Thomas A. Vilgis

Reinforcement of Elastomers

The review describes recent research about reinforcement in elastomers where the main intention is to gain insight into the relationship between disordered filler structure on different length scales and reinforcement and to microscopic mechanisms of strain enhancement. Several theoretical concepts will be discussed together with very recent experimental findings related to hydrodynamic reinforcement, rigid filler aggregates with fractal structure and polymer adsorption on heterogeneous filler surfaces. Based on the new concepts, we present several recent efforts to understand typical effects in filled rubbers that have an extraordinary practical importance (for example, stress softening of carbon black filled rubbers during repeated large strain stretching and during small strain dynamic excitations).

Theory of nematic networks

Using classical elasticity theory, the rise in free energy upon crosslinking nematogenic polymers into a network is calculated for the isotropic and nematic phases. Spontaneous strains are allowed for in the network. The consequence of network formation upon nematic–isotropic equilibria is calculated by adding these elastic contributions to a conventional Landau theory. Memory of the crosslinking conditions yields quartic and quadratic additions to the standard Landau theory. We find that crosslinking in the isotropic state lowers the nematic–isotropic phase transition temperature compared with the unlinked case and the application of suitable stress raises it again. Crosslinking in the nematic state raises the transition temperature. We recover the mechanical critical point proposed long ago by de Gennes. Our Gaussian theory encompasses both main‐ and side‐chain polymers. The hairpin limit for main chain networks yields a modulus varying exponentially with temperature. The Landau–de Gennes free energy for comb polymers is presented for the first time.

Driven polymer translocation through a nanopore: A manifestation of anomalous diffusion

We study the translocation dynamics of a polymer chain threaded through a nanopore by an external force. By means of diverse methods (scaling arguments, fractional calculus and Monte Carlo simulation) we show that the relevant dynamic variable, the translocated number of segments s(t), displays an anomalous diffusive behavior even in the presence of an external force. The anomalous dynamics of the translocation process is governed by the same universal exponent α=2/(2ν+2−γ1), where ν is the Flory exponent and γ1 the surface exponent, which was established recently for the case of non-driven polymer chain threading through a nanopore. A closed analytic expression for the probability distribution function W(s, t), which follows from the relevant fractional Fokker-Planck equation, is derived in terms of the polymer chain length N and the applied drag force f. It is found that the average translocation time scales as . Also the corresponding time-dependent statistical moments, and reveal unambiguously the anomalous nature of the translocation dynamics and permit direct measurement of α in experiments. These findings are tested and found to be in perfect agreement with extensive Monte Carlo (MC) simulations.

Polymer translocation through a nanopore: A showcase of anomalous diffusion

The translocation dynamics of a polymer chain through a nanopore in the absence of an external driving force is analyzed by means of scaling arguments, fractional calculus, and computer simulations. The problem at hand is mapped on a one-dimensional anomalous diffusion process in terms of the reaction coordinate s (i.e., the translocated number of segments at time t ) and shown to be governed by a universal exponent alpha=2(2nu+2-gamma(1), where nu is the Flory exponent and gamma(1) is the surface exponent. Remarkably, it turns out that the value of alpha is nearly the same in two and three dimensions. The process is described by a fractional diffusion equation which is solved exactly in the interval 0<s<N with appropriate boundary and initial conditions. The solution gives the probability distribution of translocation times as well as the variation with time of the statistical moments <s(t) and <s2(t)-<s(t)>2, which provide a full description of the diffusion process. The comparison of the analytic results with data derived from extensive Monte Carlo simulations reveals very good agreement and proves that the diffusion dynamics of unbiased translocation through a nanopore is anomalous in its nature.

SOME GEOMETRICAL AND TOPOLOGICAL PROBLEMS IN POLYMER PHYSICS

Abstract In this work we discuss some problems of polymer physics which require use of the geometrical and topological methods for their solution. Selection of problems is made to provide some balanced view between the real physical situations and the mathematical methods which are required for their understanding. We consider both static and dynamic properties of polymer solutions which depend on the presence of entanglements. These include: problems related to polymer collapse, statics and dynamics of individual circular polymers and concentrated polymer solutions, problems related to elasticity of rubbers and gels, motion of polymers through pores, etc. This work serves both as an introduction to the field and as a guide for further study.

