# Seminar "Material Modeling"

This interdisciplinary seminar is dedicated to the mathematical modeling of different phases of matter and their transitions, covering microscopic and macroscopic scales and using both discrete and continuum descriptions. Topics include both stationary and evolutionary processes. Techniques include, among others, adaptive computational methods, asymptotic analysis, mathematical physics, non-smooth differential equations, stochastics, thermodynamic modeling, and variational methods.

Place: | Weierstrass-Institute for Applied Analysis and Stochastics |

Mohrenstraße 39, 10117 Berlin | |

Organizers: | Thomas Eiter, Martin Heida, Elena Magnanini, Dirk Peschka, Marita Thomas, Barbara Wagner |

## Upcoming and recent talks

Title: Beyond the Born-Oppenheimer Approximation by Surface Hopping Trajectories Methods. Speaker: Leonardo Araujo (TU Munich) Time: Monday, 25.09.2023, 10:00 Location: WIAS-ESH In molecular dynamics, the Born-Oppenheimer approximation is a fundamental concept, assuming the decoupling of electronic and nuclear motions due to the significant mass difference between them. This implies that, from the perspective of the nuclei, the electrons effectively reside in an eigenstate. While this approximation is widely employed in quantum chemistry to expedite molecular wavefunction calculations, it encounters limitations, especially in scenarios involving light atoms and high-energy systems. In the first part of this presentation, we confront this limitation by going beyond the Born-Oppenheimer approximation and adding multiple electronic eigenstates into the approximation. We will discuss the theory of this approach and present the resulting equations of motion for the nuclei. Moving on to the second part, we explore computationally efficient methods for simulating such dynamics by considering a class of quantum-classical methods: the Surface Hopping Trajectories. These methods entail an ensemble of trajectories, sampled from the initial nuclear configuration, performing classical dynamics on the eigensurface of an electronic eigenstate. They further allow each trajectory to undergo transitions to different eigenstates guided by hopping probabilities. We present two famous versions of Surface Hopping: the Fewest-Switches Surface Hopping (FSSH) and the Single Switch Surface Hopping (SSSH), also recognised as Landau-Zener Surface Hopping. To culminate, we illustrate the use of these methods through a practical example using WavePacket, a Matlab software package for quantum-mechanical and quantum-classical simulations. |

Title: Slow time-scale behavior of fast microscopic dynamics. Speaker: Amit Acharya (Carnegie Mellon University) [link] Time: Tuesday, 16.05.2023, 15:15 Location: WIAS-ESH/Online This talk aims to understand and exploit the slow time-scale behavior of rapidly evolving microscopic dynamics posed in terms of systems of nonlinear ordinary differential equations, not necessarily containing an a priori split into fast and slow variables. Such a question arises naturally and ubiquitously in efforts to understand macroscopic dynamics, on engineering time-scales, of well-accepted models of microscopic dynamics that, however, are not amenable to practical computing over the much, much larger macroscopic time-scales of interest. This is because there is a vast separation of time scales involved between the dynamics of the macroscopic variables of interest and the microscopic dynamics, and evolving the microscopic dynamics directly fails to address the question of the macroscopic dynamics. The methodology employed involves a computational scheme based on fundamental mathematical theory that a) defines appropriate `coarse' variables corresponding to the microscopic dynamics that evolve in a stable manner on the coarse time scale; b) determines the equation of evolution for such variables; and c) defines a practically useful strategy for accurately initializing short bursts of microscopic runs for the evolution of the slow variables, without special requirements on the nature of the microscopic dynamics. We will illustrate the theory with examples that violate ergodicity and include both conservative and dissipative behavior. A first step towards coarse graining of discrete dislocation dynamics to a pde-based plasticity model without relying on constitutive assumptions will also be demonstrated. This is joint work with Sabyasachi Chatterjee, Zvi Artstein, Giacomo Po, Xiaohan Zhang, and Nasr Ghoniem. |

Title: From an egg to an embryo - inferring the temporal dynamics of cells during embryonic development. Speaker: Markus Mittnenzweig (Weizmann Institute ) Time: Thursday, 06.04.2023, 11:00 Location: WIAS-ESH/Online Novel single-genomics technologies allow probing biological tissues and entire embryos at unprecedented molecular resolution. In the context of developmental biology, these technologies allow measuring in a very quantitative way the emergence of spatial and molecular complexity that characterizes early embryonic development. Mouse embryonic development is a canonical model for studying mammalian cell fate acquisition. Recently, we introduced a temporal flow model for early mouse development, consisting of data from 153 individually sampled embryos spanning 36 hours of molecular diversification. Using a convex optimization framework and precise timing of embryos we infer the differentiation dynamics of individual cells along the continuum of embryonic cellular states. In the second part, I will show how to use this time-resolved model to assess the function of epigenetic regulators and intercellular signals involved in early embryonic development. |

