Polymer-brush lubrication: a review of recent theoretical advances.

This review compiles recent theoretical advances to describe compressive and shear forces of polymer-brush bilayers, which consist of two opposing brushes in contact. Such model systems for polymer-brush lubrication are frequently used as a benchmark to gain insight into biological problems, e.g., synovial joint lubrication. Based on scaling theory, I derive conformational and collective properties of polymer-brush bilayers in equilibrium and out-of-equilibrium situations, such as shear forces in the linear and nonlinear response regimes of stationary shear and under non-stationary shear. Furthermore, I discuss the influence of macromolecular inclusions and electrostatic interactions on polymer-brush lubrication. Comparisons to alternative analytical approaches, experiments and numerical results are performed. Special emphasis is given to methods for simulating polymer-brush bilayers using molecular dynamics simulations.

Polymer-brush lubricated surfaces with colloidal inclusions under shear inversion.

We characterize the response of compressed, sheared polymer-brush bilayers with colloidal inclusions to highly nonstationary inversion processes by means of molecular dynamics simulations and scaling theory. Bilayers with a simple (dimeric) solvent reveal an overshoot for the shear stress, while simulations of dry brushes without explicit solvent molecules fail to display this effect. We demonstrate that mechanical instabilities can be controlled by the inclusion of macromolecular structures, such as colloids of varying softness. Based on a recently developed theory, we suggest a scaling approach to determine a characteristic time for conformational and collective responses.

Algebraic displacement correlation in two-dimensional polymer melts.

Dense self-avoiding polymer chains in strictly two dimensions are compact objects with fractal contours. Using scaling arguments and molecular dynamics simulations (with negligible momentum conservation) it is shown that correlated amoebalike fluctuations of the (sub)chain contours dominate the relaxation dynamics on all scales. The incompressibility of the melt and the compactness of (sub)chains impose a scale-free constraint on the contour fluctuations. This leads to strong long range spatiotemporal correlations of the displacement field as shown, e.g., by the (negative) algebraic decay of the center-of-mass velocity correlation function C(t)∼-1/t(6/5) with time t.

Frictional forces between strongly compressed, nonentangled polymer brushes: molecular dynamics simulations and scaling theory.

By means of molecular dynamics simulations and scaling theory we study the response of opposing polymer brushes to constant shear motion under good solvent conditions. Model systems that contain explicit solvent molecules (Lennard-Jones dimers) are compared to solvent-free systems while varying of the distance between the grafted layers and their molecular parameters, chain length and grafting density. Our study reveals a power-law dependence of macroscopic transport properties on the Weissenberg number, W, beyond linear response. For instance, we find that the kinetic friction constant scales as mu approximately W(0.57) for large values of W. We develop a scaling theory that describes our data and previous numerical data including recent experiments.

Polymer-brush lubrication in the limit of strong compression.

By means of molecular dynamics simulations we demonstrate power laws for macroscopic transport properties of strongly compressed polymer-brush bilayers to stationary shear motion beyond the Newtonian response. The corresponding exponents are derived from a recently developed scaling theory, where the interpenetration between the brushes is taken as the relevant length scale. This allows to predict the dependence of the critical shear rate, which separates linear and non-linear behavior, on compression and molecular parameters of the bilayer. We present scaling plots for chain extension (R), viscosity (η) , and shear force (F over a wide range of Weissenberg numbers, W . In agreement with our theory, the simulation reveals simple power laws, R ∼ W (0.53), η ∼ W (-0.46), and F ∼ W (0.54), for the non-Newtonian regime.

A finite excluded volume bond-fluctuation model: static properties of dense polymer melts revisited.

