Monomer Density Profile Research Articles

Semiflexible polymers grafted to a solid planar substrate: Changing the structure from polymer brush to “polymer bristle”

Monte Carlo simulations are presented for a coarse-grained model of polymer brushes with polymers having a varying degree of stiffness. Both linear chains and ring polymers grafted to a flat structureless non-adsorbing substrate surface are considered. Applying good solvent conditions, it is shown that with growing polymer stiffness the brush height increases significantly. The monomer density profiles for the case of ring polymers (chain length N(R) = 64) are very similar to the case of corresponding linear chains (N(L) = 32, grafting density larger by a factor of two) in the case of flexible polymers, while slight differences appear with increasing stiffness. Evidence is obtained that the chain dynamics in brushes is slowed down dramatically with increasing stiffness. Very short stiff rings (N(R) ≤ 16) behave like disks, grafted to the substrate such that the vector, perpendicular to the disk plane, is oriented parallel to the substrate surface. It is suggested that such systems can undergo phase transitions to states with liquid crystalline order.

Frictional forces in polyelectrolyte brushes: effects of sliding velocity, solvent quality and salt

We report dissipative particle dynamics (DPD) simulations of grafted polymer brushes under shear at three separation distances. We investigate the impact of the shear on single monolayers and compressed bilayers formed by neutral and charged polymer chains. The dependence of the friction on the solvent quality is studied for different charge fractions in two compressed polymer brushes. The change in the frictional forces upon the addition of the salt is analyzed in terms of change in the interpenetration coefficient. We complete this study by a description of the brush structure including ordering, composition, tilt and local electroneutrality at different separation distances, salt concentrations, charge fractions and shear rates. Interestingly, some heterogeneities in the monomer density profiles of polyelectrolyte brushes appear under shear and vanish upon addition of salt.

Structure of bottle-brush brushes under good solvent conditions: a molecular dynamics study

We report a simulation study for bottle-brush polymers grafted on a rigid backbone. Using a standard coarse-grained bead–spring model extensive molecular dynamics simulations for such macromolecules under good solvent conditions are performed. We consider a broad range of parameters and present numerical results for the monomer density profile, density of the untethered ends of the grafted flexible backbones and the correlation function describing the range that neighboring grafted bottle-brushes are affected by the presence of the others due to the excluded-volume interactions. The end beads of the flexible backbones of the grafted bottle-brushes do not access the region close to the rigid backbone due to the presence of the side chains of the grafted bottle-brush polymers, which stretch the chains further in the radial directions. Although a number of different correlation lengths exist as a result of the complex structure of these macromolecules, their properties can be tuned with high accuracy in good solvents. Moreover, qualitative differences with ‘typical’ bottle-brushes are discussed. Our results provide a first approach to characterizing such complex macromolecules with a standard bead–spring model.

Monomer density profiles of real polymer chains in confined geometries

The universal density–force relation is analyzed and the correspondent universal amplitude ratio B real is obtained using the massive field theory approach in fixed space dimensions d = 3 up to one-loop order. The layer monomer density profiles of ideal chains and real polymer chains with excluded volume interaction in a good solvent between two parallel repulsive walls, one inert and one repulsive wall and for the case of wall and a mesoscopic spherical particle of a big radius are obtained using the massive field theory approach up to one-loop order. The last mentioned situation is analyzed for both cases when wall and particle are repulsive and for the mixed case of repulsive wall and inert particle. Besides, the layer monomer density profiles of polymer chains with one end fixed at the surface and other free are analyzed near the repulsive and inert wall up to two-loop order. The obtained results are in qualitative agreement with previous theoretical investigations and with the results of Monte Carlo simulations.