Title: Modeling and simulations towards the design of high performance batteries. Speaker: Alberto Salvadori (University of Brescia) [link] Time: Tuesday, 21.02.2023, 13:30 Location: WIAS-406/Online The upcoming request of renewable energy requires high performance energy storage and power delivery systems. The conventional batteries rely on liquid electrolytes, which is still the state of art due to their high ionic conductivity. These systems, though, show safety concerns in view of the flammability of toxic organic solvents. Therefore, there is a great effort to introduce novel electrolyte materials with excellent transport properties, low interfacial resistance, good mechanical strength, and safer behavior. Solid-state electrolytes are promising candidates. Upon combining with Li or Na metal anodes they have the potential to deliver higher energy densities with enhanced safety compared to liquid electrolyte batteries. However, upon charging such cells at current densities greater than a critical value, "dendrites" nucleate and grow from the metal electrode and result in short-circuiting the cell. Furthermore, SSE present contact problem with the porous electrode interfaces. For this reason, gel polymer electrolytes (GPE) can be seen as a valid alternative. It is composed by a polymer network with solvent filling the interstitial spaces, the confined liquid into the polymer matrix can boost the conductivity and can provide better adhesion at the electrode interfaces. It is well known that low ionic diffusivities cause high concentration and potential polarization across thick, porous cathodes at high current rates. Furthermore, the rapid depletion of Li+ ions at the reaction surface limits the rate capability of thick electrode-based Li-ion batteries. In a series of different research endeavours, modelling and simulations have been carried out at the m4lab towards the design of the next generation of electrochemical storage systems. In this talk, a brief overview will be presented at first, concerning dendritic growth and a multiscale compatible approach in electrolytes. Eventually, a detailed investigation of gel polymer electrolytes, optimal electrode design, and the influence of ionic additives in the porous carbon-binder network will be presented and validated against experimental evidence. see pdf file |

Title: Asymptotic stability in a free boundary PDE model of active matter. Speaker: Leonid Berlyand (Pennsylvania State University) [link] Time: Tuesday, 08.11.2022, 13:30 Location: WIAS-ESH/Online We begin with a brief overview of the rapidly developing research area of active matter (a.k.a. active materials). These materials are intrinsically out of thermal equilibrium resulting in novel physical properties whose modeling requires development of new mathematical tools. We next focus on study the onset of motion of a living cell (e.g., a keratocyte) driven by myosin contraction with focus on a transition from unstable radial stationary states to stable asymmetric moving states. We introduce a two-dimensional free-boundary PDE model that generalizes a previous one-dimensional model by combining a Keller--Segel model, Hele--Shaw kinematic boundary condition, and the Young--Laplace law with a novel nonlocal regularizing term. This nonlocal term precludes blowup or collapse by ensuring that membrane-cortex interaction is sufficiently strong. We found a family of asymmetric traveling solutions bifurcating from stationary solutions. Our main result is the nonlinear asymptotic stability of traveling wave solutions that model observable steady cell motion. We derived and rigorously justified an explicit asymptotic formula for the stability determining eigenvalue via asymptotic expansions in a small speed of cell. This formula greatly simplifies the computation of this eigenvalue and shows that stability is determined by the change in total myosin mass when stationary solutions bifurcate to traveling solutions. Our spectral analysis reveals the physical mechanisms of stability. It also leads to interesting mathematics due to non-selfadjointness of the linearized problem which is a signature of active matter out-of-equilibrium systems. If time permits, we will discuss work in progress on fingering instability in multicellular tissue spreading. This is joint work with V. Rybalko and C. Safsten published in Transactions of AMS (to appear) and Phys. Rev. B, 2022. |

