Monte carlo simulations of polydisperse polymers grafted on spherical surfaces

This article presents effects of polydispersity in polymers grafted on spherical surfaces on grafted polymer chain conformations, grafted layer thickness, and free‐end monomer distribution within the grafted layer. At brush‐like grafting densities, as polydispersity index (PDI) increases, the scaling exponent of radius of gyration of grafted chains approaches that of a single chain grafted on the same nanoparticle, because polydispersity alleviates monomer crowding within the brush. At high PDI, the chains shorter than the number average chain length, N_n, have more compressed conformations, and the chains longer than N_n overall stretch less than in the monodisperse case. As seen in polydisperse flat brushes at high grafting densities, the grafted layer thickness on spherical nanoparticle increases with PDI. Polydispersity eliminates the region near the surface devoid of free‐end monomers seen in monodisperse cases, and it reduces the width of free‐end monomer distribution and shifts the free‐end monomer distribution close to the surface. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2012     

On the monomer density of grafted polyelectrolyte brushes and their interactions

Most of the modern theories of grafted polyelectrolyte brushes are valid only for moderate stretching of the polyelectrolyte. However, particularly at low ionic strength and high grafting densities, even a moderate charge of the polyelectrolyte can generate a strong stretching. A simple mean field model for strongly stretched grafted polyelectrolyte brushes is suggested, based on an approximate calculation of the partition function of a polyelectrolyte chain. It is shown that the average Boltzmann factor of a possible chain configuration can be approximated by the Boltzmann factor of a configuration with a constant monomer distribution, for which the free energy can be readily obtained. The monomer density in the brush and the interaction between two surfaces with grafted polyelectrolyte brushes could be calculated as a statistical average over all possible configurations. Some simple analytical results are derived, and their accuracy is examined. The dependence of the brush thickness on the electrolyte concentration is investigated, and it is shown that the trapping of a fraction of counterions in the brush influences strongly the thickness of the brush. When two surfaces with grafted polyelectrolyte brushes approach each other more rapidly than the ion diffusion parallel to the surface, the trapping of the counterions between the brushes can affect the interactions by orders of magnitude.     

Interaction of bovine serum albumin and human blood plasma with PEO-tethered surfaces: influence of PEO chain length, grafting density, and temperature

Solid surfaces are modified by grafting poly(ethylene oxide), PEO, to influence their interaction with indwelling particles, in particular molecules of bovine serum albumin and human plasma proteins. As a rule, the grafted PEO layers suppress protein adsorption. The suppression is most effective when the PEO layer is in a molecular brush conformation having a reciprocal grafting density (area per grafted PEO chain) less than the dimensions of the protein molecules. Nevertheless, the protein molecules may penetrate the PEO brush to some extent. For a given grafting density, the penetration is facilitated by increasing thickness of the brush. Tenuous brushes of reciprocal grafting densities exceeding the protein molecular dimensions enhance protein adsorption. The results point to a weak attractive interaction between PEO and protein. The protein repellency of a densely PEO-brushed surface is ascribed to a high activation energy for the protein molecules to enter the brush. Varying the temperature between 22 and 38 degrees C does not significantly affect the range of grafting density over which the brush changes from protein-attractive to protein-repellent.     

Mean‐field theoretical analysis of brush‐coated nanoparticle dispersion in polymer matrices

The equilibrium dispersion of nanoparticles with grafted polymer chains into polymer matrices, of the same chemical structure as the brush, is studied through the device of mean‐field theory. Our results show that the disperion of brush‐coated nanoparticles into a matrix polymer is improved with (i) decreasing particle radius and (ii) increasing brush chain length. Both of these aspects can be understood based on the fact that, unlike the case of planar surfaces, homopolymer chains end‐grafted to spherical nanoparticle surfaces tangentially spread away from the surface thus alleviating the packing frustration that is created by the relatively high grafting densities. This permits significant brush/matrix overlap, even at high grafting densities, a regime that has only recently become experimentally available due to advances in polymer synthesis (i.e., the “grafting‐to” methods). © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 351–358, 2008     

Combinatorial study of the mushroom-to-brush crossover in surface anchored polyacrylamide

We present a method for fabricating anchored polymers with a gradual variation of grafting densities on solid substrates. The technique for generating such structures comprises (i) formation of a molecular gradient of polymerization initiator on the solid substrate and (ii) polymerization from the substrate-bound initiator centers ("grafting from"). We measure the mushroom-to-brush transition in grafted polyacrylamides and show that the mushroom and brush behavior can be described using existing scaling theories.     

