalent counterions in competitive adsorption. The theory further predicts that at modest physical conditions for a flexible polyelectrolytes such as sodium polystyrene sulfonate in aqueous solutions polymer charge is compensated and reversed with increasing divalent salt. Consequently, the polyelectrolyte shrinks and reswells. Lower temperatures and higher degrees of dielectric heterogeneity between chain backbone and solvent enhance condensation of all species of ions. Complete diagrams of states for the effective charge calculated as functions of the Coulomb strength and salt concentration suggest that (a) overcharging requires a minimum Coulomb strength and (b) progressively higher presence of salt recharges the polymer due to either electrostatic screening (for low Coulomb strengths) or coion condensation (for high Coulomb strengths). Consideration of ion-bridging by divalent counterions leads to a first-order collapse of polyelectrolytes in modest presence of divalent salts and at higher Coulomb strengths. The authors' theoretical predictions are in agreement with the generic results from experiments and simulations.