CeNoS Kolloquium im Sommersemester 2014

Liquid drops on soft solids

Jacco H Snoeijer, Fac. of Science and Technology, University of Twente
(Einladender: Prof. Dr. U. Thiele)

The wetting of a liquid on a solid usually assumes the substrate to be perfectly rigid. However, this is no longer appropriate when the substrate is very soft: capillary forces can induce substantial elastic deformations, as has been demonstrated e.g. for drops on elastomers. In this talk we discuss the fundamentals of elasto-capillary interactions. Theory, simulations and experiments reveal the surprising nature of capillary forces, which turn out to be different from anything proposed in the literature. We also discuss how the law for the contact angle (Young's law) is modified for soft substrates.

Reduced basis methods for problems involving parametrized PDEs

Dr. Laura Iapichino, Fachbereich Mathematik und Statistik, Universität Konstanz
(Einladender: Prof. Dr. M. Ohlberger)

Often a physical model is represented by a set of partial differential equation where a set of parameters characterizes the system of interest and describes physical quantities (like source terms, boundary conditions, material properties) and/or geometrical configuration, so that the system solution is parameter dependent. The reduced order methods are innovative techniques to solve parametric partial differential equations that, compared with the classical numerical methods, require a lower computational time by maintaining a suitable level of the solution accuracy.

The presented model reduction paradigms are particularly suitable for solving problems that require considerable number of input-output evaluations in realtime for many different values of the parameters, not feasible with the classical numerical techniques.

In particular, the proposed strategies combine reduced basis (RB) method with both domain decomposition and optimal control theories. The combination of these frameworks becomes a very effective tool for real applications since we have the possibility to deal with complex geometrical configurations obtained as composition of simpler geometry deformable through suitable parametric transfinite maps and with optimal control problems related to systems modeled by parametric PDEs.

Some numerical results show the effectiveness of the proposed approaches by ensuring a certain level of solution accuracy and a very low computational time.

Probabilistic solution methods for position target problems

Prof. Dr. Stefan Ankirchner, Universität Jena
(Einladender: Prof. Dr. S. Dereich)

We consider the dynamic control problem of attaining a target position at a finite time T, while minimizing a cost functional depending on the position and speed. The talk presents a probabilistic solution method based on a maximum principle and on backward stochastic differential equations possessing a singularity at the terminal time T. We illustrate our results in a financial application, where we derive optimal trading strategies for closing financial asset positions in markets with stochastic price impact.

The talk is based on joint work with Monique Jeanblanc and Thomas Kruse.

Andrew Archer, Department of Mathematical Sciences, Loughborough University
(Einladender: Prof. Dr. U. Thiele)

Over the last few years, a number of dynamical density functional theories (DDFT) have been developed for describing the dynamics of the one-body density of both colloidal and atomic fluids. The DDFT is capable of describing the dynamics of the fluid down to the scale of the individual particles. DDFT is particularly successful for colloidal fluids, for which one may assume that the particles have stochastic equations of motion and from the resulting Fokker-Plank equation one is able to derive the DDFT. I will give an overview of the DDFT and show applications to the description of solidification fronts in supercooled (colloidal) suspensions. As the solidification front advances into the unstable liquid phase, we find that the density profile behind the advancing front develops density modulations and the wavelength of these modulations is a dynamically chosen quantity. For shallow quenches, the selected wavelength is that of the crystalline phase and so well-ordered crystalline states are formed. However, when the system is deeply quenched, we find that this wavelength can be quite different from that of the crystal, so the solidification front naturally generates disorder in the system. Significant rearrangement and ageing must subsequently occur for the system to form the regular well-ordered crystal that corresponds to the free energy minimum. We illustrate these findings with simulation results from DDFT and also a more simplified so called "phase-field crystal" model. Results for a system with quasicrystaline ordering are formed from such a quench will also be presented, explaining a new mechanism for how and why such structures can form.