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.     

Theoretical study on tethered polymers with explicit grafting points in Θ-solvent

Systematic studies on the polymers chemically grafted onto a solid substrate with various grafting densities are presented based on the self-consistent mean-field theory (SCMFT). The distribution of the grafting points is explicitly included and all the three coordinates of each grafting point are fixed during the calculations. The existence of solvent molecules is also explicitly considered in the model and the case of Θ-solvent is investigated. The structure of the system is derived by solving the SCMFT equations in three-dimensional space. For the cases of low grafting density, the system is highly inhomogeneous and typical mushroom-like structures are derived. On the other hand, when the grafting density is high enough, the system is nearly homogeneous along the substrate and the polymer concentration profile is consistent with the numerical results of one dimensional SCMFT calculations. The crossover between "mushroom" regime and polymer brush is obtained by tuning the grafting density. In addition, in brush limit, while the root-mean-squared thickness of the brush is linearly dependent on the degree of polymerization, its dependency on the grafting density is in general more complicated than a simple power law.     

Diblock polyampholytes grafted onto spherical particles: effect of stiffness, charge density, and grafting density

The structure of spherical brushes formed by symmetric diblock polyampholytes end-grafted onto small spherical particles in aqueous solution is examined within the framework of the so-called primitive model using Monte Carlo simulations. The properties of the two blocks are identical except for the sign of their charges. Three different chain flexibilities corresponding to flexible, semiflexible, and stiff blocks are considered at various polyampholyte linear charge densities and grafting densities. The link between the two blocks is flexible at all conditions, and the grafted segments are laterally mobile. Radial and lateral spatial distribution functions of different types and single-chain properties are analyzed. The brush structure strongly depends on the chain flexibility. With flexible chains, a disordered polyelectrolyte complex is formed at the surface of the particle, the complex becoming more compact at increasing linear charge density. With stiff blocks, the inner blocks are radially oriented. At low linear charged density, the outer blocks are orientationally disordered, whereas at increasing electrostatic interaction the two blocks of a polyampholyte are parallel and close to each other, leading to an ordered structure referred to as a polyampholyte star. As the grafting density is increased, the brush thickness responds differently for flexible and nonflexible chains, depending on a different balance between electrostatic interactions and excluded volume effects.     

Evaluation of an atomic force microscopy pull-off method for measuring molecular weight and polydispersity of polymer brushes: effect of grafting density

The accuracy of the molecular weights Mn and polydispersities of polymer brushes, determined by stretching the grafted chains using atomic force microscopy (AFM) and measuring the contour length distribution, was evaluated as a function of grafting density sigma. Poly(N,N-dimethylacrylamide) brushes were prepared by surface initiated atom transfer radical polymerization on latex particles with sigma ranging between 0.17 and 0.0059 chains/nm2 and constant Mn. The polymer, which could be cleaved from the grafting surface by hydrolysis and characterized by gel permeation chromatography (GPC), had a Mn of 30,600 and polydispersity (PDI) of 1.35. The Mn determined by the AFM technique for the higher density brushes agreed quite well with the GPC results but was significantly underestimated for the lower sigma. At high grafting density in good solvent, the extended structure of the brush increases the probability of forming segment-tip contacts located at the chain end. When the distance between chains approached twice the radius of gyration of the polymer, the transition from brush to mushroom structure presumably enabled the formation of a larger number of segment-tip contacts having separations smaller than the contour length, which explains the discrepancy between the two methods at low sigma. The PDI was typically higher than that obtained by GPC, suggesting that sampling of chains with above average contour length occurs at a frequency that is greater than their spatial distribution.     

