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C3: Spinodal decomposition of polymer-solvent systems

The goal of the project is to obtain stable and consistent descriptions of flow dynamics on multiple scales in a class of systems exhibiting highly complex non-equilibrium dynamics, namely phase-separating polymer solutions. This is done by combining (i) the derivation, analysis, and simulation of macroscopic two-fluid models describing the dynamics of viscoelastic phase separation, (ii) the mesoscopic simulation of viscoelastic phase separation by extension of a coupled Lattice-Boltzmann / Molecular Dynamics method, and (iii) the calibration of the macroscopic models to results from mesoscopic simulations by means of parameter estimation and inverse problems methodology.

Systematic derivation of hydrodynamic equations for viscoelastic phase separation
Dominic Spiller, Aaron Brunk, Oliver Habrich, Herbert Egger, Mária Lukáčová-Medvid'ová and Burkhard Dünweg
Journal of Physics: Condensed Matter 33 (36), 364001 (2021);
URL: https://iopscience.iop.org/article/10.1088/1361-648X/ac0d17
doi:https://doi.org/10.1088/1361-648X/ac0d17

We present a detailed derivation of a simple hydrodynamic two-fluid model, which aims at the description of the phase separation of non-entangled polymer solutions, where viscoelastic effects play a role. It is directly based upon the coarse-graining of a well-defined molecular model, such that all degrees of freedom have a clear and unambiguous molecular interpretation. The considerations are based upon a free-energy functional, and the dynamics is split into a conservative and a dissipative part, where the latter satisfies the Onsager relations and the second law of thermodynamics. The model is therefore fully consistent with both equilibrium and non-equilibrium thermodynamics. The derivation proceeds in two steps: firstly, we derive an extended model comprising two scalar and four vector fields, such that inertial dynamics of the macromolecules and of the relative motion of the two fluids is taken into account. In the second step, we eliminate these inertial contributions and, as a replacement, introduce phenomenological dissipative terms, which can be modeled easily by taking into account the principles of non-equilibrium thermodynamics. The final simplified model comprises the momentum conservation equation, which includes both interfacial and elastic stresses, a convection–diffusion equation where interfacial and elastic contributions occur as well, and a suitably convected relaxation equation for the end-to-end vector field. In contrast to the traditional two-scale description that is used to derive rheological equations of motion, we here treat the hydrodynamic and the macromolecular degrees of freedom on the same basis. Nevertheless, the resulting model is fairly similar, though not fully identical, to models that have been discussed previously. Notably, we find a rheological constitutive equation that differs from the standard Oldroyd-B model. Within the framework of kinetic theory, this difference may be traced back to a different underlying statistical-mechanical ensemble that is used for averaging the stress. To what extent the model is able to reproduce the full phenomenology of viscoelastic phase separation is presently an open question, which shall be investigated in the future.

Analysis of a viscoelastic phase separation model
Aaron Brunk, Burkhard Dünweg, Herbert Egger, Oliver Habrich, Mária Lukáčová-Medvid'ová, Dominic Spiller
Journal of Physics: Condensed Matter 33 (23), 234002 (2021);
doi:10.1088/1361-648x/abeb13

A Second-Order Finite Element Method with Mass Lumping for Maxwell's Equations on Tetrahedra
Herbert Egger, Bogdan Radu
SIAM Journal on Numerical Analysis 59 (2), 864-885 (2021);
doi:10.1137/20m1318912

On the Energy Stable Approximation of Hamiltonian and Gradient Systems
Herbert Egger, Oliver Habrich, Vsevolod Shashkov
Computational Methods in Applied Mathematics 21 (2), 335-349 (2020);
doi:10.1515/cmam-2020-0025

On a Second-Order Multipoint Flux Mixed Finite Element Methods on Hybrid Meshes
Herbert Egger, Bogdan Radu
SIAM Journal on Numerical Analysis 58 (3), 1822-1844 (2020);
doi:10.1137/19m1236862

Chemotaxis on networks: Analysis and numerical approximation
Herbert Egger, Lukas Schöbel-Kröhn
ESAIM: Mathematical Modelling and Numerical Analysis 54 (4), 1339-1372 (2020);
doi:10.1051/m2an/2019069

Structure Preserving Discretization of Allen–Cahn Type Problems Modeling the Motion of Phase Boundaries
Anke Böttcher, Herbert Egger
Vietnam Journal of Mathematics 48 (4), 847-863 (2020);
doi:10.1007/s10013-020-00428-w

