Planar Polymer Brushes Research Articles

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. We present a mean-field theoretical approach 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. We derive the position-dependent free energy of the colloid insertion into the polymer brush which controls the rate of colloid transport across the brush. We show 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. Our theory should be useful to design sensing and purification devices with enhanced selectivity and to better understand mechanisms underpinning the functions of biopolymer brushes. This article is protected by copyright. All rights reserved.

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.

Stress in a Stimuli-Responsive Polymer Brush

The application of a polymer brush in sensing, actuation, self-folding, among others acutely depends on the tuneable bending of a brush-grafted substrate caused by the stress in the brush. However, the stress in a stimuli-responsive brush has not been investigated. In this work, we study the stress in the stimuli-responsive planar polymer brushes of neutral water-soluble polymers with low to very high graft densities using strong stretching theory (SST). First, SST with the Langevin force-extension relation for a polymer chain is extended to the study of stimuli-responsive brushes. Stress profile and other properties of a Poly(N-isopropylacrylamide) (PNIPAm) brush are then obtained using the extended SST and an empirical Flory-Huggins parameter. The model predicts that the stress in a PNIPAm brush is inhomogeneous and compressive at all temperatures and graft densities. The resultant stress is predicted to increase in magnitude with increasing graft density. Moreover, it decreases in magnitude with an increase in temperature before plateauing in low graft density brushes. In contrast, its magnitude increases weakly with increasing temperature in high density brushes. This contrasting behavior is traced to the minimum in interaction free energy density \emph{vs} polymer volume fraction curve for PNIPAm solution at a large volume fraction, and stiffening of chains due to finite extensibility. Furthermore, our results indicate that the ability to tune the resultant stress by changing temperature diminishes with increasing graft density.

Conformations of Macromolecules and Intramolecular Conformational Transitions

Theoretical studies concerning the conformations of macromolecules and performed by the author (or with participation of the author) in the period from the 1950s to the present time are reviewed. The review demonstrates the diversity of conformations of macromolecules and shows how the evolution of polymer science determined the formulation of successive tasks in the theoretical research of conformations and conformational transitions in macromolecules. Advancing the molecular theory of flexibility of macromolecules and the theory of intramolecular conformational transitions helix–coil and globule–coil is considered. Conformations of regularly branched (star-like and comb-shaped) macromolecules are discussed within the framework of the polymer brush concept, and the conformational transition uncharged globule–ionized star in a starlike pH-sensitive homopolyelectrolyte with amphiphilic monomer units is examined. Conformational transition coil–flower in the linear polymer chain embedded into a planar polymer brush is described. Structural peculiarities of dendron brushes are discussed, and it is shown that their use as a matrix makes it possible to elaborate the model of thermosensitive molecular switch.

Swelling of Planar Polymer Brushes in Solvent Vapors

The swelling of densely grafted planar homo- and copolymer brushes in solvent vapors was studied by the dissipative particle dynamics method. It is shown that the distribution of the solvent inside the polymer film is inhomogeneous with the explicit maximum at the polymer–vapor interface. The value of the maximum does not depend on the grafting density and is mainly defined by the concentration of solvent vapors. At the same time, the distribution function of the free ends in homopolymer brush swollen in vapors of a good solvent is closer to the respective function in dry (solvent-free) brush: it has a rather narrow maximum at the upper border of the brush. In case of copolymer brushes, the swelling in vapors of a selective solvent can be accompanied by separation, the character of which is defined by the sequence of polymer segments (random or gradient) and also by the mean fraction of the segments of different types. The distribution function of free ends in the copolymer brush is calculated.

Attraction between Opposing Planar Dipolar Polymer Brushes.

