Cover Page
The handle http://hdl.handle.net/1887/21709 holds various files of this Leiden University dissertation.
Author: Siemens, Alexander Oltmann Nicolaas
Title: Elasticity and plasticity : foams near jamming
Issue Date: 2013-09-12
Alexander O. N. Siemens
Elasticity and Plasticity:
Foams near Jamming
Copyright c Alexander O. N. Siemens
Cover design: my foray in IDL... Alexander O. N. Siemens Printed by Gildeprint, The Netherlands
Elasticity and Plasticity:
Foams near Jamming
PROEFSCHRIFT
ter verkrijging van
de graad van Doctor aan de Universiteit Leiden, op gezag van Rector Magnificus
Prof. mr. C.J.J.M. Stolker,
volgens besluit van het College voor Promoties te verdedigen op donderdag 12 september 2013
klokke 15:00 uur
door
Alexander O. N. Siemens
geboren te Washington D.C., Verenigde Staten van Amerika
in 1985
Promotiecommissie:
Promotor: Prof. dr. M.L. van Hecke
Overige leden: Dr. E.I. Corwin (University of Oregon, Eugene, USA) Dr. B.P. Tighe (Technische Universiteit Delft )
Prof. dr. E.R. Eliel
Prof. dr. ir. T.H. Oosterkamp
Casimir PhD Series, Delft-Leiden, 2013-21 ISBN 978-90-8593-162-1
Dit werk maakt deel uit van het onderzoeksprogramma van de Stichting voor Fundamenteel Onderzoek der Materie (FOM), welke financieel wordt gesteund door de Nederlandse Organisatie voor Wetenschappelijk
Onderzoek (NWO).
An electronic version of this dissertation is available at the Leiden University Repository (http://openaccess.leidenuniv.nl).
“For all that has been – Thanks.
For all that shall be – Yes.”
Dag Hammarskj¨old
To my family and friends
Contents
1 Motivation: Jamming of 2D Foams 1
2 Introduction 5
2.1 Jamming in a Simple Model . . . 7
2.1.1 Model . . . 7
2.1.2 The Jamming Point . . . 8
2.2 Jammed Materials are Not Ordinary Solids . . . 10
2.2.1 Contact Number . . . 11
2.2.2 Elastic Moduli . . . 13
2.2.2.1 Compression Modulus . . . 14
2.2.2.2 Shear Modulus . . . 15
2.3 Beyond Effective Medium Theory . . . 16
2.3.1 Nonaffine Deformations . . . 16
2.4 Experimental Review . . . 18
3 2D Foams in a Wedge Geometry: Set-up 21 3.1 Experimental Set-up . . . 22
3.1.1 Set-up . . . 24
3.1.2 Foam . . . 27
3.1.3 Temperature Control . . . 30
3.1.4 Imaging . . . 34
3.2 Leveling . . . 37
3.2.1 Protocol to Level . . . 40
3.2.2 Equilibrium Coordinates . . . 44
3.3 Compression Protocols . . . 46
3.3.1 Deflection Angle vs. Open Angle . . . 46
3.3.2 Dynamics of Different Protocols . . . 47 i
CONTENTS
3.3.3 Stress vs Strain Control . . . 48
3.3.4 Protocols . . . 49
3.3.4.1 Flat Protocol . . . 49
3.3.4.2 Corner Hole Protocol . . . 51
3.3.4.3 Slice Hole Protocol . . . 52
4 Jamming of a 2D Foam 53 4.1 Elastic Response in a Wedge . . . 55
4.2 From Gravity to Boundary Jammed States . . . 58
4.2.1 Determining T (τ2) . . . 58
4.2.2 Determining the Foam Area . . . 62
4.3 Results . . . 63
4.3.1 Using AF as Control Parameter . . . 65
4.3.2 G + K: A Comparison . . . 69
4.3.3 Residual Torque . . . 71
4.3.4 High Tilt Compressions . . . 73
4.4 Nonaffinity Near Jamming . . . 76
4.4.1 Nonaffine Deformation . . . 77
4.4.2 Variance in Compression Runs . . . 80
4.4.3 Foam Motion Signal . . . 81
4.4.4 Comparing Torque and Variance Signals . . . 85
5 Rearrangement of Foams 87 5.1 Previous Work . . . 89
5.2 Biaxial Set-up . . . 90
5.2.1 Set-up . . . 92
5.2.2 Shear and Strain . . . 96
5.2.3 Protocols . . . 99
5.2.3.1 Continuous Shear Protocol . . . 101
5.2.3.2 Start/Stop Protocol . . . 101
5.2.4 Affine Deformation and Variance . . . 102
5.2.5 Stick-slip in the Set-up . . . 104
5.3 Results . . . 107
5.3.1 Continuous Shear . . . 107
5.3.2 Start/stop Shear . . . 113
5.4 Conclusions and Outlook . . . 115
Summary 117
Samenvatting 121
ii
CONTENTS
Publication List 125
Curriculum Vitae 127
Acknowledgments 129
Bibliography 131
iii
