Born energy, acid-base equilibrium, structure and interactions of end-grafted weak polyelectrolyte layers

Solvent Isotherms and Structural Transitions in Nanoparticle Superlattice Assembly

We introduce a Molecular Theory for Compressible Fluids (MOLT-CF) that enables us to compute free energies and other thermodynamic functions for nanoparticle superlattices with any solvent content, including the dry limit. Quantitative agreement is observed between MOLT-CF and united-atom molecular dynamics simulations performed to assess the reliability and precision of the theory. Among other predictions, MOLT-CF shows that the amount of solvent within the superlattice decreases approximately linearly with its vapor pressure and that in the late stages of drying, solvent-filled voids form at lattice interstitials. Applied to single-component superlattices, MOLT-CF predicts fcc-to-bcc Bain transitions for decreasing vapor pressure and for increasing ligand length, both in agreement with experimental results. We explore the stability of other single-component phases and show that the C14 Frank-Kasper phase, which has been reported in experiments, is not a global free-energy minimum. Implications for precise assembly and prediction of multicomponent nanoparticle systems are discussed.

Explaining Giant Apparent pK_{a} Shifts in Weak Polyelectrolyte Brushes

Recent experiments on weak polyelectrolyte brushes found marked shifts in the effective pK_{a} that are linear in the logarithm of the salt concentration. Comparing explicit-particle simulations with mean-field calculations we show that for high grafting densities the salt concentration effect can be explained using the ideal Donnan theory, but for low grafting densities the full shift is due to a combination of the Donnan effect and the polyelectrolyte effect. The latter originates from electrostatic correlations that are neglected in the Donnan picture and that are only approximately included in the mean-field theory. Moreover, we demonstrate that the magnitude of the polyelectrolyte effect is almost invariant with respect to salt concentration but depends on the grafting density of the brush. This invariance is due to a complex cancellation of multiple effects. Based on our results, we show how the experimentally determined pK_{a} shifts may be used to infer the grafting density of brushes, a parameter that is difficult to measure directly.

Electroresponse of weak polyelectrolyte brushes

End-tethered polyelectrolytes are widely used to modify substrate properties, particularly for lubrication or wetting. External stimuli, such as pH, salt concentration, or an electric field, can induce profound structural responses in weak polyelectrolyte brushes, which can be utilized to further tune substrate properties. We study the structure and electroresponsiveness of weak polyacid brushes using an inhomogeneous theory that incorporates both electrostatic and chain connectivity correlations at the Debye-Hückel level. Our calculation shows that a weak polyacid brush swells under the application of a negative applied potential, in agreement with recent experimental observation. We rationalize this behavior using a scaling argument that accounts for the effect of the surface charge. We also show that the swelling behavior has a direct influence on the differential capacitance, which can be modulated by the solvent quality, pH, and salt concentration.

Weak polyelectrolyte brushes

I review experimental developments in the growth and application of surface-grafted weak polyelectrolytes (brushes), concentrating on their surface, tribological, and adhesive and bioadhesive properties, and their role as actuators.

Acid-Base Equilibrium and Dielectric Environment Regulate Charge in Supramolecular Nanofibers

Peptide amphiphiles are a class of molecules that can self-assemble into a variety of supramolecular structures, including high-aspect-ratio nanofibers. It is challenging to model and predict the charges in these supramolecular nanofibers because the ionization state of the peptides are not fixed but liable to change due to the acid-base equilibrium that is coupled to the structural organization of the peptide amphiphile molecules. Here, we have developed a theoretical model to describe and predict the amount of charge found on self-assembled peptide amphiphiles as a function of pH and ion concentration. In particular, we computed the amount of charge of peptide amphiphiles nanofibers with the sequence C₁₆ − V₂A₂E₂. In our theoretical formulation, we consider charge regulation of the carboxylic acid groups, which involves the acid-base chemical equilibrium of the glutamic acid residues and the possibility of ion condensation. The charge regulation is coupled with the local dielectric environment by allowing for a varying dielectric constant that also includes a position-dependent electrostatic solvation energy for the charged species. We find that the charges on the glutamic acid residues of the peptide amphiphile nanofiber are much lower than the same functional group in aqueous solution. There is a strong coupling between the charging via the acid-base equilibrium and the local dielectric environment. Our model predicts a much lower degree of deprotonation for a position-dependent relative dielectric constant compared to a constant dielectric background. Furthermore, the shape and size of the electrostatic potential as well as the counterion distribution are quantitatively and qualitatively different. These results indicate that an accurate model of peptide amphiphile self-assembly must take into account charge regulation of acidic groups through acid–base equilibria and ion condensation, as well as coupling to the local dielectric environment.

