institut / 2020 ) für mechanik berichte des instituts für

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Inst

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für M

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Mechanik

Stefan Descher

Modeling and Simulation of Crystallization Processes in Polymer Melt Flows

kasseluniversity

press

9 783737 608770

ISBN 978-3-7376-0877-0

Berichte des Instituts für Mechanik Bericht 2/2020

Stefan Descher


kasseluniversity

press

This work has been accepted by Faculty of Mechanical Engineering of the University of Kassel as a thesis for acquiring the academic degree of Doktor der Ingenieurwissenschaften (Dr.-Ing.).

Supervisor: Prof. Dr.-Ing. habil. Olaf Wünsch Co-Supervisor: Dr.-Ing. habil. Markus Rütten

Defense day: 28. May 2020

This document – excluding quotations and otherwise identified parts – is licensed under the Creative Commons Attribution-Share Alike 4.0 International License (CC BY-SA 4.0: https://creativecommons.org/licenses/by-sa/4.0/)

https://orcid.org/0000-0001-9031-7340 (Stefan Descher)

Bibliographic information published by Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.dnb.de.

Zugl.: Kassel, Univ., Diss. 2020 ISBN 978-3-7376-0877-0 https://doi.org/doi:10.17170/kobra-202011022043

© 2020, kassel university press, Kassel https://kup.uni-kassel.de

Printed in Germany

Abstract

The last stage of many manufacturing processes used in polymer processing industryare primary forming processes such as extrusion or injection molding. If melts of semi-crystalline plastics are subjected to such processes, temperature control opens up thepossibility of influencing solid state properties. This concerns those that depend on crys-tallinity, as it is possible to enhance crystallization by slow cooling or to suppress it byquenching. However, during the forming process the melt rarely rests, so that solidifica-tion processes in flows occur. Those complex processes can only be examined in detail bynumerical simulation. The present work contributes to this by developing a novel model-ing approach for isotactic polypropylene, detailed presentation and solution of problemsin modeling and numerics, as well as exemplary studies for the calculation of a profileextrusion and injection molding process.

Detailed calorimetric and rheometric investigations of the solidification behavior and aconsideration of molecular processes during crystallization serve as a fundament for mod-eling. The crystallization model is based on the derivation of the crystallization progressfrom data of a dynamic scanning calorimetry over a large range of cooling rates. It en-ables the consideration of suppression of crystallization and a local determination of thecrystallinity. The flow behavior of the melt is described by a thermorheological, general-ized Maxwell model with the exponential expansion of Phan-Thien and Tanner. Solidifiedregions are modeled using an adequately parameterized Newtonian law. The numericalrealization is done by implementing the modeling approaches in the open source CFDlibrary OpenFOAM. To ensure reliability of the solver, the log-conformation reformu-lation, both side diffusion stabilization and block-coupled pressure-velocity coupling areused. Detailed studies for elementary static and dynamic problems verify the methodand investigate the interaction of all modeling approaches. Parameter studies for realisticprofile extrusion and injection molding configurations in 2D and 3D results show exam-ples of application. The results show that the developed method allows to predict theinteraction between melt and solidified domains and the crystallinity in the solid.

v

Kurzfassung

Am Ende vieler Herstellungsprozesse, die in der kunststoffverarbeitenden Industrie An-wendung finden, stehen urformende Herstellungsverfahren wie Strangextrusion oder Spritz-gießen. Werden Schmelzen teilkristalliner Kunststoffe derartigen Prozessen unterzogen,ergeben sich durch die Temperaturführung Möglichkeiten der Beeinflussung von Festkör-pereigenschaften. Das betrifft solche, die von der Kristallinität abhängen, denn es istmöglich Kristallisation durch langsames Abkühlen zu begünstigen, oder sie durch Ab-schrecken zu unterdrücken. Während des Formgebungsprozesses ruht die Schmelze je-doch in den seltensten Fällen, sodass es zu Erstarrungsprozessen in Strömungen kommt.Hierbei handelt es sich um komplexe Prozesse, die nur durch Simulationen näher un-tersucht werden können. Einen Beitrag hierzu leistet die vorliegende Arbeit durch dieEntwicklung eines neuartigen Modellierungsansatzes für isotaktisches Polypropylen, diedetaillierte Darlegung und Lösung von Problemen der Modellierung und Numerik, sowieexemplarischen Studien zur Berechnung eines Strangextrusions- und Spritzgießprozesses.

Als Modellierungsgrundlage dienen sowohl ausführliche kalorimetrische und rheometrischeUntersuchungen des Erstarrungsverhaltens, als auch eine molekulare Betrachtung desKristallisationsprozesses. Das Kristallisationsmodell basiert auf der Ableitung des Kristalli-sationsfortschrittes aus Daten einer Dynamischen Differenzkalorimetrie über einen großenKühlratenbereich. Es ermöglicht die Berücksichtigung der Unterdrückung von Kristalli-sation und eine lokale Bestimmung der Kristallinität. Das Fließverhalten im Schmelzezu-stand wird durch ein thermorheologisches, verallgemeinertes Maxwell Modell mit der ex-ponentiellen Erweiterung nach Phan-Thien und Tanner beschrieben. Erstarrte Bereichedurch ein adäquat parametrisiertes newtonsches Fließgesetz. Die numerische Umsetzunggeschieht durch die Implementierung der Modellierungsansätze in die Open Source CFDBibliothek OpenFOAM. Um eine zuverlässige Berechnung zu gewährleisten, kommen dieLog-Conformation Reformulation, Both Side Diffusion Stabilisierung und eine blockgekop-pelte Druck-Geschwindigkeitskopplung zum Einsatz. Durch ausführliche Studien für el-ementare statische und dynamische Problemstellungen wird die Methode verifiziert unddas Zusammenspiel aller Modellierungsansätze untersucht. Sowohl Parameterstudien fürrealitätsnahe Profilextrusions- und Spritzgusskonfigurationen in 2D, als auch Ergebnissedreidimensionaler Berechnungen zeigen Anwendungsbeispiele auf. Die Ergebnisse zeigen,dass sich durch die entwickelte Methode die Interaktion zwischen Schmelze und erstarrtenBereichen, sowie die Kristallinität im Festkörper vorhersagen lässt.

vi

Contents

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V

1 Introduction 1

1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objective and Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Related Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Crystallization of Polypropylene 6

2.1 Thermal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Differential Scanning Calorimetry . . . . . . . . . . . . . . . . . . . 8

2.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Rheological Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Rotational Rheometer . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Polymer Physical Consideration . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.1 Molecular Configuration . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.2 Formation of Crystals . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.3 Influence of the Cooling Rate . . . . . . . . . . . . . . . . . . . . . 23

2.3.4 Influence of Deformation . . . . . . . . . . . . . . . . . . . . . . . . 25

I

II Contents

3 Modeling of Crystallization 27

3.1 Spherulite Growth Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1.1 Phase Field Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1.2 2D Growth Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Macroscopic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.1 Fixed Cooling Rate Model . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.2 Cooling Rate Dependent Model . . . . . . . . . . . . . . . . . . . . 34

3.2.3 Shear Rate Dependent Model . . . . . . . . . . . . . . . . . . . . . 38

3.2.4 Calculation of the Local Relative Crystallinity . . . . . . . . . . . . 40

4 Rheological Modeling 41

4.1 Modeling of the Molten State . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.1.1 Viscoelastic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1.2 Nonlinear Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.1.3 Thermorheological Model . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 Modeling of the Solid State . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2.1 Discontinuous Numerical Approach . . . . . . . . . . . . . . . . . . 50

4.2.2 Continuous Empirical Approach . . . . . . . . . . . . . . . . . . . . 50

4.3 Fluid-Solid Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.4 Dissipation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.5 Wall Slip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5 Numerical Treatment 61

5.1 Log-Conformation Reformulation . . . . . . . . . . . . . . . . . . . . . . . 62

5.1.1 Standard Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.1.2 Logarithmic Transport Equation . . . . . . . . . . . . . . . . . . . . 63

5.1.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.1.4 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.1.4.1 Lid Driven Cavity . . . . . . . . . . . . . . . . . . . . . . 69

5.1.4.2 Start-Up Channel Flow . . . . . . . . . . . . . . . . . . . 70

Contents III

5.1.4.3 EPTT Model . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2 Stabilization for Low Solvent Contribution Flows . . . . . . . . . . . . . . 73

5.3 Convection Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.3.1 Total Variation Diminishing Convection Discretization . . . . . . . 75

5.3.2 Low Solvent Contribution Example . . . . . . . . . . . . . . . . . . 78

5.4 Pressure Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.4.1 Comparison of Segregated and Block Coupled Pressure Correction . 82

5.5 Algorithmic Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6 Verification and Basic Testing 88

6.1 Verification of the Crystallization Model . . . . . . . . . . . . . . . . . . . 89

6.1.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.1.2.1 Fixed Rate Model . . . . . . . . . . . . . . . . . . . . . . 90

6.1.2.2 Variable Rate Model . . . . . . . . . . . . . . . . . . . . . 91

6.2 Stefan-like Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92


6.2.2 Comparison of Fixed and Variable Rate Modeling . . . . . . . . . . 92

6.2.3 Influence of the Fourier Number . . . . . . . . . . . . . . . . . . . . 94

6.2.4 Influence of Latent Heat . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2.5 Influence of the Time Step Size . . . . . . . . . . . . . . . . . . . . 96

6.3 Crystallizing Channel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 97


6.3.2 Exemplary Field Evaluation . . . . . . . . . . . . . . . . . . . . . . 97

6.3.3 Influence of the Pressure Gradient . . . . . . . . . . . . . . . . . . . 99

6.3.4 Shear Rate Dependent Crystallization Model . . . . . . . . . . . . . 100

6.4 Crystallizing Cavity Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 101


6.4.2 Mesh Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.4.3 Exemplary Field Evaluation . . . . . . . . . . . . . . . . . . . . . . 102

IV Contents

6.4.4 Influence of the Lid Velocity . . . . . . . . . . . . . . . . . . . . . . 104

7 Application to Process Engineering Examples 106

7.1 Profile Extrusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.1.1 2D Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.1.1.1 Problem Description . . . . . . . . . . . . . . . . . . . . . 108

7.1.1.2 Field Evaluation . . . . . . . . . . . . . . . . . . . . . . . 110

7.1.1.3 Influence of the Extrusion Rate . . . . . . . . . . . . . . . 111

7.1.2 Exemplary 3D Simulation . . . . . . . . . . . . . . . . . . . . . . . 112


7.1.2.2 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.2 Injection Molding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.2.1 2D Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115


7.2.1.2 Field Evaluation . . . . . . . . . . . . . . . . . . . . . . . 116

7.2.1.3 Influence of the Injection Velocity . . . . . . . . . . . . . . 118

7.2.2 Exemplary 3D Simulation . . . . . . . . . . . . . . . . . . . . . . . 119


7.2.2.2 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8 Summary and Outlook 123

8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

A EPTT Material Functions 127

List of Figures 132

Nomenclature

Greek Symbols

Ψ . . . . . . . . . . . . . . . . Numeric variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [-]Ω . . . . . . . . . . . . . . . . Rotation tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [s−1]α . . . . . . . . . . . . . . . . Molecular anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [-]αw . . . . . . . . . . . . . . . Wall heat transfer coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [m−1]γ . . . . . . . . . . . . . . . . Shear deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [-]γ . . . . . . . . . . . . . . . . Shear rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [s−1]Δhcryst . . . . . . . . . . . crystallization enthalpy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [J kg−1]η . . . . . . . . . . . . . . . . Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Pa s]η0 . . . . . . . . . . . . . . . . Zero viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Pa s]ηE . . . . . . . . . . . . . . . Elongation viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Pa s]ηp . . . . . . . . . . . . . . . . Polymer viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Pa s]ηs . . . . . . . . . . . . . . . . Solvent viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Pa s]ηv . . . . . . . . . . . . . . . . Both side diffusion viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Pa s]ηsolid . . . . . . . . . . . . . Solid viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Pa s]θ . . . . . . . . . . . . . . . . Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [K, ◦C]θm . . . . . . . . . . . . . . . Bell middle temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [K, ◦C]θmelt . . . . . . . . . . . . . Melting temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [K, ◦C]θonset . . . . . . . . . . . . . Onset temperature of crystallization . . . . . . . . . . . . . . . . . . . . . . . . [K, ◦C]θ . . . . . . . . . . . . . . . . Temperature rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [K s−1]κ . . . . . . . . . . . . . . . . Dissipation blending coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [-]λ . . . . . . . . . . . . . . . . Relaxation time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [s]ρ . . . . . . . . . . . . . . . . Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [kg m−3]τ . . . . . . . . . . . . . . . . Shear stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Pa]χ . . . . . . . . . . . . . . . . Fluid-solid indicator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [-]ψ . . . . . . . . . . . . . . . . Blending function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [-]

V

VI Nomenclature

Roman Symbols

B . . . . . . . . . . . . . . . . Stretch tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [s−1]C . . . . . . . . . . . . . . . . Conformation tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [-]D . . . . . . . . . . . . . . . . Strain rate tensor (= 1

2(L + LT )) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Pa]

L . . . . . . . . . . . . . . . . Velocity gradient tensor (= ∇v) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [s−1]T . . . . . . . . . . . . . . . . Cauchy stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Pa]Tp . . . . . . . . . . . . . . . Polymer stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Pa]Ts . . . . . . . . . . . . . . . Solvent stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Pa]v . . . . . . . . . . . . . . . . Velocity vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [m s−1]c . . . . . . . . . . . . . . . . . Specific heat capacity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [J kg−1 K−1]cr . . . . . . . . . . . . . . . . Relative crystallinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [-]D1/2 . . . . . . . . . . . . . Bell half width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [K]E0 . . . . . . . . . . . . . . . Activation energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [J mol−1]Fo . . . . . . . . . . . . . . . Fourier number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [-]G′, G′′ . . . . . . . . . . . Storage and loss modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Pa]h . . . . . . . . . . . . . . . . Channel height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [m]k . . . . . . . . . . . . . . . . Heat conductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [W m−1 K−1]M . . . . . . . . . . . . . . . Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Nm]m . . . . . . . . . . . . . . . . Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [kg]N . . . . . . . . . . . . . . . . Number of nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [-]N1 . . . . . . . . . . . . . . . First normal stress difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Pa]p . . . . . . . . . . . . . . . . Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Pa]P e . . . . . . . . . . . . . . . Peclet number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [-]q . . . . . . . . . . . . . . . . . Specific heat rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [W kg−1]Q . . . . . . . . . . . . . . . . Specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [J kg−1]R0 . . . . . . . . . . . . . . . Universal gas constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [J mol−1 K−1]Re . . . . . . . . . . . . . . . Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [-]Sp . . . . . . . . . . . . . . . Dissipation source term of polymer stress . . . . . . . . . . . . . . . . . . [W m−3]SHR . . . . . . . . . . . . . Strain hardening ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [-]St . . . . . . . . . . . . . . . Strouhal number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [-]T . . . . . . . . . . . . . . . . Characteristic time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [s]t . . . . . . . . . . . . . . . . . Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [s]Tsolid . . . . . . . . . . . . . Solidification time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [s]u, v, w . . . . . . . . . . . Cartesian velocity components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [m s−1]Uex . . . . . . . . . . . . . . Extrusion velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [m s−1]Uinj . . . . . . . . . . . . . . Injection velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [m s−1]V . . . . . . . . . . . . . . . . Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [m3]We . . . . . . . . . . . . . . Weissenberg number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [-]

CHAPTER 1

Introduction

1.1 Background and Motivation

The manufacturing process of a polymer component usually includes several thermome-chanical processes. For this purpose, the polymer, for example in the form of granulate, isfirst melted and processed in an extruder (see [1] or [2]). It is then subjected to a primaryforming process, during which cooling takes place. The melt does not necessarily have torest during this process. Often the opposite is the case, because solidification during a flowis what many processes make use of. One example is the freezing of the molecular chainsoriented in the direction of flow during injection molding [3]. The solidification of a meltis a fluid-solid phase transition. This does not happen suddenly, but extends over a tem-perature range. How large this range is, depends on whether the material is amorphousor semi-crystalline. A characteristic of amorphous thermoplastics is that solidification isa continuous process. The reason for this is mobility of the macromolecules is linked totemperature. It decreases steadily with temperature until the glass transition takes place.Before that, only a continuous increase in viscosity can be observed. Even semi-crystallinethermoplastics are amorphous in the molten state and the mobility of the macromoleculesdecreases with temperature. However, when a certain temperature range, the so-calledcrystallization range, is reached, some polymers begin to arrange themselves in a struc-tured manner. Thus, after complete solidification, ordered (crystalline) and disordered(amorphous) structures coexist in the partially crystalline material. The formation ofcrystalline structures becomes noticeable by a strong increase in viscosity [4]. An abruptincrease in viscosity during solidification of semi-crystalline thermoplastics means thatthey are more difficult to process than amorphous ones. Special precautions are there-fore taken during processing. For example, profile extrusion is executed with the use of

1

2 Chapter 1. Introduction

structure supports to maintain shape of the product, e.g. described in [5]. This meanssolidification takes place inside the shaping tool, because in case it would be extruded inthe molten state, it would just collapse under the influence of gravity.

From the moment that support is provided, the polymer should not randomly be cooleddown, as by the design of the cooling process the structure formation can be influenced.This also only applies to semi-crystalline thermoplastics, which is topic of the presentwork. For these it is possible to reduce crystallization to a minimum by shock cooling,but also to maximize it by slow cooling or tempering. This has an influence on the fi-nal crystallinity, which affects density, transparency, diffusion and mechanical propertiesof the solid. The thermal design of cooling processes is therefore of great importance.However, this is not trivial for semi-crystalline polymers, because unlike amorphous ther-moplastics, latent heat is released. The amount depends on the crystallinity that isestablished in the process and is therefore also dependent on the cooling rate. There is adependency on temperature as well. This is expressed by crystallization occurring at hightemperature for slow cooling, at low temperatures during rapid cooling. Accordingly, aheat flow occurs in the cooling process, of which the temporal and quantitative occur-rence is coupled to the cooling rate. In addition to the temperature dependency, the flowbehavior is therefore also dependent on the cooling rate, since the increase in viscosityis caused by the formation of crystalline structures. This leads to high gradients in theproperties, which causes the melt to interact with solidified areas.

Not to be neglected is also the energy released by dissipation. Depending on the charac-teristic process velocity it might even be a dominant factor, e.g. investigated theoreticallyin [6] or experimentally in [7]. In this case, a neglection as done for lowly viscous fluids isout of the question. For some cases it might even influence the crystallization as in theproximity of the fluid-solid interface there is the global peak of dissipation. Accordingly,an additional source of heat release, located right at the crystallization front, decreasesthe cooling rates occurring.

A well investigated phenomenon is also the influence deformation has on crystallization.It is known that if crystallization e.g. occurs under shearing, it is enhanced, see [8, 9].There is an influence on the onset of crystallization as well. Generally, the temperaturecrystallization sets on is increased by mechanical load. This effect will even be increasedin extensional flows, see [10]. The crystallization is therefore also depending on the typeof deformation applied.

In all the descriptions given above, the flow behavior of the melt was only described asviscous with reference to the level of viscosity. Though real materials exhibit many effectsthat are far from Newtonian behavior, best described in [11, 12]. They behave non-linearlyviscoelastic, which in case of common polymers includes shear-thinning behavior. Thismeans the shear viscosity decreases with increasing shear rates. A further effect is theelongation behavior, which differs from the shear behavior. It is common that strainhardening or softening occurs depending on the strain rate. Furthermore, normal stressdifferences occur. This means during shear, forces normal to the shear plane occur. Theexistence of these forces causes e.g. the well-known swelling of flows that are exitingnozzles. These effects are summarized in the steady state behavior. But there is also

1.2. Objective and Outline 3

a time dependence, because stresses build up and relaxate in viscoelastic fluids time-dependent. Using simple constitutive relationships, such as the generalized Newtonianmodel, will therefore not adequately describe the fluid, even for the isothermal, completelyfluid state.

As shown, describing crystallization processes in polymer melt flows involves many effects.Objective of this work is to capture a selection in a simulation environment, for a certainmaterial, starting from executing experiments on. This means aspects of experimentalinvestigations, modeling and special numerical methods used to create a simulation envi-ronment will be highlighted. Because this involves many fields, works covering all aspectsare rare1. It is intended to contribute such a work by the publication of this thesis.

1.2 Objective and Outline

The objective of this work is to create a simulation framework (solver) for simulatingcrystallization processes in polymer melt flows using OpenFOAM. It is intended to dothis for a common polypropylene including experimental investigations and modeling.To consider crystallization an empirical model based on findings from dynamic scanningcalorimetry investigations should be formulated. It is the aim to model the melt usinga Maxwell type constitutive equation, crystallized regions as a highly viscous fluid thatacts like a solid with a viscosity to be determined. As for the consideration of viscoelasticbehavior the application of special methods is mandatory, they should be implementedand/or adapted to the needs of this work. To verify the models implemented, extendedstudies should be carried out2. As the final part, it is intended to perform simulationsfor examples from process engineering. To document this project, the following structurewas chosen.

It starts with Chapter 2 where the experimental investigation of crystallization and theinterpretation of results is topic. In detail, with focus on their usage to obtain cooling ratedependent data for modeling, the equipment used for thermal and rheological analysis isexplained. The results are evaluated and transferred into a useful form. As closing ashort introduction into polymer physics is given, to formulate a crystallization model incompliance.

In Chapter 3 an empirical crystallization model is derived from data gathered in thermalanalysis under the usage of data by other authors to extend the measuring range. Modelsneglecting and considering the suppression of crystallization are formulated. For the latteran additional shear rate dependence is introduced. A way to capture the latent heatreleased during crystallization, allowing to determine crystallinity locally is presented.

Chapter 4 describes the rheological modeling approach. A nonlinear Maxwell type modelis chosen and made temperature dependent for the fluid state. Energy dissipation is taken

1Certainly works covering a selection of aspects exist. They are named in chapter 1.3.2Especially because crystallization models are used a lot in simulations, but almost never it is inves-

tigated how they actually behave in a simulation.

4 Chapter 1. Introduction

a closer look at, because the standard consideration was found out to fail. For the solidstate different modeling approaches are discussed.

The numerical treatment is topic of Chapter 5. It starts with an introduction to a loga-rithmic reformulation of constitutive equations and its implementation. Thereafter, theproper discretization of the constitutive equation is topic. As for low Reynolds numberflows pressure correction is an issue, the applied method is presented and tested. The finalpart of this chapter is the algorithmic realization for coupling all equations considered.

Verification and testing of the code is documented in Chapter 6. At first a verificationbased on a virtual thermal analysis is done. Afterwards testing for a one-dimensional,static and dynamic problem is carried out. The final part of this chapter are studies fora freezing cavity flow.

In Chapter 7 the method developed is applied for applications of process engineering. Oneis profile extrusion, based on the dimensions of a real tool. The other is a testing case forinjection molding. Studies are performed in 2D for both cases, exemplary results for thecorresponding 3D case are presented.

Chapter 8 summarizes this work and gives an outlook on continuing studies.

1.3 Related Studies

Because describing crystallization in polymer melt flows touches so many disciplines,it does not make sense to list state-of-the-art techniques at this point for every fieldwithout any introduction. In each chapter, this will be done regarding the respectivetopic. As in this work the aim is to create a whole simulation framework, the number ofpublications doing related work is limited. Especially because the topic is non-isothermalcrystallization in a flow, not isothermal crystallization in the static state, for which manymore works exist. There are however three publications close to this work, of which thecontent is named and related to the present work.

In 2014 Zhao [13] presented a paper on a simulation framework for what is called a "sus-pension of amorphous and semi-crystalline" structures. In terms of this work amorphousmeans melt and semi crystalline solid. Crystallization is modeled by a set of evolutionequations based on nucleation theory. The fluid is described using a finitely extensiblenonlinear elastic (FENE) model and the solid based on an orientation based stress tensor.All parameters used for the model applied originate from literature, constant crystalliza-tion enthalpy is used. The solver is based on the Finite Element Method and simulationsare performed for a profile extrusion tool.

In 2016 Zhao and Mu [14] presented a work regarding the die swell occurring for a hollowprofile extrusion tool. It is based on the same solver Zhao used in [13] but with a fewchanges in modeling. Just as in the present work, the solid phase is modeled highlyviscous and the melt with a Phan-Thien Tanner model. The simulations also include theexit of the profile from the shaping tool and the cooling on air in simplified manner.

1.3. Related Studies 5

In the same year Spina [15] published a work on the simulation of injection moldingbased on a two phase solver. Here the Level-Set method is applied to model the melt-air system. For crystallization a multi scale method is used that also has a constantcrystallization enthalpy. The fluid and solid state are both modeled as a non-isothermalpurely viscous fluid. When crystallization occurs, the viscosity is strongly increased tomodel solidification. The parameters chosen originate from experiments executed withpolypropylene. Simulations were performed for mold filling process. As in the works ofZhao, a Finite Element solver was used.

Approaches applied in these works can be found in the present work as well. The greatestdifference is surely the crystallization model. Here, it is derived from experiments thatinvestigate the suppression of crystallization. Therefore, the crystallization enthalpy isnot constant and it is possible to investigate the suppression numerically. In terms offluid modeling also viscoelastic constitutive equations are used and the solid is describedhighly viscous. The same cases of application are calculated, however in the present workthey have more a benchmark character. In contrast to the other works, the Finite VolumeMethod is used in the framework of OpenFOAM.

CHAPTER 2

Crystallization of Polypropylene

Thermoplastics are divided into two classes regarding their molecular orientation in thesolid state. If the molecule chains are randomly oriented the material is called amorphous,if they are ordered in substructures such as crystals they are called semi-crystalline. Thematerials this work focuses on are therefore semi-crystalline thermoplastics. But as thereis a large variety of materials, one had to be chosen as a representative which is, basedon its great relevance in industry [16], polypropylene1.

Modeling approaches used and developed in this work are based on experimental findingsand polymer physical principles. The experiments carried out deal with the thermody-namic and rheological behavior. An introduction into a common principle of thermalanalysis for polymers is therefore given at first. With the knowledge about the thermody-namic behavior of the sample, that allows predicting temperatures the onset of crystalliza-tion occurs, the reader is introduced into the rheological investigation of crystallization.For this purpose, rheometers have to be operated at their mechanical limits, because thesample becomes undeformable during investigation. Therefore, the subchapter dedicatedto this subject focuses on the feasibility of experiments with the used rheometer and inter-pretation of results considering those. Both, the thermodynamic and rheological behaviorare caused by processes on molecular level, which are classified as a topic of polymerphysics. As the final topic, for this purpose an introduction into the fundamentals ofcrystallization from this standpoint is given. From the view of a material modeler, thismight seem as going a bit too far as the working method is usually phenomenological.But many valuable findings for deriving the constitutive models will result from this, aspolypropylene can be considered unique regarding its molecular structure.

1The specific material chosen is PP 575P in granular form, produced by Sabic International. In thedata sheet [17] it is described as "homopolymer for injection molding". Its description says: "This gradecombines a high stiffness with moderate impact strength. It is suitable for production of complex articleswith long flow paths and thin walls. Typical applications are closures and garden furniture.".

6

2.1. Thermal Analysis 7

2.1 Thermal Analysis

The standard procedure for investigating crystallization processes in polymers is to per-form a differential scanning calorimetry (DSC, [18]) analysis. In simplified terms, a DSCrecords the heat flow emitted or absorbed by a sample, when passing a certain temper-ature range with a fixed cooling rate2. In a heating process of a sample with nearlyconstant heat capacity, to achieve a constant heating rate, a constant positive heat fluxis required. For cooling processes heat needs to be removed, therefore the heat flux isnegative. Everything that differs from these assumptions is, in terms of a DSC analysis,assigned to the material’s special thermodynamic behavior. Besides the caloric heat flow,from which the heat capacity can be determined, any kind of thermodynamically relevantprocess can be detected. Of course, this presupposes a certain magnitude this effect has onthe basic signal. The aim of a calorimetric characterization for common semi-crystallinepolymers is to find three essential effects. In the following enumeration they are namedin descending order, regarding the expectable magnitude.

1. Melting/crystallization: the fluid-solid phase transition is a physical processabsorbing/releasing energy. In the temperature range of melting, if the aim is tohave a constant heating rate, the heat flow entering the sample needs to be increased.For a constant cooling rate during crystallization, the magnitude of the outgoingheat flux needs to be increased. Thus, in both cases the result is a peak in heat flowfor the concerning temperature range.

2. Glass transition: a physical process that results in a transition from visco-elasto-plastic (solid) to glass-like. In a DSC scan this will result in a smooth jump in theheat capacity. A glass transition temperature θG can be determined.

3. Thermal history: the crystallization of a polymer undergoing a manufacturingprocess does usually not occur in the stress-free state. This results in mechanicalstresses being frozen in the solid. When heating up a sample containing residualstresses, entering glass transition results into relaxation of those3. This resultsinto an endothermic peak in superposition with the jump of the heat flux at glasstransition. If the cooling was fast during manufacturing, crystallization might havebeen suppressed. This will lead to recrystallization when entering the melting range.The result is an exothermic peak in superposition with the endothermic melting. Inthose cases the melting peak appears flattened out [20].

Therefore, many phenomena can be investigated using a DSC. But for this work, only thecrystallization is interesting, which by all the named processes is the easiest to investigateregarding the interference with other phenomena. Thermal history will not influence the

2The industrial and scientific standard is a temperature rate of 10 K/min. The reason for this is ofpragmatical nature. On one hand this rate is easily achievable with standard equipment and it limits theexperiment time to a reasonable timescale. On the other hand this rate is not sufficient to influence thebehavior of the sample in a way that e.g. crystallization is relevantly suppressed.

3A constitutive approach presented in [19] gives an insight into the thermomechanical effects.

8 Chapter 2. Crystallization of Polypropylene

measurements, as it is already removed by melting. For the polypropylene investigated theglass transition is below 0 ◦C which is far from the crystallization temperature range. Theonly effect that could interfere with the measurements is if the melting process was notcompleted. But this is taken care of by choosing the starting temperature high enough.

2.1.1 Differential Scanning Calorimetry

The principle of a DSC measurement is to determine a difference in heat fluxes whene.g. heating up a well known reference material in comparison to a sample. There aremany different experimental setups in which such measurements can be performed, a goodoverview is given in [21]. One of them is the heat-flux DSC which is the most widespreadcommercialized system today. It was used for the experiments presented in this work. Forlower cooling rates the TA-Instruments DSC Q1000 was used, for higher cooling rates thePerkin Elmer Pyrus DSC 6. As an example in fig. 2.1 (a) the testing chamber of the Q

1000 is shown. In its center the platforms for sample and reference is positioned.

(a) Testing chamber (b) Pans

Figure 2.1: Testing chamber and sample pans of the Q 1000 DSC. On the sample platforms(dark) the two pans are positioned, each on one platform. The four radially arranged darkholes at the bottom of the cell belong to the purge system. In the ring gap surrounding thechamber cooled nitrogen is flowing during the experiment.

A material is prepared by positioning it between two aluminum pans which are thenpressed together in a way that they are not hermetically sealed. A picture of a polymerprepared like this is shown in fig. 2.1 (b) left. On the right an empty pan is shown.The typical polymer sample mass is in the one-digit milligram range. A typical referencematerial is Indium, but it is common practice to just leave the reference pan empty.Since the dimension of the sample pans is very small and the positioning is of importancethe platforms are automatically equipped with the two pans by a robot arm. For theexperiment the lid of the testing chamber is closed automatically as well.


