Thermodynamic Modeling and Analysis of Phase Transitions

Publications

Monographs

• A.H. Erhardt, K. Tsaneva-Atanasova, G.T. Lines, E.A. Martens, eds., Dynamical Systems, PDEs and Networks for Biomedical Applications: Mathematical Modeling, Analysis and Simulations, Special Edition, articles published in Frontiers of Physics, Frontiers in Applied Mathematics and Statistics, and Frontiers in Physiology, Frontiers Media SA, Lausanne, Switzerland, 2023, 207 pages, (Collection Published), DOI 10.3389/978-2-83251-458-0 .

• B. Wagner, M. Timme, eds., Special Issue ``The Mathematics in Renewable Energy'', 34 of European Journal of Applied Mathematics, Springer Nature, Heidelberg et al., 2023, 190 pages, (Collection Published).

Articles in Refereed Journals

• L. Schmeller, D. Peschka, Gradient flows for coupling order parameters and mechanics, SIAM Journal on Applied Mathematics, 83 (2023), pp. 225--253, DOI 10.1137/22M148478X .
  Abstract
  We construct a formal gradient flow structure for phase-field evolution coupled to mechanics in Lagrangian coordinates, present common ways to couple the evolution and provide an incremental minimization strategy. While the usual presentation of continuum mechanics is intentionally very brief, the focus of this paper is on an extensible functional analytical framework and a discretization approach that preserves an appropriate variational structure as much as possible. As examples, we first present phase separation and swelling of gels and then the approach of stationary states of multiphase systems with surface tension and show the robustness of the general approach.

• R. Müller, M. Landstorfer, Galilean bulk-surface electrothermodynamics and applications to electrochemistry, Entropy. An International and Interdisciplinary Journal of Entropy and Information Studies, 25 (2023), pp. 416/1--416/27, DOI 10.3390/e25030416 .
  Abstract
  In this work, the balance equations of non-equilibrium thermodynamics are coupled to Galilean limit systems of the Maxwell equations, i.e. either to (i) the quasi-electrostatic limit or (ii) the quasi-magnetostatic limit. We explicitly consider a volume $Omega$ which is divided into $Omega^+$ and $Omega^-$ by a possibly moving singular surface S, where a charged reacting mixture of a viscous medium can be present on each geometrical entity ($Omega$^+, S, $Omega^-$). By the restriction to Galilean limits of the Maxwell equations, we achieve that only subsystems of equations for matter and electric field are coupled that share identical transformation properties with respect to observer transformations. Moreover, the application of an entropy principle becomes more straightforward and finally it helps to estimate the limitations of the more general approach based the full set of Maxwell equations. Constitutive relations are provided based on an entropy principle and particular care is taken for the analysis of the stress tensor and the momentum balance in the general case of non-constant scalar susceptibility. Finally, we summarize the application of the derived model framework to an electrochemical system with surface reactions

• A.K. Barua, R. Chew, L. Shuwang, J. Lowengrub, A. Münch, B. Wagner, Sharp-interface problem of the Ohta--Kawasaki model for symmetric diblock copolymers, Journal of Computational Physics, 481 (2023), pp. 112032/1--112032/23, DOI 10.1016/j.jcp.2023.112032 .
  Abstract
  The Ohta-Kawasaki model for diblock-copolymers is well known to the scientific community of diffuse-interface methods. To accurately capture the long-time evolution of the moving interfaces, we present a derivation of the corresponding sharp-interface limit using matched asymptotic expansions, and show that the limiting process leads to a Hele-Shaw type moving interface problem. The numerical treatment of the sharp-interface limit is more complicated due to the stiffness of the equations. To address this problem, we present a boundary integral formulation corresponding to a sharp interface limit of the Ohta-Kawasaki model. Starting with the governing equations defined on separate phase domains, we develop boundary integral equations valid for multi-connected domains in a 2D plane. For numerical simplicity we assume our problem is driven by a uniform Dirichlet condition on a circular far-field boundary. The integral formulation of the problem involves both double- and single-layer potentials due to the modified boundary condition. In particular, our formulation allows one to compute the nonlinear dynamics of a non-equilibrium system and pattern formation of an equilibrating system. Numerical tests on an evolving slightly perturbed circular interface (separating the two phases) are in excellent agreement with the linear analysis, demonstrating that the method is stable, efficient and spectrally accurate in space.