The classical bond-fluctuation model (BFM) is an efficient lattice Monte Carlo algorithm for coarse-grained polymer chains where each monomer occupies exclusively a certain number of lattice sites. In this paper we propose a generalization of the BFM where we relax this constraint and allow the overlap of monomers subject to a finite energy penalty epsilon. This is done to vary systematically the dimensionless compressibility g of the solution in order to investigate the influence of density fluctuations in dense polymer melts on various static properties at constant overall monomer density. The compressibility is obtained directly from the low-wave vector limit of the static structure factor. We consider, e.g., the intrachain bond-bond correlation function P(s) of two bonds separated by s monomers along the chain. It is shown that the excluded volume interactions are never fully screened for very long chains. If distances smaller than the thermal blob size are probed (s<>g) like swollen chains of incompressible blobs with P(s) approximately g(0)s(-3/2).

Perimeter length and form factor in two-dimensional polymer melts.

Self-avoiding polymers in two-dimensional (d=2) melts are known to adopt compact configurations of typical size R(N) approximately N;{1/d} , with N being the chain length. Using molecular-dynamics simulations we show that the irregular shapes of these chains are characterized by a perimeter length L(N) approximately R(N);{d_{p}} of fractal dimension d_{p}=d-Theta_{2}=5/4 , with Theta_{2}=3/4 being a well-known contact exponent. Due to the self-similar structure of the chains, compactness and perimeter fractality repeat for subchains of all arclengths s down to a few monomers. The Kratky representation of the intramolecular form factor F(q) reveals a strong nonmonotonous behavior with q;{2}F(q) approximately 1/(qN;{1/d});{Theta_{2}} in the intermediate regime of the wave vector q . Measuring the scattering of labeled subchains the form factor may allow to test our predictions in real experiments.

Comparison of dissipative particle dynamics and Langevin thermostats for out-of-equilibrium simulations of polymeric systems.

In this work we compare and characterize the behavior of Langevin and dissipative particle dynamics (DPD) thermostats in a broad range of nonequilibrium simulations of polymeric systems. Polymer brushes in relative sliding motion, polymeric liquids in Poiseuille and Couette flows, and brush-melt interfaces are used as model systems to analyze the efficiency and limitations of different Langevin and DPD thermostat implementations. Widely used coarse-grained bead-spring models under good and poor solvent conditions are employed to assess the effects of the thermostats. We considered equilibrium, transient, and steady state examples for testing the ability of the thermostats to maintain constant temperature and to reproduce the underlying physical phenomena in nonequilibrium situations. The common practice of switching off the Langevin thermostat in the flow direction is also critically revisited. The efficiency of different weight functions for the DPD thermostat is quantitatively analyzed as a function of the solvent quality and the nonequilibrium situation.

Nonlinear effects in charge stabilized colloidal suspensions.

Molecular-dynamics simulations are used to study the effective interactions in charged stabilized colloidal suspensions. We focus on highly charged macroions in the limit of low salt concentrations. Within this regime, nonlinear corrections to Debye-Hückel (DH) theory have to be considered. For non-bulk-like systems, such as isolated pairs or triples of macroions, we show that nonlinear effects can become relevant, which cannot be described by the charge renormalization concept [S. Alexander, J. Chem. Phys. 80, 5776 (1984)]. For an isolated pair of macroions, we find an almost perfect qualitative agreement between our simulation data and DH theory. However, on a quantitative level, neither DH theory nor the charge renormalization concept can be confirmed in detail. This seems mainly to be related to the fact that for small ion concentrations, microionic layers can strongly overlap, whereas, simultaneously, excluded volume effects are less important. In the case of isolated triples, where we compare between coaxial and triangular geometries, we find attractive corrections to pairwise additivity in the limit of small macroion separations and salt concentrations. These triplet interactions arise if all three microionic layers around the macroions exhibit a significant overlap. In contrast to the case of two isolated colloids, the charge distribution around a macroion in a triple is found to be anisotropic.

Shear-Induced Ordering in Thin Films of Diblock Copolymer Melts.