A mass-lumped mixed finite element method for acoustic wave propagation
H. Egger, B. Radu
Numerische Mathematik 145 (2), 239-269 (2020);
doi:10.1007/s00211-020-01118-y

On the transport limit of singularly perturbed convection–diffusion problems on networks
Herbert Egger, Nora Philippi
Mathematical Methods in the Applied Sciences 44 (6), 5005-5020 (2020);
doi:10.1002/mma.7084

Semiautomatic construction of lattice Boltzmann models
Dominic Spiller, Burkhard Dünweg
Physical Review E 101 (4), (2020);
doi:10.1103/physreve.101.043310

Structure preserving approximation of dissipative evolution problems
H. Egger
Numerische Mathematik 143 (1), 85-106 (2019);
doi:10.1007/s00211-019-01050-w

A hybrid mass transport finite element method for Keller–Segel type systems
J.A. Carrillo, N. Kolbe, M. Lukacova-Medvidova
J. Sci. Comp 80, 1777-1804 (2019);
doi:10.1007/s10915-019-00997-0

We propose a new splitting scheme for general reaction–taxis–diffusion systems in one spatial dimension capable to deal with simultaneous concentrated and diffusive regions as well as travelling waves and merging phenomena. The splitting scheme is based on a mass transport strategy for the cell density coupled with classical finite element approximations for the rest of the system. The built-in mass adaption of the scheme allows for an excellent performance even with respect to dedicated mesh-adapted AMR schemes in original variables.

Energy-stable linear schemes for polymer-solvent phase field models
P. Strasser, G. Tierra, B. Dünweg, M. Lukacova-Medvidova
Comp. Math. Appl. 77 (1), 125-143 (2019);
URL: https://www.sciencedirect.com/science/article/pii/S0898122118305303?via%3Dihub
doi:https://doi.org/10.1016/j.camwa.2018.09.018

We present new linear energy-stable numerical schemes for numerical simulation of complex polymer–solvent mixtures. The mathematical model proposed by Zhou et al. (2006) consists of the Cahn–Hilliard equation which describes dynamics of the interface that separates polymer and solvent and the Oldroyd-B equations for the hydrodynamics of polymeric mixtures. The model is thermodynamically consistent and dissipates free energy. Our main goal in this paper is to derive numerical schemes for the polymer–solvent mixture model that are energy dissipative and efficient in time. To this end we will propose several problem-suited time discretizations yielding linear schemes and discuss their properties.

Existence of global weak solutions to the kinetic Peterlin model
P. Gwiazda, M. Lukacova-Medvid'ova, H. Mizerova, A. Szwierczewska-Gwiazda
Nonlinear Analysis: Real World App. 44, 465-478 (2018);
URL: https://www.sciencedirect.com/science/article/pii/S1468121818305480?via%3Dihub
doi:https://doi.org/10.1016/j.nonrwa.2018.05.016

We consider a class of kinetic models for polymeric fluids motivated by the Peterlin dumbbell theories for dilute polymer solutions with a nonlinear spring law for an infinitely extensible spring. The polymer molecules are suspended in an incompressible viscous Newtonian fluid confined to a bounded domain in two or three space dimensions. The unsteady motion of the solvent is described by the incompressible Navier–Stokes equations with the elastic extra stress tensor appearing as a forcing term in the momentum equation. The elastic stress tensor is defined by Kramer’s expression through the probability density function that satisfies the corresponding Fokker–Planck equation. In this case a coefficient depending on the average length of polymer molecules appears in the latter equation. Following the recent work of Barrett and Süli (2018) we prove the existence of global-in-time weak solutions to the kinetic Peterlin model in two space dimensions.

Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange-Galerkin method. Part I: a nonlinear scheme
Lukáčová-Medviďová, M.; Mizerová, H.; Notsu, H.; Tabata, M.
ESAIM Math. Model. Numer. Anal. 51 (5), 1637–1661. (2017);
URL: https://www.esaim-m2an.org/

We present a nonlinear stabilized Lagrange–Galerkin scheme for the Oseen-type Peterlin viscoelastic model. Our scheme is a combination of the method of characteristics and Brezzi Pitkäranta’s stabilization method for the conforming linear elements, which yields an efficient computation with a small number of degrees of freedom. We prove error estimates with the optimal convergence order without any relation between the time increment and the mesh size. The result is valid for both the diffusive and non-diffusive models for the conformation tensor in two space dimensions. We introduce an additional term that yields a suitable structural property and allows us to obtain required energy estimate. The theoretical convergence orders are confirmed by numerical experiments. In a forthcoming paper, Part II, a linear scheme is proposed and the corresponding error estimates are proved in two and three space dimensions for the diffusive model.