Interactions of polyelectrolyte brushes with oppositely charged surfactants

Using Brownian dynamics simulations, we study the effect of the charge ratio, the surfactant length, and the grafting density on the conformational behavior of the complex formed by the polyelectrolyte brush with oppositely charged surfactants. In our simulations, the polyelectrolyte chains and surfactants are represented by a coarse-grained bead-spring model, and the solvent is treated implicitly. It is found that varying the charge ratio induces different morphologies of surfactant aggregates adsorbed onto the brush. At high charge ratios, the density profiles of surfactant monomers indicate that surfactant aggregates exhibit a layer-by-layer arrangement. The surfactant length has a strong effect on the adsorption behavior of surfactants. The lengthening of surfactant leads to a collapsed brush configuration, but a reswelling of the brush with further increasing the surfactant length is observed. The collapse of the brush is attributed to the enhancement of surfactants binding to polyelectrolyte chains. The reswelling is due to an increase in the volume of adsorbed surfactant aggregates. At the largest grafting density investigated, enhanced excluded volume interactions limit the adsorption of surfactant within the polyelectrolyte brush. We also find that end monomers in polyelectrolyte chains exhibit a bimodal distribution in cases of large surfactant lengths and high charge ratios.

Monomer density profiles for polymer chains in confined geometries: Massive field theory approach

Taking into account the well known correspondence between the field theoretical ϕ(4) O(n)-vector model in the limit n → 0 and the behavior of long flexible polymer chains in a good solvent, the universal density-force relation is analyzed and the corresponding universal amplitude ratio B(real) is obtained using the massive field theory approach in fixed space dimensions d < 4. The monomer density profiles of ideal chains and real polymer chains with excluded volume interaction in a good solvent between two parallel repulsive walls, one repulsive and one inert wall, are obtained in the framework of the massive field theory approach up to one-loop order. Besides, the monomer density profiles for the dilute polymer solution confined in semi-infinite space containing mesoscopic spherical particle of big radius are calculated. The obtained results are in qualitative agreement with previous theoretical investigations and with the results of Monte Carlo simulations.

New Development of Monte Carlo Techniques for Studying Bottle-brush Polymers

Due to the complex characteristics of bottle-brush polymers, it became a challenge to develop an e_cient algorithm for studying such macromolecules under various solvent conditions or some constraints in the space by using computer simulations. In the limit of a bottle-brush polymer with a rather sti_ backbone (straight rigid backbone), we generalize the variant of the biased chain growth algorithm, the pruned-enriched Rosenbluth method, for simulating polymers with complex architecture, from star polymers to bottle-brush polymers, on the simple cubic lattice. With the high statistics of our Monte Carlo results, we check the theoretical predictions of side chain behavior and radial monomer density profile. For the comparison of the experimental data for bottle-brush polymers with a flexible backbone and flexible side chains, based on the bond fluctuation model we propose another fast Monte Carlo algorithm combining the local moves, the pivot move, and an adjustable simulation lattice box. By monitoring the autocorrelation functions of gyration radii for the side chains and for the backbone, we see that for fixed side chain length there is no change in the behavior of these two functions as the backbone length increases. Our extensive results cover the range which is accessible for the comparison to experimental data and for the checking of the theoretically predicted scaling laws.

Structural properties of concave cylindrical brushes interacting with free chains

We present a self-consistent field theoretical study of the microstructure of concave cylindrical brushes as a function of the cylinder radius, grafting density, grafted chain length, and the solvent quality. We show that the results for the radial monomer density profile and the distribution of the free ends are in good agreement with the corresponding molecular dynamics results. Part of the investigation is focused on the conformational behavior of a free macromolecule in a cylindrical brush. A central result is the observed non-monotonous variation of the size of a free chain in a brush-coated tube when the tube radius is systematically changed. An interpretation of this behavior which differs qualitatively from that of a polymer in cylindric confinement is suggested in terms of scaling theory and rationalized by considering the overlap between the free polymer and the grafted chains as a function of the tube radius.

Spherical polymer brushes under good solvent conditions: Molecular dynamics results compared to density functional theory

A coarse grained model for flexible polymers end-grafted to repulsive spherical nanoparticles is studied for various chain lengths and grafting densities under good solvent conditions by molecular dynamics methods and density functional theory. With increasing chain length, the monomer density profile exhibits a crossover to the star polymer limit. The distribution of polymer ends and the linear dimensions of individual polymer chains are obtained, while the inhomogeneous stretching of the chains is characterized by the local persistence lengths. The results on the structure factor of both single chain and full spherical brush as well as the range of applicability of the different theoretical tools are presented. Finally, a brief discussion of the experiment is given.