How molecular interactions tune the characteristic time of nanocomposite colloidal sensors

HYPOTHESIS

Mass transport critically controls the performance of colloidal metal-polymer sensors. We hypothesize that molecular-level pair interactions, such as electric, steric, and specific binding effects, govern the mass transport and, in return, the characteristic time of these sensors.

THEORY

Here we present a simple theory guided by experimental data to examine the sensing performance of two usually encountered archetypal metal-polymer sensors, namely (1) core-shell and (2) yolk-shell architectures. For this purpose, we use the static reactive density functional theory framework, determining how (i) charge, (ii) size, and (iii) non-covalent binding factors modulate the characteristic time.

FINDINGS

We show how an interplay between diffusivity and partitioning governs the sensing time of the sensors, where an anti-correlation cancellation between them renders the time non-trivial. Our study demonstrates that the convoluted substrate-hydrogel shell interaction controls the characteristic time of these colloidal sensors, especially when the sensors are in a collapsed state. Notably, the substrates with a high dipole moment tend to equilibrate greatly, but undesirably, at the shell-solution interface. With this, we encourage the formation of a metastable sorption state.

Proteins Adsorbing onto Surface-Modified Nanoparticles: Effect of Surface Curvature, pH, and the Interplay of Polymers and Proteins Acid-Base Equilibrium

Protein adsorption onto nanomaterials is a process of vital significance and it is commonly controlled by functionalizing their surface with polymers. The efficiency of this strategy depends on the design parameters of the nanoconstruct. Although significant amount of work has been carried out on planar surfaces modified with different types of polymers, studies investigating the role of surface curvature are not as abundant. Here, we present a comprehensive and systematic study of the protein adsorption process, analyzing the effect of curvature and morphology, the grafting of polymer mixtures, the type of monomer (neutral, acidic, basic), the proteins in solution, and the conditions of the solution. The theoretical approach we employed is based on a molecular theory that allows to explicitly consider the acid-base reactions of the amino acids in the proteins and the monomers on the surface. The calculations showed that surface curvature modulates the molecular organization in space, but key variables are the bulk pH and salt concentration (in the millimolar range). When grafting the NP with acidic or basic polymers, the surface coating could disfavor or promote adsorption, depending on the solution's conditions. When NPs are in contact with protein mixtures in solution, a nontrivial competitive adsorption process is observed. The calculations reflect the balance between molecular organization and chemical state of polymers and proteins, and how it is modulated by the curvature of the underlying surface.

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.

How Dissociation of Carboxylic Acid Groups in a Weak Polyelectrolyte Brush Depend on Their Distance from the Substrate

A weak polyelectrolyte brush is composed of a layer of polyacids or polybases grafted by one end of their chains to a substrate surface. For such brush layers immersed in an aqueous solution, the dissociation behavior of the acidic or basic groups and the structural and physical properties of the brush layer will thus be strongly dependent on the environmental conditions. For a polyacid brush layer consisting of, e.g., poly(acrylic acid), this means that the chains in the brush layer will be charged at high pH and uncharged at low pH. However, theoretical scaling laws not only foresee the structural changes occurring in response to the pH-induced dissociation behavior but also how the dissociation behavior of the brush layer depends on the ionic strength of the aqueous solution and the density of acidic groups within the brush layer. We have herein employed spectroscopic ellipsometry and a quartz crystal microbalance with dissipation monitoring (QCM-D) to experimentally evaluate the theoretically predicted dissociation and structural behavior of PAA brushes. Spectroscopic ellipsometry allows us to study the brush thickness as a function of pH and ionic strength, while QCM-D gives us an opportunity to investigate the swelling behavior of PAA brushes at various penetration depths of propagating acoustic waves. Our studies show that the dissociation degree of the carboxylic acid groups in a PAA brush increases with increasing distance from the substrate. Moreover, the ionic strength enhances carboxylic acid dissociation, such that a higher ionic strength leads to a narrower distribution and higher average dissociation degree. In conclusion, our results provide an experimental verification of the theoretically predicted gradient in the degree of dissociation of the acid groups in weak polyacid brush layers and shows that at a pH value equal to approximately the average pK_a value of the brush, the state of the acid groups varies from being almost uncharged to almost fully dissociated depending on the ionic strength and vertical position in the brush.