When considering the testing chamber as a thermodynamic system, it is obvious thatit is much easier to add heat in a controlled manner than to remove. That’s why thechamber is constantly cooled down4 and heated electrically as required to obtain a certaintemperature or temperature rate. To prevent any oxidation inside the testing chamberit is purged with adequately tempered nitrogen. In case of the Q 1000, the source forcooling has a temperature of -90 ◦C. This not only limits the lowest possible temperature,but also the achievable cooling rates at lower temperatures, as this is dependent on thetemperature difference between chamber and cooling system.

If the chamber is loaded with a sample and e.g. an empty pan as reference, duringtemperature change there is a heat flux between both platforms. It exists because thecell is built as symmetrical as possible, see [22]. So if it is unloaded or two empty pansare placed inside, this heat flux vanishes. Any heat flux that can be detected in thisconfiguration is caused by manufacturing tolerances [18]. It should now become clear aswell, why the positioning of the pans is of importance.

1© 2©3© 4©

Sample

Reference

Figure 2.2: Arrangement of thermocouples in a DSC cell.

The heat-flux is measured by differential thermocouples as shown in fig. 2.2. A standardheat-flux DSC configuration uses only one differential thermocouple between the twoplatforms. For this the temperature measuring positions 1© and 2© are operated. Usingthe thermal resistance between these two points the heat flux is calculated. In case of theQ 1000 the determination of the heat flux is corrected using a second thermocouple basedon the measuring points 3© and 4©. This method is well-known as Tzero correction byTA-Instruments [23]. The benefit is a very flat baseline of the heat flux. What this meansis explained hereafter.

The signal delivered by any DSC is the sample heat flux. If no phase transition occursthe property it depends on is solely the heat capacity, thus

q = c∂θ

∂t. (2.1)

Therefore e.g. a constant cooling rate will result in a constant heat flux. When plottingq over temperature, the result is a straight horizontal line, also called the baseline. If a

4Based on a system using liquid nitrogen.


system is described to be able to have flat baseline it is able to show exactly this behaviorfor samples with constant heat capacity. Examples for both types of systems are given inthe next subchapter.

When it comes to higher cooling rates, heat flux systems reach their limits. There are twomain problems that also occurred during the experiments for this work. The first is eventhough the testing chamber can be considered as relatively small with its diameter of 2cm, thermal inertia combined with a limited cooling power already becomes an issue justabove industry standard cooling rates. For this purpose, usually other systems come touse. The most prominent other type is the power compensating DSC. It uses a separatetesting chamber for the sample and reference which allows the system to be much smallerthan a heat flux system. With this type faster cooling rates are possible, but a flat baselinecan not be achieved. The most recent developments work on nano scale for which theapparatus used in [24] is surely fundamental.

2.1.2 Results

In chapter 1.1 amorphous polymers were described to thermodynamically behave muchless complex than semi-crystalline polymers. For the purpose to give an example, ofwhat to expect from DSC measurements, this statement is verified hereafter. As a rep-resentative for amorphous polymers a polycarbonate (Markrolon 2405) was chosen, thesemi-crystalline material is PP 575P.

-10

0

0

0

10

20

30

50 50100 100150 150200 200250 250θ [◦C]θ [◦C]

c[k

Jkg

−1

K−

1]

PP PC 2.5 K min−1

5 K min−1

10 K min−1

Glass transitionCrystallization

Melting Exo up

Figure 2.3: DSC graphs for polypropylene (PP) and polycarbonate (PC).

Measurements were performed over a wide temperature range, their results are presentedin fig. 2.3. The so called "exo up" representation is used which means that exothermicprocesses will cause positive peaks. Therefore, the heating curve is in the negative range,the cooling in the positive. Three different cooling rates were applied in a range from 0◦C to 250 ◦C with the Q 1000. To be able to compare the rate-depended behavior, theheat capacity c = q/|θ| is plotted. This shifts the baselines on one level since the heatcapacity, visible as the base line, is not temperature rate dependent.


The statement made in the introduction, that amorphous materials are not hard to handle,is supported by the comparison to the PP curve. For PC the only phenomenon occurringis the glass transition. It is independent on the cooling rate and also independent on ifthe sample is heated or cooled. For PP there are many differences. The most obviousis that crystallization and melting are not in the same temperature range. Melting willtake place at ≈ 165 ◦C relatively independent on the heating rate, whereas crystallizationstarts after ≈ 130 ◦C clearly depended on the cooling rate. The heat capacity for bothmaterials is similar which is consistent with the corresponding data sheets. There is aweak linear dependence on temperature. For PP 575P c(200 ◦C) ≈ 2500 J kg−1K−1 andc(0 ◦C)≈ 1700 J kg−1K−1, the crystallization causes no detectable jump in between. Inthe investigated temperature range, no glass transition was detected.

100 105 110 115 120 125 130 135 140

0

1000

2000

3000

4000

8000

9000

10000

11000

12000

13000

P1

P2

θ [◦C]

q[W

/kg]

q[W

/kg]

Exo up

Endo upRaw data

Appr. baseline

Corr. data

Figure 2.4: Example of a baseline subtraction for a measurement at -10 K min−1.

As mentioned in chapter 2.1.1, for a measuring series extending over a larger range ofcooling rates, a Perkin Elmer Pyrus DSC 6 was used. The aim of this series was to gathera complete dataset for modeling. But since this DSC does not use the Tzero technology,the measured data had to be corrected by baseline subtraction as shown in fig. 2.4. Inthe top part raw data from a measurement at -10 K min−1 in "edo-up" representation isshown. First the data set was reduced to the proximity of the peak. As the baseline isnot a linear function it was reconstructed using a polynomial. For fitting, the dataset wasreduced by the peak region between the points P1 and P2. The order of the polynomialwas chosen by eye to guarantee a certain smoothness in the peak region. Degrees in therange of ten were found out to be sufficient. The procedure was implemented in Matlab

using its least squares fit function.


In this measurement series a problem regarding the sample mass arose. Generally, it isknown that there is an issue in heat transfer for thick samples that shifts and broadens thebell curves. It was already investigated theoretically very early in [25] and also showed upslightly in the measurement in fig. 2.5. However, the artefact showing up for m = 13.975mg seems to have a different origin. As can be seen a sharp drop in the curve occursjust after passing the peak heat flux. At this point the cooling power might not besufficient and the electric heating is almost deactivated. Just after passing the peak, itis reactivated overshooting the required heat flux for the cooling rate. This problem issolved by reducing the sample mass, as can be seen for m = 1.745 mg. The evaluation ofthe measuring series is therefore done using measurements for different sample masses.

115 120 125 130 1350

0

1500

1500

3000

3000

4500

4500

2.5 K/min

5 K/min

7.5 K/min

10 K/min

m = 1.745 mg

m = 13.975 mg

θ [◦C]

q[W

/kg]

q[W

/kg]

Figure 2.5: Artifacts occurring for large sample masses.

The bell curves can be reduced to three characteristic values, as shown in fig. 2.6. Thoseare the maximum heat flux qmax, the temperature of the maximum heat flux θm andthe half width of the bell curve D1/2. The crystallization enthalpy Δhcrist representsthe surface under the curves divided by the corresponding cooling rate. An automatedway of evaluation is presented in chapter 3.2.1, which was applied to the measurements.Evaluating each of these properties in fig. 2.6 shows that qmax increases with the coolingrate. Because of the artefacts shown in fig. 2.5 the slope of the curve is smaller for theheavy sample. As the crystallization enthalpy should be constant for a fixed cooling ratethe curves broaden up, decreasing qmax. This effect is responsible for the earlier increaseof D1/2 as well. The peak temperature decreases with an increasing cooling rate, for thehighest a 35 ◦C lower value was measured.


0

000.10.1 11

90

110

130

140

150

2

2

4

4

6

6

8

8

10

1010

10

12

20

40

60

80

100

100

100100

120

120θ m

[◦C

]

q max

[kW

/kg]

D1/2

[K]

Δh

cryst

[kJ/

kg]

−θ [K min−1]−θ [K min−1]

1.745 mg2.693 mg

13.975 mg

Figure 2.6: Evaluation of the measuring series performed with the DSC 6. The bell curvespeak value is qmax. Their half width is D1/2 and the peak temperature is θm. Δhcryst is thecrystallization enthalpy calculated by integration.

Extrapolating the curve for Δhcryst in a graph with linear axis to zero will lead to a valueof ≈ 115 kJ kg−1. This represents the enthalpy released when reaching the maximumpossible degree of crystallinity, which is a product specific value. For PP 575P this valueis 56% based on the melting enthalpy of pure PP crystals (207 kJ kg−1) as named in [26].It is a common value for standard PP. Now the question might arise, why a very slowcrystallized PP is not 100% crystalline. Since the answer is very complex, chapter 2.3 isamongst other things dedicated to this. The influence of the cooling rate on Δhcryst is thatcrystallization is suppressed for faster cooling. Regarding fig. 2.6 a reduction by 15% to avalue 48% can be determined. When comparing these measurements to results publishedin [27] the agreement is good. However, the investigated cooling rate range is too smallto reach a region with strong suppression of crystallization. For the investigated materialthis should happen at around 250 K s−1 = 15000 K min−1 according to the findings in[28], which is far away from the rates possible with a conventional DSC.


2.2 Rheological Analysis

The rheological investigation of polymers during crystallization is not a standard proce-dure, as the viscosity increases rapidly. However, there are publications that address theinfluence of applied shear [29], elongation [30] or both [31]. It makes sense to couple suchexperiments with DSC scans performed in advance. In this way it is known at which tem-perature crystallization sets in. Just before the onset of crystallization the melt is highlyviscous. From the moment on first crystals grow a transition from a viscoelastic fluid toa solid is started. This means during crystallization the material becomes unexaminableby rheometers since they rely on the material being infinitely deformable. However, thereis a small window of time in which measurements are possible. Which experiments havebeen performed is topic of this subchapter.

2.2.1 Rotational Rheometer

For all rheological investigations the TA-Instruments AR-G2 rotational rheometer, asshown in fig. 2.7 (a), was used. It was equipped with an Environmental Testing Chamber(ETC) and 25 mm Parallel-Plate System (PPS) as shown in fig. 2.7 (b). The ETC al-lows to set temperatures precisely to 0.1 ◦C up to 600 ◦C. And to prevent an oxidationaldecomposition of molten polymers it is purged with 10 min−1 nitrogen. A camera sur-rounded with a led ring is mounted inside to monitor the gap during experiments. It isvisible in fig. 2.7 (a) on the right half of the cell. The gap of the PPS is loaded with thesample slightly bend outwards as symbolically shown fig. 2.7 (b). During experiments thegap height h is set to 600 μm to ensure a certain stability against the sample leaving thegap.

(a) AR-G2 (b) Parallel-Plate System

Figure 2.7: Rheological measuring equipment. (a) TA-Instruments AR-G2 equipped withthe Environmental Testing Chamber and the Parallel Plate System. (b) Detailed view or theParallel Plate System. Its Diameter is 25 mm, here the gap height is 2 mm and the samplePolycarbonate. During experiments the upper plate is rotating.

2.2. Rheological Analysis 15

The AR-G2 is a Controlled Shear Stress rheometer, best explained in [32]. This means arequested shear rate is applied by controlling the shear stress. As the following equationshows, in the PPS shear stress is proportional to torque.

τ = ηγ =2M

πr3(2.2)

Torque is transmitted by a drag cup motor to the shaft the upper plate is connected to [33].The drag cup motors torque is dependent on the current applied to the stator. In case ofthe AR-G2 the maximum applicable torque is 200 mNm which means the power supplylimits the current. If the gap is loaded with a sample that is highly viscous it is possiblethat the shear rate can not be reached. However, since the motor is contact-less under nocircumstances damage can be caused. The graph for the maximum measurable viscosity,that originates from eq. (2.2), is given in fig. 2.8 left. It is compared to the viscosity curve ofPP 575P at 140 ◦C. Since this is a temperature just above the crystallization temperaturerange for low cooling rates (c.f. fig. 2.6), the comparison shows that the instrument isoperated at all times inside its measuring range as long as the polymer is fluid. But assoon as crystal growth is initiated, the instrument is quickly overloaded as the resultsshown in the next subchapter will imply. Imply, because actually measurements were notcarried out until the torque limit was reached. Because the AR-G2 has a magnetic axialbearing for the shaft the upper plate is mounted to, it is very sensitive to normal forces.If a sample crystallizes in the gap of the PPS, its density increases which generates alarge normal force drawing the upper plate down. Therefore, as soon as an exponentialgrowth of viscosity is measured the experiment is stopped and the instrument is unloadedby reducing the gap height.

1e10

1e8

1e6

1e4

1e31e21e2

1e11e01e-11e-21e-3

η[P

as]

γ [s−1]

θ[◦

C]

t [min]0 3.5 7 10.5 14 17.5 21 24.5 28

130

140

150

160

170

180

190

200

maximum measurable viscosity

PP575 at 140 ◦Cexperiment

exact cooling rates

2.5 Kmin −15K

min −

1

10K

min

−1

20K

min

−1

Figure 2.8: Operational limits for the AR-G2. (left) Maximum measurable viscosity. (right)Testing for maintaining a constant cooling rate.

Aim of the rheological experiments is to reproduce the temperature program of the DSCmeasurements. Only in this case it is possible to draw conclusions from the data sincethe state of solidification is known. A fixed cooling rate experiment is not explicitlyprovided by the AR-G2, but it is possible to perform fixed shear rate measurements fora temperature ramp. Therefore, a start temperature, an end temperature and a timein which the end temperature should be reached needs to be defined. However, becausethe ETC and the PPS contain a lot of material to be cooled down, the thermal inertia is


quite high compared to a DSC. Results of such measurements for different cooling rates arepresented in fig. 2.8 (right). Even if it seems as if the cooling rates are reached perfectly,besides a weak influence of the thermal lag at the beginning, the results presented laterhave to be examined critically. The reason is that for the PPS the temperature is measuredat the lower plate. Since PP has a low thermal conductivity and around all the metalparts flows nitrogen, the measured temperature is not equal to the temperature of thesample. Furthermore, the sample does not have an equally distributed temperature.The following observation for a sample placed on the lower plate without performing anexperiment supports this statement.

θ = 145 ◦C, t = 0 min θ = 140 ◦C, t = 1 min θ = 135 ◦C, t = 2 min

θ = 130 ◦C, t = 3 min θ = 125 ◦C, t = 4 min θ = 120 ◦C, t = 5 min

Figure 2.9: Visual cooling experiment performed with PP 575P at 5 K min−1[34].

The mass of the sample shown here is approximately 3 g, which is significantly morematerial than in an experiment with a gap height of 600 μm. In this case the samplemass is approximately 280 mg. As shown in this sequence of pictures, the crystallizationstarts from the surface in contact with nitrogen, then moves inwards. The most importantindicator for this situation is that crystallization already starts between 145 ◦C and 140 ◦Cmeasured on the bottom plate. However, for 5 K min−1 crystallization should start at124 ◦C as the DSC investigations in fig. 2.6 show. The pictures from 140 ◦C to 120 ◦Calso show a noticeable decline of transparency and increase of opacity. Of course, inthe PPS configuration during measurements the upper plate reduces the free surface,as can be seen in fig. 2.7 (b) and the mass is lower. But the problem will remain.Surfaces of the molten polymer, in direct contact with the nitrogen stream are cooleddown faster. This is important to know, since this region has the largest influence ontorque. Therefore, a significant increase of viscosity will occur before the temperatureexpected for the according cooling rate.

2.2. Rheological Analysis 17

2.2.2 Results

As PP 575P behaves shear thinning, it was investigated at a shear rate of γ = 0.01 s−1. In[34] it was found out, that this value is sufficient to be inside the first Newtonian plateau atall temperatures reached in the experiment. As shown in fig. 2.10 three cooling rates wereinvestigated. All experiments started from 200 ◦C, but they are plotted beginning from180 ◦C because before the cooling rate is not constant as shown in fig. 2.8. For comparisonthe results from the DSC are plotted on top, in the middle the measured viscosity. Whatcan be seen first, is that the viscosity increases with temperature. It is expected for allpolymers. Close to the crystallization rage, viscosity increases exponentially, as alreadyannounced. When correlated to the heat flux, it can be seen that for 2.5 K min−1 theviscosity increases with the onset of crystallization. For 10 K min−1 this statement is stilltrue, just a little increase in viscosity is already present when the crystallization shouldstart according to the DSC measurements. For 15 K min−1 this shift is visible even moreclearly. The measured viscosity is therefore not an information that represents the wholesample. Most probably it correlates with the outer surface. As explained, it is cooleddown faster by nitrogen that is used for purge. For modeling, the only thing that can beconcluded is that the viscosity will increase strongly. The slope measured should not becorrelated to the progress of crystallization.

For measuring the elastic behavior, oscillation experiments were executed. At a constantfrequency of 10 Hz and a deformation amplitude of 1 % the ratio of viscous to elastic mod-ulus G′′/G′ was measured during crystallization. The result is that due to crystallizationthe elastic modulus strongly increases, which makes G′′/G′ decrease exponentially. Be-cause a transition to a solid that is also elastic is transformed, this is expected. But againthe same shifts regarding the DSC measurements were discovered, which means that thisresult only has an informational character. No modeling of these curves should be sought.

Looking forward to a consideration of shear dependence in the crystallization model, forusage in a CFD framework it is the easiest to use formulations based on the shear rate.Therefore, at -10 K min−1 its influence was investigated. So again experiments startingfrom 200 ◦C were executed, but this time the shear rate is varied. The results are shownin fig. 2.11 left. Instead of plotting the viscosity over temperature, here the undercoolingθmelt − θ w.r.t. the melting temperature θmelt = 165 ◦C is used. In fig. 2.11 right thechange in the onset temperature is plotted over the shear rate. It was determined at thetemperature a strong increase in viscosity occurred. This shows a light drop for smallshear rates but the overall tendency is that γ increases the onset temperature. This is inline with findings from literature5.

From the curves plotted in fig. 2.11 the only thing that should be drawn is the onsettemperature of crystallization. Usually with a PPS investigations of a shear rate aboveγ = 1 s−1 are impossible because the sample will exit the measuring gap. In case of thepresented investigations however, this could be prevented with a little trick.

5However the experiments commonly executed in literature differ from the experiment executed here.Further discussion of that topic is done in chapter 2.3.4.


0

0

0

0.5

1

1

2

2.53

4

5

10

10

15

15

20

100 110 120 130 140 150 160 170

η[k

Pa

s]G

”/G

′[-

]q

[kW

kg−

1]

θ [◦C]

cooling rates in K min−1

Figure 2.10: Results of the rheological experiments compared to the corresponding DSCcurves. The shear experiments (middle) were executed at γ = 0.01 s−1. The oscillatoryexperiments at a frequency of 10 Hz and a deformation amplitude of 1%. G′′/G′ is the ratioof viscous to elastic modulus.

As the outer surface (fig. 2.7 right) is always slightly bend outwards, strong shearingcauses the sample to roll which then causes material to exit the gap. That is why under-filling was applied, which means that the sample bends slightly inwards. This enabledmeasurements up until γ = 12.5 s−1, but has to be seen critical regarding the measuredviscosity. As during cooling and crystallization the density increases, the under-filling willget stronger. This means the onset of crystallization can still be detected, as it causesthe viscosity to grow exponentially. However, the viscosities measured are smaller thanthey really are. The curves for γ > 1 s−1 therefore have a different slope. Under-fillingis not the only thing influencing the slope. As the material is shear thinning, which getsstronger with a decreasing temperature, the slope of the curves decreases naturally. Theresults are therefore plausible.

2.3. Polymer Physical Consideration 19

110

0

0

00

2

2

4

45

5 6

6

8

8

10

10

10

10 12

12

14

14

15

15

1620

20 25 30 35θmelt − θ [K]

η[k

Pa

s]

Δθ o

nse

t[K

]

γ [s−1]

0.01 0.05 0.10.5 2.5

5 7.512.5

shear rates in s−1

Figure 2.11: Shear rate dependent investigation of crystallization. (left) Viscosity plottedover the undercooling. (right) Plot of the difference in onset temperature w.r.t the onsettemperature determined for γ = 0.01 s−1.

Another effect to comment on is the loss of exponential growth for high shear rates. Mostlikely this is caused by wall slip. Due to the increase in density by crystallization, thesample contracts volumetrically and therefore might partially loose contact the upperplate. This is a topic not investigated further, as the onset could be read from the datanevertheless.

2.3 Polymer Physical Consideration

As one of the main tasks of this work is to develop a model that includes the suppressionof crystallization, it is first needed to understand the processes behind. Therefore, at thispoint an introduction into the polymer physical consideration is given. It also contributesin getting more information about the material investigated, needed to extend the dataset gathered by the DSC. Because the cooling rate range investigated was too small, asdescribed later in chapter 3.2.2, this was necessary.

2.3.1 Molecular Configuration

The property most important for crystallization of polypropylene is tacticity, c.f. theregarding chapters in [35] or [36] on polymorphism. This makes it kind of unique sincethere are not many common polymers for which this property occurs. To explain this, itis needed to take a look at its smallest constituents.

The monomer of polypropylene, propene (C3H6), is a gaseous carbon-hydrogen bond.


As shown in fig. 2.12 it consists out of three carbon atoms of which two are connectedby a double bond. The remaining free valencies are satiated by six hydrogen atoms.Considering that carbon has four free valences and hydrogen has one, the molecule iscomplete. In the process of polymerization, the double bond is broken into a single bondcreating a molecule with two free valencies, the mer (repeat unit) of polypropylene. Thisallows to form a polymer, which are multiple mers connected.

n

Polymerization

Figure 2.12: Creation of a PP mer in polymerization.

The mer itself can be divided into the backbone (CH2 − CH) and the methyl group(CH3), as e.g. described in [37]. This is where the spatial orientation comes into play sincecovalent bonds not just form in straight lines as indicated in fig. 2.12 right. For Carbon thefour valence bonds will form separated by an (bond-)angle of 109.5◦ distributed uniformin space. When considering a chain structure, the position of the methyl is not unique.It can either be on top or bottom of the backbone as visualized in the following figure.

Figure 2.13: Methyl group orientation in syndiotactic polypropylene.

The configuration shown, with an alternating position of the methyl group, is calledsyndiotactic (sPP). If the methyl group is one-sided the polypropylene is isotactic(iPP). A random distribution is called atactic (aPP). All tree configurations can beproduced by adding different catalysts during polymerization, c.f. for iPP [38]. The mostcommon type of polypropylene is iPP. In contrast to the configuration for sPP in fig. 2.13,its chains cannot be straight because the methyl groups will repulse each other. Therefore,iPP chains are helically oriented, c.f. the chapter on chain conformation in [39].

The reason why there are different configurations possible in the first place is that the PPmer is asymmetrical. Polystyrene is the only other polymer of mass production to whichtacticity is relevant.


2.3.2 Formation of Crystals

In polymers the segmental vibration of chains is associated with temperature. This allowsto define a free volume that represents the empty space blocked by the movement [40].Since lower temperatures corresponds to smaller movements the free volume decreaseswith temperature bringing the polymer chains closer together. If the distance of chainsegments falls under a certain value, intermolecular forces will bond them. Polyolefineslike PP are governed by van der Waals (vdW) forces6 that are irrelevant for distanceslarger than 7 to 10Å, see [41]. The following figure illustrates the size scales of thesedistances.

Figure 2.14: Van der Waals volumes for syndiotactic polypropylene.

What is shown here are the vdW volumes that define a thought shell of the molecule thatcannot be penetrated by other molecules. Since it is possible to display sPP energeticallycorrect and neatly arranged, it chosen as example again. The height of the structuredisplayed corresponds approximately to six vdW radii of carbon which corresponds to10.2 Å. For iPP this height would be 6.8 Å.

The importance of the position of the methyl group becomes clear when looking at thealignment of chains, e.g. treated in [42]. As shown in fig. 2.15 a) a perfectly isotactic PPcan be packed very closely. But in case there are defects in tacticity, as shown in fig. 2.15b) empty spaces occur, lowering the crystallinity of the material. As shown in fig. 2.15 c)for sPP the alternating position of the methyl group is preventing the chains to approachclose. The average distance reached in this case is too large to sufficiently bond, thereforesPP has a lower crystallinity than iPP. Atactic polypropylene is therefore amorphous, asno sufficiently strong bond is created. Regarding the molecular properties of PP 575P, nostatement is made in the datasheet. But it is possible when considering that sPP has anapproximate crystallinity of 30%, c.f. [36]. As in context to fig. 2.6 the crystallinity wasestimated with 56% it can be considered as isotactic.

The effect defects in tacticity have was shown in context of fig. 2.15 b). Therefore, acharacterization of iPP is usually done based on an isotacticity index in percent. Adefinite statement about the isotacticity index of PP 575P cannot be made, since thedata sheet does not provide this information and clear statements about the influenceof isotacticity on material parameters are rare. The work presented in [43] allows aclassification, however. Based on values for Youngs’s modulus, yield stress, yield strain,

6The van der Waals forces are ∼ 1/d6 with d being the distance between segments.


crystallization enthalpy at 10 K min−1 and the melting temperature given in dependenceof the isotacticity, the isotacticity index of PP 575P is assumed to be in the range of90-93%. Based on a classification in [42] stating 98% as high and 68% as low, PP 575Pcan be classified as a material with average isotacticity.

a) iPP b) iPP defects c) sPP

Figure 2.15: Alignment of polymers for isoatctic and syndiotactic polypropylene.

The alignment of chain segments is not a phenomenon that is equally distributed overspace. It occurs in clusters, where the density of aligned chains is high, called nuclei.Their number is generally high in presence of extra surfaces like e.g. impurities [44] orwalls. But it can also be increased by the flow type as explained in chapter 2.3.4. Underthe absence of stress, the density of nuclei N can be described by

N(t) = N0eAN (θmelt−θ) (2.3)

as given in [45]. With two positive empirical parameters N0 and AN . When definingθmelt − θ as undercooling one can see that the number of nuclei increases exponentiallywith increased undercooling. This initiates the continuous process of lamellae growth [46],described hereafter.

Adjacent re-entry no re-entry

Figure 2.16: Possible alignment of polymers in lamellae.

If a certain critical density is reached in a nucleus, a continuous recruitment of chainsinto ordered blocks called lamellae is triggered. As fig. 2.16 shows this could happen inperfect order as shown by the adjacent re-entry model, or totally unordered outside of thelamellae as shown by the no re-entry model, c.f. [47]. For high isotacticity, the lamellaetend to be more of the re-entry-type, as the chains have a stronger attraction.

In the ongoing process of crystallization different lamellae align, connected on the shortsides, and grow in direction of the temperature gradient [48]. Since during the formation


of lamellae, the the kinetic energy of the vibrating chains is transformed into heat, thegrowth direction is radially away from the nucleus. The structure formed in this wayis called spherulite, shown in fig. 2.17 as α-Crystal. Lamellae connected lengthwise arerepresented by black lines in this representation. Optical investigations of such a structurecan e.g. be found in [49].

α-Crystal β-Crystal Detail

DetailAmorphous

Crystalline

Figure 2.17: Formation of crystals by radial growth of lamellae.

In case the crystallization takes place under a high temperature gradient, which in poly-mers is related to high cooling rates, distorted structures as shown in fig. 2.17 as β-crystalform. Microscopic investigations of beta-crystals grown in high temperature gradients aree.g. shown in [48]. The detail in 2.17 indicates the chain alignment for the neighboringlamellae. Because the lamellaes structure is not adjacent re-entering, the space in betweenthe lamellae is unordered, known as amorphous. After the isotacticity defects occurringinside the lamellae, this is the second mechanism lowering crystallinity.

The largest structures lowering crystallinity are the impingement areas of crystals. Theyform between neighboring crystals when their crystallization fronts meet. Typically, theyform a Voronoi pattern as discussed in chapter 3.1.2 regarding fig. 3.2. To summarize,isotacticity defects, unordered regions between the lamellae and impingements lower thecrystallinity of a polypropylene. In case of PP 575P by 48%, as discovered in the evaluationof the DSC measurements.

2.3.3 Influence of the Cooling Rate

The influence of processing conditions on the crystallinity of PP is well known. Lowcooling rates will increase crystallinity, high cooling rates decrease crystallinity. A reasonfor this effect is given in this chapter based on an example.

As mentioned in the previously, crystallization starts from nuclei and according to eq. (2.3)their number is increasing with the undercooling defined as the difference of the meltingtemperature and the actual temperature. Assuming a constant growth speed this wouldresult in a higher number of crystals, causing a higher number of impingements. As


impingements decrease crystallinity, this could be an explanation. It actually is, but it isnot that simple.

The assumption of constant growth speed is wrong, as experiments in [50] showed. Herean iPP was cooled down from 220 ◦C to different values of undercooling with 5000 K min−1

and growth velocities were measured using a microscope. Even if this setup representsan isothermal crystallization, important conclusions can be drawn for non-isothermalprocesses. For this purpose, the data published were transformed into a graph that showsthe growth velocity in dependence of the undercooling.

0 20 40 60 80 100 1201e0

1e1

1e2

1e3

1e4

ugrow

th[n

ms−

1]

Tm − Tc [K]

0.5

1

2.5

10

25

5075

Annotated: cooling rates in K min−1

Measurements [50]

Estimated in DSC

Figure 2.18: Growth velocity of iPP spherulites in dependence of the undercooling w.r.t. themelting temperature. The solid line represents data from [50]. The triangles represent theundercooling for non-isothermal crystallization presented in fig. 2.6.

The next step is to take the data points for θm in fig. 2.6 and subtract them from themelting temperature. This leads to an undercooling for each cooling rate, which can beannotated on the curve as done in fig. 2.18. An association of cooling rates with growthspeeds is therefore possible, even if it’s just an estimation. From these points can bedrawn, that for small cooling rates growth speed and cooling rate are nearly proportional.For high cooling rates, on the contrary, a limit value is reached. In this case the effectmentioned at the beginning, described as too simple, gets relevant. In between, of coursea finer structure will develop, but the crystallinity will not get suppressed strongly. Theseare exactly the findings from the results for Δhcryst in fig. 2.6. That the estimation ofgrowth speeds is not highly inaccurate can be seen when looking at the extended curvefor θm in fig. 3.5. Here a strong suppression of crystallization sets in at θm(−250 K s−1) ≈80 ◦C which, drawn into fig. 2.18, would be right at the start of the flat region.

The development of finer structures is therefore causing suppression of crystallization. Aneffect that is also well investigated in microscopy, for iPP e.g. in [51] with a special focuson high cooling rates. There is also the possibility of substructure formation in quenchediPP, but this topic was considered as too specific. To close the subchapter on the influenceof the cooling rate, a practical example is given in fig. 2.19.


Figure 2.19: Crystal formation in the proximity of a weld seam in iPP.7

It shows the crystal structure in proximity of a weld seam, visualized with a polarizingfilter. Right of the seam the material crystallized in a high temperature gradient andcooling rate, forming β-crystals. On the left it crystallized slightly slower, as a consequenceonly fine α-crystals form. Further away from the seam, crystals are larger as the coolingrate was lower there.