Sondertermin: Vortrag im Rahmen des Physikalischen Kolloquiums

From wildly branched drying patterns to bifurcations of driven flows

Uwe Thiele, Institut für Theoretische Physik, Universität Münster

A phenomenon well known in our daily life is the coffee stain effect, where a drying drop of coffee leaves behind a well defined ring and not a uniformly distributed stain. Similar effects occur for many liquid mixtures and suspensions over a range of different length scales, where they may result in a rich variety of beautiful patterns ranging from regular and irregular lines to networks and wildly branched structures.

First, examples of such structures are shown that are created by several interacting physical phenomena: the tendency of liquids to cover or uncover a solid substrate (wettability), evaporation of volatile components, and the possible decomposition of a complex liquid into its components. Then, several types of mathematical models are introduced that describe thin layers of liquids on a surface either in a discrete or continuous way. In particular, these are a kinetic Monte Carlo model, dynamical density functional theory and thin film hydrodynamics.

The final part shows that such models can not only describe the dynamics of relaxational processes (systems approaching equilibrium) but as well driven systems that are permanently out of equilibrium as, for example, the deposition of a simple or complex liquid onto a moving plate. We show that this may result in complex behaviour manifested in a very rich bifurcation structure. The general concepts behind the models are explained, their successes are illustrated with selected results, and their limitations are discussed.

How can time-delays induce patterns?

Serhiy Yanchuk, Institute of Mathematics ,Humboldt University of Berlin
(Einladende: Dr. S. Gurevich)

Dynamical systems with time delays are common in many fields, such as optics, vehicle systems, neural networks, information processing, etc. A finite propagation velocity of the information or processing times introduce in such systems a new relevant scale, which may change the dynamics drastically. In my presentation, I would like to show how such time delays lead to spatio-temporal patterns. Firstly, I present a system with multiple hierarchically long time delays, whose dynamics "encodes" such spatio-temporal patterns as spiral defects of defect turbulence. Finally, I show a two-dimensional lattice of delay-coupled oscillators, which is capable to produce practically arbitrary periodically oscillating pattern.

Zusatztermin

Collective motion in heterogeneous media and in confined geometries: theory and experiments.

F.Peruani
(Einladender: Prof. Dr. U. Thiele)

The rapidly expanding study of active particles has focused so far almost exclusively, theoretically as well as experimentally, on the statistical description of particle motion in idealized, homogeneous spaces. However, the great majority of natural active particle systems take place, in the wild, in heterogeneous media: from active transport inside the cell, which occurs in a space that is filled by organelles and vesicles, to bacterial motion, which takes place in highly heterogeneous environments such as the soil or complex tissues such as in the gastrointestinal tract. I will show that the presence of spatial heterogeneities brings new physics unseen in homogeneous systems at the level of the transport properties and collective dynamics of such systems. I will show first that a random distribution of “obstacles” can lead to spontaneous trapping of active particles. Such “obstacles” represent undesirable areas that the active particles avoid and may correspond to a source of a repellent chemical, a light gradient, or whatever threat sensed by the moving particles. Inside traps, active particles exhibit a vortex-like motion and remain arbitrary long times. Particle motion then becomes genuinely sub-diffusive. We also find that the presence of such obstacles has a dramatic effect on the collective dynamics of usual self-propelled particle systems in two dimension. In particular, we observe: i) the existence of an optimal (angular) noise amplitude that maximizes collective motion, and ii) quasi-long range order and the existence of two critical points (cf. with the so-called Vicsek model in homogeneous spaces where order is long-range and there is a unique critical point).

Equally true and important is the fact that most active natural systems and experiments are subject to boundary conditions. How a confined geometry affects the collective properties of such active systems remains largely unexplored. I will show that often observed effect of accumulation of particles close to walls as well as the boundary-following phenomenon can both be collective effects controlled by the alignment strength. I will provide evidence that indicates that though a density-instability occurs at all system sizes, ordering vanishes in the thermodynamical limit.

This collection of results opens a new perspective for the design and control of active particle systems. Moreover, I will show that these observations are consistent with recent experiments performed with Quincke rollers in D. Bartolo’s Lab.