Fabrication and in situ functionalisation of mesoporous silica films by the physical entrapment of functional and responsive block copolymer structuring agents

Stimuli-responsive mesoporous silica films were prepared by evaporation-induced self-assembly through the physical entrapment of a functional block copolymer structuring agent, which simultaneously serves to functionalise the mesopore. These polymer-silica hybrid materials exhibit remarkable ionic permselectivity under highly filled conditions, and offer the potential for local polymer functionalisation for enhanced and tunable ionic permselectivity. This innovative and simple approach for the in situ functionalisation of mesoporous silica has the potential to improve how we approach the design of complex architectures at the nanoscale for enhanced transport, and is thus relevant for a variety of technologies based on molecular transport in nanoscale pores including separation, sensing, catalysis, and energy conversion.

Salt- and pH-induced swelling of a poly(acrylic acid) brush via quartz crystal microbalance w/dissipation (QCM-D)

We infer the swelling/de-swelling behavior of weakly ionizable poly(acrylic acid) (PAA) brushes of 2-39 kDa molar mass in the presence of KCl concentrations from 0.1-1000 mM, pH = 3, 7, and 9, and grafting densities σ = 0.12-2.15 chains per nm2 using a Quartz Crystal Microbalance with Dissipation (QCM-D), confirming and extending the work of Wu et al. to multiple chain lengths. At pH 7 and 9 (above the pKa ∼ 5), the brush initially swells at low KCl ionic strength (<10 mM) in the "osmotic brush" regime, and de-swells at higher salt concentrations, in the "salted brush" regime, and is relatively unaffected at pH 3, below the pKa, as expected. At pH 7, at low and moderate grafting densities, our results in the high-salt "salted brush" regime (Cs > 10 mM salt) agree with the predicted scaling H ∼ Nσ+1/3Cs-1/3 of brush height H, while in the low-salt "osmotic brush" regime (Cs < 10 mM salt), we find H ∼ Nσ+1/3Cs+0.28-0.38, whose dependence on Cs agrees with scaling theory for this regime, but the dependence on σ strongly disagrees with it. The predicted linearity in the degree of polymerization N is confirmed. The new results partially confirm scaling theory and clarify where improved theories and additional data are needed.

Combined Experimental and Theoretical Study of Weak Polyelectrolyte Brushes in Salt Mixtures

The swelling behavior of a hydrophobic poly(2diisopropylamino)ethyl methacrylate (PDPA) brush immersed in aqueous solutions of single and mixed salts has been investigated using ellipsometry and numerical self-consistent field (nSCF) theory. As a function of solution ionic strength, the osmotic and salted brush regimes of weak polyelectrolyte brushes as well as substantial specific anion effects in the presence of K⁺ salts of Cl^-, NO₃^-, and SCN^- are found. For solutions containing mixtures of NO₃^- and Cl^-, the brush swelling is the same as one would expect on the basis of the concentration-weighted average of the brush behavior in the single salt solutions. However, in mixtures of SCN^- and Cl^-, the swelling response is more complicated and substantial divergence from ideal behavior is observed. Mean-field theory shows excellent qualitative agreement with the ellipsometry findings. nSCF reveals that for the SCN^-/Cl^- cases the swelling behavior of the PDPA brush most likely arises from the predominant localization of the weakly hydrated SCN^- within the brush compared to the more strongly hydrated Cl^-.

The interplay of nanointerface curvature and calcium binding in weak polyelectrolyte-coated nanoparticles

When engineering nanomaterials for application in biological systems, it is important to understand how multivalent ions, such as calcium, affect the structural and chemical properties of polymer-modified nanoconstructs. In this work, a recently developed molecular theory was employed to study the effect of surface curvature on the calcium-induced collapse of end-tethered weak polyelectrolytes. In particular, we focused on cylindrical and spherical nanoparticles coated with poly(acrylic acid) in the presence of different amounts of Ca2+ ions. We describe the structural changes that grafted polyelectrolytes undergo as a function of calcium concentration, surface curvature, and morphology. The polymer layers collapse in aqueous solutions that contain sufficient amounts of Ca2+ ions. This collapse, due to the formation of calcium bridges, is not only controlled by the calcium ion concentration but also strongly influenced by the curvature of the tethering surface. The transition from a swollen to a collapsed layer as a function of calcium concentration broadens and shifts to lower amounts of calcium ions as a function of the radius of cylindrical and spherical nanoparticles. The results show how the interplay between calcium binding and surface curvature governs the structural and functional properties of the polymer molecules. This would directly impact the fate of weak polyelectrolyte-coated nanoparticles in biological environments, in which calcium levels are tightly regulated. Understanding such interplay would also contribute to the rational design and optimization of smart interfaces with applications in, e.g., salt-sensitive and ion-responsive materials and devices.