2.3.4 Influence of Deformation

Any deformation applied in a cooling process will influence crystallization. In shear flowspolymers are aligned, slipping along their neighbors, decreasing the distance betweensegments. As a result, more bonds are formed creating nuclei. The applied denominationfor this effect is shear enhanced nucleation. It is usually investigated based on a paperof Janeschitz-Kriegl [8], often mentioned as a protocol. This says the melt should firstbe undercooled to a fixed temperature, then a shear pulse is applied and its influenceon crystallization is evaluated. A paper presenting such investigations for iPP is e.g.[52]. Here it was found out that the larger the deformation brought in is, the earlier

7By courtesty of Rolf Diederich, Department of Plastics Technology, Institute of Materials Engineering,University of Kassel


crystallization starts and the shorter it takes. However, there is a plateau reached, whichmeans that no unlimited enhancement is possible. It also means, that the deformationhistory is of interest, which looking forward to the usage of a model in simulations, iscritical as it is generally not available in a CFD framework. That is why the experimentsin chapter 2.2.2 were performed as continuous shearing experiments, e.g. presented in[53] as well. It might ignore certain findings, but it was the only way to obtain data to beused in a model that works with commonly available fields. As a plateau in the increaseof the onset temperature was found in the results presented in fig. 2.11 as well, this mightbe a similar effect as it occurs for deformation.

Elongation flows are known to have a greater influence on crystallization. Looking atthe flow type of uniaxial elongation this makes sense as the polymers are stretched. Thedistances between chain segments should therefore be smaller compared to shear, whichleads to greater nucleation. A publication investigating the differences between shear andelongation on crystallization is [31]. The main statement given in this work is, that forhigh elongation rates crystallization is enhanced significantly, compared to the case ofshear with the same rate.

CHAPTER 3

Modeling of Crystallization

In this chapter the crystallization model is derived. As mentioned in the introduction, thefocus is on the cooling rate dependent behavior. The aim is to formulate a mathematicalrelationship that can be used in a CFD environment, leading to a field that enablesthe evaluation of crystallinity locally. As the influence of shear on crystallization wasinvestigated, it is intended to include these findings as well, but not as a main feature.

To introduce the topic of phase change, an example based on a phase field method istreated. Here, the concept of phase change is explained w.r.t. its consideration in micro-scale simulations. A rather unconventional view, exclusively based on the interpretationof terms appearing in this method, is presented. Finally, spherulite growth is simulatedin a confined space using a simple model. The possibilities of application are describedand a reason why it is not intended to use a phase field method in this work is given.

Following, the focus is on macroscopic modeling. A well known phenomenological modelfor these kind of approaches is formally discussed for introduction. As it is complex toimplement the experimental findings in this model, an alternative of empirical characteris suggested. For deriving a cooling rate dependent crystallization evolution equation,first a model assuming a fixed cooling rate is developed. Thoroughly an introduction tothe identification of parameters is given. The model derived thereafter is a modificationof the fixed rate approach using parameter functions dependent on the cooling rate. Asthe range of cooling rates investigated was found out to be way too small to define thoseseriously, the experimental results from chapter 2.1.2 were extended by data from litera-ture. Finally, a possibility to extend this model for shear enhanced effects is presented.Based on a limiter, for all models it is prevented that instant melting occurs right aftercrystallization, as this would not be in compliance with polymer physics. Comments onnumerical implementation are given for each model, the focus is therefore not completelyon modeling. As the last point a possibility to evaluate the relative crystallinity based ontime integration of the latent heat flux is given.

27

28 Chapter 3. Modeling of Crystallization

3.1 Spherulite Growth Simulation

An approach very popular these days for all kinds of modeling tasks is the phase fieldmethod, see [54]. In the field of crystallization, it is however used more often for metals,e.g. in dendrite growth simulations [55]. One of the most cited works in this fields is [56].They are developed as far that even several levels of dendrite arms (primary, secondary,tertiary, ...) are to be found in simulations, which can on some level be seen as the pedantto lamellae in polymers. For polymers, the growth of lamellae can be considered as well,e.g. shown in [57]. But as spherulites are not as geometrically complex as dendrites, it isalso possible just describing the movement of the outer interface e.g. done in [58]1. Thedevelopment of a phase field method for crystallization is usually done based on a freeenergy functional for the development of the (crystallization-)surface, c.f. [56]. This leadsto a more or less complex scalar transport equation for the phase field. A description ofthe resulting equations in a fluid mechanical view point is given hereinafter.

3.1.1 Phase Field Modeling

For simulating crystallization using the phase-field method usually a phase field 0 ≤ χ ≤ 1is used. The value χ = 0 represents the fluid state, χ = 1 the solid which is transportedby the phase field equation. Independent on how it was derived, a condensation to

∂χ

∂t+ ∇ · χvχ = Sχ (3.1)

is always possible. In here, a phase velocity vχ and a source term Sχ are used. Sincethe phase should grow in its normal direction, it is formulated using the interface normalvector, calculated based on the gradient of χ. Growth is therefore set using

vχ = −Uχ(χ, ∇χ, θ, ∇θ, θ, v, ...)∇χ

|∇χ| . (3.2)

The normal velocity Uχ can be formulated in dependence of all fields possibly influencingit. For example, it could be increased in regions with high temperature gradients, or setto zero when a certain temperature is reached. To understand the role of Sχ it must beknown that χ is initially seeded with regions or cells set to χ = 1. Therefore, the phasefields surface is transported away from the seeds. In case Sχ = 0 this would lead to acontinuously decreasing value of χ in the center of the spherulite, as it would occur fordiffusion2. But as χ should describe a phase transition, this is not allowed and thereforeSχ will make sure that χ = 1 is reached again. As the required production rate is mostlydependent on the growth velocity

Sχ = Sχ(χ, Uχ) (3.3)

is a sufficient formulation. The dependence on χ is mainly to ensure that the sourcevanishes for χ = 0 and χ = 1 to ensure 0 ≤ χ ≤ 1. Hence it is only non-zero at the

1In contrast to the other works named, here the level set method is used.2Diffusion is actually just gradient proportional convection against the normal direction of a field.

3.1. Spherulite Growth Simulation 29

interface. Admittedly, this might be an unconventional way to describe the phase fieldmethods developed for crystallization. But this is what is left over of the best practiceapproaches and what is important for an implementation in CFD.

To consider the latent heat released during crystallization a source term is included in theenergy equation. In the static state this is done in the following way by most authors:

ρc∂θ

∂t= kΔθ + ρΔhcryst

∂χ

∂t. (3.4)

Heat is therefore released only in the transition zone. The amount is described by thecrystallization enthalpy Δhcryst, which is a material specific independent variable. Usingthese two equation enables to perform simple spherulite growth simulations. They allowto show what to expect from a phase field simulation.

3.1.2 2D Growth Simulation

The problem solved can be considered as the typical isothermal crystallization case. Forgrowth velocity, the value belonging to 10 K min−1 in fig. 2.18 was used. Thereafter, foran undercooling of 45 K, the growth speed is Uχ = 200 nm s−1. To allow the observationof multiple spherulites impinge, the domains dimensions were set to 40 x 80 μm.

00

1 8e-3

χ [-] Δθ [K]

Figure 3.1: Spherulite growth after 15 s. Phase field (left) and overheating Δθ w.r.t the wall(right). The spherulites initial diameter was 1.5 μm, at the time shown it is 7.5 μm.

As initial condition seven seeds with a radius of 750 nm were placed centrally. The tem-perature is set uniformly to 120 ◦C in the domain. For χ at all boundaries homogeneousNeumann conditions are applied, for temperature a fixed value of 120 ◦C is set. In fig. 3.1


a selected time of this simulation is shown. It can be seen, that all spherulites grow withthe same velocity, just as it was intended. Latent heat increases the temperature locally,however the magnitude is minimal. As the dimensions are small, this has to be the case,because heat conduction is strong. During growth the spherulites impinge. This allows tofollow all impingements over time, leading to a Voronoi pattern. In fig. 3.2 its temporalevolution is shown.

0 s 45 s30 s 95 s

Figure 3.2: Temporal progress of the impingements forming a Voronoi pattern.

It is obvious that more seeds will cause more impingements and therefore a finer struc-ture, all known effects mentioned in chapter 2.3.3 as the mechanism to lower crystallinity.Therefore, it is definitely possible to simulate crystallization processes at this scale. How-ever, the effort is way too large to be practicable. If the calculation performed hereshould be applied to a 2D channel with the height of 1 cm and a length of 10 cm alreadyaround 40 billion cells would be required. Obviously calculations like this are not practi-cal. However, the usage in a method based on representative elementary volumes as FE2

is conceivable. As this is not intended in this work, a macroscopic model is formulated.

3.2. Macroscopic Model 31

3.2 Macroscopic Model

As the macroscopic model developed in this work should include the suppression of crys-tallization, the first thing important to understand is how the latent heat source q of theenergy equation works. When considering crystallization in a general process,

ρcDθ

Dt= kΔθ + ρ

(Δhcryst

Dχ

Dt

)︸︷︷︸

q

. (3.5)

A material volume should now be considered during phase change. The initial value ofthis process is χ(t0) = 0, the final χ(t1) = 1. In this case the integration over space andtime leads to

mΔhcryst =∫ V

0

∫ t1

t0

(ρΔhcryst

dχ

dt

)dt dV. (3.6)

Therefore, by the phase transition exactly Δhcryst is released. This is true for any modelusing a constant value for Δhcryst and a span of χ(t1) − χ(t0) = 1. An example for apopular method is given hereinafter.

It is possible to describe crystallization statistically, based on expanding spheres. Awork summarizing the development in this field is the book of Janeschitz-Kriegl [59]. Asstated here, this theory goes back on the independent works of Kolmogorov [60], Avrami[61] and Evans [62]. Based on those also Schneider et al. [63] developed the popularrate equations3. They are based on a growth rate of spherulites G, dependent on theundercooling θu(θ) = θmelt − θ. Using a source term Sφ3 dependent on the nucleation rateN , the evolution of the number of spherulites per unit volume is described by

Dφ3

Dt= Sφ3(N). (3.7)

Coupled to the evolution of φ3 are the evolutions of volume specific radii (φ2), surfacearea (φ1) and volume of spherulites (φ0) by

Dφi

Dt= G(θu)φi+1, i = 0, 1, 2. (3.8)

The evolution of χ is described, e.g. including corrections for impingements in Sχ, by

Dχ

Dt= Sχ(φ0). (3.9)

Finally, in the energy equation (eq. (3.5)), the rate of change is used in

q = Δh0χ

relDχ

Dt, (3.10)

based on a fixed value of χrel < 1 to describe the final relative crystallinity. The crystal-

lization enthalpy can be chosen as Δh0 = Δhcryst(θ → 0). It is therefore conceivable that

3These equations are also used in the work of Zhao[13], to which the present work was compared toin the itroduction.


in this approach suppression of crystallization can be included. Either by choosing Sχ in away that χ will not reach one, or by setting χ

∞ to a predefined value, based on the coolingrate expected in the process. The expression that would include DSC measurements isG(θu). However, no work was found that tests this model for DSC measurements w.r.t.cooling rates that suppress crystallization. Regarding that Fast Scanning Calorimetry[64] just came up in the last decade, this might still follow. However, as this model canbe described as quite complex regarding the many parameters and functions used, this issomething the present work will not provide.

As it is the aim to directly derive an evolution equation from the DSC measurements, thedeveloped model has to be classified as empirical. For this purpose, the DSC measure-ments are transformed into curves that represent the phase transition. A similar approachhas already been presented in [65], based on the Nakamura-Ziabicki [66, 67] model. Therange of cooling rates included were however much smaller.

3.2.1 Fixed Cooling Rate Model

As the simplest model, a fixed rate approach can be chosen. It assumes the crystallizationto develop at a predefined cooling rate, which can e.g. be the industrial standard 10K min−1. This means crystallization will always occur at the same temperature rangeand the same amount of crystallization enthalpy is released4. As starting point for theformulation of an evolution equation for χ, an expression that can be found in the worksof Andrej Ziabicki, e.g. [68] is used. It is a Gaussian bell function close to eq. (3.13), ofwhich the integral is

χ(θ) =12

⎡⎣1 + erf

(2√

ln 2(θm − θ)D1/2

)⎤⎦. (3.11)

The fluid solid transition is hereby described as an error function with the diameter of2D1/2 that is 50% completed at θm. As it is exclusively dependent on temperature theresulting evolution equation is

Dχ(θ)Dt

=dχ

dθ

Dθ

Dt. (3.12)

The first multiplier can be calculated analytically, which results in

dχ(θ)dθ

= − 2D1/2

√ln 2π

exp

⎛⎝−4ln 2

(θm − θ)2

D21/2

⎞⎠ , (3.13)

whereas the second multiplier is calculated in the simulation by solving eq. (3.5). Basedon eq. (3.5) it is possible to define a maximum latent heat rate qmax = q(θm) that is usefulfor parameter identification. Inserting its value in

Δhcryst = − qmax

θ

D1/2

2

√π

ln 2(3.14)

4Hence no information about crystallinity can be gathered as well, as it is always the value obtainedat the reference rate.


enables to determine the crystallization enthalpy easily. As mentioned eq. (3.13) describesa Gaussian bell curve, which is also known to occur in a DSC. This means qmax representsthe measured peak value. It can directly be read from the heat flux plot in the followingfigure on the left.

0 0110 110115 115120 120125 125130 130

20

40

60

80

100

1000

2000

3000

4000ModelModel

ExperimentExperiment

θ [◦C] θ [◦C]

q[W

/kg]

Δh

cryst

[kJ/

kg]

D1/2

D1/2

θmθm

qmax Δhcryst

Δhcryst/2

Figure 3.3: Possibilities of parameter identification at -10 K min−1.

Of course this also applies to D1/2 and θm. However, there is a more elegant way, that willdirectly lead to Δhcryst, based on integration of the heat flux. Dividing the integral bythe cooling rate leads to Δhcryst(θ) as shown in fig. 3.3 right. In this plot all parameterscan be determined as well.

500

0 0110 110115 115120 120125 125130 130

20

40

60

80

100

1000

1500

2000

2500

3000

3500

Model Model

Experiment Experiment

θ [◦C]θ [◦C]

q[W

/kg]

Δh

cryst

[kJ/

kg]

Figure 3.4: Parameter identification for measurements showing artefacts.

Ultimately it possible to use both methods. However, it must be noted that integratingthe heat flux will smooth the data. Therefore, even data that shows artefacts can easily


be processed as shown in fig. 3.4. Here the data of fig. 2.5 is used to clarify this property.As can be seen the artefact occurring in q(θ) disappears in Δhcryst(θ). Just the exact formof the error function is not met in the second half. But this does not cause problems inthe process of parameter identification. The parameters used in this work were thereforeidentified based on the integrated data. For 10 K min−1 e.g. in this way D1/2 = 3.2 K,θm = 394.15 K and Δhcryst = 93594 J kg−1 was determined.

Summarizing, one additional transport equation for χ has to be solved, given by eq. (3.12).But as it is defined using the temperature rate, it is not only valid for cooling processes.Therefore, when heating occurs during the crystallization process, this will cause χ todecrease. It can be caused by the latent heat release and dissipative heating combinedwith insufficient cooling. Independent on the cause, using eq. (3.12) will therefore violatea very important principle: this is, as can be seen in fig. 2.3, just crystallized regions willnot melt, because melting occurs at least ca. 25 ◦C above the crystallization temperature.It is caused by the polymers being trapped by van der Waals forces, c.f. chapter 2.3. Abond that, if once established, can only be broken by adding sufficient kinetic (= thermal)energy. The new evolution equation is therefore

Dχ(θ)Dt

= max

(0,

dχ

dθ

Dθ

Dt

). (3.15)

A property to consider for this equation is, if due to discretization an overshooting of χ

occurs, it will not decrease to χ = 1, as it would without limiter. It can be seen in theinvestigations presented in chapter 6.2.5. In this case too much latent heat is released.

3.2.2 Cooling Rate Dependent Model

The consideration of cooling rate dependent effects is done replacing the parameters ofeq. (3.11) with functions of θ. It is therefore redefined to

χ(θ, θ) =12

⎡⎣1 + erf

(2√

ln 2(θm(θ) − θ)

D1/2(θ)

)⎤⎦. (3.16)

As shown in fig. 2.6, in all functions, there is a great dependence on the cooling rate.Because of that, the investigated rage is way too small to formulate a meaningful model.Measurements presented in literature were therefore used to extend the data. As it waspossible to identify PP 575P as a common iPP in chapter 2.3, it was also possible tofind a sufficient number of publications focusing on high cooling rates. Using a programto retrieve data points from figures5, the presented data was digitized. If the heat fluxwas given, a baseline correction as described in chapter 2.1.2 was executed, then theparameters were identified. In some papers the parameters were already plotted directlyin dependence of the cooling rate, which made the procedure much easier. Fig. 3.5 showsthe data obtained in this way in comparison with own measurements. Parameter functionsare already included as well, commented thereafter.

5WebPlotDigitizer


1e-3 1e-2 1e-1 1e0 1e1 1e2 1e3 1e40

0

5

1015

20

20

20

253035

40

40

40

60

60

80

80

100

100

120

120

140

140

160

θ m[◦

C]

D1/2

[K]

Δh

cryst

[kJ

kg−

1]

-θ [K s−1]

Istrate [69]Own exp.De Santis [70] Rohades [71]

Schawe [72] Steinmann [73]Mileva [27] Model

Figure 3.5: Extended DSC data used for the definition of parameter functions.

The data for θm shows that it is continuously decreasing over the cooling rate. Someauthors provided this plot directly, as it is common to evaluate the function θm(−θ) inmaterial science. Because the change in θm occurs over several decades, the power law

θm(θ) = θm,0

(1 − (aθm

θ)nθm

)(3.17)

was found out to fit best. For vanishing rates, it reaches a constant value, for ratesexceeding the data it extrapolates in a useful way. The parameters are given in thefollowing table.

Table 3.1: Parameters for eq. (3.17)

θm,0 [K] aθm[K−1 s] nθm

[-]

416.15 -1.66 0.15


The determination of D1/2 is difficult for the range crystallization is starting to get sup-pressed. Here the peak in heat flux flattens out which makes it, at some extent, hard todifferentiate from the base line. The work of Schawe [72] however, allowed to make a cleardifferentiation of bell curve and base signal. This is why a Cross-like approach (c.f. [74])reaching a limit value was chosen, describes by

D1/2(θ) =D1/2,0 − D1/2,∞

1 + (kDθ)nD

+ D1/2,∞. (3.18)

As in own experimental data an increase of D1/2 might show too early, the literature datawas weighted stronger. Only nD was determined using least squares, all other parameterswere set by the eye as they have an interpretable meaning. The parameter values todescribe the plotted curve, are named in the following table.

Table 3.2: Parameters for eq. (3.18)

D1/2,0 [K] D1/2,∞ [K] kD [K−1 s] nD [-]

3.2 40 -0.047 0.95

Investigating crystallinity over the cooling rate can be considered as a standard procedurethese days. It is enabled by flash or nano calorimetry, for which one of the first machines isdescribed in [70] and a comprehensive description w.r.t. modern application is describedin [75]. Commercial equipment, such as the Mettler Toledo Flash DSC 1 allows to performremarkable experiments, as those published by Schawe in [72], who also proposed a modelfor Δhcryst(θ) in [76]. It consists of functions for generic crystallinity g(θ) and retardationv(θ), of which the latter is introducing suppression of crystallization. They are

g(θ) = 1 − g1log10(−θ) − g2log10(−θ)2 ∧ v(θ) = (1 + ev1log(−θ/v2)). (3.19)

But as for vanishing cooling rates, g(θ) was found out to turn negative, this function wasreplaced by a power-law. The modified approach of Schawe is therefore

Δhcryst(θ) = Δh01

1 + (g1θ)g2︸︷︷︸g(θ)

1

1 + ev1log10(θ/v2)︸︷︷︸v(θ)

. (3.20)

For parameter identification, only the data of Schawe was used in the range crystallinityis suppressed. The parameters identified are given in the following table.

Table 3.3: Parameters for eq. (3.20).

Δh0 [J kg−1] g1 [K−1 s] g2 [-] v1 [-] v2 [K s−1]

115000 -1.7744e-3 0.29 8 250


From this table, v2 can be interpreted as the characteristic cooling rate for PP 575P. Infig. 3.5 it can be seen, v2 represents a rate that is approximately in the center of theregion, suppression of crystallization is strong.

In all models, if a product with the cooling rate occurred, the corresponding parameteris negative. This considers the fact, that θ is negative in cooling processes. However, incase heating occurs locally, this will lead to a NaN-exception6 in the code. It is preventedby limiting θ to values smaller than -1e-6 K s−1.

As an additional dependency was introduced, the evolution equation changes. Comparedto the fixed rate models evolution equation (eq. (3.12)), a term considering changes re-garding to θ occurs, as can be seen in the following equation:

Dχ(θ, θ)Dt

=dχ

dθ

∣∣∣∣∣θ

θ +dχ

dθ

∣∣∣∣∣θ

θ. (3.21)

This makes calculating the second material time derivative of temperature necessary.According to [77] the expanded version of this expression is

θ =D2θ

Dt2=

∂2θ

∂t2+

Dv

Dt

∂θ

∂x+ 2v

∂θ

∂x∂t+ v2 ∂2θ

∂x2, (3.22)

using ∂/∂x as notation for the gradient. Regarding the numerical implementation, thereshould be no problems, as the following comments will state.

The local second time derivative is by default available in most codes. For the second term,instead of interpolating the velocity on the faces to obtain a flux, here the accelerationmust be used straightforward. The second term can therefore be treated with standardconvection procedures. The third term opens questions, as it is a mixed derivative, whichis not standard in CFD. Schwarz’s theorem says that derivatives can be executed inarbitrary order. This means, either boundary conditions are needed for the local timederivative of temperature. Or the gradient needs to exist is as an additional field to beable to calculate its time derivative. By just setting a homogeneous Neumann boundarycondition for ∂θ/∂t or adding ∂θ/∂x as a field this can be solved. This would however involvehaving to test both cases and also perform a verification. The last term is diffusive anduses v2 as a coefficient, which can be treated with standard procedures.

Regarding the need of studies for implementation of this term, that is certainly interestingfrom a continuum mechanical and numerical standpoint, it was neglected. A comparisonfor each calculation to be performed would be needed, which is seen as not expedient,even if only in this way the model is formally correct. Summarizing, the rate dependentmodel is just eq. (3.15) with rate dependent coefficients.

6"Not a Number" exceptions occur if the result of an operation is complex.


3.2.3 Shear Rate Dependent Model

The strategy to extend the cooling rate dependent model by a dependence on the shearrate is the same as applied before. From the experimental results shown in fig. 2.11, acorrection of parameters is derived. However, because in the experiments only the onsettemperature could be determined, there is no unique way of consideration. As

θonset = θm + D1/2, (3.23)

it is impossible to determine if a change in θm or D1/2 caused an increase. Non-isothermalmeasurements presented in [78], performed in a transparent flow channel, showed thatthey both depend on the deformation history and therefore also the shear rate. Hence itwas decided to make just θm shear rate dependent and as a consequence, see this approachas a test if a model like this is promising if the missing information is available.

Because the results in fig. 2.11 suggest a limit value, that is also in line with findings fromliterature as discussed in chapter 2.3.4, the correction of θm is modeled7 by

Δθm(γ) =Δθ∞

2

(tanh (aΔθγ + bΔθ) + 1

). (3.24)

A comparison with the measurement depicted in fig. 3.6 shows good agreement with thedata above γ = 1 s−1. The slight decrease for shear rates below is neglected. Whenimplementing the correction into the cooling rate dependent model, this results in

θm(θ, γ) = θm(θ) + Δθm(γ). (3.25)

As deformations in general are known to enhance crystallization8 due to increased nucle-ation, it makes sense to use a corrected cooling rate. It is reduced to a virtual value, thatwould correspond to the new value of θm. This is done rearranging eq. (3.17) which leadsto the case discrimination

θcorr =

⎧⎪⎨⎪⎩

0, if θm(θ,γ)θm,0

> 1

1aθm

(1 − θm(θ,γ)

θm,0

)1/nθm

, otherwise(3.26)

as θm(θ, γ) was modeled using an upper limit value for vanishing rates. The last step isa correction of crystallization enthalpy using the new cooling rate, therefore

Δhcryst = Δhcryst(θcorr). (3.27)

The effect these modifications have on the crystallization behavior can be seen in fig. 3.6.As intended, the top right plot shows that θm(θ) is simply increased with the shear rateby a constant offset over the cooling rate. In the bottom left graph, the corrected coolingrate is plotted. Since for the quiescent case θcorr = θ and by shearing θcorr is reduced,all corrected rates are below the curve for γ = 0 s−1. The aim of this was to shift thecrystallinity curve to higher cooling rates, which can be seen on the bottom right.

7The parameters used are: Δθ∞ = 19.9 K, aΔθ = 0.54 s and bΔθ = −2.2.8Enhancement of crystallization can also be reached using nucleation agents. They ensure obtaining

high crystallinity even at high cooling rates, as investigated by Schawe in [79]. A result was that thecharacteristic cooling rate (v2 in eq. (3.20)) could be increased by the factor of ten.


1e-4

1e-41e-4

1e-4

1e-2

1e-2 1e-2

1e-2

1e0

1e0

1e0 1e01e2 1e2

1e2

1e2

1e4 1e4

1e4

1e4

0

0

0

2 4

5

6 8 10

10

12 14

15

20

20

20

40

40

60

60

80

80

100

100

120

120

140

140

160

θ m[◦

C]

Δh

cryst

[kJ

kg−

1]

-θ [K s−1]

-θ [K s−1]-θ [K s−1]

-θco

rr

[Ks−

1]

γ [s−1]

ExperimentModel

Δθ

[K]

γ = 0 s−1

γ = 0 s−1

γ = 0 s−1

γ = 3 s−1

γ = 3 s−1

γ = 3 s−1

γ = 4 s−1

γ = 4 s−1

γ = 4 s−1

γ = 5 s−1

γ = 5 s−1

γ = 5 s−1

γ = 15 s−1

γ = 15 s−1

γ = 15 s−1

Figure 3.6: Influence on the shear rate extension on the crystallization behavior.

In comparison to the cooling rate dependent model in eq. (3.21), adding this dependencewill introduce a third term. It considers the change of the phase indicator in relation tochanges in the shear rate. The full evolution equation is therefore

Dχ(θ, θ, γ)Dt

=dχ

dθ

∣∣∣∣∣θ,γ

θ +dχ

dθ

∣∣∣∣∣θ,γ

θ +dχ

dγ

∣∣∣∣∣θ,θ

Dγ

Dt. (3.28)

As stated in the previous chapter, in principle it is possible to implement the secondterm. For the last term, it is possible to calculate the local time derivative. Whencalculating the convection term of the substantial time derivative, it is possible to sethomogeneous Neumann boundary conditions. Hence implementing this term is possible,but also extensive testing would make sense in this case. As it is not intended to dothis for this model as well, it will be neglected. This means, analogous to the coolingrate dependent model, just the parameters of eq. (3.12) are made shear and cooling ratedependent, the implementation is therefore done using eq. (3.15).


3.2.4 Calculation of the Local Relative Crystallinity

All the work put into the formulation of the cooling and shear rate dependent modelswould be for nothing if local crystallinity is not evaluated. Just as for the evaluation ofthe DSC experiments, it can be obtained by integrating the heat flux over time. Theresult is then divided by the highest possible value of the crystallization enthalpy. In caseof the models developed, this is the parameter Δh0 from eq. (3.20). The integration hasto be executed during the simulation by solving

DQ

Dt= Δhcryst(θ)

Dχ

Dt. (3.29)

Besides having to solve an equation system, because the left side is treated implicit, theadditional effort created by this is very limited. For evaluation, the relative crystallinity

cr =Q

Δh0

(3.30)

can be used as a scalar field. As already stated, this evaluation only makes sense forthe rate dependent models. For the fixed rate model, always the crystallization enthalpybelonging the reference cooling rate is released. An example of this behavior is given inchapter 6.2.2.

CHAPTER 4

Rheological Modeling

The chapter on rheological modeling includes three parts directly related to constitutivemodeling of the materials behavior. One each for fluid and solid behavior, one for modelingof the fluid-solid transition. Its purpose is to document the process of finding the approachused and to point out open and new problems. Its purpose is not to name all kinds ofpossibilities there are for modeling. Therefore, very specific problems are addressed. Foran universal introduction into the field of non-Newtonian fluids, the two books of Bird,Armstrong and Hassager [11, 12] are generally worth studying.

Aim of the first subchapter is to define a non-isothermal non-linear viscoelastic model forPP 575P. As the first part therefore the solvent contribution is determined. Then a modelis chosen from common approaches, based on elongation behavior. To implement temper-ature dependency, parameters are modified as functions. The second subchapter focuseson the modeling of the solid. Different possibilities are generally discussed for introduc-tion. A simple numerical approach is described and investigated. Since this approach hastoo many disadvantages a constitutive alternative is presented and investigated for suitedparametrization analytically. As closing of the chapters treating constitutive equations,the approach for fluid solid transformation is described.

Because in flows of polymer melts it is important to consider viscous dissipation, focusis put on this topic. It was found out that negative dissipation commonly occurs in vis-coelastic models, which was reduced using a correction from literature. To find parametersfor this approach, a simple example was considered analytically.

As the last part of this chapter, the phenomenon of wall slip is highlighted. In contextof extrusion tools, it plays an important role. But as hereby transient phenomena canbe induced, it is excluded from this work. A short introduction based on an example isgiven, to justify this decision.

41

42 Chapter 4. Rheological Modeling

4.1 Modeling of the Molten State

The molten state is described by the Elastic-Viscous-Split-Stress (EVSS) approach. Itstates that the stress is the sum of an elastic and viscous contribution [80]. Hereafter, theelastic part is called polymer (index p), the viscous solvent (index s), thus

T = Tp + Ts. (4.1)

The polymer part is modeled using a Maxwell-type evolution equation, as described inthe following subchapter. For the solvent part the Newtonian model

Ts = 2ηsD (4.2)

based on the solvent viscosity ηs is used. Independent on how the polymer part is modeled,a polymer viscosity ηp occurs. Therefore, it is common to define a solvent contribution

β =ηs

ηs + ηp(4.3)

that defines the influence of the solvent part on the zero viscosity η0 = ηs + ηp. It canactually be determined from experiments for a material. Even though in publications ofnumerical nature, it might seem that for certain models values of β have established. Thereason for this might be the dampening effect which benefits numerical stability. However,if the aim is to model a fluid, this is no valid argument since β has a large influence onthe evolution of stresses. This will generally even influence steady state solutions. A wayto identify β is by performing shear experiments over a wide rate-band. Then, from theviscosity curve the zero and infinite viscosity

limγ→0

η(γ) = η0, limγ→∞

η(γ) = η∞ (4.4)

have to be determined. The relation of the infinite viscosity to the zero viscosity is thento be chosen as the solvent contribution, thus

β = η∞/η0. (4.5)

However, measuring η∞ for polymer melts is a challenging task, e.g. described in [81]. Itwas not possible with own equipment, a literature study on iPP was therefore conducted.But as this is a very special topic, only two publications investigating similar materialswere found, named in the following table.

Table 4.1: Solvent contributions from literature for iPP

Source θ [◦C] η0 [Pa s] η∞ [Pa s] β [-] Material

[82] 200 700-1100 0.6-0.8 0.07-0.09% HP561R, 100-GA12

[83] 200 2900 0.5-0.8 0.02-0.03% 100-SA 12

4.1. Modeling of the Molten State 43

In [82] the investigated PP is described as isostatic, in [83] the addition homopolymeris made. As PP 575P is described as a homopolymer as well, c.f. [17], this publicationis also named. Even though the measured zero viscosities are admittedly different, bothworks show that the solvent contribution is neglectable. For higher temperatures thismight change, but these cases are irrelevant for this work. As a result, the solvent partwill be neglected, this means β = 0.