We use a field theory approach to study the effects of permanent dipoles on interpenetration and free energy changes as a function of distance between two identical planar polymer brushes. Melts (i.e., solvent-free) and solvated brushes made up of polymers grafted on nonadsorbing substrates are studied. In particular, the weak coupling limit of the dipolar interactions is considered, which leads to concentration-dependent pairwise interactions, and the effects of orientational order are neglected. It is predicted that a gradual increase in the dipole moment of the polymer segments can lead to attractive interactions between the brushes at intermediate separation distances. Because classical theory of polymer brushes based on the strong stretching limit (SSL) and the standard self-consistent field theory (SCFT) simulations using the Flory's χ parameter always predicts repulsive interactions at all separations, our work highlights the importance of dipolar interactions in tailoring and accurately predicting forces between polar polymeric interfaces in contact with each other.

Vertical Phase Segregation Induced by Dipolar Interactions in Planar Polymer Brushes

We present a generalized theory for studying structural properties of a planar dipolar polymer brush immersed in a polar solvent. We show that an explicit treatment of the dipolar interactions yields a macroscopic concentration dependent effective “chi” (the Flory–Huggins-like interaction) parameter. Furthermore, it is shown that the concentration dependent chi parameter promotes phase segregation in polymer solutions and brushes so that the polymer-poor phase consists of a finite/nonzero polymer concentration. Such a destabilization of the homogeneous phase by the dipolar interactions appears as vertical phase segregation in a planar polymer brush. In a vertically phase segregated polymer brush, the polymer-rich phase near the grafting surface coexists with the polymer-poor phase at the other end. Predictions of the theory are directly compared with prior reported experimental results for dipolar polymers in polar solvents. Excellent agreements with the experimental results are found, hinting that the di...

Star Brushes Under Deformation: Structure and Thermodynamics

SummaryA planar polymer brush of arm‐grafted polymer stars subjected to external deformation is studied theoretically We propose a simple mean‐field model of the star brush that takes into account segregation of the brush‐forming stars in two populations: (i) those with weakly extended arms and (ii) those with a very strongly stretched grafting arm (stem) and all free arms extended towards brush periphery. The polymer density profile in the brush is assumed to have a “two‐step” shape. Stars in the extended population are assumed to be equally stretched, the position of extended stars' branching points sets the boundary between two parts of the brush. We show that the theory quantitatively accurately describes earlier numerical results obtained with Scheutjens‐Fleer numerical self‐consistent field (SF‐SCF) approach for free (non‐deformed) star brush. We study compression of a star brush by an impenetrable plane by using two complementary models, the SF‐ SCF approach and the developed mean‐field theory, and demonstrate a very good quantitative agreement between the two models. It is shown than depending on the grafting density of the stars, compression could affect the two‐population brush structure in opposite ways. In sparsely grafted brushes the fraction of stars in the stretched population increases while in densely grafted brushes it decreases with deformation.

Self-assembly of polymer brushes in the presence of a surfactant: A system of strands

This theoretical study is focused on the formation of a cylindrical microstructure in a planar polymer brush in the presence of a surfactant. It is assumed that the brush may be nonuniform in the direction along the grafting plane and that there are regions with constant concentrations of monomer units and regions occupied only by the surfactant. The surfactant molecule is simulated by a dimer whose parts interact in a different manner with the monomer units of the polymer. At the interface between these regions, dimer molecules are oriented mainly perpendicularly to this interface and the surface tension is reduced. If the surface energy becomes negative, the formation of a structured brush is more favorable in terms of energy than that of a uniform brush. As a result, there may appear a cylindrical microstructure in which grafted macromolecules are united into strands perpendicular to the grafting plane. The stretching of macromolecules and their interaction with the solvent within the strands are described by the Alexander-de Gennes model, whereas the surface energy is calculated with allowance for the surface curvature of strands at a high degree of amphiphilicity of the surfactant molecules. It is shown that the arising strands have radii of the order of the surfactant-molecule length, while the number of macromolecules per strand is proportional to the surface density of their grafting. With an increase in the grafting density, the strand length increases initially, while the volume fraction of the polymer in a strand remains constant. Furthermore, strands start to shorten and their density grows. Structural characteristics are calculated as a function of the parameter characterizing the degree of amphiphilicity of the solvent molecules, their sizes, and their average energy of interaction with monomer units.