Behavior of Weak Polyelectrolyte Brushes in Mixed Salt Solutions

Hydrophilic and hydrophobic weak polybasic brushes immersed in aqueous solutions of mixed salt counterions are considered using a mean-field numerical self-consistent field approach. On top of the solvent quality of the polymer, the counterion–solvent interactions are accounted for by implementing Flory–Huggins interaction parameters. We show that ion specificity within the brush can bring about large changes in conformation. It is found that the collapse transition of hydrophobic, weak polyelectrolyte brushes features an intermediate two-phase state wherein a subset of chains are collapsed in a dense layer at the substrate, while the remainder of chains are well-solvated and strongly stretched away from the it. Besides pH and ionic strength, solvent quality of counterions and the composition of ions in the solvent are important control parameters for the behavior of polyelectrolyte brushes. Increasingly hydrophobic counterions penetrate deeper within the brush and stabilize the collapsed region, while hydrophilic counterions do the opposite.

Structural behavior of competitive temperature and pH-responsive tethered polymer layers

Herein, we develop a molecular theory to examine a class of pH and temperature-responsive tethered polymer layers. The response of pH depends on intramolecular charge repulsion of weakly acidic monomers and the response of temperature depends on hydrogen bonding between polymer monomers and water molecules akin to the behavior of water-soluble polymers such as PEG (poly-ethylene glycol) or NIPAAm (n-isopropylacrylamide). We investigate the changes in structural behavior that result for various end-tethered copolymers: pH/T responsive monomers alone, in alternating sequence with hydrophobic monomers, and as 50/50 diblocks with hydrophobic monomers. We find that the sequence and location of hydrophobic units play a critical role in the thermodynamic stability and structural behavior of these responsive polymer layers. Additionally, the polymers exhibit tunable collapse when varying the surface coverage, location and sequence of hydrophobic units as a function of temperature and pH. As far as we know, our results present the first molecularly detailed theory for end-tethered polymers that are both pH and temperature-responsive via hydrogen bonding. We propose that this work holds predictive power for the guided design of future biomaterials.

Single-chain-in-mean-field simulations of weak polyelectrolyte brushes

Structural properties of brushes which are composed of weak acidic and basic polyelectrolytes are studied in the framework of a particle-based approach that implicitly accounts for the solvent quality. Using a semi-grandcanonical partition function in the framework of the Single-Chain-in-Mean-Field (SCMF) algorithm, the weak polyelectrolyte is conceived as a supramolecular mixture of polymers in different dissociation states, which are explicitly treated in the partition function and sampled by the SCMF procedure. One obtains a local expression for the equilibrium acid-base reaction responsible for the regulation of the charged groups that is also incorporated to the SCMF sampling. Coupled to a simultaneous treatment of the electrostatics, the approach is shown to capture the main features of weak polyelectrolyte brushes as a function of the bulk pH in the solution, the salt concentration, and the grafting density. Results are compared to experimental and theoretical works from the literature using coarse-grained representations of poly(acrylic acid) (PAA) and poly(2-vinyl pyridine) (P2VP) polymer-based brushes. As the Born self-energy of ions can be straightforwardly included in the numerical approach, we also study its effect on the local charge regulation mechanism of the brush. We find that its effect becomes significant when the brush is dense and exposed to high salt concentrations. The numerical methodology is then applied (1) to the study of the kinetics of collapse/swelling of a P2VP brush and (2) to the ability of an applied voltage to induce collapse/swelling of a PAA brush in a pH range close to the pK_a value of the polymer.

Charge-controlled nano-structuring in partially collapsed star-shaped macromolecules

Hydrophobic polyelectrolytes exhibit intra-molecular nano-scale self-organization instead of macroscopic phase separation because of the interplay between short-range hydrophobic attraction and long-range electrostatic repulsion. We aim to unravel how the morphology of the intra-molecular nanostructures can be controlled through the topology of the macromolecule on one hand and by adjustable ionization on the other hand. Specifically, we focus on hydrophobic star-branched polyelectrolytes, composed of either strong or weak acidic monomers. While both collapse in a globule when uncharged, and expand to full stretching of arms at high ionization, they exhibit quite different intermediate scenarios. For the strong ones, we observe the formation of bundles of arms as the main structural motif, and for the weak ones the intramolecular micelle-like structure is found at the same overall charge of the macromolecule. Here intramolecular disproportionation leaves some arms in a collapsed virtually neutral core, while others are substantially ionized and stretched in the corona.