Conformational studies of bottle-brush polymers absorbed on a flat solid surface

The adsorption of a bottle-brush polymer end-grafted with one chain end of its backbone to a flat substrate surface is studied by Monte Carlo simulation of a coarse-grained model, that previously has been characterized in the bulk, assuming a dilute solution under good solvent conditions. Applying the bond fluctuation model on the simple cubic lattice, we vary the backbone chain length N(b) from N(b)=67 to N(b)=259 effective monomeric units, the side chain length N from N=6 to N=48, and set the grafting density to σ=1, i.e., parameters that correspond well to the experimentally accessible range. When the adsorption energy strength ɛ is varied, we find that the adsorption transition (which becomes well-defined in the limit N(b)→∞, for arbitrary finite N) roughly occurs at the same value ɛ(c) as for ordinary linear chains (N=0), at least within our statistical errors. Mean square end-to-end distances and gyration radii of the side chains are obtained, as well as the monomer density profile in the direction perpendicular to the adsorbing surface. We show that for longer side chains the adsorption of bottle-brushes is a two-step process, the decrease of the perpendicular linear dimension of side chains with adsorption energy strength can even be nonmonotonic. Also, the behavior of the static structure factor S(q) is analyzed, evidence for a quasi-two-dimensional scaling is presented, and consequences for the interpretation of experiments are discussed.

Shearing and compression behavior of end-grafted polyelectrolyte brushes with mono- and trivalent counterions: a molecular dynamics simulation

We investigate polyelectrolytes end-grafted on two apposing walls using molecular dynamics simulation techniques. Monovalent and trivalent counterions are explicitly treated. Under normal compression, the osmotic pressure is examined in detail by decomposing it into various virial terms. It has been found that at small wall separations the increase in the osmotic pressure can be ascribed to the increase in the short-range virial term. At large wall separations, a negative osmotic pressure is observed in trivalent systems. Moreover, we study the effect of lateral shear on the density profiles of monomers and counterions, the net charge distribution, the local pressure tensor, the degree of interpenetration and the friction coefficient. At large shear ratios, the electrostatic interactions are weakened at the interface between two brushes. It is worth noting that although the magnitudes of the normal and shear stress components for the trivalent case are significantly lesser than those for the monovalent case, the friction coefficient is larger in the trivalent systems.

Molecular dynamics simulations of end-grafted centipede-like polymers with stiff charged side chains

We use molecular dynamics simulations to investigate centipede-like polymers with stiff charged side chains, end-grafted to a planar wall. The effect of the grafting density and the Bjerrum length on the conformational behaviour of the brush is examined in detail. In addition, we make a comparison of centipede-like polyelectrolyte (CPE) brushes with neutral centipede-like polymer (NCP) and linear polyelectrolyte (LPE) brushes. At weak electrostatic interaction, the main chains of the CPE chains adopt a strongly stretched conformation, and the monomer density profiles of side chains exhibit a clear oscillatory behaviour. With increasing Bjerrum length, the CPE brush undergoes a collapse transition. Compared to the CPE brushes, the counterion condensation effect is stronger for the LPE brushes, regardless of whether the electrostatic interaction is weak or strong and of whether the grafting density is low or high. Additionally, it is shown that the architecture of the grafted chains makes a weak contribution to the counterion condensation at strong electrostatic interaction. We also find that the electrostatic repulsion between charged side chains can enhance the stiffness of the main chains and thus limit the range of movement of the free-end monomers.