• A.C. Moore, M.G. Hennessy, L.P. Nogueira, S.J. Franks, M. Taffetani, H. Seong , Y.K. Kang, W.S. Tan, G. Miklosic, R. El Laham, K. Zhou, L. Zharova, J.R. King , B. Wagner, H.J. Haugen, A. Münch, M.M. Stevens, Fiber reinforced hydrated networks recapitulate the poroelastic mechanics of articular cartilage, Acta Biomaterialia, 167 (2023), pp. 69--82, DOI 10.1016/j.actbio.2023.06.015 .

• R. Shiri, L. Schmeller, R. Seemann, D. Peschka, B. Wagner, Impact of noise on spinodal dewetting of liquid-liquid films, communications physics, 6 (2023), pp. 109/1--109/11, DOI 10.1038/s42005-023-01208-x .
  Abstract
  We investigate the spinodal dewetting of a thin liquid polystyrene (PS) film on a liquid polymethylmethacrylate (PMMA) subtrate. Following the evolution of the corrugations of the PS film via in situ measurements by atomic force microscopy (AFM) and those of the PS-PMMA interface via ex situ imaging, we provide a direct and detailed comparison of the experimentally determined spinodal wavelengths with the predictions from linear stability analysis of a thin-film continuum model for the bilayer system. The impact of rough interfaces and fluctuations is studied theoretically by investigating the impact of different choices of initial data on the unstable wavelength and on the rupture time. The key factor is the mode selection by initial data perturbed with correlated colored noise in the linearly unstable regime, which becomes relevant only for liquid bilayers to such an extent. By numerically solving the mathematical model, we further address the impact of nonlinear effects on rupture times and on the morphological evolution of the interfaces in comparison with experimental results.

• G.L. Celora, M.G. Hennessy, A. Münch, B. Wagner, S.L. Waters, The dynamics of a collapsing polyelectrolyte gel, SIAM Journal on Applied Mathematics, 83 (2023), pp. 1146--1171, DOI 10.1137/21M1419726 .

• E. Meca, A.W. Fritsch, J. Iglesias--Artola, S. Reber, B. Wagner, Predicting disordered regions driving phase separation of proteins under variable salt concentration, Frontiers in Physics, section Biophysics, 11 (2023), 1213304, DOI 10.3389/fphy.2023.1213304 .
  Abstract
  We determine the intrinsically disordered regions (IDRs) of phase separating proteins and investigate their impact on liquid-liquid phase separation (LLPS) with a random-phase approx- imation (RPA) that accounts for variable salt concentration. We focus on two proteins, PGL-3 and FUS, known to undergo LLPS. For PGL-3 we predict that an IDR near the C-terminus pro- motes LLPS, which we validate through direct comparison with in vitro experimental results. For the structurally more complex protein FUS the role of the low complexity (LC) domain in LLPS is not as well understood. Apart from the LC domain we here identify two IDRs, one near the N-terminus and another near the C-terminus. Our RPA analysis of these domains predict that, surprisingly, the IDR at the N-terminus (aa 1-285) and not the LC domain promotes LLPS of FUS by comparison to in vitro experiments under physiological temperature and salt conditions.

• M. Landstorfer, M. Ohlberger, S. Rave, M. Tacke, A modeling framework for efficient reduced order simulations of parametrized lithium-ion battery cells, European Journal of Applied Mathematics, 34 (2023), pp. 554--591, DOI 10.1017/S0956792522000353 .
  Abstract
  In this contribution we present a new modeling and simulation framework for parametrized Lithium-ion battery cells. We first derive a new continuum model for a rather general intercalation battery cell on the basis of non-equilibrium thermodynamics. In order to efficiently evaluate the resulting parameterized non-linear system of partial differential equations the reduced basis method is employed. The reduced basis method is a model order reduction technique on the basis of an incremental hierarchical approximate proper orthogonal decomposition approach and empirical operator interpolation. The modeling framework is particularly well suited to investigate and quantify degradation effects of battery cells. Several numerical experiments are given to demonstrate the scope and efficiency of the modeling framework.