4.1.1 Viscoelastic Model

For modeling the behavior of PP 575P, a generalized version of the Maxwell model

τ + λτ = ηγ (4.6)

is used. To generalize it, the shear stress τ is replaced by the stress tensor Tp, an objectivetime derivative denoted with (∇) instead of (·) is inserted and the right side is replacedby the generalized Newtonian law. This leads to

Tp + λ�

Tp = 2ηpD,�

Tp =DTp

Dt− LTp − TpL

T . (4.7)

Stand alone it is called Upper-Convected Maxwell Model [12], because of the usage ofan upper convected objective time derivative. As the time derivative of the Cauchystress tensor is not objective, the terms −LTp − TpL

T are added to restore it, see e.g.[84]. From a continuum mechanical standpoint, the model is hereby independent of thecoordinate system it is used in. But this doesn’t mean that it physically makes sense, asthe rheological material functions

η(γ) = ηp + ηs, N1(γ) = 2ηλγ2, ηE(ε) =3ηp

(1 − 2λε)(1 + λε)+ 3ηs (4.8)

show. It is uncritical that the shear viscosity is constant and the first normal stressdifference increases quadratically, as this is the actual behavior of polymeric liquids forlow shear rates [85]. That the elongation viscosity is three times the shear viscosity forlow elongation rates as well, since this is known as Trouton’s-ratio [86]. However, that ηE

has a pole at ε = 1/2λ and after that approaches zero from negative values, c.f. [87], hasto be seen very critical. This means a certain level of solvent contribution is needed to fixthis flaw. An exact value is arguable since just after the pole formally infinite negativevalues for ηE occur. In practice, meaning in the application of the Oldroyd-B1 model innumerical simulations, values of β ≈ 0.5 have established.

There are many fixes possible to the problem in elongation behavior that will also intro-duce shear thinning behavior. Some introduce additional transport equations [88], othersjust modify eq. (4.7) with additional terms. The latter approach is followed in this work,described in the following subchapter.

1The Oldroyd-B model is the upper-convected Maxwell model used with β > 0.


4.1.2 Nonlinear Extensions

This subchapter exists to describe the procedure of finding a suitable non-linear exten-sion of the upper convected Maxwell model for PP 575P. The extension fixing defects ofeq. (4.7) is based on an additional term, changing the model to

Tp + λ�

Tp +(n)

Q(Tp)Tp = 2ηpD. (4.9)

A tensorial function Q of order zero or two in dependence of Tp is therefore used, pursuingthat the additional term vanishes at least quadratically for small stresses [89]. Followingthe practice in solid mechanics, one might think that this function is just formulated independence on Tp or its invariants. Just like e.g. for yield-criteria [90], but in contrastto introduce the phenomena typical for polymer melts. However, this is not directly thecase, even though the outcome is similar.

All approaches presented hereafter are based on a molecular theory [91]. It describes thepolymers molecules statistically averaged as beads connected by springs, often referred toas the dumbbell-model. The viscous part is hereby introduced by the drag acting on thebeads due to their movement, e.g. modeled by Stokes drag. The elastic part that resultsout of the relative motion between the beads to each other, e.g. by the Hookean model.If the molecules are deformed, an anisotropic mobility is introduced, described by Q. Forselected models it is named in the following table.

Table 4.2: Nonlinear extensions of the upper-convected Maxwell model.

Giesekus [92] LPTT [93] EPTT [94]

(n)

Q αλ

ηp

Tp αλ

ηp

tr(Tp) e

(α λ

ηptr(Tp)

)− 1

α (0, 1] > 0 > 0

The extension of Giesekus [92] is exactly based on the considerations stated above, leadingto a quadratic extra term. Phan-Thien and Tanner additionally consider destruction andcreation rates of junctions in the network, leading to a mobility linear [93] or exponential[94] in Tp. All models are stated to be used as a multimode, meaning that several elementswith different parameters are connected in parallel. This means, for each element eq. (4.9)has to be solved, which consist out of six scalar equations. The overall stress is thencalculated by the sum of all elements in the sense of

Tp =n∑

i=1

Tp,i. (4.10)

Referring to, that a polymer usually has a certain molecular weight distribution, which isan indicator for chain length [95], this approach is reasonable. However, it is not possibleto determine which extension to choose, how many modes are needed or which values the


parameters should have, exclusively from a deeper knowledge about the molecules. Allthis has to be done based on experiments, essentially reducing this quite elegant theory toa very unpleasant problem of parameter identification, since the equations are generallynot explicitly solvable for their rheological functions. Especially if multiple modes areused, which is necessary for an accurate description of measurements, the effort is great.This is why the present work aims to use one mode that should mimic the materialsbehavior. Therefore, a suitable single mode had to be found.

As can be seen in fig. 5.7, all extensions introduce shear thinning behavior at λγ = 1,which also lets the slope of N1(γ) decrease. In elongation behavior however, there is amajor difference. The Giesekus and LPTT model are strain hardening, the EPTT modelis strain hardening in the region of λγ ≈ 1 and strain softening hereafter. To identify thesuited extension, elongation experiments are therefore essential.

1e21e21e2

1e2

1e01e01e0

1e0

1e-21e-21e-2

1e-2

ηP,SηP,SηP,S

ηP,EηP,EηP,E

N1N1N1 Giesekus EPTTLPTT

λγ, λε [-]λγ, λε [-]λγ, λε [-]

η P,S

/ηp,

η P,E

/ηp,

N1λ

/ηP

[-]

Figure 4.1: Comparison of the models in tab. 4.2. The shear behavior is characterized by theshear viscosity ηP,S and the first normal stress difference, both determined for planar shearflow. Elongation behavior is described by the elongation viscosity ηP,E , determined for uniaxialelongation. The abscissa shows the shear and elongation rate, γ resp. ε. Detailed explanationson how these properties are derived from the models and how they influence flows are given in[11] and [12]. The upper convected Maxwell model is plotted in black for comparison.

Own elongation experiments were not performed as PP 575P can be assumed to be alinear-PP (l-PP) and therefore has a specific elongation behavior. The reason for this is,there is another prominent type of PP that is specially treated after polymerization toobtain long branches in the polymer. These Long Chain Branched PPs (lcb-PP) characteris that they have a great melt strength [96] to enable manufacturing processes that pulloff the melt. In terms of the elongation viscosity, this means in l-PP the strain hardeningis weaker than in lcb-PP. As lcb-PPs are special purpose polymers, that are also marketedas such, PP 575P was assumed to be linear as this does not apply to it.

Independent on if PP is lcb or linear, its elongation behavior is of the EPTT type, as allpublications named in tab. 4.3 show. The only thing remaining open is the amount ofstrain hardening. To quantify strain hardening, it is possible to define the strain hardeningratio SHR = max(ηE(ε))/3η0. It was calculated for all experimental data sets presentedin the publications named in tab. 4.3.


Table 4.3: Literature values of the strain hardening ratio (SHR) for different linear and longchain branched polypropylenes.([97] measured at 190 ◦C all others at 180 ◦C.)

Sourcel-PP lcb-PP

SHR Sample SHR Sample

Bernnat [97] 1.2 Novolen 1100H 20 non-commercial [98]

Kamleitner [99] 1 HA 104E, HC 600TF, 3.75 mod. of the lin. samples

HD 601CF, HF 700 SA

Münstedt [100] 1 Novolen PP H2150 5 mod. of the lin. sample

Resch [101] 1.1 Moplen HP 556E 20 Profax PF814

To perform elongation experiments wide above the melting point is impossible for thevast majority of materials. That’s why the testing temperature in most of the literaturesources is 180 ◦C. As the temperature range relevant for this work is in this range andbelow, this is not an issue. A comparison between the data sheets of the materials namedin tab. 4.3 showed that the material investigated in [97] is the closest to PP 575P, thereforeSHR= 1.2 is assumed. To find the corresponding value of α the EPTT model was solvedfor its elongation viscosity function ηE(ε) semi-analytically as described in appendix A,which leads to the following graph for its SHR.

1e2

1e1

1e01e01e-11e-21e-3

α [-]

SH

R[-

]

Figure 4.2: Strain hardening ratio of the exponential Phan-Thien Tanner model.

Since all the extensions named in tab. 4.2 vanish for α = 0 the SHR is singular for α → 0as then, the upper convected Maxwell models defect occurs. But as the SHR for linear-PP is low, this range of values is not relevant. From fig. 4.2 the value of α = 0.5 wasdetermined for SHR ≈ 1.2. The next step is to fit the model to shear data since α has aminor influence on the start of transition to shear thinning behavior as well.


4.1.3 Thermorheological Model

As the anisotropic mobility α is known for PP 575P, in this subchapter η0 and λ aredetermined. But as the model will be used in non-isothermal processes, they need to bedefined as functions of temperature. As foundation of this thermorheological model, alldata presented in chapter 2.2.2 and steady state experiments, especially performed forthis purpose, are used. The steady state shear experiments were evaluated as specified forthe application of the Time-Temperature Superposition (TTS), c.f. the regarding chapterin [102]. Therefore, for each temperature the viscosity data points are divided by theidentified value of η0. All shear rate data points are multiplied with the identified valueof λ. By this procedure a master curve can be assembled as shown in the following figure.

1e31e2

1e1

1e1

1e0

1e0

1e-1

1e-1

1e-2

1e-21e-3

1e-31e-4

η/η

0(θ

)[-

]

λ(θ)γ [-]

140 ◦C155 ◦C170 ◦C185 ◦C200 ◦C

measurements at:

EPTT

Figure 4.3: Fitting of the EPTT model to the master curve of PP 575P.

In comparison, the master curve of the EPTT model is plotted. This shows the modelhas a deviation in the transition regime from Newtonian to shear thinning behavior. Itrepresents the major downside of Maxwell-type equations, used in single mode. They havea fixed transition behavior and a fixed slope in the shear thinning regime. That the slopein the shear thinning regime fits in this case, is just coincidence. It could also be, that ifmeasurements are performed for higher shear rates, a deviation will appear in this region.A different approach might be, to just shift the models curve to lower shear rates, byusing an offset for λ. This might lead to a smaller overall deviation, but would introducestrong shear thinning at way too small rates. The only possible way to heal this defect forsingle modes, is to use a different model that allows changing the transition. Of course, asstated before, the EPTT model could be used as a multi-mode, but as this increases theeffort in modeling and simulation drastically, it was not realized. Using flexible empiricalapproaches like the Carreau-Yasuda model [103], fitting those data would be no problemas they are specially designed for this task. However, they do not show normal stressdifferences and stress relaxation.


The basis for parameter identification of the shifting function, in this case chosen as

φArr(θ) = exp[E0

R0

(1θ

− 1θ0

)], (4.11)

known as the Arrhenius [104] approach, is to plot the parameters used to normalize theviscosity curve over temperature. But as for η and λ the same shifting function is used, itmakes sense to normalize these values again, here to the corresponding value at 155 ◦C.Hence at this temperature φArr = 1, as the following figure shows.

400 410 420 430 440 450 460 470 480

130 140 150 160 170 180 190 200

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

θ [K], θ [◦C]

cooling experiments

steady state experiments

Arrhenius

10 K min−1

15 K min−1

2.5 K min−1

obtained from η0(θ)

obtained from λ(θ)

φA

rr

Figure 4.4: Relative shifting function used for time temperature superposition.

As some of the cooling experiments of chapter 2.2.2 were performed at a low shear rate(γ = 0.01 s−1), they can be used for parameter identification too. They are normalizedat 155 ◦C as well, which leads to a great density of data points that fills the space inbetween the values obtained from the steady state experiments. However, in the regionjust below 200 ◦C the point density is a little lower, as the thermal lag region describedin chapter 2.2.1 was cut. But as the good overall agreement of φArr(θ) shows, this did notcause any problems in the identification, performed with the least squares method. Thefunction is then used in a simulation to explicitly adjust the parameters. For the sake ofcompleteness, the solvent term is also included in this equation. The thermorheologicalmodification of the EPTT model used in this work is therefore

Ts = 2βη0(θ)D ∧ λ(θ)�

Tp +[e

α(1−β)

λ(θ)η0(θ)

tr(Tp)]

Tp = 2(1 − β)η0(θ)D, (4.12)

but as the introduction (c.f. tab. 4.1) showed, with β = 0. In these equations, based oneq. (4.11), the temperature shifted parameters

λ(θ) = λrefφArr(θ) ∧ η0(θ) = η0,refφArr(θ) (4.13)

are in use. The corresponding parameters are given in the following table.

4.2. Modeling of the Solid State 49

Table 4.4: Parameters for time temperature superposition.

θref [K] η0,ref [Pa s] λref [s] E [J mol−1] R [J K−1 mol−1]

428.15 5800 0.16 26000 8.31

The concept of TSS assumes all characteristic times undergo the same change in depen-dence of temperature. To consequently apply it, α should be made temperature dependenttoo. But out of unknown reasons, this is not commonly done in literature. Not even in afundamental work regarding to non-isothermal viscoelastic fluids [105], focused more onin chapter 4.4. Therefore, these dependencies are not considered in this work as well.

If α should be made temperature depended a multiplication with φArr(θ) would not makesense. As intermolecular forces correlate to free volume, explained in chapter 2.3, fordecreasing temperature they will increase. But when increasing α, shown in fig. 4.2,the strain hardening ratio will be decreased. Concerning that in uniaxial elongation flowsintermolecular forces play an even bigger role than in all other flows, see chapter 2.3.4, thiscan not be physical. Therefore, one approach could be to just use α(θ) = αref/φArr(θ).

4.2 Modeling of the Solid State

As the measurements in chapter 2.2.2 showed, solidification is causing a singularity inviscosity measurements. The oscillation experiment revealed that by solidification, thedominance of elastic behavior is increasing singularly as well. Just as it is expectedfor a solid that undergoes small deformations. To correctly describe this behavior, themomentum equation must be formulated in deformations and in Lagrangian specification.Something that is the opposite of a CFD framework, which is based on velocities and theEulerian specification. A solution could be to apply Fluid Structure Interaction (FSI)based on a fluid solid indicator that divides the regions. This requires constant remeshingto have a clear interface at which the coupling conditions can be satisfied. Surely thiswould be the most promising approach, but it is not a state of the art procedure in CFDwith the Finite Volume Method. Regarding approaches like this, the development inFinite Element Method codes is much further, most certainly because many developmentsof the much larger solid mechanics community find their way in. As an example, it is bestpractice in FEM to solve multiphase flows based on meshes that follow the interface as e.g.described in [106]. In FVM however, these problems are solved best using GeneralizedInternal Boundaries (GIB) [107], a method aligning the mesh locally with the interface.But as this approach is not yet available in OpenFoam, it is not possible to use it. Apartfrom implementing an FSI method capable of using GIB is a very complex matter, itwould be very costly to solve such a problem. Therefore, methods to be used in standardsolvers were investigated.


4.2.1 Discontinuous Numerical Approach

To model solidification numerically is an easy but inflexible approach. The aim is to forcefixed values of velocity in solidified cells. If processes as form filling should be simulated,e.g. v = 0 can be set. Therefore, the coefficient matrix created for the momentumequation is manipulated before using it in pressure correction. Its general form is

aP vP +∑N

aN vN = rP , (4.14)

which means cell (P ) and neighboring (N) values of velocity are multiplied with theircorresponding weights a to form a matrix and a vector of unknowns. The right side isformed by the equations source terms, the applied discretization schemes and boundaryconditions. A fixed value vfix can be obtained by the application of two steps for thedesired cell. Firstly, all the neighboring correspondences aN must be set to zero. Secondly,the right side is set to aP vfix, thus the row entry is

aP vP = aP vfix, (4.15)

to which the solution is vP = vfix. Even if this approach seems very elegant from anumerical standpoint, it does not allow a smooth fluid-solid transition as shown in thefollowing figure.

Figure 4.5: Example of cells to be set to v = 0 in a solidifying channel flow.

This method works binary. Above a certain value of the phase indicator, the velocityis set. This can only be done discretely, resulting in non-smooth surfaces containingsteps. For temporal progress the same effect occurs, meaning that from one time step toanother a cell completely solidifies in the rheological sense. In every case, this leads tojumps in global quantities over time. Another downside is, that special velocity boundaryconditions can not be applied to the solidified region any more, as the cell value does notdepend on boundary conditions any more. Considering these properties, it was decidedto use a continuous approach.

4.2.2 Continuous Empirical Approach

The idea behind this approach is to assign a large viscosity to solidified regions and tomodel the transition smoothly. It is demanded that, just as for a solid, a no-slip condition

4.2. Modeling of the Solid State 51

is realizable at the interface and melt should be deflected. The stress in solidified regionsis therefore described by

Tsolid = 2ηsolidD, (4.16)

based on the solid viscosity. The open question in this approach is, how large the viscosityincrease, in reference to the fluid region, has to be. To find this out, a pressure driven, twofluid channel flow was solved analytically. As shown in fig. 4.6, the fluids are separatedat y = 0. They have the same density, are Newtonian and different viscosities. At theinterface, the velocities and shear stress are supposed to be equal.

x

y ρ, η1

ρ, η2

τ1 = τ2

U1 = U2

Figure 4.6: Sketch of the pressure driven two fluid channel flow problem.

To solve this problem, two solutions have to be coupled. They are obtained from thesimplification of the Navier-Stokes equation for a 1D channel flow. Double integrationover the y-coordinate will lead to

U1(y) =1

2η1

∂p

∂xy2 + C1,1y + C1,2, U2(y) =

12η2

∂p

∂xy2 + C2,1y + C2,2. (4.17)

The equality of interface stress and velocity leads to the following boundary conditions:

U1(y = 0) = U2(y = 0) ∧ η1U′1(y = 0) = η2U ′

2(y = 0). (4.18)

As they are also not allowed to slip at the channel walls, the additional conditions

U1(y = h/2) = 0 ∧ U2(y = −h/2) = 0 (4.19)

are considered. By inserting these conditions into eq. (4.17) the coefficients

C1,1 =1

2η1

∂p

∂x

(η1 − η2

η1 + η2

)h

2, C1,2 =

h

2η2

(η1C1,1 − ∂p

∂x

h

4

), (4.20)

C2,1 =η1

η2

C1,1, C2,2 = C1,2 (4.21)

can be determined. In fig. 4.7 the solution is plotted for different viscosity ratios η1/η2.It shows that if the lower viscosity is increased, the velocity decreases in this half of thechannel. When carrying this to extreme ratios, the velocity in the lower part decreasesso far, that it could as well be replaced by a wall located at y = 0. A condition that istargeted with the continuous approach.

To find out a recommended value for the viscosity jump, an investigation of the velocitiesin relation to each other was carried out for the case η2 > η1. As can be seen in fig. 4.7for U2 in this case the maximum velocity always occurs at the interface. The maximumvalue of U1 moves towards the center of the upper half with a deceasing viscosity ratio.


1/1

1/2

1/4

1/8

1/1024

0 1 2 3 4

0.25

0.50

-0.25

-0.50

0.00

annotated: η1/η2

y/h

[-]

u/u0 [-]

Figure 4.7: Solutions of the two phase channel flow for different viscosity ratios.

By differentiation of eq. (4.17), shown in eq. (4.22) and eq. (4.23), the maximum occurringvelocity Umax and the interface velocity U0 can be determined.

U ′1 = 0 → ymax,1 = −C1,1

4→ U1(ymax,1) = Umax = C1,2 − η1C2

1,1

2 ∂p∂x

(4.22)

U ′2 = 0 → ymax,2 = 0 → U2(ymax,2) = U0 = C2,2 (4.23)

The plot in fig. 4.9 can be seen as an error, because in this case the desired value wouldbe U0 = 0. To expect an error of 1%, the viscosity in the region that is assumed to besolidified needs to be 792 times higher than in the fluid region.

1e-3

1e-2

1e-1

1e+0

1e+0

1e+1 1e+2 1e+3 1e+4

U(y

=0)

/Um

ax

η2/η1

Figure 4.8: Error of the interface velocity in comparison to a fixed wall.

4.3. Fluid-Solid Transition 53

The conclusion from this analytical consideration is, in simulations the solid regions vis-cosity is chosen 1000 times higher than the viscosity of the melt. But as the fluid ismodeled thermorheologically and shear thinning, there are many possible viscosities. Asall simulations in this work start at 200 ◦C or have this temperature set at the inlet, it ischosen as η0(200 ◦C) ≈ 2500 Pa s. The solid viscosity is therefore set to 2.5e6 Pa s.

When looking at the analytical solution for η2/η1 = 1/1024, the question might arise ifthis profile can be reached numerically. The following figure shows a clear answer.

0

0

1

0.25

-0.25

0.2 0.4

0.5

0.6 0.8-0.5

y/h

[-]

u/u0 [-]

Pressure correction residual

Analytical

1e-3 1e-4 1e-5

1e-6 1e-8 1e-10

Figure 4.9: Numerical study on trying to reach the solution for η1/η2=1/1024.

It was not possible to reach it with any standard solver. However, it was found out thatthe pressure correction residual plays an important role. A finding that will reoccur inchapter 5.4. It is suspected that the reason for this disturbance lays in the velocity fieldnot being continuously differentiable. But in combination with the crystallization model,such a jump over one cell will never occur, if the spatial resolution is chosen reasonable.

4.3 Fluid-Solid Transition

In the work of Pantani [45], many possibilities of transition modeling are given for gener-alized Newtonian fluids. They are mostly singular formulations, based on the fluid-solidindicator χ, describing a viscosity ratio to increase. But as the modeling approach for thefluid chosen in this work is of another type, it was decided not to apply these equations.The main reason for this was, inserting great viscosities into a shear thinning model wasseen as not useful. Depending on the shear rate, this might result in a viscosity thatis decades lower and possibly deflection of melt and no slip can not be realized at theinterface. That is why the approaches described in the first two sub chapters were cho-sen. Here, the fluid behavior exclusively depends on temperature, the solid behavior is


described by a fixed value of viscosity, capable of introducing the desired effects, as provenin chapter 4.2.2. As the fluid solid transition is already described by a smooth function,c.f. chapter 3.2.1, it was seen as unnecessary introducing another dependency. Therefore,both models are equally weighted using

T = Tfluid + Tsolid (4.24)

The fluid stress, as described in chapter 4.1, is based on EVSS leading to

Tfluid = (1 − χ)(Tp + Ts), (4.25)

including polymer and solvent stress. For solid stress the Newtonian model

Tsolid = χ2ηsolidD (4.26)

is inserted. To show the behavior of this approach in solidification experiments, it iscompared to the measurements already presented in chapter 2.2.2 in the following figure.

0

2.5

5

10

1015 15

20

100 110 120 130 140 150 160 170

η[k

Pa

s]

θ [◦C]

cooling rates in K min−1

Model

Figure 4.10: Transition model compared to experiments.

It shows that the general behavior matches the experiments. But the increase is slightlyshifted to lower temperatures. As in the experiment crystallization starts earlier at theouter surface, thoroughly discussed in chapter 2.2.2, this is expected, as the crystallizationmodel was fitted to DSC experiments. Additionally, as shown in chapter 3.2.2, the modelfor θm deviates from the measurements in this region of cooling rates as well. In conclusion,it can be stated that a linear interpolation is also capable describing singularities.

As the consideration of the solid part as a highly viscous fluid is just a workaround, it willnot be considered as a source of dissipation. So instead of formulating a source term basedon the overall stress T, just the fluid stress is used. The energy equation, c.f. eq. (3.5), istherefore extended as follows:

ρcDθ

Dt= kΔθ + ρq + S(Tfluid). (4.27)

The dissipation source term is not further specified, as using S(Tfluid) = tr(TfluidD) isnot a reasonable choice for viscoelastic fluids, as the following subchapter will show.

4.4. Dissipation of Energy 55

4.4 Dissipation of Energy

As described briefly in chapter 4.1.2, the models for viscoelastic fluids are mainly de-veloped based on network theory. Independent on if one believes that this theory leadsto equations that describe the phenomena, or the phenomena lead to this theory. Theyhave to be assigned to the class of phenomenological constitutive equations. This means,in contrast to the classic derivation of constitutive equations in mechanics, they are notdeveloped by the usage of the Clausius-Duhem inequality. As a result, they are by defini-tion not thermodynamically consistent.2 The consequence can be observed when havinga closer look at tr(TpD), the common dissipation source term of the energy equation. Asan example, in fig. 4.11 the result of a simulation of the flow around a cylinder in a channelis displayed. The model used is the EPTT with α = 0.5 and the solvent contribution setto β = 0.1. As the flow is pressure driven, the Deborah number De = Uλ/h = 8.9 wascalculated using the averaged inlet velocity.

1e-4 1e0λ |tr(TpD),Sc,Sp|d

ηpu0[-]

tr(TpD)

Scorr

Sp

Figure 4.11: Different sources of dissipation refereed to for the flow around a cylinder.

All regions surrounded by black lines represent regions of negative dissipation. Thismeans, here the melt will be cooled due to model behavior. An artefact just recentlynamed in [109], to the knowledge of the author of the present work never discussed beforein context of numerical simulations. As stated in [109], this defect will be solved by thecontribution of the solvent term tr(TsD) = ηstr(DD), as this is always positive. However,

2That there are issues with non-Newtonian fluids and thermodynamics is well known. In his chapteron rational thermodynamics in [108], Müller focuses historically on that topic in a very entertaining way.He describes, that after Giesekus came up with the concept of objectivity, some authors put this conceptover every other principle. Possibly, the result of that can be observed in this subchapter.


as the solvent contribution for real materials is small to non existent, c.f. tab. 4.1, thiscan’t be the solution to this problem. Therefore, dissipation has to be considered in thesense of the work of Peters and Baaijens [105].

They insert the models into the Clausius Duhem inequality and suggest a correction termfor dissipation. An example of application is carried out by them for the LPTT model,allowing to formulate a correction term for all other models using a scalar molecularanisotropy Q. To apply this method, first eq. (4.9) must be rewritten as

λ�

Tp + [1 + Q(Tp)] Tp = 2ηD. (4.28)

In this formulation temperature dependencies were only dropped for clarification. It isneedless to say that for temperature dependent coefficients this methodology is still valid.In the energy equation, based on the expression in square brackets, an additional sourceterm Scorr is introduced in the following way:

ρcDθ

Dt= λΔθ + tr(TsD) + κtr(TpD) + (1 − κ)

tr(Tp)2λ

[1 + Q(Tp)

]︸︷︷︸

Scorr

. (4.29)

It is linearly weighted with the conventional dissipation term using 0 ≤ κ ≤ 1 accordingto [105] interpolating between the energy (κ = 0) and entropy elastic (κ = 1) case. Theparameter κ is a material constant that should be determined experimentally. For theequations named in tab. 4.2, therefore Scorr can be determined easily as given in tab. 4.5.

This reveals, for the Phan-Thien Tanner models Scorr is nonlinear in tr(Tp). As alreadydiscovered in the work of Peters and Baajens, for low Deborah numbers the value of κwill not have a great impact. However, for large De it will. They also state findingexperiments for identifying κ is a problem to be solved. But until today, it remainsunsolved. Therefore, no best practice approach for the choice of κ is present. However, amethodology based on a simple flow type to estimate a value was developed.

Table 4.5: Corrective dissipation source for models based on a scalar anisotropic mobility.

Oldroyd-B LPTT EPTT

Scorrtr(Tp)

2λ

tr(Tp)2λ

[1 + αλ

ηptr(Tp)

]tr(Tp)

2λe

(α λ

ηptr(Tp)

)

Subject of this methodology is an oscillation test as performed with rheometers. It repre-sents a channel with an oscillating top plate, condensed to a scalar problem. The applieddeformation is described by γ(t) = γ0sin(ωt). Oscillations are executed with a sufficientlysmall amplitude, to let the nonlinear term in eq. (4.9) disappear.

4.4. Dissipation of Energy 57

For the polymer stress, therefore the transport equation can be simplified to

τp,xy + λ∂τp,xy

∂t= ηγ0ωcos(ωt), (4.30)

τp,xx + λ∂τp,xx

∂t= 2τp,xy(t)λγ0ωcos(ωt). (4.31)

The equation for shear stress can be solved by integration, leading to

τp,xy(t) =ηγ0ω

(cos(ωt) + λωsin(ωt) − e−t/λ

)λ2ω2 + 1

. (4.32)

Just one extra term for the start up from τp,xy(t) = 0 is included. With this solution theoriginal dissipation term is known since tr(TpD) = τp,xyγ for this special type of flow.To calculate Scorr eq. (4.31) must be solved, which is more complex as eq. (4.32) is in itssource. However, integration and sorting of terms leads to:

τxx(t) =B1cos(2ωt) + B2sin(2ωt) + B3 e−t/λsin(ωt) + B4 − (B1 + B4)e

−t/λ

N, (4.33)

B1 = ληγ20ω3(1 − 2λ2ω2), B2 = 3λ2ηγ2

0ω4,

B3 = −2ηγ20ω2(4λ2ω2 + 1), B4 = ληγ2

0ω3(4λ2ω2 + 1), (4.34)

N = ω(λ2ω2 + 1)(4λ2ω2 + 1).

In this case the exponential terms introduce the start up from τxx(t) = 0. The plot ofthose fields over time is presented in fig. 4.13, on the bottom of next page.

2 3 40

0

1

-0.2

0.6

0.2

0.4

0.80

t/T [-]

λS

p/η

pγ

ω[-

]

κ = 1 0.250.50.75

Figure 4.12: Influence of elasticity blending by κ on the dissipation in an oscillatory flow.Energy elastic behavior corresponds to κ = 0, entropy elastic behavior to κ = 1.

In fig. 4.13 it can be observed, that the shear stress τp,xy has a slight phase shift comparedto the shear rate γ. This causes tr(TpD) to turn negative, as it is just the product of


these two functions. In contrast, as τp,xx is always positive, Scorr is always positive aswell. As the next step therefore a study over κ, c.f. eq. (4.29), was performed of whichthe result is plotted in fig. 4.12 on the previous page.

It can be seen, that for κ = 0.25 the dissipation is just positive in the quasi steadystate. Therefore, this value was chosen in all calculations. This will however not solvethe problem of negative dissipation in all possible cases. In fig. 4.11 the lower two figuresshow Scorr and the overall source term

Sp = κtr(TpD) + (1 − κ)Scorr. (4.35)

As can be seen, Sp still shows regions of negative dissipation, but the magnitude and sizewas decreased. As for higher process velocities Scorr will strongly decrease, as it growsexponentially with Tp, usually the dissipation is globally positive.

0

0

0

0

0

0

11

-1-1

-0.5

0.5

0.3

0.6

0.2

0.4

0.4

-0.4

0.8

00 11 22 33 44t/T [-]t/T [-]

γ /γ

0[-

]

γ /γ

0ω

[-]

τ p,x

y/η

pγ

0ω

[-]

τ p,x

x/η

pγ

0ω

[-]

λtr

(TpD

) /η

pγ

0ω

[-]

λS

co

rr/η

pγ

0ω

[-]

Figure 4.13: Evaluation of a 0D oscillating flow w.r.t. its dissipative behavior.

