Polyelectrolyte Brushes with Protein-Like Nanocolloids.

Electrostatic interaction of ampholytic nanocolloidal particles (NPs), which mimic globular proteins, with polyelectrolyte brushes is analyzed within mean-field Poisson-Boltzmann approximation. In accordance with experimental findings, the theory predicts that an electrostatic driving force for the particle uptake by the brush may emerge when the net charge of the particle in the buffer and the charge of the brush are of the same sign. The origin of this driving force is change in the ionization state of weak cationic and anionic groups on the NP surface provoked by interaction with the brush. In experimental systems, the ionic interactions are complemented by excluded-volume, hydrophobic, and other types of interactions that all together control NP uptake by or expulsion from the brush. Here, we focus on the NP-brush ionic interactions. It is demonstrated that deviation between the buffer pH and the NP isoelectric point, considered usually as the key control parameter, does not uniquely determine the insertion free energy patterns. The latter depends also on the proportion of cationic and anionic groups in the NPs and their specific ionization constants as well as on salt concentration in the buffer. The analysis of the free energy landscape proves that a local minimum in the free energy inside the brush appears, provided the NP charge reversal occurs upon insertion into the brush. This minimum corresponds either to a thermodynamically stable or to a metastable state, depending on the pH offset from the IEP and salt concentration, and is separated from the bulk of the solution by a free energy barrier. The latter, being fairly independent of salt concentration in height, may strongly impede the NP absorption kinetically even when it is thermodynamically favorable. Hence, change reversal is a necessary but insufficient condition for the uptake of the NPs by similarly charged polyelectrolyte brushes.

Surface-Immobilized Interpolyelectrolyte Complexes Formed by Polyelectrolyte Brushes.

A scaling theory of interaction and complex formation between planar polyelectrolyte (PE) brush and oppositely charged mobile linear PEs is developed. Counterion release is found to be the main driving force for the complexation. An interpolyelectrolyte coacervate complex (IPEC) between the brush and oppositely charged mobile PEs is formed at moderate grafting density and low salt concentration. At higher grafting density mobile chains penetrate the brush, but the brush structure is controlled by the balance between entropic elasticity and nonelectrostatic short-range interactions, as happens in a neutral brush. An increase in salt concentration beyond the theoretically predicted threshold leads to the release of the guest polyions from the brush. For brushes with moderate grafting density, complexation with oppositely charged guest polyions is predicted to trigger lateral microphase separation and formation of the finite-size surface IPEC clusters. Power law dependencies for the cluster dimensions on the brush grafting density, PE length, and salt concentration are provided.

Interaction of Polyanionic and Polycationic Brushes with Globular Proteins and Protein-like Nanocolloids.

A large number of experimental studies have demonstrated that globular proteins can be absorbed from the solution by both polycationic and polyanionic brushes when the net charge of protein globules is of the same or of the opposite sign with respect to that of brush-forming polyelectrolyte chains. Here, we overview the results of experimental studies on interactions between globular proteins and polycationic or polyanionic brushes, and present a self-consistent field theoretical model that allows us to account for the asymmetry of interactions of protein-like nanocolloid particles comprising weak (pH-sensitive) cationic and anionic groups with a positively or negatively charged polyelectrolyte brush. The position-dependent insertion free energy and the net charge of the particle are calculated. The theoretical model predicts that if the numbers of cationic and anionic ionizable groups of the protein are approximately equal, then the interaction patterns for both cationic and anionic brushes at equal offset on the "wrong side" from the isoelectric point (IEP), i.e., when the particle and the brush charge are of the same sign, are similar. An essential asymmetry in interactions of particles with polycationic and polyanionic brushes is predicted when fractions of cationic and anionic groups differ significantly. That is, at a pH above IEP, the anionic brush better absorbs negatively charged particles with a larger fraction of ionizable cationic groups and vice versa.

Cylindrical brushes with ionized side chains: Scaling theory revisited.

We revisit the classic scaling model of a cylindrical polyelectrolyte (PE) brush focusing on molecular brushes with stiff backbones and dispersions of polymer-decorated nanorods. Based on the blob representation we demonstrate that similarly to the case of planar PE brushes, separation of intra- and intermolecular repulsions between charges leads to novel scaling regimes for cylindrical PE brushes in salt-added solution and a sharper decrease in its thickness versus salt concentration dependence. These theoretical predictions may inspire further comprehensive experimental research and computer simulations of synthetic and biopolyelectrolyte cylindrical brushes.

Polyelectrolyte Cylindrical Brushes in Hairy Gels.

We considered dispersions of cylindrical polyelectrolyte (PE) brushes with stiff backbones, and polymer-decorated nanorods with tunable solubility of the brush-forming PE chains that affected thermodynamic stability of the dispersions. We focused on thermo-induced and deionization-induced conformational transition that provokes loss of aggregative dispersion stability of nanorods decorated with weakly ionized polyions. A comparison between theoretical predictions and experiments enabled rationalization and semi-quantitative interpretation of the experimental results.