• A.H. Erhardt, E. Wahlén, J. Weber, Bifurcation analysis for axisymmetric capillary water waves with vorticity and swirl, Studies in Applied Mathematics, 149 (2022), pp. 904--942, DOI 10.1111/sapm.12525 .
  Abstract
  We study steady axisymmetric water waves with general vorticity and swirl, subject to the influence of surface tension. This can be formulated as an elliptic free boundary problem in terms of Stokes' stream function. A change of variables allows us to overcome the generic coordinate-induced singularities and to cast the problem in the form ?identity plus compact,? which is amenable to Rabinowitz's global bifurcation theorem, whereas no restrictions regarding the absence of stagnation points in the flow have to be made. Within the scope of this new formulation, local curves and global families of solutions, bifurcating from laminar flows with a flat surface, are constructed.

• A.H. Erhardt, S. Solem, Bifurcation analysis of a modified cardiac cell model, SIAM Journal on Applied Dynamical Systems, 21 (2022), pp. 231--247, DOI 10.1137/21M1425359 .

• G.L. Celora, M.G. Hennessy, A. Münch, B. Wagner, S.L. Waters, A kinetic model of a polyelectrolyte gel undergoing phase separation, Journal of the Mechanics and Physics of Solids, 160 (2022), pp. 104771/1--104771/27, DOI 10.1016/j.jmps.2021.104771 .
  Abstract
  In this study we use non-equilibrium thermodynamics to systematically derive a phase-field model of a polyelectrolyte gel coupled to a thermodynamically consistent model for the salt solution surrounding the gel. The governing equations for the gel account for the free energy of the internal interfaces which form upon phase separation, as well as finite elasticity and multi-component transport. The fully time-dependent model describes the evolution of small changes in the mobile ion concentrations and follows their impact on the large-scale solvent flux and the emergence of long-time pattern formation in the gel. We observe a strong acceleration of the evolution of the free surface when the volume phase transition sets in, as well as the triggering of spinodal decomposition that leads to strong inhomogeneities in the lateral stresses, potentially leading to experimentally visible patterns.

• G. Shanmugasundaram, G. Arumugam, A.H. Erhardt, N. Nagarajan, Global existence of solutions to a two-species predator-prey parabolic chemotaxis system, International Journal of Biomathematics, 15 (2022), pp. 2250054/1--2250054/23, DOI 10.1142/S1793524522500541 .

• M. Landstorfer, R. Müller, Thermodynamic models for a concentration and electric field dependent susceptibility in liquid electrolytes, Electrochimica Acta, 428 (2022), pp. 140368/1--140368/19, DOI 10.1016/j.electacta.2022.140368 .
  Abstract
  The dielectric susceptibility $chi$ is an elementary quantity of the electrochemical double layer and the associated Poisson equation. While most often $chi$ is treated as a material constant, its dependency on the salt concentration in liquid electrolytes is demonstrated by various bulk electrolyte experiments. This is usually referred to as dielectric decrement. Further, it is theoretically well accepted that the susceptibility declines for large electric fields. This effect is frequently termed dielectric saturation. We analyze the impact of a variable susceptibility in terms of species concentrations and electric fields based on non-equilibrium thermodynamics. This reveals some non-obvious generalizations compared to the case of a constant susceptibility. In particular the consistent coupling of the Poisson equation, the momentum balance and the chemical potentials functions are of ultimate importance. In a numerical study, we systematically analyze the effects of a concentration and field dependent susceptibility on the double layer of a planar electrode electrolyte interface. We compute the differential capacitance and the spatial structure of the electric potential, solvent concentration and ionic distribution for various non-constant models of $chi$.

