Martin Kröger

Rheology and structural changes of polymer melts via nonequilibrium molecular dynamics

Results of nonequilibrium molecular dynamics computer simulations of a planar Couette flow are presented for the multibead anharmonic‐spring model. The finitely extensible nonlinear elastic force law is used to connect the up to 100 beads of a chain molecule. Rheological data (shear viscosity, normal pressure differences) are discussed and compared with quantities describing the chain conformation (e.g., alignment tensor, static structure factor). This renders possible a test of the theoretical approaches which connect these quantities. In agreement with recent experiments, the static strucure factor exhibits characteristic elliptical distortions of the polymer coil whose magnitude depends on the distance from the gyration center. In our simulations the zero‐shear‐rate viscosity is found to scale linearly with the number of beads N up to chains with N=60. A weak upturn of the viscosity per bead for N=100 is found which may indicate the onset of the reptation regime.

Topological analysis of polymeric melts: chain-length effects and fast-converging estimators for entanglement length.

Primitive path analyses of entanglements are performed over a wide range of chain lengths for both bead spring and atomistic polyethylene polymer melts. Estimators for the entanglement length N_{e} which operate on results for a single chain length N are shown to produce systematic O(1/N) errors. The mathematical roots of these errors are identified as (a) treating chain ends as entanglements and (b) neglecting non-Gaussian corrections to chain and primitive path dimensions. The prefactors for the O(1/N) errors may be large; in general their magnitude depends both on the polymer model and the method used to obtain primitive paths. We propose, derive, and test new estimators which eliminate these systematic errors using information obtainable from the variation in entanglement characteristics with chain length. The new estimators produce accurate results for N_{e} from marginally entangled systems. Formulas based on direct enumeration of entanglements appear to converge faster and are simpler to apply.

Tuning Polymer Thickness: Synthesis and Scaling Theory of Homologous Series of Dendronized Polymers

The thickness of dendronized polymers can be tuned by varying their generation g and the dendron functionality X. Systematic studies of this effect require (i) synthetic ability to produce large samples of high quality polymers with systematic variation of g, X and of the backbone polymerization degree N, (ii) a theoretical model relating the solvent swollen polymer diameter, r, and persistence length, lambda, to g and X. This article presents an optimized synthetic method and a simple theoretical model. Our theory approach, based on the Boris-Rubinstein model of dendrimers predicts r approximately n(1/4)g(1/2) and lambda approximately n(2) where n = [(X - 1)(g) - 1]/(X - 2) is the number of monomers in a dendron. The average monomer concentration in the branched side chains of a dendronized polymer increases with g in qualitative contrast to bottle brushes whose side chains are linear. The stepwise, attach-to, synthesis of X = 3 dendronized polymers yielded gram amounts of g = 1-4 polymers with N approximately = 1000 and N approximately = 7000 as compared to earlier maxima of 0.1 g amounts and of N approximately = 1000. The method can be modified to dendrons of different X. The conversion fraction at each attach-to step, as quantified by converting unreacted groups with UV labels, was 99.3% to 99.8%. Atomic force microscopy on mixed polymer samples allows to distinguish between chains of different g and suggests an apparent height difference of 0.85 nm per generation as well as an increase of persistence length with g. We suggest synthetic directions to allow confrontation with theory.

A thermodynamically admissible reptation model for fast flows of entangled polymers. II. Model predictions for shear and extensional flows

Numerical predictions of a previously proposed thermodynamically consistent reptation model for linear entangled polymers are presented for shear and extensional flows. Comparisons with experimental data and two alternative molecular-based models are given in detail. The model studied in this paper incorporates the essence of double reptation, convective constraint release, and chain stretching, and it avoids the independent alignment approximation. Here, no use is made of the ingredient of anisotropic tube cross sections of the previously proposed model. Simulation results reveal that the model at a highly simplified level with few structural variables, i.e., four degrees of freedom, is able to capture qualitatively all features of the available experimental observations and is highly competitive with recently proposed models in describing nonlinear rheological properties of linear entangled polymers.

The Largest Synthetic Structure with Molecular Precision: Towards a Molecular Object

Pushing the limits: A 200A - 10 Da structurally defined, linear macromolecule (PG5) has a molar mass, cross-section dimension, and cylindrical shape that are comparable to some naturally occurring objects, such as amyloid fibrils or certain plant viruses. The macromolecule is resistant against flattening out on a surface; the picture shows PG5 embracing the tobacco mosaic virus (TMV).

Viscoelastic flows studied by smoothed particle dynamics

Abstract A viscoelastic numerical scheme based on smoothed particle dynamics is presented. The concept goes a step beyond smoothed particle hydrodynamics (SPH) which is a grid-free Lagrangian method describing the flow by fluid-pseudo-particles. The relevant properties are interpolated directly on the resulting movable grid. In this work, the effect of viscoelasticity is incorporated into the ordinary conservation laws by a differential constitutive equation supplied for the stress tensor. In order to give confidence in the methodology we explicitly consider the non-stationary simple corotational Maxwell model in a channel geometry. Without further developments the scheme is applicable to ‘realistic’ models relevant for three-dimensional (3D) viscoelastic flows in complex geometries.

Effect of charge, hydrophobicity, and sequence of nucleoporins on the translocation of model particles through the nuclear pore complex.