Polyelectrolyte Brush Interacting with Ampholytic Nanoparticles as Coarse-Grained Models of Globular Proteins: Poisson-Boltzmann Theory.

The self-consistent field Poisson-Boltzmann framework is applied to analyze equilibrium partitioning of ampholytic nanoparticles (NPs) between buffer solution and polyelectrolyte (PE) polyanionic brush. We demonstrate that depending on pH and salt concentration in the buffer solution, interactions between ionizable (acidic and basic) groups on the NP surface and electrostatic field created by PE brush may either lead to the spontaneous uptake of NPs or create an electrostatic potential barrier, preventing the penetration of NPs inside PE brush. The capability of PE brush to absorb or repel NPs is determined by the shape of the insertion free energy that is calculated as a function of NP distance from the grafting surface. It is demonstrated that, at a pH value below or slightly above the isoelectric point (IEP), the electrostatic free energy of the particle is negative inside the brush and absorption is thermodynamically favorable. In the latter case, the insertion free energy exhibits a local maximum (potential barrier) at the entrance to the brush. An increase in pH leads to the shallowing of the free energy minimum inside the brush and a concomitant increase in the free energy maximum, which may result in kinetic hindering of NP uptake. Upon further increase in pH the insertion free energy becomes positive, making NP absorption thermodynamically unfavorable. An increase in salt concentration diminishes the depth of the free energy minimum inside the brush and eventually leads to its disappearance. Hence, in accordance with existing experimental data our theory predicts that an increase in salt concentration suppresses absorption of NPs (protein globules) by PE brush in the vicinity of IEP. The interplay between electrostatic driving force for NP absorption and osmotic repelling force (proportional to NP volume) indicates that for large NPs with relatively small number of ionizable groups osmotic repulsion overcomes electrostatic attraction preventing thereby absorption of NPs by PE brush.

Selective Colloid Transport across Planar Polymer Brushes.

Polymer brushes are attractive as surface coatings for a wide range of applications, from fundamental research to everyday life, and also play important roles in biological systems. How colloids (e.g., functional nanoparticles, proteins, viruses) bind and move across polymer brushes is an important yet under-studied problem. A mean-field theoretical approach is presented to analyze the binding and transport of colloids in planar polymer brushes. The theory explicitly considers the effect of solvent strength on brush conformation and of colloid-polymer affinity on colloid binding and transport. The position-dependent free energy of the colloid insertion into the polymer brush which controls the rate of colloid transport across the brush is derived. It is shown how the properties of the brush can be adjusted for brushes to be highly selective, effectively serving as tuneable gates with respect to colloid size and affinity to the brush-forming polymer. The most important parameter regime simultaneously allowing for high brush permeability and selectivity corresponds to a condition when the repulsive and attractive contributions to the colloid insertion free energy nearly cancel. This theory should be useful to design sensing and purification devices with enhanced selectivity and to better understand mechanisms underpinning the functions of biological polymer brushes.

Bovine Serum Albumin Interaction with Polyanionic and Polycationic Brushes: The Case Theoretical Study.

We apply a coarse-grained self-consistent field Poisson-Boltzmann framework to study interaction between Bovine Serum Albumin (BSA) and a planar polyelectropyte brush. Both cases of negatively (polyanionic) and positively (polycationic) charged brushes are considered. Our theoretical model accounts for (1) re-ionization free energy of the amino acid residues upon protein insertion into the brush; (2) osmotic force repelling the protein globule from the brush; (3) hydrophobic interactions between non-polar areas on the globule surface and the brush-forming chains. We demonstrate that calculated position-dependent insertion free energy exhibits different patterns, corresponding to either thermodynamically favourable BSA absorption in the brush or thermodynamically or kinetically hindered absorption (expulsion) depending on the pH and ionic strength of the solution. The theory predicts that due to the re-ionization of BSA within the brush, a polyanionic brush can efficiently absorb BSA over a wider pH range on the "wrong side" of the isoelectric point (IEP) compared to a polycationic brush. The results of our theoretical analysis correlate with available experimental data and thus validate the developed model for prediction of the interaction patterns for various globular proteins with polyelectrolyte brushes.

Hairy Gels: A Computational Study.

We present results of MD and MC simulations of the equilibrium properties of swelling gels with comb-like or bottlebrush subchains and compare them to scaling-theory predictions. In accordance with theory, the simulation results demonstrate that swelling coefficient of the gel increases as a function of the polymerization degree of the main chains and exhibits a very weak maximum (or is virtually constant) as a function of the polymerization degree and grafting density of side chains. The bulk osmotic modulus passes through a shallow minimum as the polymerization degree of the side chains increases. This minimum is attributed to the onset of overlap of side chains belonging to different bottlebrush strands in the swollen gel.