4.5. Wall Slip 59

4.5 Wall Slip

Wall slip plays a big role in polymer processing. As described in [110], there are twotheories about the molecular mechanism that causes it, much debated in literature. Thefirst proposes that the molecules themselves are sliding at the wall. The second proposesthat molecules stick to the wall and adjacent molecules are getting pulled out of theirproximity, where intermolecular forces act strongly. Rather simple however, is the phe-nomenological mechanism. This just states, if certain wall shear stress is exceeded, slipwill occur. The stress wall slip sets in and the behavior, in case load is further increasedwhen wall slip is already present, highly depend on the material.

There are many simple wall slip models available, for which usually the geometrical gen-eralization is challenging. Though, an elegant and very practical approach, regarding itstransferability into a CFD framework, is developed in [111]. There are also several wallslip models developed on network theory, e.g. [112], based on both PTT models. But ofcourse, they are limited to the usage of these exact models.

However, introducing observable effects into a simulation is possible using simple models.If e.g. after a certain wall shear stress τcrit, the slip velocity should increase linearly, thiscan be implemented using

Uslip(τwall) =

⎧⎨⎩0, for τwall ≤ τcrit

Aslip(τwall − τcrit), for τwall > τcrit

. (4.36)

If additionally, an elastic character should be implemented, this condition is not set in-stantaneously, but with a certain retardation. As described in [110], this goes back on thework of Pearson and Petrie [113]. It is known as the memory slip model, described by itsown differential equation

Uwall + λslipDUwall

Dt= U(τwall). (4.37)

If used in combination with a viscoelastic model describing stress in the fluid, the wellknown slip-stick melt fracture effect (c.f. [114]) will occur for certain configurations. Fordemonstration, a pressure driven channel flow is considered.

The channels height is h, its length is L and a pressure difference Δp is applied. Thefluid is described by the Oldroyd-B model with the relaxation time λ, and the solventcontribution is β = 0.5. For this configuration, the reference velocity is u0 = Δph2/8η0L andthe Deborah number De = λU0/h. The critical wall shear stress is chosen that τcrit/η0u0 = 1/8,the slope coefficient in eq. (4.36) is chosen that Aslipη0/h = 1. A low-Re flow accordingto the implementation in chapter 5.4 is considered. The channel is planar in the x-yplane and therefore the wall shear stress is considered to be |τxy|. At the inlet T = 0is set to introduce an inlet length effect. This means stress will increase to a saturatedvalue over the length, dependent on the wall shear rate. Therefore, after a certain length,proportional to De, τcrit is reached. Here, the boundary condition will introduce wall slip.Hence a no slip region just after the inlet is followed by the slip region.


If the parameters are chosen as named above, an instability will occur. It is caused by ahysteresis effect, that is initiated because the expected wall shear stress is just above τcrit.The result is shown in the following figure.

0

2

|v|/u

0[-

]

traveling downstream

De = 4, λ/λwall = 50

De = 8, λ/λwall = 100

De = 16, λ/λwall = 200

Figure 4.14: Slip stick instability in a pressure driven channel flow.

The mechanism behind this pattern is as follows. In the inlet region, just after τcrit isreached, slipping is introduce delayed in time. This delay will allow stresses larger thanτcrit to build up. If the slip has reached a sufficient value, the wall velocity gradientdecreases and stresses relaxate. When dropping under τcrit, due to the delayed no slip,the build up is delayed. Then the process starts from the beginning. As in the moment slipoccurs, stress is transported downstream, this makes the whole effect travel downstream,as shown in fig. 4.14. If the flow would now exit a nozzle, noticeable defects of the surfacewould occur, well known in practice. The study over De shows that lowering De decreasesthe distance between the slip regions, increasing will increase it. A window of instabilitiestherefore exists.

It was shown, that wall slip can have a great influence on the process. To model it correctly,first there needs to be an experimental basis, which was left out completely above. Thena correct model must be chosen and parameters have to be identified. A numericalconsideration is possible, as shown in this subchapter. For some configurations, this willintroduce wall slip in simulations and in rare cases even instabilities. It was decided,that all the named topics would lead too far in the context of this work. Especially thepossible occurrence of instabilities, which might interact with crystallization was foundto be too specific. In modeling and simulation, therefore wall slip as a result of thepolymer structure is neglected. Slip as a result of solidification will however play a rolein chapter 7.1.

CHAPTER 5

Numerical Treatment

In this chapter special numerical topics regarding the handling of viscoelastic fluids andlow Reynold number flows are discussed. Furthermore, a detailed description of stepsexecuted by the code composed for the present work is presented. The intention is tocover implementations made and methods used. It is not to give an introduction to CFD,as the environment chosen for realization (foam-extend-4.0) is not self developed. Forthis purpose, the book of Moukalled, Mangani, and Darwish [115] or Peric and Ferziger[116] is a good choice. This chapter should be seen as a documentation of the importantmethods and settings used.

As it is the standard method for viscoelastic fluids in CFD, first an introduction in thelog-conformation reformulation is given. It covers a comparison to the standard method,detailed information about the implementation and a validation of the code for threeexamples. As for the constitutive equations discretization of the convection term is a welldiscussed topic, it is also addressed here. This part covers a short introduction to stateof the art schemes and a numerical example to justify the choice. Applying standardmethods of pressure correction to low Reynolds number flows can be very inefficient.Therefore, an alternative choice is presented and tested. A representative situation duringa simulation is used for a comparison to the standard method. The last part is thealgorithmic realization used to solve problems including crystallization. All the modelsand methods are summarized in this subchapter, based on a flow chart.

61

62 Chapter 5. Numerical Treatment

5.1 Log-Conformation Reformulation

The Log-Conformation Reformulation (LCR) is the state of the art method applied in nu-merical simulation of viscoelastic fluids. It was developed by Kupferman and Fattal [117]in 2004 to overcome the High Weissenberg Number Problem. A blow up in time, occur-ring independent on the method used, applied discretization schemes or spatial resolution.The cause is an insufficient resolution of convection due to large gradients occurring inthe stress field. In contrast to many authors trying to solve this problem numerically,they proposed an ultimate solution by using a logarithm of the constitutive equation,eliminating any large gradients. In their second work on the topic [118], they used a 2Dfinite difference method to solve a cavity flow, as also presented later for validation. Sincethen, many researchers implemented this method, mainly improving coupling.

The most advanced method up to the date this thesis was realized, is described in the workof Knechtges [119]. A fully implicit FEM solver avoiding eigen-decomposition. As usuallyduring the application of the LCR eigenvectors have to be calculated, this can be describedas a real breakthrough, because this allows to solve the system of equations non-iterative.The first work describing an LCR implementation in OpenFOAM [120] was published in2014. Based on descriptions made in this publication, the LCR implementation used inthe present work was realized in 2015. There are however differences in stress coupling andpressure correction, which are also topic of this subchapter. After excessive testing, e.g. in[121], the present solver was further developed to be used for crystallization simulation. Inthis time Pimenta released the RheoTool extension for OpenFOAM, based on his paperwith Alves [122]. With this tool, a framework to use the LCR with a wide range ofviscoelastic models was provided to the public. As the interpolation approach presentedin chapter 4.3 is used and the RheoTool just provides isothermal solvers, it was nevertried to use it with the crystallization model.

To describe the solver used in the present work, first an introduction into the LCR andits benefits compared to the standard method is given. Then the exact implementationis described, followed by the results of code validation.

5.1.1 Standard Method

Any of the standard constitutive models for viscoelastic fluids can just be treated assix additional transport equations. Those are coupled to the momentum equation, theprocedure is the same as e.g. for calculating turbulent flows. On a top level, in one timestep, multiple times a solution is generated for each variable, which is thereafter updatedin other equations it occurs in as a source.

5.1. Log-Conformation Reformulation 63

The set of equations for this partitioned approach is therefore

∇ · v = 0, (5.1)∂v

∂t+ ∇ · (vv) = −∇p + ∇ · (T p + 2ηsD), (5.2)

∂T p

∂t+ ∇ · (vT p) =

1λ

(2ηpD − f(T p)

)+ LT p + T pL

T , (5.3)

using the right hand sides listed in tab. 5.1 for the different models. Another possibilityis to use the dimensionless conformation tensor

C =λ

ηpT p + 1 (5.4)

instead of stress, which is a common choice e.g. done in [123]. When for example substi-tuting T p in eq. (5.3) for the Oldroyd-B Model this will lead to

ηp

λ

⎛⎝DC

Dt− D1

Dt= 2D − 1

λ(C − 1) + LC − L + CL

T − LT

⎞⎠, (5.5)

which makes it easy to see that D is canceled out. The new transport equation is therefore

∂C

∂t+ ∇ · (vC) = −g(C)

λ+ CL + L

TC, (5.6)

based on a new model dependent function g(C) given in the following table.

Table 5.1: Source terms for eq. (5.3) and eq. (5.6).

Model Oldroyd-B Giesekus LPPT EPTT

f(T p) T p T p+αλ

ηp

T2p T p+

αλ

ηp

tr(T p)T p e

(αληp

tr(T p)

)T p

g(C) C-1 C-1+α(C-1)2 (1+α(tr(C)-3))(C-1) eα(tr(C)-3)(C-1)

The code available to solve viscoelastic flows in OpenFOAM goes back to the work ofFavero [124]. It was first released in 2009 and is still part of the foam-extend-4.0 packageused in this work. In this solver the equations are solved in stress formulation.

5.1.2 Logarithmic Transport Equation

The starting point of the logarithmic reformulation are the constitutive models in confor-mation tensor formulation. As it is already described step-by-step in [125], here just theessential steps, results and benefits are shown.


The core of the LCR is a change of variables into the logarithmic space, performed foreq. (5.6). A new transport equation based on a numeric variable1

Ψ = log(C) (5.7)

is the result. The transport equation for Ψ is then solved instead. A transformation backinto the linear space of C is performed by using the matrix exponential

C = eΨ . (5.8)

Calculating the matrix exponential can not be done by just taking the exponential com-ponent wise. The correct way is to use the series expansion of the exponential

C =∞∑

n=0

1n!

Ψn. (5.9)

It usually delivers accurate results in a limited number of steps. But it is not used, as atransformation of the equation can not be performed for a series. Hence, the exponentialis calculated via diagonalization. The exponential of any diagonalized matrix can becalculated by just taking the exponential of the diagonal entries

C = R

(exp(RT

ΨR))

RT . (5.10)

In this equation Ψ is diagonalized using the change of basis tensor R. It has the theeigenvectors of Ψ sorted in vertically. Using the R

T only works, because C is a symmetricpositive definite tensor, otherwise R

−1 would have to be used.

The outcome of the reformulation is, that for the Oldroyd-B model

∂Ψ

∂t+ ∇ · (vΨ ) − (ΩΨ − ΨΩ) − 2B = −1

λ(1 − e−Ψ ) (5.11)

has to be solved. Here the stretch and rotation tensor B resp. Ω come into play. Tocalculate them, first the velocity gradient has to be transformed by

M = RT LR. (5.12)

Furthermore, the diagonalized exponential of Ψ has to be calculated using

Λ = exp(RTΨR). (5.13)

If both fields are present, the rotation and stretch tensor are calculated using

Ω = R

⎡⎢⎢⎣

0 ω12 ω13

ω21 0 ω23

ω31 ω32 0

⎤⎥⎥⎦R

T ∧ B = R

⎡⎢⎢⎣

m11 0 0

0 m22 0

0 0 m33

⎤⎥⎥⎦R

T . (5.14)

1As the logarithm of T p is not defined for the stress free state, C is used.


in which components of Ω are calculated in the following way:

ωij =Λjjmij + Λiimji

Λii − Λjj. (5.15)

Only in case of the stress free state, in this expression a floating point exception has tobe prevented. In [120] it is suggested to set C = 1, Ω = 0 and B = D in this case.

If a non-linear model should be used, just the right hand side has to be changed. Since themodels used here are common, the related source terms were already derived by others,see e.g. [126] or [127]. A new source term h(Ψ ) is introduced to be used in

∂Ψ

∂t+ ∇ · (vΨ ) − (ΩΨ − ΨΩ) − 2B = h(Ψ). (5.16)

For all models that were topic of this work, it is presented in the following table.

Table 5.2: Source terms for eq. (5.16)

Model Old.-B Giesekus LPPT EPTT

λh(Ψ ) e-Ψ-1 e-Ψ -1-αeΨ(e-Ψ-1

)2 [1+α

(tr(eΨ )-3

)](e-Ψ-1) eα(tr(eΨ)-3)(e-Ψ-1)

The benefit of logarithmization can be seen when comparing Tp and Ψ for increasingprocess velocities. The example chosen for this is the lid driven cavity flow used byKupferman and Fattal in [118]. Its definition is shown in the following figure.

L

L 00

00

16

116

1

1

0.2

0.2

0.4

0.4

0.6

0.6

0.8

0.8

f[-

]g

[-]

x/L [-]

t/λ [-]

u0f(x)g(t)

x

y

Figure 5.1: Definition of the lid driven cavity case used in [118]


As the left sketch shows, it is a basic cavity flow with the coordinate system located inthe bottom left corner. In this system the lid velocity is set using x = x/L and t = t/λ as

u(x, t) = u0

(x2(1 − x)2

)︸︷︷︸

f(x)

(8(1 + tanh

[8(t − 0.5)

]))︸︷︷︸

g(t)

(5.17)

to eliminate singularities in the upper corners and introduce a smooth startup2. For thiscase the Weissenberg number is defined as

We :=u0λ

L. (5.18)

Regarding stability, it represents a critical value for the standard procedure when reachingWe ≈ 1. The reason for this is the occurrence of large stress gradients, which the followingexample shows.

|Ψ | [-]|Ψ | [-] |Ψ | [-]

λ|Tp|/ηp [-]λ|Tp|/ηp [-] λ|Tp|/ηp [-]0

0

0

0

0

01 3 4.5

1.5 20 90

a) We = 0.1 b) We = 0.33 c) We = 0.66

Figure 5.2: Comparison of Ψ and Tp in the steady state of the cavity flow.

As can be seen, for We = 0.1 the fields of Tp and Ψ are very similar. This is no surpriseas for |Ψ | < 1 the transformation is still linear in the largest part of the domain. When

2The startup is irrelevant for the steady state solution. Nevertheless it was used in the calculations.


increasing We stress increases drastically close to the lid and therefore large gradientsoccur in linear space. In contrast |Ψ | is still a smooth function that should not cause anyproblems with any standard method of approximation.

5.1.3 Implementation

A good documentation of the steps needed for an LCR implementation, based on thepressure-velocity-stress coupling of the standard solver viscoelasticFluidFoam by Favero[124], is already given by Habla in [120]. At this point however, it is described in moredetail and in contrast for the usage in a PIMPLE loop, c.f. chapter 5.5. Here the stepscan be performed outside the pressure correction. Furthermore, a different method forcalculating the change of basis tensor is described. For the following description, it isimportant to know, that v, Ψ , Λ and R are already known, either from the initialcondition, previous time step or previous iteration.

calculate L = ∇v

for All Cells do

M = RTLR

ωij = Λjjmij+Λiimji

Λii−Λjj

Ω = RΩRT

B = Rdiag(M)RT

eΨ = diag(λ1, λ2, λ3)˜e-Ψ = diag(λ-1

1 , λ-12 , λ-1

3 )

h(Ψ) = RT h(eΨ , ˜e-Ψ )R

ΓΨ = (ωΨ − ΨΩ) + 2B + h(Ψ )end

solve∂Ψ

∂t+ ∇ · (vΨ ) = ΓΨ

for All Cells do

R = R(Ψ )

Λ = exp(diag(λΨ,1, λΨ,2, λΨ,3))

Tp =ηp

λ

(RΛR

T − 1)

end

Algorithm 1: LCR stress update

The first step is to calculate the velocity gradient as it is exclusively needed for theconstitutive equation. After that, a loop over all cells is performed to calculate the rightside for the transport equation of Ψ . In here, a floating point exception is prevented


according to the description of eq. (5.15). As it is possible to calculate the source h(Ψ )diagonalized, denoted with a tilde, and perform the retransformation afterwards, it istaken advantage of this. Finally, the right side of the Ψ -equation is composed.

After solving for a new Ψ the final loop for updating the stress is executed. Here, forevery Ψ the block matrix of eigenvectors R and corresponding eigenvalues λΨ,1, λΨ,2, λΨ,3

are calculated. After taking the exponential, they are transformed into stresses, based onthe definition of the conformation tensor, given in eq. (5.4).

For calculating the eigenvalues, there are many possibilities. A direct way is Cardanosmethod, which is implemented in the tensor class of OpenFOAM as standard and alsoused in [120]. However, in the present work it turned out not to be a reliable method. Forthat reason, the iterative Jacobi method was applied. It is also used by other authors,e.g. [128] or [122]. In early versions of the code it was self-implemented, in the versionused for the calculations of the present work JACOBI_EIGENVALUE3 is used.

initialize R = 1, A = Ψ , r = 1while r > 1e − 12 do

find maximum off-diagonal element of A → k, l

B =

⎡⎣Akk Akl

Alk All

⎤⎦

λ+ = B11+B22

2+√

B212 + (B11−B22)2

4

C = B − λ+1

U = 1√C2

11+C212

⎡⎣ C12 −C11

−C11 −C12

⎤⎦

G = 1

Gkk = U11 Gkl = U12 Glk = U21 Gll = U22

A = GT AG

R = RG

r =√

2(A212+A2

13+A223)

A211+A2

22+A233

end

return λΨ,1 = A11, λΨ,2 = A22, λΨ,3 = A33, R

Algorithm 2: Jacobi eigenvector iteration

As shown in algo. 2, it is easily implementable. The algorithm presented here is an essen-tial version of an algorithm presented in [129], used like this also to perform simulations.

3Provided by John Burkardt under: people.sc.fsu.edu/~jburkardt/cpp_src/


The principle of working is that the matrix A = Ψ is diagonalized iteratively. That isalso why the residual is based on the ratio of off-diagonal elements to diagonal elements.To show the convergence behavior, the eigenvalues of a test matrix4 are calculated.

1e+00

1e-04

1e-08

1e-12

1e-16

1 2 3 4 5 6 7 8 9 10iterations [-]

resi

dual

[-]

Figure 5.3: Residual plot of the Jacobi method as presented in algo. 2 for the symmetricmatrix: A11 = 2, A12 = −4, A13 = −1, A22 = 5, A23 = 1, A33 = 2.

As can be seen, after six iterations the desired residual is reached and after eight thesolution is not changing any more because machine accuracy is reached. The additionaleffort is therefore limited, even though this calculation is done for every cell.

5.1.4 Validation

As implementing a procedure like the LCR is a rather extensive task, some time is wellspent validating. For the code used, it is documented in this subchapter. For validationtwo cases for the general implementation and one for the EPTT model is used. The steadystate is validated using the lid driven cavity case, as already presented in chapter 5.1.2. Asviscoelastic fluids have a distinct transient behavior, the analytical solution of a pressuredriven start-up channel flow is used to test the correct temporal scaling of the right sideof eq. (5.16). The EPTT implementation is validated using a semi analytical solutionsfor the steady state rheological functions.

5.1.4.1 Lid Driven Cavity

When thinking about a general validation of the implementation, one could easily comeup with the idea to just compare results with a stress based solver in linear space. Butas this formulation usually shows too high stresses above a certain We and additionally

4Also provided in [129], however the convergence for this example is not discussed in this work.


blows up, this is not possible. Therefore, component plots of v and Ψ over certain cut-lines provided in [118], calculated with a finite difference method, are used. As can beseen in the following figure, the results achieved are showing good agreement.

0 1

11

2 3 4 5 6 7

1.0

1.0

1.0

1.2

1.6

0.8

0.8

0.8

0.8

0.8

0.8

0.6

0.6

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0.4

0.4

0.2

0.2

0.2

0.2

0.2

0.2

0.1

0.0

0.0

0.0

0.0

0.0

0.00.0

-0.1

-0.2-0.2

-0.4

Kupferman & Fattal [128x128]Mesh [128x128]

Mesh [96x96]Mesh [64x64]

y/h

y/h

u(y, x = h/2)/u0

v(x

,y=

3h/4

)/u

0

x/h

x/h Ψxx(x = h/2, y)

Ψx

y(x

,y=

3h/4

)

Figure 5.4: Comparison for the lid-driven cavity at We = 1 with [118].

However, there are regions in which are minor differences. The reason for this is Kupfer-man and Fattal used a specific higher order convection scheme. In the calculations withthe solver of the present work, the upwind scheme was used. This topic is further discussedin chapter 5.3.

5.1.4.2 Start-Up Channel Flow

Solving transient problems for viscoelastic fluids analytically is mostly not possible. How-ever, for a start-up channel flow it is. An analytical solution for a start-up channel flowwith an Oldroyd-B model was presented by Waters [130] in 1970. This solution recentlygot more attention as CFD with viscoelastic models got popular. Therefore, it is countedas a standard test, also performed e.g. in [131], [122] and [120]. As this solution also


contains the solvent free case (β = 0) it is already possible to investigate how the codebehaves for this case.

The start-up case represents a channel flow in the x − y plane with the height h undera temporally constant pressure gradient ∂p

∂xthat starts from the static state. A well

summarized version of Walters solution that is programmable straightforward is presentedin [132] and translates in this works notation as follows.

u(y) = u0

⎡⎣4

y

h

(1 − y

h

)− 32

∞∑n=1

sin(

Nyh

)N3

GN(t)

⎤⎦ (5.19)

GN(t) =

⎧⎪⎪⎨⎪⎪⎩

12

[aN exp

(pN ν0t

h2

)+ bN exp

(qN ν0t

h2

)], β2

N ≥ 0

exp(−α∗

Nν0t

h2

) [cos

(β∗

Nν0t

h2

)+ SN

βNsin

(β∗

Nν0t

h2

)], β2

N < 0(5.20)

u0 = -∂p

∂x

h2

8η0

(5.21)

Re :=u0h

ν0(5.22)

η0 = ηS+ηP (5.23)

Wi :=λu0

h(5.24)

β =ηS

η0

(5.25)

E :=Wi

Re(5.26)

ν0 =η0

ρ(5.27)

N = (2n-1)π (5.28)

β2N = α2

N-4N2E (5.29)

β∗N =

βN

2E(5.30)

pN = -α∗N+β∗

N (5.31)

SN = 1-(2-β)N2E (5.32)

βN =√

abs(β2N ) (5.33)

aN = 1+SN

βN

(5.34)

qN = -α∗N -β∗

N (5.35)

αN = 1+βN2E (5.36)

α∗N =

αN

2E(5.37)

bN = 1-SN

βN

(5.38)

As can be seen this solution allows to perform a comparison for the whole channel height.But in fig. 5.5 just the velocity at the centerline is evaluated. Several values of solventcontribution are tested. As can be seen, the general agreement is good, but when reachingthe solvent free case there are some deviations, especially at the extrema of the curve,that have a kink.

The reason is, there is a kink in the velocity profile as well, shown in fig. 5.6. Naturallyit is hard to capture such profiles numerically. Obviously the result could be improvedusing a finer mesh. But this was not the intention of this study. It was to show up thatthere are effects which are hard to resolve in the zero solvent case. However, in no othercalculation performed for this work profiles like this could be observed.


β = [0, 0.01, 0.05, 0.15, 0.325, 0.625, 1]

-10

0

1

2

2

3

4

4

6 8 10

analyticalnumerical

t/λ [-]

u(y

/h=

0.5)

/u0

[-]

β = 1

β = 0spatial resolution: 40 cells over h

time step: λ/Δt = 1000

Figure 5.5: Centerline velocity validation for E = 1 in dependence of the solvent contribution.

-10

0

0.2 0.4 0.6 0.8 1

1

2

3

4

analyticalnumerical

t/λ = 0.25t/λ = 0.5

t/λ = 0.95

t/λ = 1.35

t/λ = 1.5

y/h [-]

u/u

0[-

]

Figure 5.6: Validation of the centerline velocity for β = 0 at E = 1.

5.1.4.3 EPTT Model

To validate the EPTT model no analytical solution in used. As it is possible to solveit semi-analytically for its rheological functions, this was seen as more useful than usingan analytical solution of similar complexity as presented for the start-up flow. How sucha semi-analytical solution can be generated, is described in appendix A. For validation,in the simulation just a single cell was used. Instead of solving for velocities, just thevelocity gradient for a shear and uniaxial elongation flow was set for different rates. Forevaluation the stress components were transformed into values for shear viscosity, firstnormal stress difference and elongation viscosity. The result can be seen in fig. 5.7.

5.2. Stabilization for Low Solvent Contribution Flows 73

1e31e2

1e2

1e1

1e1

1e0

1e0

1e-1

1e-1

1e-2

1e-2

1e-31e-3

ηP,S

ηP,E

N1

λγ, λε [-]

η P,S

/ηp,

η P,E

/ηp,

N1λ

/ηP

[-]

semi-analytical

numerical

Figure 5.7: Validation of the EPTT implementation for α = 0.1.

As can be seen, the numerical implementation meets the semi analytical results exactly.The implementation of the LCR and the EPTT model is therefore completed with apositive result.

5.2 Stabilization for Low Solvent Contribution Flows

It is known that coupling the polymer stress into the momentum equation, regardlessof the loop structure used in a solver, is problematic for low/zero solvent contributioncases. As in the momentum equation in this case there is low/no diffusion, usually thereare problems with convergence of the stress coupling. The solution originates from [133]often called Discrete Elastic Viscous Split Stress, later works [125] call it Both Side Dif-fusion (BSD), latest works [122] Improved Both Side Diffusion. Independent on the exactimplementation, they all are based on a zero addition to the stress tensor

T = 2ηsD + Tp + 2ηvD︸︷︷︸implicit

− 2ηvD︸︷︷︸explicit

(5.39)

that differs in the discretization. As can be seen, the first part is treated implicit thesecond explicit. The idea is, that during an iteration procedure, at first diffusion is addedand due to the decreasing differences in the solution for v from iteration to iterationthe additional terms cancel out each other. As for zero solvent contributions ηs = 0the momentum equation is hyperbolic, it is very sensitive to the propagation of numericalartifacts. As they are more probable at the beginning of a new time step, they are dampedaway by the additional diffusion, making the momentum equation elliptic.

The simplest way to implement BSD is to just add two laplacian terms to the momentumequation, as done in current versions of viscoelasticFluidFoam in foam-extend-4.0. Whenwriting all the implicit terms on the left and all explicit terms to the right for a low


Reynolds number flow, see chapter 5.4, this corresponds to:

∂v

∂t− ηvΔv = −∇p + ∇ · Tp − ηvΔv. (5.40)

Unfortunately, this treatment will uncouple, found out by Pimenta in [122]. However, asolution was just delivered in the same work. Formally it can be written as

∂v

∂t− ηvΔv = −∇p + ∇ · Tp + ηv∇ · (∇v), (5.41)

which corresponds to the implementation in RheoTool, of which Pimenta is the author.The trick with this formulation is, that the separate calculation of divergence and gradientincludes an interpolation on the surface. This has the same effect as the well known Rhie-Chow interpolation [134]. It prevents decoupling that will usually show as a checkerboardin the solution.

As the problems to be solved in this work are zero solvent contribution cases, the solverhad to be updated, since it also used the implementation that decouples. To test it, thecavity case was calculated at We = 1 with β = 0 using both implementations. The steadystate results are shown in the following figure.

0

1

u/u0 [-]

old updated

Figure 5.8: Comparison of the BSD implementations on cell level.

The comparison shows, the old formulation clearly suffers of decoupling. A strong checker-board pattern forms. The method presented by Pimenta is stable. Therefore, the codewas successfully updated to the state of the art BSD stabilization.

5.3 Convection Discretization

Discretization of the convective term in the constitutive equations is an important topicin simulation practice of viscoelastic fluids. As shown in a fundamental work of Alves,Oliveira and Pinho [135], that proposes the state of the art CUBISTA scheme, the influ-ence can be large. Among other things, they discovered that for a 4:1 contraction higherorder discretization can strongly influence the corner vortex. However, as it is a higher

5.3. Convection Discretization 75

order scheme, its stability in application is conditional, even if it works perfectly in typicalone dimensional problems like the convection of a step function. Therefore, some authorswork with stability charts, e.g. [136], to state if a steady state was reached.

As described in chapter 4.1, the solvent contribution for the material modeled in this workis effectively zero. This introduces problems regarding the possible choices of a convectionscheme, as documented in a study at the end of this subtopic. But to be able to interpretthe results, first a small introduction to TVD schemes is needed that follows hereafter.

5.3.1 Total Variation Diminishing Convection Discretization

Independent on the constitutive model used, the equation for Ψ 5 can be summarized as

∂Φ∂t

+ ∇ · (vΦ) = SΦ. (5.42)

As common for deriving the FVM, applying Gauss’s theorem to the convective term allowsto reformulate this equation in integrals over the surface and volume, leading to∫

V

∂Φ∂t

dV +∫

SvΦdS =

∫V

SΦdV. (5.43)

The focus is now put on the convective term. Dividing the surface integral into a sumover all faces and applying the midpoint rule simplifies this expression to∫

SvΦdS =

∑faces

(∫Sf

vΦ dSf

)≈ ∑

faces

(vf · Sf )︸︷︷︸φf

Φf . (5.44)

Here the product of velocity and surface vector at the face centroid, both known inadvance, are combined as the face flux φf . This leaves the unknown face value Φf , thathas to be formulated in dependence of the centroid values of the neighboring cells, to beable to set up an equation system. To formulate a scheme therefore an upwind (U) anddownwind (D) formulation has established, based on the flow direction as shown in thefollowing figure.

v

Δx Δx

U P Df

Figure 5.9: Location of the upwind and downwind points in dependence of the flux.

Key of this formulation is the upwind-downwind gradient ratio, or consecutive gradient

r =ΦP − ΦU

ΦD − ΦP. (5.45)

5Also for TP in case the models are used in linear space.


In this way any scheme can be written as a blending function ψ(r) in

Φf = ΦP︸︷︷︸diffusive

+12

ψ(r)[ΦD − ΦP

]︸︷︷︸

anti-diffusive

. (5.46)

Because using the point value ΦP as the face value, known as upwind scheme, producesnumerical diffusion, it is assigned to that property. Using the values ΦU and ΦD cor-responds to a higher order approximation that has no diffusion and therefore is calledanti-diffusive as it is usually just blended in.

The dependence on the upwind node in this equation might not be obvious, but whentaking e.g. ψ(r) = r in eq. (5.46), this results in

Φf =32

ΦP − 12

ΦU , (5.47)

known as the Linear Upwind Differencing Scheme (LUDS). This means, as r is based onΦU , by choice of the limiter any function of all three nodes can be expressed as shown inthe following table.

Table 5.3: Limiter formulation of standard schemes. Limiters denoted with 1 are TVD.

Scheme ψ(r) Scheme ψ(r)

upwind 0 downwind 2

CDS 1 LUDS r

QUICK3 + r

4MUSCL1[137] max

(0, min

(2r,

r + 12

, 2))

van Leer1[138]r + |r|1 + |r| CUBISTA1[135] max

(0, min

(1.5r, 0.75 +

r

4, 1.5

))

Herein basic schemes are not denoted with 1. They interpolate the face value based ona simple function in dependence of neighboring centroid values. Upwind and downwindassume a constant and are therefore the simplest schemes. The upwind scheme is knownto produce numerical diffusion, but is unconditionally stable when treated implicit. Thedownwind scheme is unconditionally unstable, as it uses only information of the down-wind direction. In the CDS scheme, a linear function is assumed which means that thecontribution of up- and downwind information is equal. In practice this will lead to zerodiffusion but high dispersion which is often interpreted as numerical oscillations. Whereasthe first property is desired, the second is not wanted. As downwind information is knownto cause problems, a second order interpolation exclusively based on upwind information,the LUDS exists. The QUICK is following a similar approach, by using two upwind nodesand one downwind node to perform a quadratic interpolation. However, both are knownto have stability issues, which are more serious for the QUICK scheme.


