Thomas Schwager

Coefficient of restitution for viscoelastic spheres: the effect of delayed recovery.

The coefficient of normal restitution of colliding viscoelastic spheres is computed as a function of the material properties and the impact velocity. From simple arguments it becomes clear that, in a collision of purely repulsively interacting particles, the particles lose contact slightly before the distance of the centers of the spheres reaches the sum of the radii, that is, the particles recover their shape only after they lose contact with their collision partner. This effect was neglected in earlier calculations, which leads erroneously to attractive forces and thus to an underestimation of the coefficient of restitution. As a result we find a different dependence of the coefficient of restitution on the impact rate.

Coefficient of tangential restitution for viscoelastic spheres

The collision of frictional granular particles may be described by an interaction force whose normal component is that of viscoelastic spheres while the tangential part is described by the model by Cundall and Strack (Géotechnique 29, 47 (1979)) being the most popular tangential collision model in Molecular Dynamics simulations. Albeit being a rather complicated model, governed by 5 phenomenological parameters and 2 independent initial conditions, we find that it is described by 3 independent parameters only. Surprisingly, in a wide range of parameters the corresponding coefficient of tangential restitution, εt, is well described by the simple Coulomb law with a cut-off at εt = 0. A more complex behavior of the coefficient of restitution as a function on the normal and tangential components of the impact velocity, gn and gt, including negative values of εn, is found only for very small ratio gt/gn. For the analysis presented here we neglect dissipation of the interaction in normal direction.

Coefficient of tangential restitution for the linear dashpot model

The linear dashpot model for the inelastic normal force between colliding spheres leads to a constant coefficient of normal restitution, epsilonn=const, which makes this model very popular for the investigation of dilute and moderately dense granular systems. For two frequently used models for the tangential interaction force we determine the coefficient of tangential restitution, epsilont, both analytically and by numerical integration of Newtons equation. Although epsilonn=const for the linear-dashpot model, we obtain pronounced and characteristic dependences of the tangential coefficient on the impact velocity, epsilont=epsilont(g). The results may be used for event-driven simulations of granular systems of frictional particles.

Computational Lipidology: Predicting Lipoprotein Density Profiles in Human Blood Plasma

Monitoring cholesterol levels is strongly recommended to identify patients at risk for myocardial infarction. However, clinical markers beyond “bad” and “good” cholesterol are needed to precisely predict individual lipid disorders. Our work contributes to this aim by bringing together experiment and theory. We developed a novel computer-based model of the human plasma lipoprotein metabolism in order to simulate the blood lipid levels in high resolution. Instead of focusing on a few conventionally used predefined lipoprotein density classes (LDL, HDL), we consider the entire protein and lipid composition spectrum of individual lipoprotein complexes. Subsequently, their distribution over density (which equals the lipoprotein profile) is calculated. As our main results, we (i) successfully reproduced clinically measured lipoprotein profiles of healthy subjects; (ii) assigned lipoproteins to narrow density classes, named high-resolution density sub-fractions (hrDS), revealing heterogeneous lipoprotein distributions within the major lipoprotein classes; and (iii) present model-based predictions of changes in the lipoprotein distribution elicited by disorders in underlying molecular processes. In its present state, the model offers a platform for many future applications aimed at understanding the reasons for inter-individual variability, identifying new sub-fractions of potential clinical relevance and a patient-oriented diagnosis of the potential molecular causes for individual dyslipidemia.

Fractal Substructure of a Nanopowder

The structural evolution of a nanopowder by repeated dispersion and settling can lead to characteristic fractal substructures. This is shown by numerical simulations of a two-dimensional model agglomerate of adhesive rigid particles. The agglomerate is cut into fragments of a characteristic size l, which then are settling under gravity. Repeating this procedure converges to a loosely packed structure, the properties of which are investigated: (a) The final packing density is independent of the initialization, (b) the short-range correlation function is independent of the fragment size, (c) the structure is fractal up to the fragmentation scale l with a fractal dimension close to 1.7, and (d) the relaxation time increases linearly with l.

Coefficient of restitution for viscoelastic disks

The dissipative collision of two identical viscoelastic disks is studied. By using a known law for the elastic part of the interaction force and the viscoelastic damping model an analytical solution for the coefficient of restitution is given. The coefficient of restitution depends significantly on the impact velocity. It approaches 1 for small velocities and decreases for increasing velocities.

Giant fluctuations at a granular phase separation threshold

We investigate a phase separation instability that occurs in a system of nearly elastically colliding hard spheres driven by a thermal wall. If the aspect ratio of the confining box exceeds a threshold value, granular hydrostatics predict phase separation: the formation of a high-density region coexisting with a low-density region along the wall that is opposite to the thermal wall. Event-driven molecular dynamics simulations confirm this prediction. The theoretical bifurcation curve agrees with the simulations quantitatively well below and well above the threshold. However, in a wide region of aspect ratios around the threshold, the system is dominated by fluctuations, and the hydrostatic theory breaks down. Two possible scenarios of the origin of the giant fluctuations are discussed.

Fractal Substructures due to Fragmentation and Reagglomeration

Cohesive powders form agglomerates that can be very porous. Hence they are also very fragile. Consider a process of complete fragmentation on a characteristic length scale l, where the fragments are subsequently allowed to settle under gravity. If this fragmentation‐reagglomeration cycle is repeated sufficiently often, the powder develops a fractal substructure with robust statistical properties. The structural evolution is discussed for two different models: The first one is an off‐lattice model, in which a fragment does not stick to the surface of other fragments that have already settled, but rolls down until it finds a locally stable position. The second one is a simpler lattice model, in which a fragment sticks at first contact with the agglomerate of fragments that have already settled. Results for the fragment size distribution are shown as well. One can distinguish scale invariant dust and fragments of a characteristic size. Their role in the process of structure formation will be addressed.

Contact of granular particles and the simulation of rapid flows using event-driven molecular dynamics

ABSTRACT We discuss several models for granular particles commonly used in Molecular Dynamics simulations of granular materials, including spheres with linear dashpot force, vis-coelastic spheres and adhesive viscoelastic spheres. Starting from the vectorial interaction forces we derive the coefficients of normal and tangential restitution as functions of the vectorial impact velocity and of the material constants. We review the methods of measurements of the coefficients of restitution and characterize the coefficient of normal restitution as a fluctuating quantity. Moreover, the scaling behavior and the influence of different force laws on the dynamical system behavior are discussed. The powerful method of event-driven Molecular Dynamics is described and the algorithmic simulation technique is explained in detail. Finally we discuss the limitations of event-driven MD.