Title: From Maxwell to Mitochondria. Speaker: Bob Eisenberg (Rush University, Chicago) Time: Thursday, 29.09.2022, 16:00 Location: Online Remark: Zoom link by mail. Applying Maxwell equations to atoms and ions in a mitochondrion or chloroplast seems a hopeless task: there are so many ions ($\sim 10^{18}$) and such complexity. But nerve axons, computers and their chips are as complicated. Analysis of nerve conduction is now an old story, so successful it is not known to young scientists. Analysis of the propagating nerve signal, from nerve, to membrane, to conductance, to channel protein, to atoms within the channel is done by computing the flow of ions, using Kirchhoff's law. Charges are not dealt with explicitly: there are just too many to deal with. Kirchhoff's Current Law is also the chief design tool of the circuits that make our computers and phones. Charge is hardly mentioned in circuit design. Kirchhoff's Current Law seems to imply the conservation of ion flux $\mathbf{J}$, but Maxwell's equations do NOT conserve ion flux. Maxwell's equations conserve only the total current $\mathbf{J}_{\rm total}=\mathbf{J}+\epsilon_0\partial\mathbf{E}/\partial t$. The total current equals ion flux $\mathbf{J}$ plus the ethereal displacement current $\epsilon_0\partial\mathbf{E}/\partial t$ which 'flows' everywhere including in a vacuum, where it propagates light. Maxwell and Kirchhoff's discovery of $\epsilon_0\partial\mathbf{E}/\partial t$ as an unavoidable component of total current allows simple circuit laws to compute nerve conduction from atoms to axons. It allows design of systems made of $>10^{12}$ semiconductor devices, that switch in $10^{-10}$ sec and function reliably over a wide range of conditions: error rate $10^{-14}$ per second. Oxidative phosphorylation provides the chemical energy of animal life. Its components are embedded in the membranes of mitochondria. Cytochrome c Oxidase is a crucial component that can be analyzed using circuit laws, as we (Shixin Xu, Zilong Song, and I, led by Huaxiong Huang) have shown. Diffusion, current flow, water flow, and chemical reactions are included combining the universal conservation of total current with Chun Liu's theory of complex fluids. Our approach applies to active transport systems of mitochondria, chloroplasts, and other cells. It includes the classical theory of nerve conduction. The analysis allows easy inclusion of more molecular and chemical details of oxidative phosphorylation as we learn of them, and they are discovered. In particular, atomic scale chemical reactions are coupled by Kirchhoff's current law with important biological and chemical consequences. Sodium and potassium channels are coupled in an action potential by Kirchhoff's current law. So are the enzyme complexes I, II, III, IV and V in a mitochondrion and chloroplast. |

Title: Examples of hydrodynamic behaviour in two-species exclusion processes. Speaker: Robert Jack (University of Cambridge) [link] Time: Monday, 19.09.2022, 13:30 Location: WIAS-406/Online We discuss several different results for simple exclusion processes with two species of particles. We first show numerical results where inhomogeneous states appear in two-dimensional systems where the species are driven in opposite directions, and we explain how these results can be rationalised by considering hydrodynamic PDEs for the density [1]. We then discuss how hydrodynamic equations in such models can be characterised, including a systematic analysis based on the method of matched asymptotics [2]. Finally, I will present some results [3] for large deviations in the hydrodynamic limit, associated with fluctuations of the entropy production in a simple model of active matter [4]. [1] H. Yu, K. Thijssen and R. L. Jack, arXiv:2204.08863 |

Title: A cell-fluid-matrix model to understand how aggressive cancer cell behavior possibly is linked to elevated fluid pressure. Speaker: Steinar Evje (University of Stavanger, Norway) [link] Time: Tuesday, 12.07.2022, 13:30 Location: Online Recent research, preclinical (mouse) and clinical (patient) observations, has suggested that high interstitial fluid pressure in a tumor is associated with aggressive cancer cell behavior, i.e., cancer cells tend to detach from the primary tumor and are able to escape to lymphatics vessels outside the tumor (Hompland et al, Cancer Research 2012). This is referred to as metastasis and is the main reason why cancer becomes a deadly disease. How to explain this phenomenon? Our objective has been to formulate a mathematical cell-fluid-matrix model that can account for two experimentally observed cancer cell migration mechanisms, as reported by Polacheck et al (PNAS 2011 and 2014). Both mechanisms are sensitive to fluid flow and therefore represent natural candidates when we formulate a model that can explore the possible relation between aggressive cancer cell behavior and interstitial fluid pressure (IFP). In the presentation I will relate the proposed model to the celebrated chemotaxis-(Navier)-Stokes model, proposed by Tuval et al (PNAS 2005), which has attracted many applied mathematicians and generated a considerable amount of results in mathematical analysis (well-posedness, as well as qualitative characterization). Our approach relies on using a two-phase formulation where cells and fluid are described in terms of two separate mass and momentum equations where different interaction forces between cells, fluid, and matrix can be accounted for in the momentum balance law. This type of two phase (Navier)-Stokes model (compressible version) represents another active area of research within applied mathematics for those interested in fluid mechanical models. Finally, an example of a reduced version of our cell-fluid model is mentioned and a mathematical result obtained for this model (Winkler and Evje, 2020). |