To improve stability, every standard scheme can be made Total Variation Diminishing(TVD). This means if variation of the field is caused by dispersion it will diminish tobe lower in the next time step. By reducing the order of discretization locally to creatediffusion, this is applied based on the value of r. The works of Harten [139] and Sweby[140] are fundamental regarding these methods. Here, it is key to concern certain regionsin a flux diagram, as the following figure shows.

ψψ

rr

LUDSQUICK

downwind

upwind

CDS

1

1

1

1

2

2

2

2

3 3

2r

1st order TVD region 2nd order TVD region

MUSCLvan LeerCUBISTA

Figure 5.10: Limiters of tab. 5.3 plotted in Sweby diagrams.

As can be seen in the left plot, all the none TVD schemes from tab. 5.3 leave the greyarea at some point. This is the first order TVD region defined as

0 ≤ ψ(r)r

≤ 2 ∧ 0 ≤ ψ(r) ≤ 2 ∧ ψ(r < 0) = 0 (5.48)

in [140]. In here, there is one point that all schemes higher than first order cross. Itconnects the second order TVD regions, as shown in the right figure, e.g. in [140] aswell. Both regions can technically be used to make non-TVD schemes TVD, simply bythe usage of if-conditions. However, there are many approaches available that are notjust based on standard schemes. Some are e.g. specifically defined to follow the boardersof the second order TVD region, such as the Minmod or Superbee scheme, see [141].Some are defined to have a certain distance to the boundaries. For the MUSCL scheme,as can be seen in fig. 5.10 right, this is the case in the bottleneck region of the secondorder point. Also shown is the scheme of Van Leer that is smoothly defined and forr → ∞ approaches ψ = 2. Summarizing, there is a broad variety of schemes to chose forconvection discretization.

For viscoelastic fluids however, over time another method called CUBISTA (Convergentand Universally Bounded Interpolation Scheme for the Treatment of Advection) has es-tablished, proposed by Alves, Oliveira and Pinho in [135]. Table 5.3 and fig. 5.10 rightshows that for the bottleneck region it is, despite being formulated using conditions, fairlysimilar to the Van Leer scheme and just limited to ψ = 1.5 for r > 3. It is not a standardmethod and is therefore not available in OpenFOAM releases, but easy to implement ina flux formulation as r is available.

As shown in this subchapter, there are many schemes available and the obvious choicewould be the CUBISTA scheme. But as already mentioned in the introduction to this


topic, there are stability issues. And they are increased when the solvent contributionis reduced, because additional damping in the momentum equation is not present. Todemonstrate the performance of the schemes described, a small study was carried out.But the purpose of this is not to show any advantages of certain scheme. It is to make areasonable choice for a convection scheme to be used in this work.

5.3.2 Low Solvent Contribution Example

Topic of this study is the lid driven cavity flow as already described in chapter 5.1.2 withthe numerical restrictions presented in chapter 5.4. They ensure a pressure correctionresidual of 1e-5 to be reached in every time step and the Courant number to be below0.1. Regarding the explicit character, the calculation of ψ has for TVD schemes, this isneeded for a fair comparison anyway. In contrast to the solvent contribution used in thebenchmark (β = 50%), here it is set to β = 5%, which comes closer to real materials6. Theevaluation is to observe the velocity field at the positions x1 = (0.5, 0.5), x2 = (0.75, 0.75)and x3 = (0.25, 0.75). The result can be seen in the following figure.

LUDSQUICK

upwindCDS

MUSCLvan Leer

CUBISTA

0

0

0

0

0

0

1010 102020 203030 304040 405050 506060 60

-0.04

-0.08

-0.12

-0.16

-0.1

-0.1-0.2

0.1

0.2

0.3

t/λ [-]t/λ [-] t/λ [-]

u/u

0,v

/u0

[-]

v

v

v

u

u

u

x1 x2 x3

Figure 5.11: Local evaluation of velocity for different schemes.

All schemes, except upwind, induce oscillations. For the CDS they are even causingthe simulation to crash at approximately 20 s, due to a blowup of velocities. Regardingthe magnitude of oscillations, the LUDS is in the range of all TVD schemes. It actuallydelivers results closest to CUBISTA. Using the one order higher QUICK scheme obviouslydoesn’t benefit stability. The disturbances are not as large as for the CDS scheme, butstill substantial. However, the simulation does not crash. MUSCL and van Leer schemedeliver similar results regarding magnitude of oscillation and average value. The vanLeer Scheme however shows a marginally smaller magnitude of oscillation. As alreadydescribed, the CUBISTA scheme delivers similar results as the LUDS. It could thereforenot convince for this case of usage despite counting as the state of the art scheme to use.

6It must be noted at this point, that most of the schemes will work without any instabilities occurringfor β = 0.5. For β = 0 instabilities for all higher order schemes are severe, most will even cause thecalculations to crash. Hence the solvent viscosity was increased slightly to show tendencies.


As in the discretization of the Ψ -field the schemes are applied, it is the cause of instabilitiesand therefore displayed in fig. 5.12. But similar to the work of Kupferman and Fattal[118], just Ψxx is considered. It shows the time step of 17.3 s, at which the instabilitiesfor each scheme are represented best, using a color bar that makes them easy to spot.

LUDS QUICKupwind

MUSCL van Leer CUBISTA

Ψxx

-1

2

Figure 5.12: Comparison of the Ψxx-field for different schemes.

Here, two types of oscillations can be observed. Large disturbances that travel downwardsfrom the top right corner as occurring for the LUDS and CUBISTA. Small disturbancestraveling in flow direction in the whole domain as for QUICK, MUSCL and vanLeer.Regarding stability, obviously the van Leer scheme performs best. As the result is closeto upwind the reason for this might be, that it releases the most diffusion or in thiscase, upwind is not as bad as its reputation. But this is a question that must remainunanswered, as it is firstly not a matter to be investigated in this work and secondly ψ isnot easily accessible in the framework of OpenFOAM.

One might come to a different conclusion when evaluating vortex sizes, but this is not thefocus of this work. It is stability, as should have become clear with this small study. If ahigher order scheme is applied it must be clear that it does not e.g. induce fluctuationsor oscillations which are possible to be mistaken for physics. This aspect only allows theusage of the upwind scheme for constitutive equations. Since for the energy equationsimilar observations were made, it is applied for it as well.


5.4 Pressure Correction

For incompressible fluids, the absolute level of pressure does not have an influence on thesolution. There is no contribution of the continuity equation in the momentum equationwhich uncouples the solutions. To solve this problem, typically for transient flows thePISO pressure correction method is applied. For low Reynolds number flows it is veryinefficient however, which makes the application of an advanced method necessary. It istopic of the following subchapter.

To find out the role of Reynolds number in the momentum equation, it has to be non-dimensionalized. This formulation can be found when writing it as a transformation fromnormalized fields in normalized space and time, all denoted with a tilde, using

x = U x, t = T t, v = U v, p =ηU

Lp. (5.49)

Inserting these scaling functions into the momentum equation enables to identify pre-factors for every term, as shown in the following equation:

ρ

[U

T

∂v

∂t+

U2

L∇ · (vv)

]= −ηU

L2∇p +

ηU

L2Δ˜v. (5.50)

As both terms of the right side have the same pre-factor, the equation can be divided byit, allowing to insert

Re :=ρUL

η, St :=

L

UT(5.51)

known as the Reynolds and Strouhal number leading to the final form

Re

[St

∂v

∂t+ ∇ · (vv)

]= −∇p + Δv. (5.52)

From this, the final set of equations, to be fulfilled together, for pressure velocity couplingcan be derived as ∣∣∣∣∣∣∣

∂v

∂t+ νΔv + ∇p

∇ · v

=

=

s

0

∣∣∣∣∣∣∣ . (5.53)

This problem can easily be written in matrix-vector formulation, that represents a blockcoupled system. Using the coefficient matrix N that merges the time derivative and thediffusion term ⎡

⎣ N ∇

−∇· 0

⎤⎦

︸︷︷︸A

⎡⎣ v

p

⎤⎦

︸︷︷︸x

=

⎡⎣ r

0

⎤⎦

︸︷︷︸b

(5.54)

represents the result. This system is obviously ininvertable, because the lower right ma-trix carries only zeros, therefore special treatment is needed. It is known as PressureImplicit with Splitting of Operators (PISO) proposed by Issa in [142]. It relies on a LU

5.4. Pressure Correction 81

decomposition of the coefficient matrix in eq. (5.54), which is described hereinafter closelyto the work of Lübke [143], that goes further into detail. It is given as⎡

⎣ N ∇

−∇· 0

⎤⎦

︸︷︷︸A

=

⎡⎣ N 0

−∇· −∇ · N−1∇

⎤⎦

︸︷︷︸L

⎡⎣ 1 N−1

∇

0 1

⎤⎦

︸︷︷︸U

(5.55)

that allows to formally solve the system segregated in four steps that follow out of

Lx = b ∧ Ux = x. (5.56)

As the upper right and lower left submatrices of L resp. U are zero, this is possible.The only problem that is left, is the inverse of N, of which the calculation is impossibleregarding numerical cost. It is therefore approximated based on the split up

N = diag(N)︸︷︷︸Nii

+ offdiag(N)︸︷︷︸Nij

, (5.57)

by neglecting the off-diagonal elements which leads to the easily calculable expression

N−1 ≈ 1Nii

. (5.58)

Every step of the iteration procedure is therefore known and eq. (5.54) will be fulfilledwhen applying eq. (5.56) multiple times in a row. However, it is beneficial regardingcomputation time to solve the momentum equation once before starting the PISO loopto get an initial guess v0, known as the momentum predictor step. Starting from k = 1the iteration sequence is given as follows.

1. vk =1

Nii(r − Nijvk−1) (5.59)

2. ∇ ·( 1

Nii

∇p)

= ∇ · vk (5.60)

3. vk = vk − 1Nii

∇p (5.61)

4. pk = pk (5.62)

The only operation an equation system is solved in, is step 2, known as the Poisson equa-tion for pressure. Excluding the optional momentum corrector step, all other operationsare explicit. Regarding a break criterion for the loop, it is good practice to use the initialresidual of the Poisson equation.

For medium to high Re flows, the PISO-algorithm will converge fast and is reliable. Forlow Re flows however, it has to be literally trimmed to enable it finding a solution. Duringthis work the following findings that are more of a practical nature were made.


1. It is most important to ensure a relatively low residual of approximately 1e-5 inevery time step. Otherwise simulations might blow up spontaneously.

2. A limited time step based on keeping the Courant number smaller than 0.1 willlimit the iterations per time step. For large time steps the method might diverge.

3. Leaving out the momentum predictor will generally contribute to stability. Allowingthe velocity of the old time step to be replaced by a new solution of the momentumequation initially will at best increase the needed iterations but might also lead tostagnation of the residual.

Following these points is costly, because the Poisson equation needs to be solved numeroustimes to reach the stated residual. The study performed in the following subchapter willgive a taste on the expectable effort for a simple problem.

A more efficient procedure is using block coupled PISO based on [144], that was intro-duced in foam-extend-3.1. It extends the continuity equation by the Poisson equation forpressure,7 which transforms eq. (5.56) into

⎡⎣ N ∇

∇· − 1Nii

Δ

⎤⎦⎡⎣ v

p

⎤⎦ =

⎡⎣ r

− 1Nii

∇ · (∇p)

⎤⎦ . (5.63)

The residual used for the block coupled iteration procedure is the initial residual of theequation system. In contrast to the segregated PISO, this also contains information aboutthe velocities. It also makes clear that the effort in one iteration is much greater as alinear equation of the dimension8 4N × 4N has to be solved. But as the coupling of p andv is stronger, the convergence rate is higher and less iterations are needed. However, therestrictions 1 and 2 from the segregated PISO were applied in every calculation neverthe-less. As the time needed solving the linear equation systems is the greatest part of thecalculation, the question might arise, if it is really faster to solve a much larger systemjust a few times than a small system multiple hundred times. A simple test representingthe worst possible case was executed to find this out.

5.4.1 Comparison of Segregated and Block Coupled Pressure

Correction

For a comparison of the segregated and block coupled PISO, the cavity case presentedin chapter 5.1.2 was solved for a Newtonian fluid. The time dependence of the upperwall was removed by setting g(t) = 16 and to prevent having to execute a transientcalculation the time derivative was removed from the momentum equation. The aim istherefore to find the steady state solution. This might seem far from the cases occurring

7Just as described for BSD, for the explicite diffusion term, first the gradient is interpolated on thesurfaces of the cell.

8N is the number of unknowns that corresponds to the number of cells.

5.4. Pressure Correction 83

in application, but it is not. As the reaction time of low-Re flows to changes is very low,the changes in the velocity field can be quite large between two time steps. The caseinvestigated here is however more relevant for a too largely chosen initial time step thatwill e. g. not resolve the start up. The results are documented in the following figure. Forinterpretation it is important to know that the convergence criterion of the linear equationsystem solvers9 were set to 1e-6 which represents the minimum reachable residual of thepressure correction. In both methods, the residual is defined as the initial residual of thelinear equation system.

1e4 1e31e3 1e21e2 1e11e1 1e01e0

1e0

1e-2

1e-4

1e-6

Iterations [-] CPU time [s]

resi

dual

[-]

PISO

Block coupled

Figure 5.13: Convergence plots for a low-Re Newtonian cavity flow. PISO and block coupledmethod are compared over iterations and computation time.

On the left, the comparison regarding needed iterations is shown. The block coupledmethod clearly outperforms the segregated method here. After four iterations its residualis in the region of 1e-6 whereas the segregated method needs approximately 6000 iter-ations. But as already stated before, this comparison is not fair, thus the residuals areplotted over time as well. It shows that in this case the benefit of block coupled treat-ment becomes relevant for residuals lower than 1e-3. At this point, the segregated methodalready performed 200 iterations whereas the block coupled method just completed itssecond. As the residuals to be reached need to be lower than 1e-5 it is clear which methodto use.

Experienced users that perform mainly calculations for higher Reynolds numbers, mightfind this low residual to be reached in each time step too conservative. This is howevernot the case, as can be seen for the velocity field displayed for a decreasing residual infig. 5.14.

As the block coupled method converges too fast, it was not possible to show the fieldat this density of different residuals. Therefore, results from the segregated method areused. The progress clearly shows that the field is indeed nonphysical until r ≈ 1e-5 isreached. This is important against the background, that its gradient will be inserted intoa constitutive equation. It is easy to imagine that doing so not contributes to the stabilityof a code. Following from the results of this chapter, the block coupled method is used.

9In both cases a preconditioned Conjugate Gradient solver was used.


0 1v/u0 [-]

4. Iteration (r ≈ 1e-1) 21. Iteration (r ≈ 1e-2) 130. Iteration (r ≈ 1e-3)

630. Iteration (r ≈ 1e-4) 2140. Iteration (r ≈ 1e-5) 10000. Iteration (r ≈ 1e-6)

Figure 5.14: Evolutions of the velocity field over the PISO residual.

5.5 Algorithmic Realization

The algorithmic realization chosen in this work consists out of three major loops. Theouter can be seen as the run time loop, as it contains only operations executed onceper time step. The middle loop is based on the PIMPLE method. Here the coupling ofpressure, velocity and stress is executed. The inner loop is for pressure-velocity coupling,based on the block coupled PISO. As they contain many operations, in this chapter also ashort summary of the main equations is given. However, most parameters are not namedagain, just references to the associated tables is given.

The general steps executed in the algorithm are shown in algo. 3. This might look com-plicated, but the idea behind it is very simple. Everything related to crystallization issolved once per time step, in between the flow is adjusted. It is important to know, thatthe solidification model is solved after the flow. This means during the steps before, χ isconstant and known. Either as initial condition or from the time step before. The firststep executed is an update the Arrhenius coefficient using

φArr(θ) = exp[E0

R0

(1θ

− 1θ0

)], (5.64)

5.5. Algorithmic Realization 85

based on the coefficients in tab. 4.4. This field is used in the temperature dependentadjustment of parameters via time temperature superposition. As the next step thePIMPLE loop starts using the following equations:

∇ · v = 0, (5.65)

ρ∂v

∂t− ∇ · (χTsolid + χTs) − χηvΔv = −∇p + ∇ · (χTp) − χηv[∇ · (∇v)], (5.66)

∂Ψ

∂t+ ∇ · (vΨ ) =

eα(tr(eΨ)-3)(e-Ψ -1)λφArr

+ (ΩΨ − ΨΩ) + 2B. (5.67)

They are solved based on block coupled PISO surrounded by a loop for stress correction.To keep the equations as short as possible, the abbreviation χ = 1 − χ is used. It isneeded as the stresses in fluid and solid are weighted equally, c.f. chapter 4.3. In thisarrangement of equations, the implicit threated terms are written on the left, the expliciton the right. According to the constitutive relations

Tsolid = 2ηsolidD, Ts = 2βφArrηref D, Tp = (1 − β)ηref(eΨ − 1)

λref. (5.68)

This means, all the stresses named on the left are diffusion terms. However, as in chap-ter 4.1 it was found out, that β = 0, the solvent term will not contribute to the matrix. Thediffusion term appearing on both sides originates from the BSD, explained in chapter 5.2.Because the viscosity used in this approach should always be in the same magnitude asthe zero viscosity, ηv = ηref.. As first step in the PISO loop, eq. (5.65) and eq. (5.66) aretransformed into ⎡

⎣ N ∇

∇· − 1Nii

Δ

⎤⎦

︸︷︷︸A

⎡⎣ v

p

⎤⎦

︸︷︷︸x

=

⎡⎣ r

− 1Nii

∇ · (∇p)

⎤⎦

︸︷︷︸b

. (5.69)

During the loop, the large 4N × 4N10 system of linear equations (SLE) is solved multipletimes, followed by a convergence check. It checks if the initial residual of the SLE wasbelow 1e-5. If this is the case, the PISO loop is broken, if not the loop starts again withan update of coefficients for the SLE. As PISO is the first step of the stress corrector loop,following the right side of eq. (5.67), named ΓΨ , is updated via algo. 1. The next step is tosolve the constitutive equations. In 3D, this is a system of six SLE to be solved segregated,in 2D four. Because the solution is in log space, it needs to be retransformed using thechange of basis tensor R, calculated using the Jacobi method as shown in chapter 5.1.3.It is applied in the calculation of polymer stress to calculate the matrix exponential ineq. (5.68). The final step of the PIMPLE loop is to check if the maximum residual ofthe six/four SLE for Ψ was below 1e-5. If this is not the case, the loop starts again withexecuting the PISO loop.

10N is the number of cells resp. unknowns.


The next step in the outer loop is to solve the evolution equation of the fluid-solid indicatorχ. For the cooling rate dependent model, which is the core of this work, this equation is

∂χ

∂t+ ∇ · (vχ) = max

⎛⎝0, − 2

D1/2(θ)

√ln 2π

exp

(−4ln 2

(θm(θ) − θ)2

D1/2(θ)2

)Dθ

Dt

⎞⎠ . (5.70)

It contains the parameter functions formulated in chapter 3.2.2. Based on the change ofχ, the source term for the equation that sums up latent heat

DQ

Dt= Δhcryst(θ)

Dχ

Dt= q, (5.71)

formulated in chapter 3.2.4, is solved. By using Q the relative crystallinity, also introducedin chapter 3.2.4, can be calculated by cr = Q/Δh0. In this equation, the maximumpossible crystallization enthalpy, named in tab. 3.3 is used. At this point, all fields thatcontribute to the heating of the melt are known, the energy equation is solved. Here,the source term of eq. (5.71) can be used again. Additionally, according to chapter 4.4, adissipation source term is considered in the fluid, hence

ρc(

∂θ

∂t+ ∇ · (vθ)

)− kΔθ = ρq + χ

(tr(TsD) + Sp

). (5.72)

As the EPTT model is used to describe the behavior in the molten state,

Sp = κtr(TpD) + (1 − κ)tr(Tp)

2λe

(α λ

ηptr(Tp)

). (5.73)

The blending factor was set to κ = 0.25, described in chapter 4.4 as well. Because aninfluence on the physical properties of the polymer was not modeled, density, heat capacityand heat conductivity are constant. They are set as ρ = 905 kg m−3, c = 2300 J kg−1 K−1

and k = 0.2 W m−1 K−1 based on own measurements and the data sheet of PP 575P. Ascan be seen in algo. 3, for each of the last three transport equations, one SLE was solved.With the update of temperature, the time step is completed, and time stepping continuesconsidering that the Courant number is smaller or equal to 0.1, see chapter 5.4.

This procedure was implemented in foam-extend-4.0. The basis of solver development isthe pimpleFoam solver. To be able to efficiently perform a pressure correction, parts ofthe solver pUCoupledFoam were imported. For calculating the change of basis tensor, anexternal solver based on the Jacobi method, see chapter 5.1.3, was used.

As the algorithm diagram shows, many SLE are solved. The most expensive of those is theblock coupled system, since it is four times as large as the other SLE. Unfortunately, it wasnot possible to solve it using a multigrid solver. It seemed as the geometric agglomeratedalgebraic multigrid solver couldn’t cope with the unusual conditioned equation system, asthe pressure correction didn’t converge with any setting provided in tutorials. The reasonis suspected in the great changes in diffusion, ranging from zero for χ = 0 and somethingproportional to 2.5e6 for χ = 1, c.f. the value of ηsolid in chapter 4.2.2. Because thepressure correction converged if a conjugate gradient solver was used, the source of thisproblem might be in the agglomeration.

5.5. Algorithmic Realization 87

New time step

Arrhenius update

Update A and b

Solve Ax = b

Convergence?

Update ΓΨ

Solve for Ψ

Calculate R

Update Tp

Convergence?

Solve for χ

Solve for Q

Enery equation

4N x 4N SLE

6 N x N SLE

Jacobi Method

N x N SLE

N x N SLE

N x N SLE

Yes

No

No

Yes

Algorithm 3: PIMPLE-based crystallization algorithm.

CHAPTER 6

Verification and Basic Testing

This chapter’s purpose is to verify and test the method developed. As it is not possible tosolve a flow problem including all modeling approaches analytically, first the crystallizationmodel is verified for the static state. At least in this way, it can be determined if themodel was implemented correctly. Following, the degree of complexity is increased toinvestigate the models behavior exactly. To knowledge of the author of the present work,a study like this has never been provided. The common procedure found in literatureis, that a solver is developed and immediately calculations are performed for complexgeometries. This also meets the publications [13, 14, 15] compared to the present work inthe introduction. The cause of this might however be, that a paper is the wrong formatfor publishing extensive results like they are presented in this chapter.

Purpose of the verification part is to find out dependencies and to check the plausibilityof results. As first case of basic testing, a 1D static cooling process is investigated. Theinfluence of cooling rate dependent modeling, domain dimensions on relative crystallinity,latent heat release, and time step is presented. Hereafter, a flow is introduced in the sameproblem. For this case all the relevant fields are evaluated in an exemplary manner and theinfluence of the flow on crystallization is investigated for the cooling rate dependent modeland its shear rate dependent extension. As last point the crystallization of a cooled cavityflow is investigated. A mesh study, an exemplary field evaluation and an investigation onthe influence of the lid velocity is presented.

88

6.1. Verification of the Crystallization Model 89

6.1 Verification of the Crystallization Model

To ensure the crystallization model was implemented correctly, a simple test based onthe investigation of a sample in a DCS is performed. Its aim is to apply a homogeneouscooling rate through the boundaries to a virtual sample. Homogeneous means, that thecooling rate in the sample is not affected by the heat released. The gradient of the coolingrate should therefore be small to non-existent. As the latter condition can not be met bydefinition, a small gradient can be ensured by choosing a large Fourier number, which isdefined as

Fo :=kTchar

ρcL2. (6.1)

The question on how large it should be chosen, is not obvious as it does only considercaloric heat, not latent heat. But as it represents the ratio of heat conduction rate to heatstorage rate and is inverse proportional to the cooling time, it is a good idea to choose itfar above one. Because the tests are performed around a cooling rate of θ = −10 K min−1

and the temperature range from 200 ◦C to 100 ◦C is investigated, the characteristic timeis chosen as Tchar = 600 s. To ensure a Fourier number as stated above, the length wasset to L = 1 μm. This value can be considered as very small considering the real samplesize for a standard DSC. However, it serves the purpose well and limitations given in anexperiment are not of interest in a simulation. With these parameters chosen, Fo ≈ 5.77e7in the investigations shown hereafter.

6.1.1 Problem Description

The simulation domain is a cube with a side length of 1 μm that is cooled from allboundaries. A transient boundary condition is set at the walls using

θwall = θ0 + θwallst. (6.2)

As initial condition θ = θ0 is set uniformly. For local evaluation the fields in the centercell are used. Here the latent heat source

q = Δhcryst(θ)∂θ

∂t(6.3)

is evaluated over time according to eq. (3.5). It is compared with the expected curves thatfollow analytically from the crystallization model combining eq. (3.12) and eq. (3.13). Inparallel, a global evaluation is performed. It is based on the wall heat flux calculatedusing

qwall = −∫

Ak

∂θ

∂ndA. (6.4)

Integrating qwall over time will sum up the overall enthalpy difference

Δh =∫

qwalldt. (6.5)

90 Chapter 6. Verification and Basic Testing

The analytical expression is easily calculated by adding a caloric part and the latent heatfollowing out of the combination of eq. (3.16) and eq. (3.20), hence

Δh(t) = cθwallt + Δhcryst(θwall)χ(θwalls, T = θwall). (6.6)

This allows a verification of the implementation in the energy equation.

6.1.2 Results

Results are presented for the two static modeling approaches. The investigated tempera-ture range starts at θ0 = 473.15 K and ends at = 373.15 K, only the temperature rangeof crystallization is evaluated however. A mesh of 5 x 5 x 5 cells was used and the timestep is fixed to 1e-3 s in all simulations. The test of the fixed rate model should not onlybe seen as a verification of the code, but also as an opportunity to see what it physicallymeans, if crystallization enthalpy is modeled as a constant. In contrast to the variablecooling rate model, that reproduces the measurements already discussed in chapter 2.1.2,it might not be obvious how it behaves regarding the occurring latent heat flux.

6.1.2.1 Fixed Rate Model

The parameters used for the fixed rate model are those for θ = −10K min−1, see fig. 3.5.Therefore, the first test was performed for θwall = −10K min−1 as well. The expected curveof the heat flux, as shown in fig. 6.1 left, is met exactly. A comparison with measurementspresented in fig. 2.4 also shows, that the expected value of qmax is met. But this is just ameasure for the accuracy of the model, not of the numerical implementation.

q[k

Wkg

−1]

θ [K]0

00

2

4

6

8

10

388 392 396 400400

50

100

100

150

200

200

250

300 500

5 K min-1

10 K min-1

20 K min-1

analytical

Δh

[kJ

kg−

1]

t [s]

Δh

cryst

Figure 6.1: Comparison of the simulated DSC to the analytical solutions for the fixed ratemodel. The time integration to obtain the enthalpy was started at t(θ = 423.15 K) = 300 s.

6.1. Verification of the Crystallization Model 91

The specific enthalpy released, evaluated at the boundaries, plotted on the right of fig. 6.1,is met exactly as well. The time integration starts at θ = 423.15 K, however. In this wayonly the relevant region is evaluated and Δh starts from zero. This plot illustrates well,that crystallization causes a jump by the value of crystallization enthalpy.

Core of the fixed rate modeling approach is, that crystallization enthalpy, bell width andpeak temperature are fixed. This means e.g. for a faster cooling rate, in a shorter time thesame amount of latent heat is released. Because of the constant width this relationship islinear. That is why for θwall = −20 K min−1 the bell magnitude is doubled, for θwall = −5K min−1 halved. The comparison in the plot of Δhcryst(t) illustrates this best. Regardingthe verification, it could be shown, that the analytical solution is met here exactly aswell.

6.1.2.2 Variable Rate Model

A comparison between analytical and numerical results for the variable rate model showsthat it was implemented correctly. Because in programming implementation, the fixedrate model is a special case of the variable rate model, this was an expected outcome. Asthe heat flux evaluation occurred in the domains center and the enthalpy evaluation overthe surface, two independent tests were passed. The verification of the implementation istherefore succeeded.

386 390 394 398 402

q[k

Wkg

−1]

θ [K]0

00

2

4

6

8

400

50

100

100

150

200

200

250

300 500

5 K min-1

10 K min-1

20 K min-1

analytical

Δh

[Jkg

−1]

t [s]

Δh

cryst

Figure 6.2: Comparison of the simulated DSC to the analytical solutions for the variable ratemodel. The time integration to obtain the enthalpy was started at t(θ = 423.15 K) = 300 s.

Regarding model behavior, it can be seen, that the bells are shifted to lower temperatureswith an increasing cooling rate. As the crystallization enthalpy and the bell diameter aremodeled rate dependent, the peak value does not linearly depend on the cooling rate. Asθm and D1/2 control the onset temperature of crystallization, the jump caused in the rightplot of fig. 6.2 does not start at the same level of enthalpy released. The same applies for


the completion of crystallization. Therefore, when connecting these points for all coolingrates, in contrast to the fixed rate model, the crystallization range is slightly at an angle.

6.2 Stefan-like Problem

The initial and boundary value problem studied in this subchapter is based on the Stefanproblem going back to the work of Josef Stefan [145]. But as the exact formulation usedhere differs from the original, that considers the formation of ice, it is just a Stefan-likeproblem. The interpretation is that an infinitely large plate with a constant height h andan initial temperature θ0 is cooled from the bottom and is isolated at the top. Due toits infinite dimensions normal to the height, it can be condensed to one dimension, whichmakes it a useful case for testing the crystallization model.


If the static case is considered, only the energy equation and crystallization model aresolved. The one dimensional domain is oriented in y-direction with the geometric originat the lower boundary, the initial condition is therefore

θ(y, t = 0) = θ0 ∧ χ(y, t = 0) = 0. (6.7)

For the energy equation, to let the crystallization front move in positive y-direction, theboundary conditions are

θ(y = 0, t) = θw ∧ ∂θ

∂y

∣∣∣∣y=h

= 0. (6.8)

Technically the crystallization model doesn’t need boundary conditions in the staticstate. The codes used for evaluation however request a declaration. Therefore, homo-geneous Neumann boundary conditions are set. As the material modeling is specific toPP 575P, the temperatures can not be set arbitrary. The initial temperature is thereforeθ0 = 200 ◦C, which ensures having a certain distance to the earliest temperature phasetransitions occurs. Referring to the model for θmax, this is 145 ◦C, c.f. eq. (3.17). Thelower boundaries temperature is set to θw = 10 ◦C, which can be considered as a commoncold water supplies temperature.

6.2.2 Comparison of Fixed and Variable Rate Modeling

The comparison of both models is done for a domain with h = 5 mm. The fixed ratemodels cooling rate was set to −10 K s−1. Different stages are evaluated based on theprogress of overall crystallization. For this purpose, the field of the phase indicator χ,relative crystallinity cr and relative overheating w.r.t the cooled boundary are evaluatedin steps of 20% in progress.

