Systematic derivation of hydrodynamic equations for viscoelastic phase separation

Abstract

Abstract 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.

Autoren

Dominic Spiller
Aaron Brunk
Oliver Habrich
Herbert Egger
Mária Lukáčová-Medvid’ová
Burkhard Dünweg

DOI

10.1088/1361-648x/ac0d17

eISSN

1361-648X

ISSN

0953-8984

Ausgabe der Veröffentlichung

36

Zeitschrift

Journal of Physics: Condensed Matter

Online publication date

2021

Paginierung

364001 - 364001

Datum der Veröffentlichung

2021

Status

Published

Herausgeber

IOP Publishing

Herausgeber URL

http://dx.doi.org/10.1088/1361-648x/ac0d17

Datum der Datenerfassung

2021

Titel

Systematic derivation of hydrodynamic equations for viscoelastic phase separation

Ausgabe der Zeitschrift

33

Data source: Crossref

Abstract

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.

Addresses

Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany.

Autoren

Dominic Spiller
Aaron Brunk
Oliver Habrich
Herbert Egger
Mária Lukáčová-Medvid'ová
Burkhard Dünweg

DOI

10.1088/1361-648x/ac0d17

eISSN

1361-648X

Externe Identifier

PubMed Identifier: 34153954

Funding acknowledgements

Deutsche Forschungsgemeinschaft: 233630050-TRR 146

Open access

false

ISSN

0953-8984

Ausgabe der Veröffentlichung

36

Zeitschrift

Journal of physics. Condensed matter : an Institute of Physics journal

Sprache

eng

Medium

Electronic

Online publication date

2021

Datum der Veröffentlichung

2021

Status

Published

Publisher licence

CC BY

Datum der Datenerfassung

2021

Titel

Systematic derivation of hydrodynamic equations for viscoelastic phase separation.

Sub types

Journal Article

Ausgabe der Zeitschrift

33

Data source: Europe PubMed Central

Abstract

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.

Date of acceptance

2021

Autoren

Dominic Spiller
Aaron Brunk
Oliver Habrich
Herbert Egger
Mária Lukáčová-Medvid'ová
Burkhard Dünweg

Autoren-URL

https://www.ncbi.nlm.nih.gov/pubmed/34153954

DOI

10.1088/1361-648X/ac0d17

eISSN

1361-648X

Ausgabe der Veröffentlichung

36

Zeitschrift

J Phys Condens Matter

Schlüsselwörter

GENERIC
Poisson brackets
coarse-graining
rheology
two-fluid model
viscoelastic phase separation

Sprache

eng

Country

England

Datum der Veröffentlichung

2021

Status

Published online

Titel

Systematic derivation of hydrodynamic equations for viscoelastic phase separation.

Sub types

Journal Article

Ausgabe der Zeitschrift

33

Data source: PubMed

Systematic derivation of hydrodynamic equations for viscoelastic phase separation

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