Title: Neuromorphic device development: from modification of surfaces to modification of functions. Speaker: Giovanni Ligorio (Humboldt-Universität zu Berlin) Time: Thursday, 23.06.2022, 14:00 Location: WIAS-ESH/Online The versatility offered by organic molecules and polymers holds great functional and economic potential for optoelectronic devices. The chemical modification of electrodes with organic materials is a common approach in our group for tuning the electronic landscape between interlayers in devices, thus facilitating charge carrier injection/extraction and improving device performance. A common tool to modify the electrode interfaces is to use covalently bound molecules carrying a permanent electric dipole group. Beside employing molecules with constant dipole, we also investigated the possibility to use switchable photochromic molecules which undergo structural modification when illuminated with light. This allows a dynamic and reversable control of the electronic properties. The gained control over the device performance enables additional functionalities in devices, such as neuromorphic applications. Neuromorphic engineering takes inspiration from the functionalities and structure of the brain to solve complex tasks and enable learning. Yet, hardware realization that simulates the synaptic activities realized with electrical devices is still not as advanced as the common implementation in computer software. In my talk I will present different approaches to emulate synapses and to fabricate both (i) optical and (ii) electronic synaptic devices. The first (i) were achieved by employing the above-mentioned photochromic molecules. This allowed control of the optical properties of gold films by reversibly modulating the surface plasmon resonance. The second (ii) were achieved by fabricating two-terminal devices based on mixed ionic-electronic conducting polymers that serve as active layer for ions and charge carrier conduction. The precise understanding and modeling of device functions will allow to fully exploit the potential of these synthetic synapses. For this purpose, a mathematical description of their behaviour might allow to move from single devices to networks and enable in materia computing. |

Title: Geometric Numerical Methods for Models from Plasma Physics. Speaker: Eric Sonnendrücker (Max Planck Institute for Plasma Physics) [link] Time: Tuesday, 31.05.2022, 13:30 Location: Online Many kinetic and fluid plasma models, like the Vlasov-Maxwell-Landau or the Magnetohydrodynamics (MHD) models feature a hamiltonian part that can be described by a Poisson bracket and a Hamiltonian and a dissipative part. The hamiltonian part features in general different kinds of invariants that play a fundamental role in the evolution of the system. On the other hand the dissipative part strictly dissipates an entropy. For an accurate long time numerical simulation of these models, keeping as much as possible from the structure of the original equations is essential, which will automatically enforce the preservation of some important invariants like for example div B = 0. In this talk we will give an overview of structure preserving discretisation of such models based on Discrete Exterior Calculus. In particular we will show how a Particle In Cell discretisation of the Vlasov Equation coupled with a Finite Element Exterior Calculus discretisation for Maxwell's equations involving a discrete de Rham complex and an appropriate Finite Element approximation of each field leads to a Finite Dimensional Hamiltonian system, which then can be discretised in time with geometric integrators like hamiltonian splitting or exact energy preserving discrete gradient methods. |

Title: Stationary non-equilibrium solutions for coagulation equations. Speaker: Alessia Nota (Universitá degli studi dell'Aquila) [link] Time: Tuesday, 26.04.2022, 13:30 Location: Online Smoluchowski's coagulation equation, an integro-differential equation of kinetic type, is a classical model for mass aggregation phenomena extensively used in the analysis of problems of polymerization, particle aggregation in aerosols, drop formation in rain and several other situations. In this talk I will present some recent results on the problem of existence or non-existence of stationary solutions to coagulation equations, for single and multi-component systems, under non-equilibrium conditions which are induced by the addition of a source term for small cluster sizes. The most striking feature of these stationary solutions is that, whenever they exist, the solutions to multi-component systems exhibit an unusual "spontaneous localization" phenomena. More precisely, the stationary solutions to the multi-component coagulation equation asymptotically localize into a direction determined by the source term. The localization is a universal property of these multicomponent systems. Indeed, it has been recently proved that localization phenomenon occurs with a great degree of generality also for time dependent solutions to mass conserving coagulation equations. (Joint work with M.A. Ferreira, J. Lukkarinen and J.J.L. Velázquez) |