6.2. Stefan-like Problem 93

The comparison of χ shows, that the fixed rate model crystallizes faster than the variablerate model. Looking at the relative crystallinity, the reason for this can be found easily. Asfor the variable rate crystallinity is lower at each position of the domain after the process,this means crystallization took place at higher cooling rates than the rate inserted intothe fixed rate model. This means for the adaptive rate model crystallization takes placeat lower temperatures, which explains the lag.

0

0

0

0.2

0.2

0.2

0.2

0.4

0.4

0.4

0.4

0.6

0.6

0.6

0.6

0.8

0.8

0.8

0.8

1.0

1.0

1.0

1.0

fixed rate

variable rate

c r[-

]θ−

θ wθ 0

−θ w

[-]

χ[-

]

y/h [-]

10 s

10 s

10 s

40 s

40 s

40 s

75 s

75 s

75 s

105 s

105 s

105 s

180 s

180 s

180 s

Figure 6.3: Comparison of the fixed and variable rate model for the Stefan problem.

It can also be recognized that the fluid solid transition is much smoother for the variablerate model, meaning that it occupies more space in the domain. This effect is increasedwith the progress of crystallization. For t = 105 s the transition region even occupiesabout 25% of the domain. As the transition regions spatial expansion correlates with thelocal temperature gradient, this is expected because for the fixed rate model D1/2 is afixed value. As already discussed, crystallization takes place at a higher cooling rate, thanthe rate used in the fixed rate model, for the variable rate model D1/2 is always larger.This means the spatial expansion of the transition region will increase with a smoothingof the temperature field, which is exactly the effect observable.


The plot of final crystallinity at t = 180 s shows that the fixed rate model released aconstant amount of latent heat over the whole domain. As it is formulated for exactly thatpurpose, this is no surprise. In contrast, the adaptive rate model shows that crystallizationwas suppressed stronger in proximity to the cooled boundary. This is a well known effectin manufacturing processes, that is predicted here correctly. A physical explanation tothis effect is given in chapter 2 in relation to fig. 2.19.

Comparing the temperature field at the different stages of progress shows just small dif-ferences. This means the latent heat release does not dominate the process, it’s the cooledwall that does. A small tendency of the fixed rate model to show higher temperaturescan be spotted anyway, which is of course to be connected of the larger amount of latentheat that is released.

6.2.3 Influence of the Fourier Number

To study the influence of process velocity on crystallinity, a study varying the domainheight was performed. As it is known, that crystallinity is suppressed with an increasingcooling rate, this is also expected to be seen in the results. The dimensionless numberchanged by h is the Fourier number. It correlates with the cooling time, as alreadydiscussed in chapter 6.1. For the characteristic time, the time it took for the domain tofully crystallize was used. It is given for each thickness in the table included in fig. 6.4and leads to the values for Fo given in tab. 6.1.

0 0 0

0.2

0.2

0.2

0.2

0.4

0.4

0.4

0.4

0.6

0.6

0.6

0.6

0.8

0.8

0.8

0.8

1.0

1.0

1.0

1.0

h tcryst h tcryst

100 μm 0.32 s 2.5 mm 41 s

500 μm 2.85 s 5 mm 2.4 min

1 mm 8.51 s 1 cm 9 min

100 μm500 μm

1 mm2.5 mm

5 mm1 cm

y/h [-]

y inte

rface

/h[-

]

c r[-

]

t/tcryst [-]

Figure 6.4: Influence of the Fourier number on the temporal progress of the fluid-solid inter-face and final relative crystallinity.

For evaluation, the relative position of the fluid-solid interface, in relation to the domainheight is used. It is plotted over the relative time of crystallization, as shown in fig. 6.4left. It shows there is a small lag in the initiation of crystallization, that relative to theoverall crystallization time is large for large Fourier numbers. It is caused by the shiftingof θm to lower temperatures and the small crystallization time.

6.2. Stefan-like Problem 95

The right side of fig. 6.4 shows that for small Fo, close to the wall the relative crystallinityis smaller than in the remaining domain. When increasing Fo this effect vanishes, whichmeans thin domains have a constant crystallinity. For the fixed rate model, such a studycould be performed as well, but as the relative crystallinity is constant in this case, it wasseen as not useful.

Table 6.1: Fourier numbers for fig. 6.4

h 100 μm 500 μm 1 mm 2.5 mm 5 mm 1 cm

Fo 3.38 1.2 0.9 0.7 0.61 0.57

6.2.4 Influence of Latent Heat

If latent heat has an influence on a process can be calculated by the Stefan number, seee.g. [59]. For processes considered in this work, it is the relation of caloric heat releaseto latent heat release during cooling. If specified only for the temperature range phasechange occurs, it can be derived directly from the model developed. Using

Ste(θ) :=2cD1/2(θ)

Δhcryst(θ), (6.9)

based on the models defined in chapter 3.2.2, it can be calculated as a material function.As shown in fig. 6.5, for slow cooling processes the influence is high, for fast cooling it islow. As the relative crystallinity reaches zero for large cooling rates, the influence is lost.

1e-31e-2

1e-2

1e-1

1e-1

1e0

1e0

1e1

1e1

1e2

1e2 1e3

Ste

,c r

[-]

−θ [K s−1]

Ste cr

Figure 6.5: Influence of the cooling rate on the Stefan number and relative crystallinity.

The result of these basic thoughts can be seen when varying the domain height of theStefan problem and evaluating the influence of latent heat on the crystallization time. To


realize this study, for three cases shown in the previous subchapter, additional calculationswere performed just with the latent heat source deactivated. The results can be seen inthe following figure, all normalized to the case latent heat is considered.

0000

0.2

0.20.20.2

0.4

0.40.40.4

0.6

0.60.60.6

0.8

0.80.80.8

1.0

1.01.01.0

y in

terf

ace

/h[-

]

t/tcryst [-]t/tcryst [-]t/tcryst [-]

with source

w/o source

h = 100μm h = 1 mm h = 1 cm

Figure 6.6: Influence of latent heat on the crystallization progress for different domain heights.

Again, the interface progress is evaluated over time. As predicted, latent heat has a greatinfluence on the crystallization progress for thick domains, as the cooling rates are smallin this case. The influence is lost with a decreasing height, as this increases the coolingrates. For the 1 cm domain, a speed up of 35% is determined if latent heat is neglected.For h = 1 mm it is 26% and for h = 100 μm 20%. In summary, this study shows that fortypical dimensions of technical processes latent heat has to be considered.

6.2.5 Influence of the Time Step Size

For static processes just the local time derivative occurs in the evolution equation for χ,c.f. eq. (5.70). Therefore, a study on the influence of the time step was carried out forh = 5 mm. The evaluation is done using the progress of the fluid-solid interface over timeand the final field of χ over the domain height, shown in the following figure.

The plot on the left shows that there is no influence on the progress of the fluid-solidinterface over time in the investigated range of time steps. For the χ-field however thereis, as the right plot shows. When the time step is chosen larger than Δt = 1e − 3 svalues greater than one will occur, which has to be seen very critical, as this means morelatent heat is released. Therefore, it should be chosen carefully. In simulations performedincluding a flow, the restriction given for the pressure correction, which is that the Courantnumber is kept below 0.1 (c.f. chapter 5.4), was found out to be sufficient.

6.3. Crystallizing Channel Flow 97

000

0.2

0.20.2

0.4

0.40.4

0.6

0.60.6

0.8

0.80.8

0.9

1.0

1.0

1.0

1.0

1.1

1.2

1.3

1.4

1.5

1.6

y/h [-]

y inte

rface

/h[-

]

t/tcryst [-]χ

[-]

1e-3 1e-31e-2 1e-21e-1 1e-15e-1 5e-1

timestep [s] timestep [s]

Figure 6.7: Influence of the time step size on the solidification of the Stefan problem.

6.3 Crystallizing Channel Flow

As a simple test case for the interaction between fluid and solid, a pressure driven channelflow is simulated. Just as in the Stefan problem, the lower boundary is cooled, the upperadiabatic. This causes the channel height to decrease over time, which causes the velocityto decrease, as the solution for a planar channel flow of a Newtonian fluid predicts. Coolingthe melt will additionally increase the viscosity which increases the deceleration of theflow. In this subchapter fields are evaluated and the influence of the pressure gradient oncrystallization is investigated. Furthermore, the shear rate dependent model is tested.


For temperature the same boundary conditions as for the Stefan problem were set, theinitial condition is the static state. For velocity no slip boundary conditions are appliedat both boundaries. Homogeneous Neumann conditions are set for polymer stress. As forthe Stefan problem, 100 equidistant cells were used over the domains height. All resultspresented are for h = 5 mm. A one dimensional setup could not be realized for solvingthe flow, the channel length was therefore set to L = 1 mm and the pressure gradient isrealized by the setting a pressure difference Δp between in- and outlet.

6.3.2 Exemplary Field Evaluation

This chapter exists to give a detailed insight into the relevant fields during crystallization.The results presented are for the case that Δp/ρ = 25 m2 s2 which corresponds to a pressuregradient of 226.25 bar m−1. As can be seen in fig. 6.8, at the time step presented theprogress of crystallization is 33%. In the fluid part the maximum velocity is approximately1 cm s−1, in the solid part it is well below this value as it was intended by the modeling


approach chosen. The overall viscosity, defined as

η = (1 − χ)ηrefφArr(θ) + χηsolid, (6.10)

shows that in the fluid part the viscosity increases towards the fluid-solid interface, whichcauses the velocity profile to be asymmetric and smoothed out strongly towards the in-terface. This will turn out to be of great importance for the shear rate dependent crys-tallization model formulated. As the solidified region is modeled viscous, it is possible tocalculate a solid stress, according to eq. (4.26). Comparing the shear components of Tsolid

and Tfluid, according to eq. (4.25), shows that the fluid stress fades out at the interfaceand the solid stress fades in, just as intended by the weighted approach. It is interestingto see that the fluid stress level is approximately continued in its curve. As the viscosityis much greater in the solidified part, this leads to much smaller velocity gradients.

0

0

0

0

0

00.2 0.4 0.6 0.8 1.0

300

600

-40

60

10

1

1

χ[-

]

y/h [-]

u[c

ms−

1]

τ xy

[kP

a]η

[kP

as]

Sp

[kW

m−

3]

fluidsolid

Figure 6.8: Exemplary evaluation of relevant fields for a crystallizing channel flow.

6.3. Crystallizing Channel Flow 99

Dissipation of energy will play a key role in following studies, therefore it is plotted hereas well. Its maximum is at the top wall. Close to the fluid-solid interface Sp is notas large because velocity and polymer stress are smoothed out here by the temperaturedependence of the viscosity.

6.3.3 Influence of the Pressure Gradient

When increasing the pressure gradient, interesting states can be created. They show theimportance of energy dissipation. To realize this study, the pressure difference was variedas fig. 6.9 shows. For evaluation, the fluid-solid interface location and the maximumvelocity in the fluid part is plotted over time.

000

0.2

0.4

0.50.5

0.6

0.8

1.01.0

1.0

1.51.5 2.0 2.0

y inte

rface

/h[-

]

t/tcryst,stat [-]t/tcryst,stat [-]

um

ax

[ms−

1]

1e0

1e-1

1e-2

1e-3

1e-4

1e-5

static

2526.5

26.62526.667

26.726.8

Δp/ρ [m2 s−2]

Figure 6.9: Influence of the pressure difference on the crystallizing channel flow.

It was found out that this case reacts very sensitive on changes in the pressure gradient,just above the pressure difference investigated in the previous subchapter. As can be seenin fig. 6.9 left, slight increases in the pressure difference will cause great retardation ofthe solidification process. When passing Δp/ρ = 26.667 m2 s−2 the process even stops aty/h = −0.28 for Δp/ρ = 26.7 m2 s−2. The plot on the right shows, for the cases that fullycrystallize a lower value of velocity is reached. This has to exist since the solid phase ismodeled highly viscous and not as an actual solid. For the cases stagnating, the velocitywill remain at the level reached. Both phenomena can be easily explained.

What is happening when increasing the pressure gradient is that energy dissipation isincreased as well. In the stagnating cases just enough energy is dissipated to feed thecooled boundary completely. This represents a stable state and therefore the process isstopped. The answer to why the system reacts so sensitive lays in the material model.When taking the approximate start value of all presented results as u ≈ 0.1 m s−1 and therelaxation time as λ ≈ 0.16 s (see tab. 4.4), the Deborah number De = uλ/h ≈ 3.3. Thisis just at the transition to the power-law-like region of the viscosity curve, c.f. fig. 4.3.Since the kinetic energy is not limited for pressure driven flows, passing this point only


slightly will increase the occurring velocities strongly. Therefore, small changes in thepressure difference cause a great retardation of crystallization.

6.3.4 Shear Rate Dependent Crystallization Model

To investigate the influence of the shear rate on crystallization, a small study using theshear rate dependent extension formulated in chapter 3.2.3 was executed. For threepressure differences, the case discussed in the previous subchapter was calculated. Theoutcome was, that it would not matter if the extension is used or not. The reason for thisis documented in the following graphs.

0000

0.2

0.20.2

0.4

0.40.4

0.6

0.60.6

0.8

0.80.8

1.0

1.01.0 1.2 5 10 15 20 25 30

y/h

[-]

γ [s−1]u [cm s−1]χ [-]

25

26.5

26.7

Δp/ρ [m2 s−2]

Figure 6.10: Evaluation of velocity and shear rate for a crystallizing channel flow.

Left, the phase indicator is plotted over the channel height for every pressure difference.The channel is therefore frozen to one third, which is marked with the black line in alldiagrams. In the middle diagram, the velocity profile is plotted over the channel height.As already presented in chapter 6.3.2, for all pressure differences, the profile flattens outtowards the fluid-solid interface. This results in the shear rate decreasing in this directionas well, as shown in the right diagram. In the process-zone of crystallization, therefore noinfluence of the shear rate is present. This was found out to be the case in all simulationsperformed in this work.

This clearly has to do with the rheological modeling of solidification chosen in this work.As the weighted approach described in chapter 4.3 is used, no sharp interface exists. Itis rather a diffuse region that has a certain dimension itself. Additionally, just in frontof the crystallization zone, the viscosity is increased due to cooling. Together this causesthe velocity to fade out instead of showing a kink, as it would be expected for a wall.

A possible study that could be performed at this point is the influence of the blendingfunction. Sharper formulations might increase the shear rates in the crystallization zone.But this makes no sense if there is no experimental background. That the measurementspresented in chapter 2.2.2 are certainly not suitable for finding this function, was already

6.4. Crystallizing Cavity Flow 101

discussed. Therefore, no investigations going deeper into this topic were carried out. Astarting point of those could however be an implementation of functions given in [45],as discussed in chapter 4.3. But the blending function will definitely not be exclusivelyresponsible for the known effects not to occur.

As already stated in chapter 2.3.4, there is also a strong dependence on the deformationhistory of the melt. Clearly this is given since the fluid-solid interface moves normal tothe flow direction. The question is how to model this dependence. From the experimentsperformed in this work, it is not possible to get information about the memory behaviorwhen shear rates are applied. Experiments from literature just focus on shear deformation.The shear rate dependent approach was therefore not investigated further.

6.4 Crystallizing Cavity Flow

The freezing cavity case relies on the existence of a steady state solution, for which thedomain is not fully crystallized. Because the solid is modeled as a highly viscous fluid,the model is not capable of handling contact with the moving lid right. So just as forthe channel flow, this means energy dissipation will feed the cooling source completely.It is dealt with to investigate the interaction between solidified regions and the melt in amultidimensional flow. First, a mesh study to find a sufficient spatial resolution is carriedout. Then all important fields are presented for one value of the lid velocity. Thereafter,the lid velocity is varied to show the influence on crystallization.


Just as for the validation of the LCR, a cavity flow without singularities in the uppercorners is used, see fig. 5.1. The domains dimension is 1 cm x 1 cm to create a coolingprocess that is slow. Boundary conditions for velocity, pressure and polymer stress corre-spond to those of the LCR benchmark. For temperature, the lower wall is set to a fixedvalue of θwall = 283.15 K, all other walls are adiabatic. Simulations start from the staticstate with the overall domain set to θ0 = 473.15 K.

6.4.2 Mesh Study

To find a proper mesh for the crystallizing cavity case, a setup with the lid velocity peakvalue set to u0 = 12.5 cm s−1 was studied. The resolution was increased in steps of 25cells in each direction, starting from 50 x 50 cells. All cases were evaluated based on theposition of the fluid-solid interface. It is plotted using the isocontour of χ = 0.5. Theresult of this study is plotted in fig. 6.11. It shows that after 125 x 125 cells the positionof the interface does not change noticeably any more. This mesh was consequently usedin all computations to generate the results presented in this subchapter.


50x50 75x75 100x100

125x125 150x150

Figure 6.11: Mesh study for the crystallizing cavity flow with u0 = 12.5 cm s−1 .

6.4.3 Exemplary Field Evaluation

Subject of this chapter is to take a closer look at the relevant fields during crystallization atu0 = 12.5 cm s−1. For this purpose, three stages of crystallization progress are comparedto each other. As a stage at the beginning of the process, results are presented at t = 5s. For the middle at t = 40 s and for the end at t = 120 s. First the influence of thesolidified region on velocity and streamlines is examined.

0 0.125|v| [m s−1]

Figure 6.12: Streamlines and velocity field at t = 5, 40, 120 s for u0 = 12.5 cm s−1.

As can be seen, the growth of the solidified phase squeezes the vortex into direction of thelid, just as expected for an actual wall. By the effective reduction of the fluid domainsheight, the velocity in the lower half of the main vortex is slightly increased. In eachbottom corner one small vortex exists, both decrease in size during the process. The nextstep is the examination of the temperature field.


10 200θ [◦C]

Figure 6.13: Temperature field at t = 5, 40, 120 s for u0 = 12.5 cm s−1.

Clearly at the beginning of the process most of the melt still has the initial temperature.There is a small layer above the fluid-solid interface that shows clear signs of coolingby heat conduction. Because the velocity in this region is small, only weak effects ofconvection are visible in the bottom left. Here cooled melt is transported upwards asthe direction of rotation is clockwise. Therefore, at this position also the growth of χ isstronger, since there is the coldest region of the fluid domain. Looking at the next timeshows that this effect is increased, thus the height of the solidified region is noticeablygreater. In general, heat convection is driving the cooling process in the fluid part. Itscounterpart is energy dissipation that stops cooling after 120 s. Why this is the case isshown in the following figure.

1e1 1e7Sp [W m−3]

Figure 6.14: Energy dissipation at t = 5, 40, 120 s for u0 = 12.5 cm s−1.

As shown regarding fig. 6.12, in the lower part of the main vortex velocity increases duringthe crystallization process. This increases velocity gradients and stresses, that results intoa greater energy dissipation rate. But as the vortex moves upwards, this is also true tothe region close to the lid, which is the main source of energy. As this region is passed bythe melt first after being transported from the lower left corner upwards, for the steady


state in the top right a larger region that is approximately isothermal exists, c.f. fig. 6.13.The effect of increased stresses can be seen in the following figure as well.

-0.5 2Ψxx [-]

Figure 6.15: First normal component of Ψ at t = 5, 40, 120 s for u0 = 12.5 cm s−1.

As can be seen, close to the lid due to the reduction of the channel height, Ψxx is increasedby approximately 0.5. Because Ψ is scaling Tp logarithmically, there is a strong increasein stresses, that lead to enough dissipative heating to feed the cooled boundary.

6.4.4 Influence of the Lid Velocity

To study the influence of the lid velocity, the result for u0 = 12.5 cm s−1 is compared withresults for u0 = 10 cm s−1 and u0 = 20 cm s−1 in fig. 6.16.

1e1 1e7Sp [W m−3] cr [-] 0.850.4

Figure 6.16: Influence of the lid velocity on the crystallization process. The studied valuesare u0 = 10, 12.5, 20 cm s−1 shown from left to right.

This shows the thickness of the solid region at stagnation decreases with an increase in lidvelocity. The reason for this is obviously the energy dissipation rate, which is increased


by the process velocity. As the energy dissipation is also increased by the reduction of thefluid domain, the time until stagnation is decreased with an increasing lid velocity. Therelative crystallinity in the solid domain shows that close to the wall it is reduced for allcases in the same manner. Because in the lower part of the cavity the velocity is alwaysinfinitesimal, c.f. fig. 6.12, this is an expectable outcome.

CHAPTER 7

Application to Process Engineering Examples

The previous chapter’s purpose was to verify and test the method developed on exem-plary cases. As this was successfully for the cooling rate dependent model, it should betested using cases relevant to process engineering. Therefore, in this chapter two commonmanufacturing processes are considered. However, there is one obvious limitation thatmakes a reduction of complexity necessary. As the method is implemented in a singlephase solver, meaning that no second fluid phase like air is considered, flows containingfree surfaces can not be considered. To processes like mold filling or cooling in a waterbath it is therefore not applicable. The purpose of this chapter is to find out how themodels behave in those applications. It is not to do large numerical studies that involveexact modeling of real processes.

As first case investigated, a profile extrusion tool is chosen. Its dimensions are based ona real tool, it is however considered in a simplified manner. For a 2D case the steadystate fields are evaluated exemplary to get an insight into the crystallization process.Thereafter the influence of the process velocity is investigated in a small study. For oneprocess velocity results of a 3D simulation are presented as closing of this topic.

The second process investigated is a simplified injection molding case. As for profileextrusion, for one case the fields are evaluated exemplary. Then a study on the influenceof the injection velocity is presented. For one case a 3D extension is presented as well.As closing, a well known phenomenon discovered is presented.

106

7.1. Profile Extrusion 107

7.1 Profile Extrusion

The process this numerical model is inspired by, is an extrusion tool for manufacturinghollow profiles1. A tool like this is operated connected to an extruder, that supplies itwith a constant mass flux of melt. As shown in fig. 7.1, it consists of a conditioning andcooling section. Because the melt passes a rather complex flow channel before enteringthis tool, the conditioning section exists to allow stresses to relaxate and the flow tobecome uniform. As crystallization should not take place in this state, this section isinsulated and optionally heated.

Heating

Polymer melt

Conditioning Cooling

Water

Figure 7.1: Cut presentation of the shaping tool. The outer wall consist of connected plateshaving different purposes. An insulation layer is applied to the conditioning section. The innerwalls only purpose is to support the profile extruded.

At the final position of the tool a cooling plate driven by a water supply is mounted. As itis relatively short, the material is therefore exiting the tool only crystallized at the surface.At the position of the support, there is a void in the product. Realistic wall thicknessof products to be manufactures with such a tool are in the single digit millimeter range.Extrusion velocities in the single digit centimeter per second range are possible. In case ofthe real tool, this study is motivated by, the length of the cooling section is approximately5 cm. Based on this information the numerical model was created.

7.1.1 2D Simplification

The actual process the geometry in fig. 7.1 originates from, is a hollow square profile,that is exemplary simulated in chapter 7.1.2. For performing studies, it was simplified totwo dimensions, as 3D calculations are very costly. As a template for 2D simplificationthe geometry as drawn can be used directly. It is however not needed to calculate thewhole conditioning section. As studies for the extrusion velocity are intended, it is bestto set a block profile at the inlet of the tool. This means a certain length is needed for

1The dimensions are chosen similar to a tool that was used in the IGF-Project 19652 N. A projectregarding the extrusion of Wood-Plastic Composites the Chair of Fluid Mechanics of the University ofKassel was engaged in from 2017-2019 performing simulations.

108 Chapter 7. Application to Process Engineering Examples

the artifacts, that usually occur for this boundary condition, to fade. It can be estimatedby the product of velocity and relaxation time. Because of the limitations named, themaximum velocity that makes sense is 10 cm s−1. The inlet temperature chosen is 200 ◦Cfor which PP 575P has a relaxation time of approximately λ = 0.1 s. As an estimationfor the inlet length, therefore Uexλ(θinlet) ≈ 1 cm is used. Based on this estimation, theconditioning sections length was set to 2 cm. To have a representative wall thicknesses,the center of the range named before is chosen. The height is therefore set to 5 mm. Justas in the real geometry, the cooling length is set to 5 cm. All the named dimensions arealso annotated in fig. 7.2.

7.1.1.1 Problem Description

Fig. 7.2 shows there are three walls to be differentiated between. The top wall of theconditioning section is adiabatic and a no slip boundary condition is set. This results inthe melt entering the cooling section with a fully developed velocity profile and a constanttemperature θ0, set at the inlet.

θ,y = 0, u = 0, v = 0 θ,y = αw(θc − θ), uΓ = χuP , v = 0

θ,y = 0, uΓ = χuP , v = 0

θ = θ0

v = 0

u = Uex

p,x = 0

θ,x = 0

v,x = 0

u,x = 0

p = 0

20 mm70 mm

(conditioning) (cooling)

(support)

x

y

Figure 7.2: Boundary conditions for the 2D profile extrusion case. The height is 5 mm.

For the upper wall of the cooling section two special conditions are used. The first one isto avoid a numerical singularity. When setting a constant temperature, e.g. 10 ◦C, a jumpin temperature occurs between the upper walls of the conditioning and cooling section.At this point the convective term of the substantive time derivative will be singular, hencethe cooling rate is artificially increased. To prevent this behavior, a continuous transitionis created by mimicking a water cooling. This is done by setting the temperature gradientin reference to a virtual coolant temperature θc. It is calculated by applying the rules ofstatic thermal transmittance to the steel wall. The heat flux going through the wall, isset equal to the wall (surface) heat flux, therefore

k∂θ

∂y

∣∣∣∣∣Γ

!=kwall

dwall

(θc − θ) (7.1)


The wall material is assumed to be standard steel which means2 kwall ≈ 50 W m−1 K−1.Choosing a value for dwall is not as simple because this is the wall thickness. In real tools itis not a constant. It will locally change as there are usually many cooling channels insidea tool. Therefore, a value that makes sense had to be set. Considering that the pressureoccurring in such a tool is high and the possible number of channels is limited, it isimpossible to have thin walls. A range for dwall in the lower centimeter range, e.g. 2.5 cm,is therefore realistic. With this information, dividing eq. (7.1) by the heat conductivityof the polymer leads to

∂θ

∂y

∣∣∣∣∣Γ

=kwall

kdwall︸︷︷︸αw

(θc − θ) (7.2)

for which αw ≈ 10000 m−1 was decided to be a reasonable value. Regarding there areactually flows on both sides of the wall, this might seem highly inappropriate. Theother choice would however be to actually perform calculations for a real tool, based onconjugate heat transfer, which was considered as going too far.

For velocity a non-standard condition is set as well. Due to crystallization the materialtransforms into a solid, which is known to slip at the boundary. Generally, this is thereason why profile extrusion works. If the profile would not slip at the walls, it is easyto imagine that the profile would deform strongly when exiting the tool. The tangentialvelocity is therefore formulated as an interpolation between no-slip and full-slip using thephase indicator. An implementation of this condition is done by using the cell centersvalue uP as noted in fig. 7.2. In this way, just as for temperature, a jump in conditionsis avoided. There is actually a benchmark formulated for the sudden change in boundaryconditions, known as the stick-slip flow, see [147] or [148]. But this was seen as not theright condition. As described in chapter 2, during crystal growth fluid and solid regionsco-exist. Assuming that crystals slip and the fluid does not, is represented by the chosenboundary condition in a better way than by a sudden change.

At the profile support, the partial slip condition is set as well. In the following studies alsovery slow extrusion velocities are investigated. This means the fluid-solid interface willhave contact with the lower wall, therefore slip is considered here as well. Furthermore,because the support is not heated or cooled, it is set adiabatic.

As already mentioned, a block profile is set at the inlet and temperature is set to 200 ◦C.At the outlet a standard outflow boundary condition and the pressure reference is set.For Tp and χ a homogeneous Neumann condition is used at every boundary. The initialcondition is the static state at 200 ◦C. A uniform mesh with 50 cells over the channelheight is used. For cases there were issues resolving the thickness of the solidified layer,it was graded by the factor of two in positive y-direction. In the following study this isthe case after Uex = 1 mm s−1.

2Value taken from [146].


7.1.1.2 Field Evaluation

To take a closer look at the crystallization process in this tool, a relatively low processvelocity of 0.1 mm s−1 is chosen. For this value, the crystallization will be completedinside the tool and not just in a layer, as shown in the study of the next subchapter. Infig. 7.3 the most important fields are shown for the region around the inlet of the coolingchannel.

1

χ [-]

1.5e-4

0

0

|v| [m s−1]

473.15

283.15

θ [K]

τxy,fluid

850

-650

[Pa]

Figure 7.3: Evaluation of fields for the 2D profile extrusion case at Uex = 0.1 mm s−1.

The top figure shows the stream lines in color of the fluid-solid indicator. As this isa steady state result, in this way material points can be tracked during crystallization.This shows for particles closer to the top wall, the dimension of the crystallization zoneis smaller. The streamlines show melt is slightly deflected by crystallization taking placeat the beginning of the cooling plate. For the region the interface has contact with thesupport this is valid as well. This effect can also be seen in the velocity field. First melt isdeflected downwards, then upwards as the maximum is shifted. By the change in velocityboundary conditions due to crystallization, in this case a slow development of slip occurs.The temperature field looks typical for a flow that is cooled one sided. Due to low processvelocity, the melt is almost cooled down to coolant temperature. In the conditioningchannel the shear component of fluid stress is uniform, when reaching the crystallizationzone due to the deflection it is slightly increased. By definition, in the solid it is zero.


7.1.1.3 Influence of the Extrusion Rate

Increasing the extrusion rate will have strong influence on the crystallization process. Thereason for this is a rise in convective transport of heat. Therefore, the depth of coolingreached inside the cooling section is decreased drastically. The dimensionless measuredescribing the influence of convection is

P e :=ρcUexh

k, (7.3)

the Peclet number. In polymer processing it is generally large, because polymers have alow heat conductivity. That is why the velocity chosen in the last subchapter was so small.Only in this case crystallization can take place inside the cooling section. The followingfigure shows what happens, when the extrusion velocity is increased until a more realisticvalue. The corresponding P e-values are given in tab. 7.1.

1

0

cr [-]

0.01 cm s−1

0.03 cm s−1

0.075 cm s−1

0.1 cm s−1

1 cm s−1

Figure 7.4: Steady state crystallinity reached inside the cooling section for different extrusionvelocities Uex. The corresponding Peclet numbers are documented in tab. 7.1

The crystallized region will not have contact to the support any more and is stronglydecreased in thickness. However, this is a basic relationship known for thermal boundarylayers, c.f. the Graetz problem [149]. If anything, latent heat release and energy dissi-pation will decrease the effect of cooling and therefore let the solidified layer decrease inthickness faster than the temperature boundary layer in a normal flow. This would be aninteresting topic to study, can however not be provided in this work.

The influence Uex has on crystallinity is clear. When comparing cr at the upper wall for0.01 cm s−1 and 1 cm s−1 a decrease from 0.85 to 0.725 can be determined. The reasonfor this is the increased cooling rate, due to convection.


Table 7.1: Peclet numbers for fig. 7.3

Uinj [cm s−1] 0.01 0.03 0.075 0.1 1

P e [-] 5.2 15.6 39 52 520

7.1.2 Exemplary 3D Simulation

To test the solver in a 3D flow, the full geometry is calculated. But as the crystallized layeris very thin for realistic extrusion velocities and therefore not resolvable with a reasonablenumber of cells, only the case with Uex = 0.075 cm s−1 is simulated. The effort for thiscase is already large for the reasons named in chapter 5.4. Those were that only blockcoupled pressure correction was found out to reliably execute the pressure correction anda multigrid solver can not be used. In addition, the calculations were executed transient,which in this case means at least 400 s need to be calculated. Together with the time steprestrictions given for pressure correction and the crystallization equation, effort increasesdrastically with the cell number.