An improved dissipative coupling scheme for a system of Molecular Dynamics particles interacting with a Lattice Boltzmann fluid
Nikita Tretyakov, Burkhard Dünweg
Computer Physics Communications 216, 102-108 (2017);
doi:10.1016/j.cpc.2017.03.009

Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange-Galerkin method, Part II: A linear scheme
M. Lukacova-Medvidova, H. Mizerova, H. Notsu, M. Tabata
Mathematical Modelling and Numerical Analysis , (2017);
doi:10.1051/m2an/2017032

This is the second part of our error analysis of the stabilized Lagrange–Galerkin scheme applied to the Oseen-type Peterlin viscoelastic model. Our scheme is a combination of the method of characteristics and Brezzi–Pitkäranta’s stabilization method for the conforming linear elements, which leads to an efficient computation with a small number of degrees of freedom. In this paper, Part II, we apply a semi-implicit time discretization which yields the linear scheme. We concentrate on the diffusive viscoelastic model, i.e. in the constitutive equation for time evolution of the conformation tensor diffusive effects are included. Under mild stability conditions we obtain error estimates with the optimal convergence order in two and three space dimensions. The theoretical convergence orders are confirmed by numerical experiments.

Global existence result for the generalized Peterlin viscoelastic model
Maria Lukacova - Medvidova, Hana Mizerova, Sarka Necasova, Michael Renardy
SIAM J. Math. Anal., 1-14 (2017);
URL: https://www.siam.org/journals/sima.php

We consider a class of differential models of viscoelastic fluids with diffusive stress. These constitutive models are motivated by Peterlin dumbbell theories with a nonlinear spring law for an infinitely extensible spring. A diffusion term is in- cluded in the constitutive model. Under appropriate assumptions on the nonlinear constitutive functions, we prove global existence of weak solutions for large data. For creeping flows and two-dimensional flows, we prove global existence of a classical solution under stronger assumptions.

Energy-stable numerical schemes for multiscale simulations of polymer-solvent mixtures
M. Lukacova-Medvidova, B. Duenweg, P. Strasser, N. Tretyakov
in Mathematical Analysis of Contimuum Mechanics and Industrial Applications II , Editor: Patrick van Meurs, Masato Kimura, Hirofumi Notsu, Chapter Chap5: Interface Dynamics , Pages 1-12, Springer International Publishing AG/ Eds. Patrick van Meurs, Masato Kimura, Hirofumi Notsu (2017);
URL: https://link.springer.com/chapter/10.1007/978-981-10-6283-4_13

We present a new second order energy dissipative numerical scheme to treat macroscopic equations aiming at the modeling of the dynamics of complex polymer-solvent mixtures. These partial differential equations are the Cahn-Hilliard equation for diffuse interface phase fields and the Oldroyd-B equations for the hydrodynamics of the polymeric mixture. A second order combined finite volume / finite difference method is applied for the spatial discretization. A complementary approach to study the same physical system is realized by simulations of a microscopic model based on a hybrid Lattice Boltzmann / Molecular Dynamics scheme. These latter simulations provide initial conditions for the numerical solution of the macroscopic equations. This procedure is intended as a first step towards the development of a multiscale method that aims at combining the two models.

The Cassie-Wenzel transition of fluids on nanostructured substrates: Macroscopic force balance versus microscopic density-functional theory
Nikita Tretyakov, Periklis Papadopoulos, Doris Vollmer, Hans-Jürgen Butt, Burkhard Dünweg, Kostas Ch. Daoulas
The Journal of Chemical Physics 145 (13), 134703 (2016);
doi:10.1063/1.4963792

Energy dissipative characteristic schemes for the diffusive Oldroyd-B viscoelastic fluid
Maria Lukacova-Medvidova, Hirofumi Notsu, Bangwei She
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS 81, 523-557 (2016);
URL: wileyonlinelibrary.com
doi:10.1002/fld.4195

In this paper, we propose new energy dissipative characteristic numerical methods for the approximation of diffusive Oldroyd-B equations that are based either on the finite element or finite difference discretization. We prove energy stability of both schemes and illustrate their behavior on a series of numerical experiments. Using both the diffusive model and the logarithmic transformation of the elastic stress, we are able to obtain methods that converge as mesh parameter is refined.

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