Directed motion of a polyelectrolyte micelle along tethered chains of oppositely charged polyelectrolyte brush

The complexation of different single polyelectrolyte (PE) micelles formed by linear diblock copolymers with oppositely charged brushes with varying grafting densities and charge content was studied by means of molecular dynamics simulations using the primitive model. We found that all micelles perform a directed motion along the vertical z axis on the grafted surface where they trapped while on the other axes the motion is restricted in a circle in which the diameter decreases with the increase of the hydrophilic length of the linear diblock copolymer. The motion of micelles is characterized as super diffusion inside brushes with low densities and low charge content. At high grafting densities and charge content the diffusion becomes Fickian or slightly subdiffusive. The number of the absorbed brush chains on the micelle corona increases almost monotonically with the increase of brush grafting density or with the decrease of charge content. The distance from the surface in which the micelle is trapped can be controlled by the charge density along the grafted PE chain. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019 , 57, 621–631     

Molecular dynamics simulations of polyelectrolyte brushes: from single chains to bundles of chains

Using molecular dynamics simulations in combination with scaling analysis, we have studied the effects of the solvent quality and the strength of the electrostatic interactions on the conformations of spherical polyelectrolyte brushes in salt-free solutions. The spherical polyelectrolyte brush could be in one of four conformations: (1) a star-like conformation, (2) a "star of bundles" conformation in which the polyelectrolyte chains self-assemble into pinned cylindrical micelles, (3) a micelle-like conformation with a dense core and charged corona, or (4) a conformation in which there is a thin polymeric layer uniformly covering the particle surface. These different brush conformations appear as a result of the fine interplay between electrostatic and monomer-monomer interactions. The brush thickness depends nonmonotonically on the value of the Bjerrum length. This dependence of the brush thickness is due to counterion condensation inside the brush volume. We have also established that bundle formation in poor solvent conditions for the polymer backbone can also occur in a planar polyelectrolyte brush. In this case, the grafted polyelectrolyte chains form hemispherical aggregates at low polymer grafting densities, cylindrical aggregates at an intermediate range of the grafting densities, and vertically oriented ribbon-like aggregates at high grafting densities.     

Attractive bridging interactions in dense polymer brushes in good solvent measured by atomic force microscopy

Using an atomic force microscope (AFM), we have investigated the interaction forces exerted by latex particles bearing densely grafted polymer brushes consisting of poly(N,N-dimethylacrylamide) (PDMA), poly(methoxyethylacrylamide) (PMEA), poly(N-isopropylacrylamide) (PNIPAM), and PMEA-b-PNIPAM in aqueous media (good solvent). The brushes were prepared by controlled surface-initiated atom transfer radical polymerization, and the hydrodynamic thicknesses were measured by dynamic light scattering. The molecular weight (Mn), grafting density (sigma), and polydispersity (PDI) of the brushes were determined by gel permeation chromatography and multiangle laser light scattering after cleaving the polymer from the latex surface by hydrolysis. Force profiles of PDMA (0.017 nm(-2) < or = sigma < or = 0.17 nm-2) and PMEA (sigma = 0.054 nm-2) brushes were purely repulsive upon compression, with forces increasing with Mn and a, as expected, due to excluded volume interactions. At a sufficiently low grafting density (sigma = 0.012 nm-2), PDMA exhibited a long-range exponentially increasing attractive force followed by repulsion upon further compression. The long-range attractive force is believed to be due to bridging between the free chain ends and the AFM tip. The PNIPAM brush exhibited a bridging force at a grafting density of 0.037 nm(-2), a value lower than the sigma needed to induce bridging in the PDMA brush. Bridging was therefore found to depend on grafting density as well as on the nature of the monomer. The grafting densities of these polymers were larger than those typically associated with bridging. Bridging interactions were used to confirm the presence of PNIPAM in a block copolymer PMEA-b-PNIPAMA brush given that the original PMEA homopolymer brush produced a purely repulsive force. The attractive force was first detected in the block copolymer brush at a separation that increased with the length of the PNIPAM block.     