Universal properties in the dynamical deformation of filled rubbers

Starting from the observation that the filler particles in filled rubbers form fractal clusters which are connected to each other, a theoretical model is developed which establishes a connection between the amplitude dependence of the elastic deformation of filled rubbers and the structural properties of the filler. It is assumed that the elastic modulus is dominated by the rigid filler network at very low strain, whereas at higher strain the rubber matrix provides the main contribution because of cluster break-up. Within this model the exponent arising in an empirical description of the amplitude dependence is derived from the connectivity of the filler clusters.

Evaluation of Self-Affine Surfaces and Their Implication for Frictional Dynamics as Illustrated with a Rouse Material

Abstract We present a theory of hysteresis friction of sliding bulk rubber networks using the dynamic Rouse model for the rubber–glass transition region. The hard substrate is described by a self-affine rough surface that is a good representative for real surfaces having asperities within different length scales of different orders of magnitude. We find a general solution of the friction coefficient as a function of sliding velocity and typical surface parameters (e.g. surface fractal dimension, correlation lengths of surface profile). Further, we show the correlation with the viscoelastic loss modulus of the bulk rubber and the applicability of the Williams–Landel–Ferry transform to the velocity and temperature dependence of the frictional force as found experimentally. We demonstrate how the succesive inclusion of relaxation Rouse modes p =1,2,3,… into the final expression for the frictional force leads to a superposition of the contributions of the different modes and, as a consequence, to a broad, bell-shaped frictional curve as observed in the pioneering experiments of Grosch. We show how the theory simplifies for the special case of a Rouse slider interacting with a Brownian surface.

FRACTAL PROPERTIES AND SWELLING BEHAVIOR OF POLYMER NETWORKS

The swelling pressure of randomly crosslinked polymer networks is related to the structural properties of the crosslink topology. Using the assumption that the network structure exhibits fractal properties within the correlation length ξ, a scaling relation between the swelling pressure and the polymer volume fraction has been derived. The exponent obtained depends on the internal fractal dimension di of the network and is in general different from the corresponding exponent for linear chains. The latter can be obtained as the special case di=1. As a consequence, a significant difference in mixing entropy between the networks and the corresponding uncrosslinked system is predicted. This explains the experimental results obtained by several authors, which are in contradiction to the Flory–Rehner assumption. Computer simulations based on the bond fluctuation model support the scaling predictions presented. The exponents obtained for the density dependence of the osmotic or swelling pressure are somewhat lar...

Forced translocation of a polymer: Dynamical scaling versus molecular dynamics simulation

We suggest a theoretical description of the force-induced translocation dynamics of a polymer chain through a nanopore. Our consideration is based on the tensile (Pincus) blob picture of a pulled chain and the notion of a propagating front of tensile force along the chain backbone, suggested by Sakaue [Phys. Rev. E 76, 021803 (2007); Phys. Rev. E 81, 041808 (2010); Eur. Phys. J. E 34, 135 (2011)]. The driving force is associated with a chemical potential gradient that acts on each chain segment inside the pore. Depending on its strength, different regimes of polymer motion (named after the typical chain conformation: trumpet, stem-trumpet, etc.) occur. Assuming that the local driving and drag forces are equal (i.e., in a quasistatic approximation), we derive an equation of motion for the tensile front position X(t). We show that the scaling law for the average translocation time 〈τ〉 changes from ∼ N2ν/f1/ν to ∼ N^1+ν/f (for the free-draining case) as the dimensionless force f[over ̃]R=aNνf/T (where a, N, ν, f, and T are the Kuhn segment length, the chain length, the Flory exponent, the driving force, and the temperature, respectively) increases. These and other predictions are tested by molecular-dynamics simulation. Data from our computer experiment indicate indeed that the translocation scaling exponent α grows with the pulling force f[over ̃]R, albeit the observed exponent α stays systematically smaller than the theoretically predicted value. This might be associated with fluctuations that are neglected in the quasistatic approximation.

Polymer adsorption on heterogeneous surfaces

Abstract:The adsorption of a single ideal polymer chain on energetically heterogeneous and rough surfaces is investigated using a variational procedure introduced by Garel and Orland (Phys. Rev. B 55, 226 (1997)). The mean polymer size is calculated perpendicular and parallel to the surface and is compared to the Gaussian conformation and to the results for polymers at flat and energetically homogeneous surfaces. The disorder-induced enhancement of adsorption is confirmed and is shown to be much more significant for a heterogeneous interaction strength than for spatial roughness. This difference also applies to the localization transition, where the polymer size becomes independent of the chain length. The localization criterion can be quantified, depending on an effective interaction strength and the length of the polymer chain.