• M. Landstorfer, M. Ohlberger, S. Rave, M. Tacke, A modelling framework for efficient reduced order simulations of parametrised lithium-ion battery cells, European Journal of Applied Mathematics, published online on 29.11.2022, DOI 10.1017/S0956792522000353 .
  Abstract
  In this contribution we present a new modeling and simulation framework for parametrized Lithium-ion battery cells. We first derive a new continuum model for a rather general intercalation battery cell on the basis of non-equilibrium thermodynamics. In order to efficiently evaluate the resulting parameterized non-linear system of partial differential equations the reduced basis method is employed. The reduced basis method is a model order reduction technique on the basis of an incremental hierarchical approximate proper orthogonal decomposition approach and empirical operator interpolation. The modeling framework is particularly well suited to investigate and quantify degradation effects of battery cells. Several numerical experiments are given to demonstrate the scope and efficiency of the modeling framework.

Contributions to Collected Editions

• A.H. Erhardt, K. Tsaneva-Atanasova, G.T. Lines, E.A. Martens, Editorial: Dynamical systems, PDEs and networks for biomedical applications: Mathematical modeling, analysis and simulations, 10 of Front. Phys., Sec. Statistical and Computational Physics, Frontiers, Lausanne, Switzerland, 2023, pp. published online on 12.01.2023, DOI 10.3389/fphy.2022.1101756 .

• B. Wagner, M. Timme, Editorial Announcement, in: Special Issue ``The Mathematics in Renewable Energies'', B. Wagner, M. Timme, eds., 34 of European Journal of Applied Mathematics, Cambridge University Press, 2023, pp. 425--428, DOI 10.1017/S0956792523000013 .

Preprints, Reports, Technical Reports

• G.L. Celora, R. Blossey, A. Münch, B. Wagner, Counterion-controlled phase equilibria in a charge-regulated polymer solution, Preprint no. 3031, WIAS, Berlin, 2023.
  Abstract, PDF (6550 kByte)
  We study phase equilibria in a minimal model of charge-regulated polymer solutions. Our model consists of a single polymer species whose charge state arises from protonation-deproto- nation processes in the presence of a dissolved acid, whose anions serve as screening counteri- ons. We explicitly account for variability in the polymers' charge states. Homogeneous equilibria in this model system are characterised by the total concentration of polymers, the concentration of counter-ions and the charge distributions of polymers which can be computed with the help of analytical approximations. We use these analytical results to characterise how parameter values and solution acidity influence equilibrium charge distributions and identify for which regimes uni- modal and multi-modal charge distributions arise. We then study the interplay between charge regulation, solution acidity and phase separation. We find that charge regulation has a significant impact on polymer solubility and allows for non-linear responses to the solution acidity: re-entrant phase behaviour is possible in response to increasing solution acidity. Moreover, we show that phase separation can yield to the coexistence of local environments characterised by different charge distributions and mixture com

• C.S. Clemente, D. Davino, O. Klein, C. Visone, On forward and inverse uncertainty quantification for a model for a magneto mechanical device involving a hysteresis operator, Preprint no. 3009, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3009 .
  Abstract, PDF (982 kByte)
  Modeling real world objects and processes one may has to deal with hysteresis effects but also with uncertainties. Following D. Davino, P. Krejčí, and C. Visone: Fully coupled modeling of magneto-mechanical hysteresis through `thermodynamic' compatibility. Smart Mater. Struct., 22(9), (2013) 0950099, a model for a magnetostrictive material involving a generalized Prandtl- Ishlinskiĭ-operator is considered here. Using results of measurements, some parameters in the model are determined and inverse Uncertainty Quantification (UQ) is used to determine random densities to describe the remaining parameters and their uncertainties. Afterwards, the results are used do perform forward UQ and to compare the results with measured data. This extends some of the results from O. Klein, D. Davino, and C. Visone. On forward and inverse uncertainty quantification for models involving hysteresis operators. Math. Model. Nat. Phenom. 15 (2020) 53.