Characteristic Length Scales and Radial Monomer Density Profiles of Molecular Bottle-Brushes: Simulation and Experiment

Extensive Monte Carlo simulations are presented for bottle-brush polymers under good solvent conditions, using the bond fluctuation model on the simple cubic lattice. Varying the backbone length (from Nb = 67 to Nb = 259 effective monomers) as well as the side chain length (from N = 6 to N = 48), for a physically reasonable grafting density of one chain per backbone monomer, we find that the structure factor describing the total scattering from the bottle-brush provides an almost perfect match for some combinations of (Nb, N) to experimental data of Rathgeber et al. [J. Chem. Phys. 2005, 122, 124904], when we adjust the length scale of the simulation to reproduce the experimental gyration radius of the bottle-brush. While in the experiment other length scales (gyration radius of side chains, backbone persistence length, scale characterizing the radial monomer density profile in the plane normal to the backbone) can be extracted only via fitting to a complicated and approximate theoretical expression deriv...

A Molecular Dynamics Study of Two Apposing Polyelectrolyte Brushes with Mono‐ and Multivalent Counterions

AbstractThe conformational characteristics of polyelectrolytes end‐grafted on two apposing walls are studied. The monovalent and multivalent counterions are explicitly treated. It is found that the polyelectrolyte brushes undergo a collapse transition in the presence of multivalent counterions at large wall separations. Moreover, the brush thickness is linearly proportional to the grafting density and independent of the counterion valence. The effects of normal compression on the density profiles of monomers and counterions, the pressure, the net charge density and the degree of interpenetration are also studied. The density profiles of counterions exhibit an asymmetrical distribution in systems with a mixture of monovalent and trivalent counterions.magnified image

Spherical brushes within spherical cavities: A self-consistent field and Monte Carlo study

We present an extensive numerical study on the behavior of spherical brushes confined into a spherical cavity. Self-consistent field (SCF) and off-lattice Monte Carlo (MC) techniques are used in order to determine the monomer and end-chain density profiles and the cavity pressure as a function of the brush properties. A comparison of the results obtained via SCF, MC, and the Flory theory for polymer solutions reveals SCF calculations to be a valuable alternative to MC simulations in the case of free and softly compressed brushes, while the Flory's theory accounts remarkably well for the pressure in the strongly compressed regime. In the range of high compressions, we have found the cavity pressure P to follow a scale relationship with the monomer volume fraction v, P approximately v(alpha). SCF calculations give alpha=2.15+/-0.05, whereas MC simulations lead to alpha=2.73+/-0.04. The underestimation of alpha by the SCF method is explained in terms of the inappropriate account of the monomer density correlations when a mean field approach is used.

Adsorption of self-avoiding tethered membranes: A Monte Carlo simulation study

Adsorption of self-avoiding tethered membranes of hexagonal orientation on a solid impenetrable plane is studied by means of Monte Carlo computer simulations of a coarse-grained continuum model, varying the membrane linear size L and the strength epsilon of the short-range attractive wall potential. A second-order adsorption transition is found to take place at a critical strength of the adsorption potential epsilon(c), as predicted earlier for binding manifolds in the so-called strong fluctuating regime. By means of finite-size scaling analysis for membranes of size 5<or=L<or=30, containing 61<or=N<or=2611 monomers, we find that the critical crossover exponent for adsorption phi approximately 0.60+/-0.01. Thus the fraction m of adsorbed segments at epsilon(c) is found to scale as m proportional to N(phi-1)=N(-0.4). The membrane thickness lambda(min) decreases with growing strength epsilon of the adhesive potential as lambda(min) proportional to mid R:epsilon/epsilon(c)-1mid R:(-psi), where psi approximately 0.58+/-0.02. The monomer density profiles of adsorbed membranes decay exponentially with the distance z from the substrate rho(z) proportional to exp[-(z/xi)], where the correlation length xi proportional to mid R:epsilon/epsilon(c)-1mid R:(-psi) with psi approximately 0.70+/-0.01. We also investigate the kinetics of adsorption of a polymerized membrane in the regime of strong adsorption and find that the order parameter variation with elapsed time during the adsorption process is given by a power law m(t) proportional to t(omega), where omega approximately 1.0, regardless of the strength of the adsorbing potential epsilon>>epsilon(c). The characteristic time for complete adsorption in this regime scales as tau(ads) proportional to L(2). Regarding the strength epsilon of the substrate potential, tau(ads) is found to diminish linearly as the respective equilibrium value of the order parameter increases. A simple analytic model yields also tau(ads) proportional to L(2) while suggesting that the process of adsorption cannot be identified by simple "unrolling."