The molecular structure of the yeast nuclear pore complex (NPC) and the translocation of model particles have been studied with a molecular theory that accounts for the geometry of the pore and the sequence and anchoring position of the unfolded domains of the nucleoporin proteins (the FG-Nups), which control selective transport through the pore. The theory explicitly models the electrostatic, hydrophobic, steric, conformational, and acid-base properties of the FG-Nups. The electrostatic potential within the pore, which arises from the specific charge distribution of the FG-Nups, is predicted to be negative close to pore walls and positive along the pore axis. The positive electrostatic potential facilitates the translocation of negatively charged particles, and the free energy barrier for translocation decreases for increasing particle hydrophobicity. These results agree with the experimental observation that transport receptors that form complexes with hydrophilic/neutral or positively charged proteins to transport them through the NPC are both hydrophobic and strongly negatively charged. The molecular theory shows that the effects of electrostatic and hydrophobic interactions on the translocating potential are cooperative and nonequivalent due to the interaction-dependent reorganization of the FG-Nups in the presence of the translocating particle. The combination of electrostatic and hydrophobic interactions can give rise to complex translocation potentials displaying a combination of wells and barriers, in contrast to the simple barrier potential observed for a hydrophilic/neutral translocating particle. This work demonstrates the importance of explicitly considering the amino acid sequence and hydrophobic, electrostatic, and steric interactions in understanding the translocation through the NPC.

Endocytosis of PEGylated nanoparticles accompanied by structural and free energy changes of the grafted polyethylene glycol

Nanoparticles (NPs) are in use to efficiently deliver drug molecules into diseased cells. The surfaces of NPs are usually grafted with polyethylene glycol (PEG) polymers, during so-called PEGylation, to improve water solubility, avoid aggregation, and prevent opsonization during blood circulation. The interplay between grafting density σp and grafted PEG polymerization degree N makes cellular uptake of PEGylated NPs distinct from that of bare NPs. To understand the role played by grafted PEG polymers, we study the endocytosis of 8 nm sized PEGylated NPs with different σp and N through large scale dissipative particle dynamics (DPD) simulations. The free energy change Fpolymer of grafted PEG polymers, before and after endocytosis, is identified to have an effect which is comparable to, or even larger than, the bending energy of the membrane during endocytosis. Based on self-consistent field theory Fpolymer is found to be dependent on both σp and N. By incorporating Fpolymer, the critical ligand-receptor binding strength for PEGylated NPs to be internalized can be correctly predicted by a simple analytical equation. Without considering Fpolymer, it turns out impossible to predict whether the PEGylated NPs will be delivered into the diseased cells. These simulation results and theoretical analysis not only provide new insights into the endocytosis process of PEGylated NPs, but also shed light on the underlying physical mechanisms, which can be utilized for designing efficient PEGylated NP-based therapeutic carriers with improved cellular targeting and uptake.

Combined Molecular Algorithms for the Generation, Equilibration and Topological Analysis of Entangled Polymers: Methodology and Performance

We review the methodology, algorithmic implementation and performance characteristics of a hierarchical modeling scheme for the generation, equilibration and topological analysis of polymer systems at various levels of molecular description: from atomistic polyethylene samples to random packings of freely-jointed chains of tangent hard spheres of uniform size. Our analysis focuses on hitherto less discussed algorithmic details of the implementation of both, the Monte Carlo (MC) procedure for the system generation and equilibration, and a postprocessing step, where we identify the underlying topological structure of the simulated systems in the form of primitive paths. In order to demonstrate our arguments, we study how molecular length and packing density (volume fraction) affect the performance of the MC scheme built around chain-connectivity altering moves. In parallel, we quantify the effect of finite system size, of polydispersity, and of the definition of the number of entanglements (and related entanglement molecular weight) on the results about the primitive path network. Along these lines we approve main concepts which had been previously proposed in the literature.

Quantifying chain reptation in entangled polymer melts: Topological and dynamical mapping of atomistic simulation results onto the tube model

The topological state of entangled polymers has been analyzed recently in terms of primitive paths which allowed obtaining reliable predictions of the static (statistical) properties of the underlying entanglement network for a number of polymer melts. Through a systematic methodology that first maps atomistic molecular dynamics (MD) trajectories onto time trajectories of primitive chains and then documents primitive chain motion in terms of a curvilinear diffusion in a tubelike region around the coarse-grained chain contour, we are extending these static approaches here even further by computing the most fundamental function of the reptation theory, namely, the probability psi(s,t) that a segment s of the primitive chain remains inside the initial tube after time t, accounting directly for contour length fluctuations and constraint release. The effective diameter of the tube is independently evaluated by observing tube constraints either on atomistic displacements or on the displacement of primitive chain segments orthogonal to the initial primitive path. Having computed the tube diameter, the tube itself around each primitive path is constructed by visiting each entanglement strand along the primitive path one after the other and approximating it by the space of a small cylinder having the same axis as the entanglement strand itself and a diameter equal to the estimated effective tube diameter. Reptation of the primitive chain longitudinally inside the effective constraining tube as well as local transverse fluctuations of the chain driven mainly from constraint release and regeneration mechanisms are evident in the simulation results; the latter causes parts of the chains to venture outside their average tube surface for certain periods of time. The computed psi(s,t) curves account directly for both of these phenomena, as well as for contour length fluctuations, since all of them are automatically captured in the atomistic simulations. Linear viscoelastic properties such as the zero shear rate viscosity and the spectra of storage and loss moduli obtained on the basis of the obtained psi(s,t) curves for three different polymer melts (polyethylene, cis-1,4-polybutadiene, and trans-1,4-polybutadiene) are consistent with experimental rheological data and in qualitative agreement with the double reptation and dual constraint models. The new methodology is general and can be routinely applied to analyze primitive path dynamics and chain reptation in atomistic trajectories (accumulated through long MD simulations) of other model polymers or polymeric systems (e.g., bidisperse, branched, grafted, etc.); it is thus believed to be particularly useful in the future in evaluating proposed tube models and developing more accurate theories for entangled systems.