• L. Schmeller, D. Peschka, Sharp-interface limits of Cahn--Hilliard models and mechanics with moving contact lines, Preprint no. 2990, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.2990 .
  Abstract, PDF (5979 kByte)
  We construct gradient structures for free boundary problems with moving capillary interfaces with nonlinear (hyper)elasticity and study the impact of moving contact lines. In this context, we numerically analyze how phase-field models converge to certain sharp-interface models when the interface thickness tends to zero. In particular, we study the scaling of the Cahn--Hilliard mobility with certain powers of the interfacial thickness. In the presence of interfaces, it is known that the intended sharp-interface limit holds only for a particular range of powers However, in the presence of moving contact lines we show that some scalings that are valid for interfaces produce significant errors and the effective range of valid powers of the interfacial thickness in the mobility reduces.

Talks, Poster

• CH. Keller, Continuum-based modeling of calcium-selective ion channels, Conference on Mathematical Life Sciences, April 17 - 20, 2023, Universität Bonn, April 20, 2023.

• CH. Keller, Development of an ion channel model-framework, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), August 20 - 25, 2023, Waseda University, Tokyo, Japan, August 23, 2023.

• CH. Keller, Mathematical modeling of biological ion channels, 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), May 29 - June 2, 2023, Technische Universität Dresden, May 31, 2023.

• CH. Keller, Numerical simulation of biological ion channels, Third Conference of Young Applied Mathematicians (YAMC23), September 18 - 22, 2023, Siena, Italy, September 21, 2023.

• L. Schmeller, Gel models for phase separation at finite strains, 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), May 29 - June 2, 2023, Technische Universität Dresden, June 2, 2023.

• L. Schmeller, Gel models for phase separation at finite strains, Conference ``Calculus of Variations and Applications'', June 19 - 21, 2023, Université Paris-Cité (Campus des Grands Moulins), France, June 19, 2023.

• L. Schmeller, Gradient flows and moving contact lines, Seminar von Prof. Sebastian Aland, Technische Universität Bergakademie Freiberg, Institut für Numerische Mathematik und Optimierung, February 8, 2023.

• L. Schmeller, Phase-field systems coupled with large deformations, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), August 20 - 25, 2023, Waseda University, Tokyo, Japan, August 25, 2023.

• A.H. Erhardt, Bifurcation analysis for axisymmetric capillary water waves, 29th Nordic Congress of Mathematicians with EMS, July 3 - 7, 2023, Aalborg Universiteit, Department of Mathematical Sciences, Denmark, July 4, 2023.

• A.H. Erhardt, Mathematical modelling and simulation of cell-hydrogel interactions, Leibniz MMS Days 2023, April 17 - 19, 2023, Leibniz Institute for Agricultural Engineering and Bioeconomy (ATB), Potsdam, April 19, 2023.

• A. Selahi, A finite-element-solver for coupled domains in rust, SIAM Conference on Computational Science and Engineering (CSE23), Amsterdam, Netherlands, February 26 - March 3, 2023.

• A. Selahi, Recovering battery ageing dynamics with invertible neuronal networks, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), August 20 - 25, 2023, Waseda University, Tokyo, Japan, August 23, 2023.

• A. Selahi, Recovery of battery ageing dynamics using Bayesian inference, 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), May 29 - June 2, 2023, Technische Universität Dresden, June 1, 2023.

• B. Wagner, On disordered regions driving phase separation of proteins under variable salt concentration, Cellular Matters Conference 2023, Ascona, Switzerland, June 4 - 7, 2023.

• B. Wagner, On disordered regions driving phase separation of proteins under variable salt concentration, Cellular Matters Conference 2023, June 4 - 7, 2023, Ascona, Switzerland.

• O. Klein, On a model for a magneto mechanical device: Forward and inverse uncertainty quantification, 13th International Symposium on Hysteresis Modeling and Micromagnetics (HMM 2023), June 5 - 7, 2023, Technische Universität Wien, Austria, June 6, 2023.

• M. Landstorfer, A model framework for Lithium-ion intercalation cells, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Minisymposium 01140 ``Modelling and simulation of electro-chemo-mechanical processes in batteries and fuel cells'', August 20 - 25, 2023, Waseda University, Tokyo, Japan, August 25, 2023.