How Does a Polymer Brush Repel Proteins？

The captioned question has been addressed by the steric effect; namely, the adsorption of proteins on a surface grafted with linear polymer chains decreases monotonically as the grafting density increases. However, there is no quantitative and satisfactory explanation why the adsorption starts to increase when the grafting density is sufficiently high and why polyethylene glycol（PEG） still remains as one of the best polymers to repel proteins. After considering each grafted chain as a molecular spring confined inside a ＂tube＂ made of its surrounding grafted chains, we estimated how its free energy depends on the grafting density and chain length, and calculated its thermal energy-agitated chain conformation fluctuation, enabling us to predict an adsorption minimum at a proper grafting density, which agrees well with previous experimental results. We propose that it is such a chain fluctuation that slows down the adsorption kinetically.     

Properties of star-branched polymer brushes

A simple model of a polymer brush was constructed. The star polymers with three arms were terminally attached with one arm (the stem) to an impenetrable surface with the other two arms (branches) free. The excluded volume effect was included into the model as the only interaction. Therefore, the system was studied in good solvent conditions. The simulations were carried out by means of the dynamic Monte Carlo method using the local changes of chain conformations to sample efficiently the conformational space. The influence of both the number of chains (the grafting density) and the length of chains on the static properties of the polymer brush was studied. The internal and local structure of a formed polymer layer was determined. It was shown that the size of the stems increased rapidly with the increase of the grafting density, while the size of the branches diminished. The changes of the spatial orientations of the stems and the branches for different grafting densities were shown and discussed.     

Adsorption of polymers on a brush: Tuning the order of the wetting phase transition

We develop a computational methodology for the direct measurement of a wetting transition and its order via the effective interface potential. The method also allows to estimate contact angles in the nonwet state and to study adsorption isotherms. The proposed methodology is employed in order to study the wetting behavior of polymers on top of a brush consisting of identical polymers. In the absence of long-range forces, the system shows a sequence of nonwet, wet, and nonwet states as the brush density is increased. Including attractive long-range interactions we can make the polymer liquid wet the bush at all grafting densities, and both first- and second-order wetting transitions are observed. The latter case is limited to a small interval of grafting densities where the melt wets the brush in the absence of long-range interactions. Second-order wetting transitions are preceded by a first-order surface transition from a thin to a thick adsorbed layer. The interval of second-order wetting transitions is limited at low grafting densities by a surface critical end point and at high grafting densities by a tricritical wetting point. Our study highlights the rich wetting behavior that results when competing adsorbent-substrate interactions of different scales are tuned over a broad range.     

PNIPAM chain collapse depends on the molecular weight and grafting density

This study demonstrates that the thermally induced collapse of end-grafted poly(N-isopropylacrylamide) (PNIPAM) above the lower critical solution temperature (LCST) of 32 degrees C depends on the chain grafting density and molecular weight. The polymer was grafted from the surface of a self-assembled monolayer containing the initiator (BrC(CH3)2COO(CH2)11S)2, using surface-initiated atom transfer radical polymerization. Varying the reaction time and monomer concentration controlled the molecular weight, and diluting the initiator in the monolayer altered the grafting density. Surface force measurements of the polymer films showed that the chain collapse above the LCST decreases with decreasing grafting density and molecular weight. At T > LCST, the advancing water contact angle increases sharply on PNIPAM films of high molecular weight and grafting density, but the change is less pronounced with films of low-molecular-weight chains at lower densities. Below the LCST, the force-distance profiles exhibit nonideal polymer behavior and suggest that the brush architecture comprises dilute outer chains and much denser chains adjacent to the surface.     

Static and dynamic properties of the interface between a polymer brush and a melt of identical chains

Molecular-dynamics simulations of a short-chain polymer melt between two brush-covered surfaces under shear have been performed. The end-grafted polymers which constitute the brush have the same chemical properties as the free chains in the melt and provide a soft deformable substrate. Polymer chains are described by a coarse-grained bead-spring model, which includes excluded volume and backbone connectivity of the chains. The grafting density of the brush layer offers a way of controlling the behavior of the surface without altering the molecular interactions. We perform equilibrium and nonequilibrium molecular-dynamics simulations at constant temperature and volume using the dissipative particle dynamics thermostat. The equilibrium density profiles and the behavior under shear are studied as well as the interdigitation of the melt into the brush, the orientation on different length scales (bond vectors, radius of gyration, and end-to-end vector) of free and grafted chains, and velocity profiles. The obtained boundary conditions and slip length show a rich behavior as a function of grafting density and shear velocity.     

