Energy-Stable Numerical Schemes for Multiscale Simulations of Polymer–Solvent Mixtures

Lukáčová-Medviďová, Mária; Dünweg, Burkhard; Strasser, Paul; Tretyakov, Nikita

doi:10.1007/978-981-10-6283-4_13

Cited by 5 publications

(6 citation statements)

References 21 publications

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“…Due to the construction via a dissipative Hamiltonian system, whose equation of motion can be found as a by-product of the GENERIC formalism [31][32][33][34], we automatically make sure that the second law is satisfied. Fully in line with the arguments put forward in reference [18], we consider this as very important both from the point of view of fundamental physical consistency, but also from the point of view of mathematical analysis and derivation of stable and convergent numerical algorithms [35,36].…”

Section: Basic 'Philosophy' Of Our Modelmentioning

confidence: 84%

Systematic derivation of hydrodynamic equations for viscoelastic phase separation

Brunk

Habrich

Egger

et al. 2021

J. Phys.: Condens. Matter

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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.

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Section: Basic 'Philosophy' Of Our Modelmentioning

confidence: 84%

Systematic derivation of hydrodynamic equations for viscoelastic phase separation

Brunk

Habrich

Egger

et al. 2021

J. Phys.: Condens. Matter

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“…Thus the Ginzburg-Landau potential allows the derivation of schemes that are energy-stable even though they are linear. It is also possible to derive schemes that are more generally applicable; one of these latter schemes is applied in our numerical experiments, in which we use the Flory-Huggins potential in order to facilitate comparisons with computer simulations of a quasiatomistic model [16]. Now, the Cahn-Hilliard equation can be derived from the mass balance law…”

Section: Mathematical Modelsmentioning

confidence: 99%

“…The numerical scheme ( 16) is linear and first order in time. We propose a linear second order numerical scheme by using a second order extrapolation for the explicit terms, arriving at a two-step numerical scheme, obeying an analogous discrete energy law as scheme (16). The proposed linear second order numerical scheme reads…”

Section: Schemes For the Simplified Model 130mentioning

confidence: 99%

“…The purpose of the present paper is to derive energy-stable and runtimeefficient numerical schemes to solve the above-mentioned equations. This task has already been tackled by us in a preliminary fashion before [16], from which paper we have also taken most of the wording of the present introduction. Compared to Reference [16], we have significantly improved the results, and also provide a much broader context and far more details.…”

Section: Introductionmentioning

confidence: 99%

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Energy-stable linear schemes for polymer–solvent phase field models

Strasser

Tierra

Dünweg

et al. 2019

Computers & Mathematics with Applications

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We present new linear energy-stable numerical schemes for numerical simulation of complex polymer-solvent mixtures. The mathematical model proposed by Zhou, Zhang and E (Physical Review E 73, 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 polymersolvent 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. 45 generalization using the log-transformation of the elastic stress tensor and the Lagrange-type approximation of the convective term is possible.The purpose of the present paper is to derive energy-stable and runtimeefficient numerical schemes to solve the above-mentioned equations. This task has already been tackled by us in a preliminary fashion before [16], from which 50 paper we have also taken most of the wording of the present introduction. Compared to Reference [16], we have significantly improved the results, and also provide a much broader context and far more details.The paper is organized in the following way. In Section 2 we present a mathematical model for the polymer-solvent mixture consisting of the Cahn- 55Hilliard equation for the interface dynamics and the Oldroyd-B equations for the hydrodynamics. We also introduce a simplified model modelling only interface

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“…We refer to [3,7,21] for results concerning the extension to multi-component systems and multiphysical problems. See also our recent works where logarithmic potentials and concentration dependent mobility functions have been used in the context of a complex model for viscoelastic phase separation [11,29,31]. Let us note that approximations in space and/or time and energy estimates are typically used to establish existence of solutions in rather general cases.…”

Section: Motivationmentioning

confidence: 99%

Relative energy estimates for the Cahn-Hilliard equation with concentration dependent mobility

Brunk¹,

Egger²,

Habrich³

et al. 2021

Preprint

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Based on relative energy estimates, we study the stability of solutions to the Cahn-Hilliard equation with concentration dependent mobility with respect to perturbations. As a by-product of our analysis, we obtain a weak-strong uniqueness principle on the continuous level under realistic regularity assumptions on strong solutions. We then show that the stability estimates can be further inherited almost verbatim by appropriate Galerkin approximations in space and time. This allows us to derive sharp bounds for the discretization error in terms of certain projection errors and to establish order-optimal a-priori error estimates for semi-and fully discrete approximation schemes.

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Energy-Stable Numerical Schemes for Multiscale Simulations of Polymer–Solvent Mixtures

Cited by 5 publications

References 21 publications

Systematic derivation of hydrodynamic equations for viscoelastic phase separation

Systematic derivation of hydrodynamic equations for viscoelastic phase separation

Energy-stable linear schemes for polymer–solvent phase field models

Relative energy estimates for the Cahn-Hilliard equation with concentration dependent mobility

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