• M. Landstorfer, Modeling and validation of material and transport models for electrolytes, Energetic Methods for Multi-Component Reactive Mixtures Modelling, Stability, and Asymptotic Analysis (EMRM 2023), September 13 - 15, 2023, WIAS, Berlin, September 15, 2023.

• M. Landstorfer, Thermodynamic modelling of aqueous and aprotic electrode-electrolyte interfaces and their and double layer capacitance, Bunsen-Tagung 2023 - Physical Chemistry of the Energy Transition, 122nd Annual Conference of the German Bunsen Society for Physical Chemistry, June 5 - 7, 2023, Berlin, June 7, 2023.

• CH. Keller, J. Fuhrmann, M. Landstorfer, B. Wagner, Development of an ion-channel model-framework for in-vitro assisted interpretation of current voltage relations, MATH+-Day 2022, Technische Universität Berlin, November 18, 2022.

• L. Schmeller, K. Remini, Equilibrium droplets between experiment and theoretical predictions, SPP 2171 Workshop ``Wetting of Flexible, Adaptive, and Switchable Substrates'', December 5 - 8, 2022, Tagungszentrum an der Sternwarte, Göttingen, December 7, 2022.

• L. Schmeller, Gradient flows for coupling order parameters and mechanics, Universitá di Brescia, Mathematical Analysis, Italy, June 14, 2022.

• L. Schmeller, Multi-phase dynamic systems at finite strain elasticity, Summer School: Mathematical Models for Bio-Medical Sciences, Lake Como, Italy, June 20 - 24, 2022.

• L. Schmeller, Multiphase systems coupled with large deformations, MathBio22 -- Mathematical Models for Biological Multiscale Systems, September 12 - 14, 2022, WIAS Berlin, September 14, 2022.

• L. Schmeller, Multiphase systems coupled with large deformations, Seminar ``Interfaces: Modeling, Analysis, Numerics'', November 21 - 25, 2022, Mathematisches Forschungsinstitut Oberwolfach, November 21, 2022.

• R. Müller, Non-equilibrium thermodynamics modeling of polycrystalline electrode liquid electrolyte interface, 31st Topical Meeting of the International Society of Electrochemistry, Meeting topic: ``Theory and Computation in Electrochemistry: Seeking Synergies in Methods, Materials and Systems'', Session 2: ``Theory and Computation of Interfacial and Nanoscale Phenomena'', May 15 - 19, 2022, Rheinisch-Westfälische Technische Hochschule Aachen, May 17, 2022.

• A. Selahi, Homogenization in PDE-based battery models and recovery of ageing dynamics via Bayesian inference (online talk), Second Conference of Young Applied Mathematicians (Hybrid Event), September 18 - 22, 2022, Arenzano, Italy, September 21, 2022.

• O. Klein, On forward and inverse uncertainty quantification for a model for a magneto mechanical device involving a hysteresis operator (joint work with Carmine Stefano Clemente and Daniele Davino, Universitá degli Studi di Sannio, Italy), MURPHYS 2022 -- Interdisciplinary Conference on Multiple Scale Systems, Systems with Hysteresis, May 29 - June 3, 2022, Silesian University, Ostravice, Czech Republic, May 31, 2022.

• M. Landstorfer, A. Selahi, M. Heida, M. Eigel, Recovery of battery ageing dynamics with multiple timescales, MATH+-Day 2022, Technische Universität Berlin, November 18, 2022.

External Preprints

• A. Erhardt, E. Wahlén, J. Weber, Bifurcation analysis for axisymmetric water waves, Preprint no. arXiv:2202.01754, arXiv, Cornell University, 2022, DOI 10.48550/arXiv.2202.01754 .
  Abstract
  We study steady axisymmetric water waves with general vorticity and swirl, subject to the influence of surface tension. This can be formulated as an elliptic free boundary problem in terms of Stokes' stream function. A change of variables allows us to overcome the generic coordinate-induced singularities and to cast the problem in the form "identity plus compact", which is amenable to Rabinowitz' global bifurcation theorem, while no restrictions regarding the absence of stagnation points in the flow have to be made. Within the scope of this new formulation, local and global solution curves, bifurcating from laminar flows with a flat surface, are constructed.