References:

O. Chepizhko, E. Altmann, F. Peruani, PRL 110, 238101 (2013)
O. Chepizhko, F. Peruani, PRL 111, 160604 (2013)

Multi-scale analysis of nonlocal Fokker-Planck equations

Prof. Dr. Michael Herrmann, Angewandte Mathematik, Universität Münster
(Einladender: Prof. Dr. M. Ohlberger)

The hysteretic behaviour of certain many-particle systems (e.g., Lithium-ion batteries) can be modeled by nonlocal
Fokker-Planck equations which involve two small parameters and are driven by a dynamical constraint. In this talk we study two
small-parameter limits by asymptotic analysis and derive reduced models for the effective dynamics of the macroscopic quantities. In the fast react regime, the limit dynamics turns out to be rate-independent and phase transitions can be described by a variant of Kramers formula. In the slow reaction regime, however, the small-parameter evolution is more involved and governed by a subtle interplay of the kinetic and the parabolic terms.

(joint work with Barbara Niethammer and Juan J.L. Velazquez)

Mathematical modelling of cell motility and antibody optimisation in germinal centres

Prof. Dr. Michael Meyer-Hermann, Helmholtz Centre for Infection Research, Braunschweig
(Einladende: Dr. O. Kamps, Prof. Dr. A. Telschow)

The generation of high-affinity antibodies in germinal centres (GC) happens in the course of an in-body evolutionary process which involves high-speed division, high-frequency somatic hypermutation and affinity-dependent selection of B cells. 40 years of GC research revealed many mechanisms and factors controlling division, mutation and selection. In the last decade, intravital multi-photon imaging of secondary lymphoid tissues during affinity maturation offered the opportunity to actually see the cells in action and to validate previously developed theories like competitive collection of antigen from follicular dendritic cells or the B cell recycling hypothesis. In addition, quantitative information were added to GC research allowing for an improved level of mathematical modelling.

With the help of agent-based modelling it is shown that tanszone B cell migration data support a high fraction of recycling of selected B cells. Furthermore, the models show that affinity maturation is optimised if T cell help in GCs is affinity-dependent and limiting. Mathematical predictions of the impact of limiting T cell help were confirmed by experimental data. Finally, the agent-based models are used to reproduce recent data of asymmetric distribution of collected antigen onto the daughters of dividing B cells. The model reveals that if the presence or absence of antigen in the daughters is used as fate decision criterion for the final differentiation of B cells to plasma and memory cells, this leads not only to an improved information processing in the GC but also leads to a 10-fold increased number of generated GC output cells provided T cell help is limiting. This prediction is not yet confirmed by experiments but supported by some experimental observations.

Modeling and simulation of Lithium intercalation and conversion batteries

Prof. Dr. Arnulf Latz , Deutsches Zentrum für Luft- und Raumfahrt e.V. Ulm
(Einladender: Prof. Dr. A. Heuer)

The electrodes of the majority of modern Li Ion batteries have very complex porous microstructures. Even if all chemical equirements for constructing a working battery as e.g. mutual chemical compatibility of electrolytes and materials, used for the electrodes, separator and the binder are fulfilled, it is not guaranteed that the final battery can reach its theoretical capacity or the required power density under operation condition. These properties are strongly influenced by the interplay of the morphological properties of the porous electrodes and the transport and reaction mechanisms of the chemical active ions. Even the extent and the type of degradation mechanisms are likely to be influenced by the morphology of the electrodes. For being able to better understand, evaluate and optimize these dependencies, proper electrochemical and physical reaction and transport models as well as efficient numerical algorithms are needed to study the resulting coupled nonlinear partial differential equations in the complex microstructure of Li ion intercalation and conversion batteries.

After giving a short general introduction on Li based intercalation and conversion batteries, a validated fully thermodynamically consistent model for transport and chemical reactions in the microstructure of the battery is presented. Simulation examples are used to demonstrate the importance of the microstructure on the distribution of ions, currents and heat sources in the microstructure of Li ion intercalation batteries. The necessary conditions for the onset of Lithium plating at low temperatures and high current densities are discussed. In the second part the challenges of developing theories for the next generation Li based conversion batteries like Li air and Li – Sulfur batteries are outlined. Results of an elementary kinetic modeling approach developed at the DLR and the HIU in a simple 1D effective porous media setting are shown for the example of Lithium sulfur batteries.