Structure of bottle-brush polymers in solution: A Monte Carlo test of models for the scattering function

Extensive Monte Carlo results are presented for the structure of a bottle-brush polymer under good solvent or theta solvent conditions. Varying the side chain length, backbone length, and the grafting density for a rigid straight backbone, both radial density profiles of monomers and side chain ends are obtained as well as structure factors describing the scattering from a single side chain and from the total bottle-brush polymer. To describe the structure in the interior of a very long bottle brush, a periodic boundary condition in the direction along the backbone is used, and to describe effects due to the finiteness of the backbone length, a second set of simulations with free ends of the backbone is performed. In the latter case, the inhomogeneity of the structure in the direction along the backbone is carefully investigated. We use these results to test various phenomenological models that have been proposed to interpret experimental scattering data for bottle-brush macromolecules. These models aim to extract information on the radial density profile of a bottle brush from the total scattering via suitable convolution approximations. Limitations of this approach and the optimal way to perform the analysis of the scattering data within this approach are discussed.

Stimuli-Responsive Interfaces Using Random Polyampholyte Brushes

We study the pH-responsive behavior of random polyampholyte brushes synthesized by surface-initiated Atom Transfer Radical Polymerization. We define the “annealed” polyampholyte as the random copolymer that contains weak acid (methacrylic acid) and weak base (2-(dimethylamino)ethyl methacrylate) units. The “semi-annealed” polyampholyte is defined as the random copolymer that contains the methacrylic acid and a strong base unit (the quaternized amine). The structure of the polyampholyte brushes in aqueous solutions at different pH is determined using neutron reflectivity. The monomer density profile and therefore the brush thickness are obtained. Polymer brushes with various grafting densities are compared. We demonstrate that the structure of the polyampholyte brush depends strongly on the net charge of the chains. With an excess of charge, the brush is as stretched as a polyelectrolyte brush, and the swelling behavior is in good agreement with the scaling laws. At the isoelectric point corresponding to a...

Compressible or incompressible blend of interacting monodisperse star and linear polymers near a surface

We consider a lattice model of a mixture of repulsive, attractive, or neutral monodisperse star (species A) and linear (species B) polymers with a third monomeric species C, which may represent free volume. The mixture is next to a hard, infinite plate whose interactions with A and C can be attractive, repulsive, or neutral. These two interactions are the only parameters necessary to specify the effect of the surface on all three components. We numerically study monomer density profiles using the method of Gujrati and Chhajer that has already been previously applied to study polydisperse and monodisperse linear-linear blends next to surfaces. The resulting density profiles always show an enrichment of linear polymers in the immediate vicinity of the surface due to entropic repulsion of the star core. However, the integrated surface excess of star monomers is sometimes positive, indicating an overall enrichment of stars. This excess increases with the number of star arms only up to a certain critical number and decreases thereafter. The critical arm number increases with compressibility (bulk concentration of C). The method of Gujrati and Chhajer is computationally ultrafast and can be carried out on a personal computer (PC), even in the incompressible case, when simulations are unfeasible. Calculations of density profiles usually take less than 20 min on PCs.

Transition from Highly to Fully Stretched Polymer Brushes in Good Solvent

The stretching of brushes of long polymers grafted to a planar surface is investigated by Monte Carlo simulations in the limit of very high grafting densities, as achieved in recent experiments. The monomer density profiles are shown to deviate considerably from the parabolic limiting form predicted by self-consistent field theory. A rapid transition is observed from parabolic to fully stretched polymers, characterized by a dramatic change in the end-monomer height distribution and by a clear crossover in the slope of the brush height versus scaled grafting density.