## Past talks

- 23.11.2021, Silvia Budday (Friedrich-Alexander-Universität (FAU), Erlangen), Brain mechanics across scales.
- 21.04.2020, Alfonso Caiazzo (WIAS), Modeling of biological flows and tissues.
- 03.12.2019, Michal Pavelka (Charles University, Prague), Symmetric Hyperbolic Thermodynamically Compatible (SHTC) equations within GENERIC.
- 25.07.2019, Robert Style (ETH Zürich), Arresting phase separation with polymer networks.
- 09.07.2019, Carsten Graeser (Freie Universität Berlin), Truncated nonsmooth Newton multigrid for nonsmooth minimization problems.
- 25.06.2019, Luca Heltai (SISSA mathLab, Trieste), Unconventional frameworks for the simulation of coupled bulk-interface problems.
- 18.06.2019, Amit Acharya (Carnegie Mellon University Pittsburgh), Line Defect dynamics and solid mechanics.
- 04.06.2019, Giselle Monteiro (Czech Academy of Sciences , Prague), On the convergence of viscous approximation for rate-independent processes with regulated inputs.
- 14.05.2019, Mirjam Walloth (TU Darmstadt), Reliable, efficient and robust a posteriori estimators for the variational inequality in fracture phase-field models.
- 07.05.2019, Rainer Falkenberg (Bundesanstalt für Materialforschung und -prüfung BAM), Aspects on the modelling of material degradation.
- 23.04.2019, Marijo Milicevic (Uni. Freiburg), The alternating direction method of multipliers with variable step sizes for the iterative solution of nonsmooth minimization problems and application to BV-damage evolution.
- 28.02.2019, Uwe Thiele (Westfälische Wilhelms-Universität Münster), Gradient dynamics models for films of complex fluids and beyond - dewetting, line deposition and biofilms.
- 29.01.2019, Vittorio Romano (University of Catania), Charge and phonon transport in graphene.
- 13.11.2018, Alex Christoph Goeßmann (Fritz Haber Institute of the Max Planck Society), Representing crystals for kernel-based learning of their properties.
- 16.10.2018, Ch. Kuhn and A. Schlüter (Technische Universität Kaiserslautern), Phase field modelling of fracture -- From a mechanics point of view.
- 16.10.2018, Arik Yochelis (Ben-Gurion University of the Negev, Israel)), From solvent free to dilute electrolytes: A unified continuum approach.
- 08.05.2018, Simon Praetorius (TU Dresden), From individual motion to collective cell migration.
- 27.03.2018, Esteban Meca (Agronomy Department, University of Cordoba, Spain), Localized Instabilities in Phase-Changing Systems: The Effect of Elasticity.
- 07.03.2018, Matthias Liero (WIAS), Modeling and simulation of charge transport in organic semiconductors via kinetic and drift-diffusion models.
- 21.02.2018, Marco Morandotti (TU München), Dimension reduction in the context of structured deformations.
- 23.01.2018, Jan Giesselmann (RWTH Aachen), Modelling error estimates and model adaptation in compressible flows.
- 14.12.2017, Bartlomiej Matejczyk (University of Warwick), Macroscopic models for ion transport in nanoscale pores.
- 16.11.2017, Andreas Münch (University of Oxford), Asymptotic analysis of models involving surface diffusion.
- 24.10.2017, Anna Zubkova (Karl-Franzens-Universität Graz), Homogenization of the generalized Poisson-Nernst-Planck system with nonlinear interface conditions.
- 12.07.2017, Rodica Toader (SISSA, Trieste), Existence for dynamic Griffith fracture with a weak maximal dissipation condition.
- 30.05.2017, Ciro Visone (University of Sannio, Benevento), The applicative challenges of Smart Materials: from Sensing to Harvesting.
- 17.05.2017, Riccarda Rossi (University of Brescia), In Between Energetic and Balanced Viscosity solutions of rate-independent systems: the Visco-Energetic concept, with some applications to solid mechanics.
- 09.05.2017, Mathias Schäffner (TU Dresden), Stochastic homogenization of discrete energies with degenerate growth.
- 09.05.2017, Martin Slowik (TU Berlin), Random conductance model in a degenerate ergodic environment.
- 25.04.2017, Ian Thompson (University of Bath, Department of Physics), Modelling Device Charge Dynamics on the Microscopic Scale.
- 11.04.2017, Luca Heltai (SISSA mathLab, Trieste), A numerical framework for optimal locomotion at low Reynolds numbers.