Electro‐osmotic flow in nanochannels with voltage‐controlled polyelectrolyte brushes: Dependence on grafting density and normal electric field

Molecular dynamics simulations were performed for electro‐osmotic flow (EOF) confined in a polyelectrolyte‐grafted nanochannel under variable grafting density and normal electric field. With decreasing the value of the normal electric field, the brush undergoes a collapse transition, and the ion distribution is changed significantly. The brush thickness increases on increasing the grafting density at positive and weak negative electric fields, whereas a reduced brush thickness is observed at strong negative electric field. Our results further reveal that the flow velocity is not only dependent on conformational transition of the brush but also related to the cation and anion distributions. At low grafting density, the EOF is almost completely quenched at high electric field strength due to strong surface friction between ions and walls. For the case of very dense grafting, the flow velocity is influenced weakly within the brush when varying the grafting density. Additionally, a bidirectional flow occurs at an intermediate electric field. The investigation on fluid flux indicates that the fluid flux is insensitive to the grafting density, when the normal electric field is removed. For nonzero normal electric fields, a significant change in the fluid flux is observed at low grafting densities. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2012     

Controlled pore functionalization of poly(ethylene terephthalate) track-etched membranes via surface-initiated atom transfer radical polymerization

A new method for surface-initiated atom transfer radical polymerization (ATRP) on the technical polymer poly(ethylene terephthalate) (PET) has been developed which allows controlling and estimating the layer thickness of the grafted polymer in the isocylindrical pores of track-etched membranes. After PET surface treatment by oxidative hydrolysis, the bromoalkyl initiator was immobilized on the PET surface in a two-step solid-phase reaction; the isoporous membrane structure was preserved, and the pore diameter was increased from 760 to 790 nm. Poly(N-isopropylacrylamide) (PNIPAAm) was grafted under ATRP conditions from a methanol/water mixture at room temperature. Both monomer concentration and reaction time could be used as parameters to adjust the degree of grafting. Effective grafted layer thickness and its response to temperature were estimated from pure water permeability. All data, especially the high polymer densities (0.37 g/cm3) in the swollen layers at 25 degrees C, indicate that grafted PNIPAAm with a "brush" structure has been achieved. For dry PNIPAAm layer thicknesses on the PET pore walls of up to 80 nm, a temperature-induced swelling/deswelling ratio of approximately 3 had been observed. Reduction of the brush grafting density, via composition of the reaction mixture used in solid-phase synthesis for initiator immobilization, led to an increase of that swelling/deswelling ratio. Further, density and temperature response of the grafted PNIPAAm layers synthesized via ATRP were compared with those obtained in the same membranes by less controlled photografting, leading to lower grafting density and larger gradients in grafted layer density and, consequently, much higher swelling/deswelling ratios (>15).     

Grafting acrylic polymers from flat nickel and copper surfaces by surface-initiated atom transfer radical polymerization

Acrylic polymers, including poly(methyl methacrylate), poly(2,2,2-trifluoroethyl methacrylate), poly( N,N'-dimethyaminoethyl methacrylate), and poly(2-hydroxyethyl methacrylate) were grafted from flat nickel and copper surfaces through surface-initiated atom transfer radical polymerization (ATRP). For the nickel system, there was a linear relationship between polymer layer thickness and monomer conversion or molecular weight of "free" polymers. The thickness of the polymer brush films was greater than 80 nm after 6 h of reaction time. The grafting density was estimated to be 0.40 chains/nm2. The "living" chain ends of grafted polymers were still active and initiated the growth of a second block of polymer. Block copolymer brushes with different block sequences were successfully prepared. The experimental surface chemical compositions as measured by X-ray photoelectron spectroscopy agreed very well with their theoretical values. Water contact angle measurements further confirmed the successful grafting of polymers from nickel and copper surfaces. The surface morphologies of all samples were studied by atomic force microscopy. This study provided a novel approach to prepare stable functional polymer coatings on reactive metal surfaces.