Theoretical advances in molecular bottlebrushes and comblike (co)polymers: solutions, gels, and self-assembly.

We present an overview of state-of-the-art theory of (i) conformational properties of molecular bottlebrushes in solution, (ii) self-assembly of di- and triblock copolymers comprising comb-shaped and bottlebrush blocks in solutions and melts, and (iii) cross-linked and self-assembled gels with bottlebrush subchains. We demonstrate how theoretical models enable quantitative prediction and interpretation of experimental results and provide rational guidance for design of new materials with physical properties tunable by architecture of constituent bottlebrush blocks.

Colloidal particles interacting with a polymer brush: a self-consistent field theory.

The interaction of colloidal particles with a planar polymer brush immersed in a solvent of variable thermodynamic quality is studied by a numerical self-consistent field method combined with analytical mean-field theory. The effect of embedded particle on the distribution of polymer density in the brush is analyzed and the particle insertion free energy profiles are calculated for variable size and shape of the particles and sets of polymer-particle and polymer-solvent interaction parameters. In particular, both cases of repulsive and attractive interactions between particles and brush-forming chains are considered. It is demonstrated that for large particles the insertion free energy is dominated by repulsive (osmotic) contribution and is approximately proportional to the particle volume in accordance with earlier theoretical predictions [Halperin et al., Macromolecules, 2011, 44, 3622]. For the particles of smaller size or/and large shape asymmetry the adsorption or depletion of a polymer from the particle surface essentially contributes to the insertion free energy balance. As a result, depending on the set of polymer-solvent and polymer-particle interaction parameters and brush grafting density the insertion free energy profile may exhibit complex patterns, i.e., from a pure repulsive effective potential barrier to an attractive well. The results of our study allow for predicting equilibrium partitioning and controlling diffusive transport of (bio)nanocolloids across (bio)polymer brushes of arbitrary geometry including polymer-modified membranes or nanopores.

Bottlebrush polymer gels: architectural control over swelling and osmotic bulk modulus.

Swelling behaviour and bulk moduli of polymer gels comprising of crosslinked bottlebrush subchains enable fine tuning by varying polymerization degrees of the main and side chains of the bottlebrush strands as well as their grafting densities. By using scaling approach we predict power law dependences of structural and elastic properties of swollen bottlebrush gels on the set of relevant architectural parameters and construct phase diagrams consisting of regions corresponding to different power law asymptotics for these dependences. In particular, our theory predict that bulk elastic modulus of the gel exhibits non-monotonous dependence on the degree of polymerization of side chains of the bottlebrush strands.

Injectable bottlebrush hydrogels with tissue-mimetic mechanical properties.

Injectable hydrogels are desired in many biomedical applications due to their minimally invasive deployment to the body and their ability to introduce drugs. However, current injectables suffer from mechanical mismatch with tissue, fragility, water expulsion, and high viscosity. To address these issues, we design brush-like macromolecules that concurrently provide softness, firmness, strength, fluidity, and swellability. The synthesized linear-bottlebrush-linear (LBL) copolymers facilitate improved injectability as the compact conformation of bottlebrush blocks results in low solution viscosity, while the thermoresponsive linear blocks permit prompt gelation at 37°C. The resulting hydrogels mimic the deformation response of supersoft tissues such as adipose and brain while withstanding deformations of 700% and precluding water expulsion upon gelation. Given their low cytotoxicity and mild inflammation in vivo, the developed materials will have vital implications for reconstructive surgery, tissue engineering, and drug delivery applications.

Micelles Formed by an AB Copolymer with Bottlebrush Blocks: Scaling Theory.

We present a scaling theory describing the equilibrium properties of spherical micelles formed by a diblock copolymer with bottlebrush blocks in a selective solvent. The theory predicts a number of new thermodynamic regimes inherent for copolymers with relatively short main chains (long side chains) in the bottlebrush blocks. These regimes with a novel set of scaling exponents for the micelle properties are characterized by limiting extension of the main chains of the core or/and corona-forming blocks and do not exist for micelles of conventional linear block copolymers. The theoretical predictions are confronted to experiments.

Polymer Brush in a Nanopore: Effects of Solvent Strength and Macromolecular Architecture Studied by Self-Consistent Field and Scaling Theory.