By means of molecular dynamics simulations we investigate the response of thin, symmetric diblock copolymer melts under shear in the limit of strong segregation with nonselective substrates, where vertically oriented lamellae form. Under small shear perpendicular to the lamellar orientation, we observe an inclination of the lamellar layers. At a critical shear rate, the lamellar layers become distorted and, for very large shear, recombine with a new orientation along the direction of shear. Our simulations are accompanied by a novel, easily understandable theoretical approach to predict the critical shear rate, at which the interfaces become distorted and shear-induced reorientation sets in. This allows one to calculate quantities such as the inclination angle or the pair interaction energy as a function of applied shear rate. Our results are relevant for many technical applications, where defect-free, long-range ordered structures are needed.

Strongly Compressed Polyelectrolyte Brushes under Shear.

By means of molecular dynamics simulations we provide evidence for pronounced counterion immobilization in strongly compressed polyelectrolyte-brush bilayers, where the counterions represent the vast majority of mobile solvent particles. As a consequence, hydrodynamic effects are strongly suppressed and semidilute bilayers can respond to shear motion like electrically neutral bilayers at melt density. For large, time-independent shear rates, γ̇, the shear force scales as f(γ̇) ∼ γ̇0.69, in agreement with scaling theory. In this regime, polyelectrolyte-brush bilayers can stabilize highly nonstationary processes, such as the instantaneous inversion of the shear direction. The absence of hydrodynamic flow leads to a suppression of the overshoot for the shear force, which is found for electrically neutral bilayers with the same molecular parameters. We suggest that nature uses this mechanism to optimize biolubrication, for instance in synovial joints.

Scaling Theory for Compressed Polymer-Brush Bilayers.

We present a novel scaling theory to describe the interaction free energy within a compressed polymer-brush bilayer. For semidilute brushes at intermediate and strong compressions, we predict that the interaction free energy scales with distance, D, between the grafting surfaces as A(D) ∼ D-2.5; i.e., the repulsive force, f(D) ∼ D-3.5, measured at the surfaces increases more strongly upon compression than predicted by the classical theory of Milner, Witten, and Cates. We find good agreement with experimental data and excellent agreement with numerical results, which follow our analytical predictions over a wide range of surface separations. The theory is based on the assumption of a strong correlation of the repulsive force and the interpenetration between the brushes. Using our numerical data, we can demonstrate this correlation with great precision.

Static Rouse modes and related quantities: corrections to chain ideality in polymer melts.

Following the Flory ideality hypothesis intrachain and interchain excluded-volume interactions are supposed to compensate each other in dense polymer systems. Multichain effects should thus be neglected and polymer conformations may be understood from simple phantom chain models. Here we provide evidence against this phantom chain, mean-field picture. We analyze numerically and theoretically the static correlation function of the Rouse modes. Our numerical results are obtained from computer simulations of two coarse-grained polymer models for which the strength of the monomer repulsion can be varied, from full excluded volume ("hard monomers") to no excluded volume ("phantom chains"). For nonvanishing excluded volume we find the simulated correlation function of the Rouse modes to deviate markedly from the predictions of phantom chain models. This demonstrates that there are nonnegligible correlations along the chains in a melt. These correlations can be taken into account by perturbation theory. Our simulation results are in good agreement with these new theoretical predictions.

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.

Static properties of end-tethered polymers in good solution: a comparison between different models.

We present a comparison between results, obtained from different simulation models, for the static properties of end-tethered polymer layers in good solvent. Our analysis includes data from two previous studies--the bond fluctuation model of Wittmer et al. [J. Chem. Phys. 101, 4379 (1994)] and the off-lattice bead-spring model of Grest and Murat [Macromolecules 26, 3108 (1993)]. Additionally, we explore the properties of a similar off-lattice model simulated close to the Theta temperature. We show that the data for the bond fluctuation and the Grest-Murat model can be analyzed in terms of scaling theory because chains are swollen inside the Pincus blob. In the vicinity of the Theta point the structure of the chains is essentially Gaussian in the Pincus blob. Therefore, the data for the second off-lattice model can be compared quantitatively to the self-consistent field theory. Different ways to determine the parameters of the self-consistent field theory are discussed.