Colloids in Brushes: The Insertion Free Energy via Monte Carlo Simulation with Umbrella Sampling

The insertion of spherical, nonadsorbing colloidal particles into a swollen planar polymer brush is characterized using Monte Carlo simulations with umbrella sampling focusing on small particles whose radius Rp is smaller than the unperturbed brush thickness, h0. This process plays a key role in the modeling antifouling poly(ethylene glycol) brushes that repress protein adsorption. Two properties are studied as a function of Rp and the altitude within the brush z: (i) The particle induced perturbation of the monomer concentration profile and (ii) The insertion free energy penalty, Fins. The perturbation of the concentration profile is short ranged involving pure depletion at high z and depletion followed by a maximum at low z. When the particle does not experience the surface depletion layer Fins grows with the monomer volume fraction ϕ and Rp. Fins ∼ Rp3 for the larger Rp reflects an osmotic insertion penalty at the high ϕ range of the brush. At the lower ϕ range a surface tension correction plays a role. In the range explored there is no evidence for Fins ∼ Rp4/3 as was suggested for small particles.

Normal and Lateral Deformation of Lyotropically Ordered Polymer Brush

Summary: Planar polymer brush formed by semirigid chains of freely jointed rigid segments and immersed into a solvent is considered. Brush collapse induced by deterioration of the solvent quality and its deformation by external normal or lateral force is studied. It is demonstrated that these three different situations can be described in the framework of the common approach. It is shown that the collapse is accompanied by liquid-crystalline (LC) ordering within the brush. The LC transition can be jump-like (the first order) or continuous, depending on the segment's aspect ratio and grafting density. Transition point is investigated in detail, the corresponding phase diagrams are calculated. It is shown that the phase diagrams of a normally deformed brush have different structures, with a narrow ‘leg’ in the good solvent region for sparsely grafted brush, with two coexistence regions and a triple point, in addition, for shorter segment length or without these features if the chains are densely grafted. For the laterally deformed brush, phase diagrams have similar structures with a critical point in the good solvent regime. Polymer brush subjected to deformation by normal (top) and lateral (bottom) external force.

Phase transitions in polymer brushes

AbstractLiquid crystalline ordering in planar polymer brushes is investigated theoretically by numerical calculations within a self‐consistent field approximation. The brushes are formed by macromolecules with mesogenic groups in main chain and immersed in a solvent. Existence of a microphase segregated brush (MSB) regime with a collapsed orientationally ordered intrinsic sublayer and a swollen external sublayer is shown. At small grafting density, the transition from a conventional brush state to the microphase segregated state is a jump‐wise first order phase transition for a finite chain length (N). The magnitudes of the jumps in the average characteristics of the brush tend to zero in the limit N → ∞ since this transition occurs only in a vanishingly small part (∝ N−1/2) of the brush. High compressibility of MSB brush is demonstrated. The origin of phase transition in planar brushes is discussed.

Self Consistent Field Calculations of Interactions between Chains Tethered to Spherical Interfaces

Self consistent mean field (SCF) theory is used to investigate the structure of and interactions between layers of polymer chains tethered to a spherical interface. Traditional methods have combined flat geometry results with the Derjaguin approximation to calculate the interaction potentials between curved surfaces, leading to a great overestimation of both the range and steepness of the pair interaction potential. We have improved upon this approach by utilizing the chain configurational statistics from the curved geometry with a modified Derjaguin approximation to calculate pair interaction potentials. By varying the core radius, the degree of polymerization, and the tethering density in an athermal solvent, the tethered layer assumes structures ranging from those in star polymer systems to planar polymer brushes. The interaction potentials are found to be strictly repulsive with varying degrees of range and steepness. These changes in the interaction potential are related to changes in the tethered la...

Moduli of polymer brushes, contact mechanics, and atomic force microscope experiments

We present two new simple derivations of the elastic moduli of planar polymer brushes in the melt regime. We find moduli which are somewhat deformation dependent. We also examine the problem of the penetration of such a brush by the tip of an atomic force microscope for the case of weak deformations