To study conformational transition occuring upon inferior solvent strength in a brush formed by linear or dendritically branched macromolecules tethered to the inner surface of cylindrical or planar (slit-like) pore, a self-consistent field analytical approach is employed. Variations in the internal brush structure as a function of variable solvent strength and pore radius, and the onset of formation of a hollow channel in the pore center are analysed. The predictions of analytical theory are supported and complemented by numerical modelling by a self-consistent field Scheutjens-Fleer method. Scaling arguments are used to study microphase segregation under poor solvent conditions leading to formation of a laterally and longitudinally patterned structure in planar and cylindrical pores, respectively, and the effects of confinement on "octopus-like" clusters in the pores of different geometries.

Self-Assembly of Bottlebrush Block Copolymers in Selective Solvent: Micellar Structures.

Block copolymers comprising chemically different bottlebrush blocks can self-assemble in selective solvents giving rise to micellar-like solution nanostructures. The self-consistent field theoretical approach is used for predicting relation between architectural parameters of both bottlebrush blocks (polymerization degrees of the main and side chains, density of grafting of the side chains to the backbone) and structural properties of micelles as well as critical micelle concentration (CMC). As predicted by the theory, replacement of linear blocks by bottlebrush ones with the same degrees of polymerization results in a decrease in the micellar core size (in aggregation number) and extension of the corona, whereas the CMC increases. These theoretical findings are in good agreement with results of computer simulations.

Proteins and Polyampholytes Interacting with Polyelectrolyte Brushes and Microgels: The Charge Reversal Concept Revised.

Weak polyampholytes and globular proteins among them can be efficiently absorbed from solutions by polyelectrolyte brushes or microgels even if the net charge of the polyampholyte is of the same sign as that of the brush/microgel. We use a mean-field approach for calculating the free energy of insertion of a probe polyampholyte molecule into a polyelectrolyte brush/microgel. We anticipate that the insertion of the polyampholyte into similarly charged brush/microgel may be thermodynamically favorable due to the gain in the cumulative re-ionization free energy of the pH-sensitive acidic and basic residues. Importantly, we demonstrate that the polyampholyte (protein) charge sign inversion upon transfer from the bulk of the solution to the brush/microgel does not provide sufficient conditions to assure negative re-ionization free energy balance. Thus (in the absence of other driving or stopping mechanisms), charge sign inversion does not necessarily provoke spontaneous absorption of the polyampholyte into the brush/microgel.

Brushes and lamellar mesophases of comb-shaped (co)polymers: a self-consistent field theory.

Theory describing equilibrium structural properties of solvent-free brushes formed by comblike polymers tethered by end segment of backbone to planar surface is developed using strong-stretching self-consistent field (SS-SCF) analytical approach and supported by numerical self-consistent field calculations based on the Scheutjens-Fleer (SF-SCF) method. The explicit dependence of self-consistent molecular potential on architectural parameters of comblike polymers is analyzed. It is demonstrated that distribution of local tension in backbones of long comblike polymers approaches that for linear chains. The star-to-comblike transition in solvent-free lamellas which occurs upon increase of backbone length of graft-polymer is analyzed.

Electroresponsive Polyelectrolyte Brushes Studied by Self-Consistent Field Theory.

End-grafting of polyelectrolyte chains to conducting substrates offers an opportunity to fabricate electro-responsive surfaces capable of changing their physical/chemical properties (adhesion, wettability) in response to applied electrical voltage. We use a self-consistent field numerical approach to compare the equilibrium properties of tethered strong and weak (pH-sensitive) polyelectrolytes to applied electrical field in both salt-free and salt-containing solutions. We demonstrate that both strong and weak polyelectrolyte brushes exhibit segregation of polyions in two populations if the surface is oppositely charged with respect to the brush. This segregation gives rise to complex patterns in the dependence of the brush thickness on salt concentration. We demonstrate that adjustable ionization of weak polyelectrolytes weakens their conformational response in terms of the dependence of brush thickness on the amplitude of the applied voltage.

Morphological Transitions in Patchy Nanoparticles.

Nanoparticles (NPs) decorated with topographically or chemically distinct surface patches are an emerging class of colloidal building blocks of functional hierarchical materials. Surface segregation of polymer ligands into pinned micelles offers a strategy for the generation of patchy NPs with controlled spatial distribution and number of patches. The thermodynamic nature of this approach poses a question about the stability of multiple patches on the NP surface, as the lowest energy state is expected for NPs carrying a single patch. In the present work, for gold NPs end-grafted with thiol-terminated polymer molecules, we show that the patchy surface morphology is preserved under conditions of strong grafting of the thiol groups to the NP surface (i.e., up to a temperature of 40 °C), although the patch shape changes over time. At higher temperatures (e.g., at 80 °C), the number of patches per NP decreases, due to the increased lateral mobility and coalescence of the patches as well as the ultimate loss of the polymer ligands due to desorption at enhanced solvent quality. The experimental results were rationalized theoretically, using a scaling approach. The results of this work offer insight into the surface science of patchy nanocolloids and specify the time and temperature ranges of the applications of patchy NPs.