Dendrimers in Nanoscale Confinement: The Interplay between Conformational Change and Nanopore Entrance

Hyperbranched dendrimers are nanocarriers for drugs, imaging agents, and catalysts. Their nanoscale confinement is of fundamental interest and occurs when dendrimers with bioactive payload block or pass biological nanochannels or when catalysts are entrapped in inorganic nanoporous support scaffolds. The molecular process of confinement and its effect on dendrimer conformations are, however, poorly understood. Here, we use single-molecule nanopore measurements and molecular dynamics simulations to establish an atomically detailed model of pore dendrimer interactions. We discover and explain that electrophoretic migration of polycationic PAMAM dendrimers into confined space is not dictated by the diameter of the branched molecules but by their size and generation-dependent compressibility. Differences in structural flexibility also rationalize the apparent anomaly that the experimental nanopore current read-out depends in nonlinear fashion on dendrimer size. Nanoscale confinement is inferred to reduce the protonation of the polycationic structures. Our model can likely be expanded to other dendrimers and be applied to improve the analysis of biophysical experiments, rationally design functional materials such as nanoporous filtration devices or nanoscale drug carriers that effectively pass biological pores.

Anion-specific effects on the behavior of pH-sensitive polybasic brushes

The anion-specific solvation and conformational behavior of weakly basic poly(2-dimethylamino)ethyl methacrylate (poly(DMA)), poly(2-diethylamino)ethyl methacrylate (poly(DEA)), and poly(2-diisopropylamino)ethyl methacrylate (poly(DPA)) brushes, with correspondingly increasing inherent hydrophobicity, have been investigated using in situ ellipsometric and quartz crystal microbalance with dissipation (QCM-D) measurements. In the osmotic brush regime, as the initial low concentration of salt is increased, the brushes osmotically swell by the uptake of solvent as they become charged and the attractive hydrophobic inter- and intrachain interactions are overcome. With increased ionic strength, the brushes move into the salted brush regime where they desolvate and collapse as their electrostatic charge is screened. Here, as the brushes collapse, they transition to more uniform and rigid conformations, which dissipate less energy, than similarly solvated brushes at lower ionic strength. Significantly, in these distinct regimes brush behavior is not only ionic strength dependent but is also influenced by the nature of the added salt based on its position in the well-known Hofmeister or lyotropic series, with potassium acetate, nitrate, and thiocyanate investigated. The strongly kosmotropic acetate anions display low affinity for the hydrophobic polymers, and largely unscreened electrosteric repulsions allow the brushes to remain highly solvated at higher acetate concentrations. The mildly chaotropic nitrate and strongly chaotropic thiocyanate anions exhibit a polymer hydrophobicity-dependent affinity for the brushes. Increasing thiocyanate concentration causes the brushes to collapse at lower ionic strength than for the other two anions. This study of weak polybasic brushes demonstrates the importance of all ion, solvent, and polymer interactions.

Hydrophobic effects within the dynamic pH-response of polybasic tertiary amine methacrylate brushes

Monomer hydrophobicity dominates the kinetics of the pH-response of tertiary amine methacrylate brushes as determined by in situ ellipsometry and QCM-D kinetic and equilibrium measurements.

Self-organized polyelectrolyte end-grafted layers under nanoconfinement

Layers of end-grafted weak polyelectrolytes in poor solvent self-organize into a rich variety of structures (such as micelles, micelles coexisting with nonaggregated chains, stripes and layers with solvent-filled holes) due to the subtle competition among hydrophobic, electrostatic and steric interactions and the chemical acid-based equilibria of the weak polyelectrolyte. In this work, a molecular theory has been used to systematically study how nanoconfinement modulates the competition among these interactions and, therefore, dictates the morphology of the self-assembled layer. Two different types of confinement were considered and compared: (i) soft lateral confinement due to increasing surface coverage in a planar polyelectrolyte brush and (ii) hard vertical confinement due to the interaction of a planar polyelectrolyte brush with an opposing surface, as typically found in AFM-colloidal-tip and surface-force-apparatus experiments. It is shown that increasing the surface coverage (soft lateral confinement) or compressing the layer with an opposing wall (hard vertical confinement) have a similar qualitative effect on the morphology of the system: both types of nanoconfinement increase the stability of morphologies that extend in one or two dimensions (such as the homogeneous brush, holes and stripes) over nonextended aggregates (such as hemispherical micelles). However, vertical confinement can also lead to pillar-like structures that are not observed in the absence of the opposing wall. Interestingly, the pillar structures, which bridge the grafting and opposing surfaces, may coexist with metastable structures collapsed to the grafting surface only. This coexistence may help to understand the hysteresis commonly observed in surface-force experiments.