The 3D geometry uses the same boundary conditions and dimensions as named for the2D case in fig. 7.2. It is just extended to obtain a square hollow profile. As such a profileis symmetric, only one quarter is calculated, as can be seen in fig. 7.5.

15 mm

15m

m

10 mm

10m

m

(sym

m)

(symm)

(coo

ling)

(cooling)

(sup

por

t)

(support)

y

z

Figure 7.5: Boundary conditions for the 3D profile extrusion case.


The boundary conditions named in brackets correspond to those named in fig. 7.2. Onecondition that can be added to all walls is w = 0. As fig. 7.5 shows the outer sides lengthis 30 mm, the inners 20 mm, again resulting in a wall thickness of 5 mm. A uniform mesh,graded towards the outer walls, resolving this space with 25 cells is used. The overall cellcount using this resolution is 937,500 cells.

7.1.2.2 Evaluation

As for the 2D case, the solidified region does not have contact to the support. There is nospecial effect occurring in the corners regarding the positioning of the fluid-solid interface.Looking at the crystallinity however, there is, because it is increased. This is howeveran artifact of the relatively coarse mesh. There are issues resolving the crystallization atthe beginning of the cooling channel, that lead to an interruption of crystallization. Asslip is initiated at the walls this defect is just transported downstream. In the cornershowever the thickness of the crystallization zone is increased faster and there is no issueof resolution. When comparing cr close to the fluid-solid interface, the value is similar tothe 2D case at ≈ 81 %.

1

0

cr [-]

Figure 7.6: 3D distribution of relative crystallinity for Uex = 0.075 cm s−1.

The velocity component in extrusion direction is shown in fig. 7.7. Beginning from theinlet, again a slight deflection in direction of the support can be determined. In the coolingchannel, the clearance in the corners is higher. Therefore, the velocity is increased. Fora normal flow this would be expected as well, this relationship is therefore not influencedby the occurring slip.


1

0

u [mm s−1]

Figure 7.7: 3D distribution of u for Uex = 0.075 cm s−1.

The distribution of temperature is shown in fig. 7.8. Because of the increased velocity inthe corners, the effectiveness of cooling is decreased here. Therefore higher temperaturesreach further into the cooling section.

200

10

θ [◦ C]

Figure 7.8: 3D Temperature distribution for Uex = 0.075 cm s−1.

7.2. Injection Molding 115

The conclusion to be drawn from this simulation is, that 3D simulations based on thismodel are generally possible. However, performing reliable simulations is a costly matter.For the case calculated P e = 39, the relevant range is from 520-5200. A much finerresolution is therefore needed, if only in the wall proximity.

7.2 Injection Molding

In contrast to profile extrusion, injection molding is not a continuous process. As indicatedin fig. 7.9, the flow channel is a closed geometry containing vents that allow the air toescape. Once the channel is filled, the melt supply is stopped. During the filling process, incontrast to profile extrusion, only small pressure gradients occur. Wall slip will thereforenot be an issue, the melt is just crystallizing at the walls and a molten core passes. Thisis a well known phenomenon called fountain flow, well described in [1] and numericallyinvestigated e.g. in [3].

polymer melt vent

Figure 7.9: Principle sketch of an injection molding process.

Because the solver is not developed to consider a second phase like air3, to test it forsuch processes the problem must be redefined. A simple way to do this is to calculate achannel that is cooled from both sides. To enable a molten core to pass solidified regions,a no slip boundary condition is used at the walls.

7.2.1 2D Simplification

Injection molding is a process to manufacture very complex geometries. Obviously it isa bad idea to test the method using such a case. Therefore, just as for profile extrusion,a straight channel is simulated. To be able to compare it with the previous results, thesame dimensions and process velocities are used.

3It is actually no problem implementing this method in a multiphase solver like interFoam. But themixing approach is not suited for the modeling approaches used in this work.



As indicated in fig. 7.10, the walls boundary conditions are chosen symmetrically. Again,there is a conditioning section that is 2 cm long, followed by a cooling section of 5 cm.The inlet condition is a block profile with the magnitude of the injection velocity Uinj andof θ0 = 200 ◦C. In the conditioning section the walls are adiabatic and a no slip conditionis set.

θ,y = 0, u = 0, v = 0 θ,y = αw(θc − θ), u = 0, v = 0θ = θ0

v = 0

u = Uinj

p,x = 0

θ,x = 0

v,x = 0

u,x = 0

p = 0

20 mm

70 mm

(conditioning) (cooling)x

y

Figure 7.10: Boundary conditions for the 2D injection molding case. The height is 5 mm.

For the cooling section the wall temperature gradient is set to mimic a water coolingwith θc = 10 ◦C as described in chapter 7.1.1.1. At the walls, no slip is allowed. Thepressure reference is set at the outlet, standard outflow conditions are used. HomogeneousNeumann conditions are set for Tp and χ at all boundaries, for p at all remaining. Theinitial condition is the static state at θ0 = 200 ◦C. To perform studies Uinj is varied. Onlyvalues are covered, for which the channel does not fully crystallize, as in this chase wallslip needs to be considered for correct modeling.

7.2.1.2 Field Evaluation

For an extended field evaluation, the steady state results of a simulation with Uinj = 1cm s−1 are presented. In fig. 7.11 the region around the inlet of the cooling channel isshown. In the top figure, again it can be seen that the approach chosen for the solidis capable of deflecting melt. The stream lines are just as expected for a contractionflow. Due to the contraction also the horizontal velocity is increased. At the beginningof the cooling channel, there are stronger vertical components as well. The temperaturedistribution shows that the core of melt is not cooled yet. Two things contribute to this.The most important factor is the Peclet number, see tab. 7.2. Energy dissipation willalso contribute to this effect. Latent heat however not, as this is the steady state and thevelocity in the solid phase are infinitesimal.

The stress components τxx and τxy generally developed as expected for a contraction flow.The only thing to be mentioned is that they decrease towards the fluid-solid interface. It is


caused by the transition model, that smooths the velocity gradients in the crystallizationprocess zone, due to the high solid viscosity, see chapter 6.3.2 and 6.3.4.

u [m s−1]

v [m s−1]

θ [K]

473.15

283.15

0

3e-2

2.5e-3

-2.5e-3

τxx [Pa]

τxy [Pa]

Sp [W m−3]

9e4

-2.5e3

4e4

-4e4

1e2

1.5e6

Figure 7.11: Evaluation of the 2D injection molding case for Uinj = 1 cm s−1: streamlines,velocity components, temperature, normal stress, shear stress and dissipation rate.

As for 2D flows, τxy is the dominating source term for the transport equation for τxx, theeffect is stronger for this component. The dissipation rate exclusively depends on stresses,therefore it is affected as well. To eliminate this behavior, a non-smooth transition modelcould be used. But as stated before, this is seen as not representing the physics as thetransition zone is indeed not a singular geometry.


7.2.1.3 Influence of the Injection Velocity

Simulating the injection molding case with the viscous solidification approach only makessense if the domain does not fully crystallize. If this is the case, immediately a Newtonianprofile appears as continuity must be fulfilled. Therefore, the first step of this study was tofind an injection velocity for which the channel is barely free. It is the top case presented infig. 7.12. When increasing the injection velocity from this value, the crystallized domainssize is decreased. As both walls are cooled, this happens for higher Peclet numbers thanfor profile extrusion, c.f. tab. 7.2 and tab. 7.1.

0

1

cr [-]

0.1 cm s−1

0.25 cm s−1

0.5 cm s−1

1 cm s−1

5 cm s−1

Figure 7.12: Influence of the injection velocity on the final relative crystallinity.

The existence of two cooling boundaries is also the cause for the overall lower crystallinity,compared to profile extrusion. The values of the highest crystallinity reached are for 0.1cm s−1 83% and for 5 cm s−1 75%. The wall crystallinity is not affected by the injectionvelocities investigated. It is always around 66%. As no-slip boundary conditions are used,the span of velocities investigated is not large and the temperature boundary conditionis the same, this makes sense.

Table 7.2: Peclet numbers for fig. 7.12

Uinj [cm s−1] 0.1 0.25 0.5 1 5

P e [-] 52 130 260 520 2600


7.2.2 Exemplary 3D Simulation

For the 3D injection molding case, the same statements regarding numerical effort asfor profile extrusion are valid, see chapter 7.1.2. The average simulation time is shorterhowever. Nevertheless, just an exemplary calculation is presented. The geometry chosenis a square channel based on the 2D example.


As the square channel is symmetrical, it is possible to simulate one quarter under theapplication of symmetry boundary conditions, see fig. 7.13. At the outer walls the sameboundary conditions as in the 2D case were set, c.f. the expressions in brackets withfig. 7.10. The lengths of conditioning and cooling sections were chosen in the same manner.

z

2.5 mm

2.5

mm

(sym

m)

(symm)

(coo

ling)

(cooling)

y

Figure 7.13: Boundary conditions for the 3D injection molding case.

Regarding the resolution of the process, the same as in 2D could be achieved. The flowchannel is meshed with 25 x 25 cells in normal direction, resulting in 1,200,500 cells.

7.2.2.2 Evaluation

To evaluate the crystallization process over time, the fields at the outlet of the coolingsection are presented in fig. 7.14. On top the velocity component in channel direction isshown. Due to the reduction of the channel by the crystallization, the velocity increasedjust in the same manner as in the 2D case. In 3D however the reduction of the flowchannel is greater. In 2D, for the steady state the clearance is 4 mm. For the 3D caseit is 2.8 mm. When comparing the peak velocity with fig. 7.11, this can be seen as well.The reason for this difference is in 3D cooling in a second dimension is added.


u[m

s−1]

θ[◦

C]

Sp

[Wm

−3]

200

10

0

4e-2

2e6

1e5

t = 5 s t = 10 s t = 20 s

Figure 7.14: Velocity, dissipation rate and temperature at the outlet of the 3D injectionmolding case. The injection velocity is Uinj = 1 cm s−1

As the velocity is increased by the reduction of the channel diameter, stresses increaseand therefore the dissipation power increases as well. In the center no shearing exists,that is why the source value is low at this position. It is not zero however, as normalstresses contribute to it as well. By the reduction of the flow channel diameter over itslength, there are always normal stresses that correspond to elongation. But they are muchsmaller than the shear stresses, Sp is therefore a few magnitudes below the shear regions.

The temperature field shows a hot core of melt exist until the exit of the cooling channelis reached. In the center still 194 ◦C are reached.

To give an impression of the progress of crystallization over time, in fig. 7.15 left, theprogress of the fluid-solid interface is plotted every 2.5 s. It can be seen that it is sloweddown with time until the final contour is reached. As the cooling process in 3D is fasterthan in 2D, higher cooling rates are reached. This enlarges the regions of low crystallinityin wall proximity noticeably, c.f. fig. 7.12.


cr [-]

0

1

Figure 7.15: Left: χ = 0.5 isocontour at the outlet in 2.5 s steps. The steady state is reachedafter 20 s. Right: Final distribution of relative crystallinity at the outlet.

As last point the flow was investigated on elongation, since it is known to influence crys-tallization strongly, c.f. chapter 2.3.4. This means crystallization can occur faster inelongation dominant regions. It is an effect that is often associated to so called conver-gent flows, see [150]. This is exactly created by the crystallization process. It can beinvestigated using a field presented in [89]. The flow type indicator

ξ =IIID

−II3/2D

3√

32

, (7.4)

based on the principle invariants of the deformation rate tensor D. In equibiaxial elon-gation flows it takes the value ξ = −1, in planar flows ξ = 0 and for uniaxial elongationξ = 1. It was used to create the isovolume presented in the following figure.

ξ [-]

0.25

1

Figure 7.16: Region of uniaxial elongation created by the convergent reduction of the channel.

Hereby a large region of uniaxial elongation in front of the cooling section can be discov-ered. Furthermore, on the centerline melt will be stretched through the whole domain.This shows the importance to include elongation in crystallization models. Due to thereduction of the cross section, shear rates are increasing. This means, if shear enhanced


crystallization is included in a model, the fluid-solid interaction will lead to a faster growthof the solid region. The stretch region however, might be predicted wrongly, as stretchenhances crystallization in a greater manner, see chapter 2.3.4. In the right conditions,crystallization could therefore be induced on the centerline. For the case presented in thissubchapter this is however unlikely, as the temperature in the stretch region is high.

CHAPTER 8

Summary and Outlook

8.1 Summary

For semi-crystalline polymers it is possible to influence material properties in the manufac-turing step of products. This can be done influencing the development of crystallinity byslow or rapid cooling. Generally, crystallization can be enhanced by low cooling rates andsuppressed by high. For modeling and simulation of such processes considering thermalinfluences, many fields come together. Those are experimental investigations, materialmodeling and state of the art numerical techniques. Every of these points was covered inthis work, summarized below.

Chapter 2

Crystallization is treated experimentally and theoretically. Results from DSC analysisexecuted for cooling rates up until 75 K min−1 are presented. A reduction of crystallinityby 15% could be detected at the highest rate. From an investigation of the rheological be-havior could be drawn that during crystallization the viscosity increases strongly. A smallstudy on the influence of shear on the crystallization showed that it enhances crystalliza-tion. Measuring a limit value was not possible. Theoretical consideration of molecularprocesses during crystallization showed that impingements of crystals are the main causefor reduced crystallinity at high cooling rates. The sample used for investigations werefound out to be an isostatic polypropylene.

Chapter 3

Novel empirical models were formulated to be considered in simulations. A simple modelassuming a constant cooling rate was first formulated and a strategy to identify parameters

123

124 Chapter 8. Summary and Outlook

presented. As the self investigated cooling rates were too small to formulate a modelcontaining suppression of crystallization, the data was extended by results from literature.To adapt the fixed rate model parameter functions covering the range 1e-3 - 1e4 K s−1

were introduced. An experimental based shear rate extension of this model was proposed.For closing, a method to evaluate relative crystallinity in simulations is presented. It isbased on the latent heat released.

Chapter 4

Rheological modeling is performed using a weighted approach. Based on the fluid-solidindicator fluid and solid stresses are linearly blended. The fluid is described viscoelasticusing a temperature dependent Phan-Thien Tanner model. Based on own experimentsand values from literature given for isotactic polypropylene, the parameters were deter-mined. It was found out solvent contribution does not play a role and only minimal stainhardening has to be considered. The solid is modeled as a highly viscous Newtonian fluid.An analytical study showed, it is necessary to set the solid viscosity approximately 1000times higher than the fluids viscosity. An exemplary simulation showed that viscoelasticmodels will cause negative dissipation when considered the standard way in the energyequation. For a model fixing this behavior, parameters were identified. As closing a smallstudy was used to show that wall slip is not considered in the modeling as it might disturbthe flow and therefore hinder to perform studies concerning crystallization.

Chapter 5

Numerical methods used in the solver composed in foam-extend-4.0 were presented. Theimplementation of the log-conformation formulation was described and validated. An up-date of a stabilizing method for low solvent contribution flows was described and verified.It was found out that it is not possible to use state of the art higher order convectionsschemes in low solvent contribution flows. Therefore, the upwind scheme is applied in allconvection terms. The influence of pressure correction on results was fond out to be great.For this purpose, a block coupled method is applied to efficiently solve the flow. The al-gorithmic realization is that once per time step a full stress-velocity-pressure correctionis executed, followed by solving the crystallization model and the energy equation.

Chapter 6

The implementation of the crystallization model could be verified performing a virtualDSC. For a Stefan problem the fixed rate model was found out to fulfill its purposeto always release constant crystallization enthalpy. The variable rate model predicteddecreased crystallinity towards cooled walls correctly. Another study showed that anincrease in the Fourier number decreases overall crystallinity. Latent heat was found outto play a greater role in processes with lower Fourier numbers. In a following study themodeling approach was found out to work well in a channel flow. Interesting states inwhich crystallization is stopped due to dissipation of energy were discovered. The shearrate dependent model was found out not to have any influence as the mixed fluid-soldstress approach smooths velocities at the interface. Tests performed for a 2D cavity flowshowed that the interaction between fluid and solid regions works well.

8.1. Summary 125

Chapter 7

As final topic the method was applied to simplified cases of profile extrusion and injectionmolding. For profile extrusion it was found out that only thin crystallized layers exist forrealistic process conditions. Generally, the crystallinity of this layer is decreased with anincrease in process velocity. 3D simulations showed that the numerical effort is great andspatial resolution in the crystallization zone is circuital. For the injection molding case,because of the increased cooling, for higher process velocities greater layer thicknesseswere detected. The interaction between fluid and solid regions was found out to workwell for this case. A 3D simulation showed that due to cooling in a second dimension thecrystallinity is decreased further than in the 2D case. The findings were that the resultsobtained were promising and realistic problems will cause great numerical effort.

126 Chapter 8. Summary and Outlook

8.2 Outlook

There are certainly many points in this work, alternatives were pointed out or simplifi-cations were made. Following them or deciding not to simplify will open new topics tostudy. There are however two points that focus could be put on with priority. The topiccan be classified in a modeling and simulation part.

Modeling

For modeling crystallization, it is important to consider the influence of shear and elon-gation. As this was not the main topic of the present work it was not followed further. Aspointed out, smoothing caused by the linearly mixed rheological consideration decreasesshear rates strongly. Therefore, different approaches could be investigated to obtain arheologically sharp interface. If that represents physics is a question to be answered.Another possibility is to assume a fading memory on the deformation history, similar toviscoelastic models. This is however not possible without an experimental foundation.

Simulation

As the method developed is single phased, the cases of usage are limited. Therefore, itshould be implemented in a suited multiphase solver. This would allow to calculate moldfilling processes which are of great relevance. An implementation was already done into theinterFoam solver. However, the volume of fluid method it used introduces typical artifactsknown for highly viscous fluids. It is based on the definition of the phase indicator, thatallows mixed cells. Therefore, cells that might contain only 1% of melt might crystallizeand suddenly show a high viscosity. An example is given in the following figure.

Figure 8.1: Simulation of a fountain flow based on the model developed in this work.

As can be seen, a typical fountain flow develops in this simulation. The melt howeverdisconnects from the solid. What is happening here is that in cells that only contain afraction of melt, crystallization occurs. When continuing this process, the molten core willbe surrounded by highly viscous, solidified "air". An effect that can also be discovered inone of the publications this work was compared to in the introduction, see [15]. In orderto handle such processes a different treatment of the interface is needed.

APPENDIX A

EPTT Material Functions

Just a minority of the viscoelastic constitutive equations can be solved explicitly fortheir material functions. The vast majority have to be solved using the Newton-Raphsonmethod, which is described here. The starting point of linearization is to adapt theequation to the steady state. This means the substantial time derivative is set to zero,which in case of the EPTT model results in

0 = 2ηD + λ(LT T + TL

)− e(αλ

ηtr(T))T. (A.1)

Note that the indices were dropped for improved clarity in contrast to chapter 4. Then,if the shear viscosity and normal stress differences should be determined, a planar shearflow is used. For determining the elongation viscosity an uniaxial elongation flow is used.

As the procedure is always the same, independent of the flow type, here it is applied fora shear flow to investigate η(γ), N1(γ) and N2(γ). As can be seen in the eqns. A.2- A.5the non-zero entries of the equation are sorted into a vector f .

f1 =1λ

⎡⎣ − exp

(α

λ

η0(τ11 + τ22 + τ33)τ11

)⎤⎦ + 2γτ12 (A.2)

f2 =1λ

⎡⎣ − exp

(α

λ

η0(τ11 + τ22 + τ33)τ22

)⎤⎦ (A.3)

f3 =1λ

⎡⎣ − exp

(α

λ

η0

(τ11 + τ22 + τ33)τ33

)⎤⎦ (A.4)

f4 =1λ

⎡⎣η0γ − exp

(α

λ

η0(τ11 + τ22 + τ12)τ12

)⎤⎦ + γτ22 (A.5)

127

128 Appendix A. EPTT Material Functions

As according to eq. (A.1) this vector has to vanish, for a given γ the equilibrium has todetermined. This can be achieved by a linearization of the system. It is not hard to dothis analytically, it is just lengthy as this requires to calculate 16 differentials, thus theyare not listed here. The result is the following iteration procedure:

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

τ11

τ22

τ33

τ12

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

n+1

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

τ11

τ22

τ33

τ12

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

n

−

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂f1

∂τ11

∂f1

∂τ22

∂f1

∂τ33

∂f1

∂τ12

∂f2

∂τ11

∂f2

∂τ22

∂f2

∂τ33

∂f2

∂τ12

∂f3

∂τ11

∂f3

∂τ22

∂f3

∂τ33

∂f3

∂τ12

∂f4

∂τ11

∂f4

∂τ22

∂f4

∂τ33

∂f4

∂τ12

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

−1

·

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

f1

f2

f3

f4

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

n

. (A.6)

To find a proper residual the procedure should be viewed as a correction procedure inwhich the second term on the right side is the correction vector. In this case

r =

√√√√(Δτ11

τ11

)2

+

(Δτ22

τ22

)2

+

(Δτ33

τ33

)2

+

(Δτ12

τ12

)2

(A.7)

which should be lower than 1e-6 for each investigated shear rate.

To find out the material functions a discrete shear rate band has to be investigated andevaluated for each value of this list. But as the Newton-Raphson method will only convergeinto the right minimum if it’s starting value is in a certain proximity of the solution, fourrules should be followed.

1. The list should contain increasing values that start from a shear rate in the Newto-nian range.

2. Start values for the first investigated shear rate are: τ11 = τ22 = τ33 = 0 andτ12 = ηγ.

3. Start values for the iteration process of the next shear rate is the solution of theprevious shear rate.

4. The list values should not be too far apart. A promising density is 100 log-equidistant points per decade.

The outcome is a list of stresses that can be transformed into the Material functions by

η(γ) =τ12(γ)

γ, N1(γ) = τ11(γ) − τ22(γ) N2(γ) = τ22(γ) − τ33(γ) (A.8)

as done several times in this work to obtain plots as e.g. in fig. 5.7.

List of Figures

2.1 Testing chamber and sample pans of the Q 1000 DSC. . . . . . . . . . . . 8

2.2 Arrangement of thermocouples in a DSC cell. . . . . . . . . . . . . . . . . 9

2.3 DSC graphs for polypropylene (PP) and polycarbonate (PC). . . . . . . . 10

2.4 Example of a baseline subtraction for a measurement at -10 K min−1. . . . 11

2.5 Artifacts occurring for large sample masses. . . . . . . . . . . . . . . . . . 12

2.6 Evaluation of the measuring series performed with the DSC 6. . . . . . . . 13

2.7 TA-Instruments AR-G2 and Parallel Plate System . . . . . . . . . . . . . . 14

2.8 Operational limits of the AR-G2 . . . . . . . . . . . . . . . . . . . . . . . . 15

2.9 Visual cooling experiment performed with PP 575P at 5 K min−1[34]. . . . 16

2.10 Results of the rheological experiments compared to DSC curves. . . . . . . 18

2.11 Shear rate dependent investigation of crystallization. . . . . . . . . . . . . 19

2.12 Creation of a PP mer in polymerization. . . . . . . . . . . . . . . . . . . . 20

2.13 Methyl group orientation in syndiotactic polypropylene. . . . . . . . . . . . 20

2.14 Van der Waals volumes for syndiotactic polypropylene. . . . . . . . . . . . 21

2.15 Alignment of polymers for isoatctic and syndiotactic polypropylene. . . . . 22

2.16 Possible alignment of polymers in lamellae. . . . . . . . . . . . . . . . . . 22

2.17 Formation of crystals by radial growth of lamellae. . . . . . . . . . . . . . 23

2.18 Growth velocity of iPP spherulites. . . . . . . . . . . . . . . . . . . . . . . 24

129

130 List of Figures

2.19 Crystal formation in the proximity of a weld seam in iPP. . . . . . . . . . . 25

3.1 Spherulite growth after 15 s. . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Temporal progress of the impingements forming a Voronoi pattern. . . . . 30

3.3 Possibilities of parameter identification at -10 K min−1. . . . . . . . . . . . 33

3.4 Parameter identification for measurements showing artefacts. . . . . . . . . 33

3.5 Extended DSC data used for the definition of parameter functions. . . . . . 35

3.6 Influence on the shear rate extension on the crystallization behavior. . . . . 39

4.1 Comparison of the models in tab. 4.2. . . . . . . . . . . . . . . . . . . . . . 45

4.2 Strain hardening ratio of the exponential Phan-Thien Tanner model. . . . 46

4.3 Fitting of the EPTT model to the master curve of PP 575P. . . . . . . . . 47

4.4 Relative shifting function used for time temperature superposition. . . . . 48

4.5 Example of cells to be set to v = 0 in a solidifying channel flow. . . . . . 50

4.6 Sketch of the pressure driven two fluid channel flow problem. . . . . . . . . 51

4.7 Solutions of the two phase channel flow for different viscosity ratios. . . . . 52

4.8 Error of the interface velocity in comparison to a fixed wall. . . . . . . . . 52

4.9 Numerical study on trying to reach the solution for η1/η2=1/1024. . . . . . . 53

4.10 Transition model compared to experiments. . . . . . . . . . . . . . . . . . 54

4.11 Different sources of dissipation refereed to for the flow around a cylinder. . 55

4.12 Influence of elasticity blending. . . . . . . . . . . . . . . . . . . . . . . . . 57

4.13 Evaluation of a 0D oscillating flow w.r.t. its dissipative behavior. . . . . . . 58

4.14 Slip stick instability in a pressure driven channel flow. . . . . . . . . . . . . 60

5.1 Definition of the lid driven cavity case . . . . . . . . . . . . . . . . . . . . 65

5.2 Comparison of Ψ and Tp in the steady state of the cavity flow. . . . . . . 66

5.3 Residual plot of the Jacobi method . . . . . . . . . . . . . . . . . . . . . . 69

5.4 Comparison for the lid-driven cavity at We = 1 with [118]. . . . . . . . . . 70

5.5 Centerline velocity validation for E = 1. . . . . . . . . . . . . . . . . . . . 72

5.6 Validation of the centerline velocity for β = 0 at E = 1. . . . . . . . . . . . 72

5.7 Validation of the EPTT implementation for α = 0.1. . . . . . . . . . . . . 73

List of Figures 131

5.8 Comparison of the BSD implementations on cell level. . . . . . . . . . . . . 74

5.9 Location of the upwind and downwind points in dependence of the flux. . . 75

5.10 Limiters of tab. 5.3 plotted in Sweby diagrams. . . . . . . . . . . . . . . . 77

5.11 Local evaluation of velocity for different schemes. . . . . . . . . . . . . . . 78

5.12 Comparison of the Ψxx-field for different schemes. . . . . . . . . . . . . . . 79

5.13 Convergence plots for a low-Re Newtonian cavity flow. . . . . . . . . . . . 83

5.14 Evolutions of the velocity field over the PISO residual. . . . . . . . . . . . 84

6.1 Comparison of the simulated DSC to the analytical solutions for the fixedrate model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.2 Comparison of the simulated DSC to the analytical solutions for the vari-able rate model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.3 Comparison of the fixed and variable rate model for the Stefan problem. . 93

6.4 Influence of the Fourier number on the temporal progress of the fluid-solidinterface and final relative crystallinity. . . . . . . . . . . . . . . . . . . . . 94

6.5 Influence of the cooling rate on the Stefan number and relative crystallinity. 95

6.6 Influence of latent heat on the crystallization progress for different domainheights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.7 Influence of the time step size on the solidification of the Stefan problem. . 97

6.8 Exemplary evaluation of relevant fields for a crystallizing channel flow. . . 98

6.9 Influence of the pressure difference on the crystallizing channel flow. . . . . 99

6.10 Evaluation of velocity and shear rate for a crystallizing channel flow. . . . 100

6.11 Mesh study for the crystallizing cavity flow with u0 = 12.5 cm s−1 . . . . . 102

6.12 Streamlines and velocity field at t = 5, 40, 120 s for u0 = 12.5 cm s−1. . . . 102

6.13 Temperature field at t = 5, 40, 120 s for u0 = 12.5 cm s−1. . . . . . . . . . . 103

6.14 Energy dissipation at t = 5, 40, 120 s for u0 = 12.5 cm s−1. . . . . . . . . . . 103

6.15 First normal component of Ψ at t = 5, 40, 120 s for u0 = 12.5 cm s−1. . . . 104

6.16 Influence of the lid velocity on the crystallization process. . . . . . . . . . . 104

7.1 Cut presentation of the shaping tool. . . . . . . . . . . . . . . . . . . . . . 107

7.2 Boundary conditions for the 2D profile extrusion case. The height is 5 mm. 108

7.3 Evaluation of fields for the 2D profile extrusion case at Uex = 0.1 mm s−1. . 110

132 List of Figures

7.4 Steady state crystallinity for different extrusion rates. . . . . . . . . . . . . 111

7.5 Boundary conditions for the 3D profile extrusion case. . . . . . . . . . . . . 112

7.6 3D distribution of relative crystallinity for Uex = 0.075 cm s−1. . . . . . . . 113

7.7 3D distribution of u for Uex = 0.075 cm s−1. . . . . . . . . . . . . . . . . . 114

7.8 3D Temperature distribution for Uex = 0.075 cm s−1. . . . . . . . . . . . . 114

7.9 Principle sketch of an injection molding process. . . . . . . . . . . . . . . . 115

7.10 Boundary conditions for the 2D injection molding case. The height is 5 mm.116

7.11 Evaluation of the 2D injection molding case for Uinj = 1 cm s−1 . . . . . . 117

7.12 Influence of the injection velocity on the final relative crystallinity. . . . . . 118

7.13 Boundary conditions for the 3D injection molding case. . . . . . . . . . . . 119

7.14 Velocity, dissipation rate and temperature at the outlet of the 3D injectionmolding case. The injection velocity is Uinj = 1 cm s−1 . . . . . . . . . . . 120

7.15 χ = 0.5 isocontour at the outlet and final distribution of relative crystallinity121

7.16 Region of uniaxial elongation created by the convergent reduction . . . . . 121

8.1 Simulation of a fountain flow based on the model developed in this work. . 126

List of Tables

3.1 Parameters for eq. (3.17) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Parameters for eq. (3.18) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3 Parameters for eq. (3.20). . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1 Solvent contributions from literature for iPP . . . . . . . . . . . . . . . . . 42

4.2 Nonlinear extensions of the upper-convected Maxwell model. . . . . . . . . 44

4.3 Literature values of the strain hardening ratio. . . . . . . . . . . . . . . . . 46

4.4 Parameters for time temperature superposition. . . . . . . . . . . . . . . . 49

4.5 Corrective dissipation source . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.1 Source terms for eq. (5.3) and eq. (5.6). . . . . . . . . . . . . . . . . . . . . 63

5.2 Source terms for eq. (5.16) . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.3 Limiter formulation of standard schemes. . . . . . . . . . . . . . . . . . . . 76

6.1 Fourier numbers for fig. 6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.1 Peclet numbers for fig. 7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.2 Peclet numbers for fig. 7.12 . . . . . . . . . . . . . . . . . . . . . . . . . . 118

133

List of Algorithms

1 LCR stress update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2 Jacobi eigenvector iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3 PIMPLE-based crystallization algorithm. . . . . . . . . . . . . . . . . . . . 87

134

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