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DISS. ETH NO. 18698
On a Coarse-Graining Concept in Colloidal Physics with Application to Fluid and Arrested Colloidal Suspensions in Shearing Fields A dissertation submitted to ETH ZURICH
for the degree of Doctor of Sciences
presented by
Alessio Zaccone Master of Science in Chemical Engineering, Politecnico di Torino born 7th of September, 1981 citizen of Italy
accepted on the recommendation of Prof. Dr. M. Morbidelli (ETH Zurich), examiner Prof. Dr. H. J. Herrmann (ETH Zurich), co-examiner Dr. E. Del Gado (ETH Zurich), co-examiner Zurich 2009
Acknowledgements I am very grateful to Prof. Morbidelli for having given me the opportunity to work at ETH Zurich, for his guidance, and for the freedom accorded to me in the choice of the research topics. I am deeply indebted to Dr. Hua Wu and Dr. Emanuela Del Gado for having guided me throughout my PhD research activities with their precious advice. Finally, a good part of this thesis would not be there without the valid assistance and hard work by Daniele Gentili. Special thanks go also to Dr. Marco Lattuada for many discussions.
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The present work is dedicated to the memory of my grandfather, Cav. Uff. Guido Benzi.
“Hunc igitur terrorem animi tenebrasque necessest Non radii solis neque lucida tela diei Discutiant, sed naturae species ratioque. Principium cuius hinc nobis exordia sumet, Nullam rem e nilo gigni divinitus umquam.” T. Lucretius Caro, De Rerum Natura I, 146-150
“Contraddictio est regula veri” G. F. W. Hegel, Habilitationsschrift
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Abstract We poorly understand the macroscopic properties of complex fluids and of amorphous bodies in general. This is mainly due to the interplay between phenomena at different levels and length-scales. In particular, it is not necessarily true that the microscopic level (dominated by direct interactions) coincides with the level where the continuum description comes into play. This is typically the case in the presence of structural inhomogeneities which are inherent to all structurally disordered states of matter below close packing. As a consequence, the macroscopic response to external fields of either fluid or arrested disordered states is not well understood. In order to disentangle this complexity, in this work we build upon a simple yet seemingly powerful concept. This can
be
summarized
as
follows:
the
mesoscopic
length-scale
of
structural
inhomogeneities is assumed to be the characteristic length-scale of the effective building blocks, while the degrees of freedom of the primary particles are integrated out. Theoretical results are derived, in the present work, for the macroscopic response of fluid and dynamically arrested model colloidal states in fields of shear. The predictions of the coarse-grained theories and the applicability of the principle are tested in comparison with original simulation and experimental data.
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Sommario Il corrente grado di comprensione delle proprieta’ macroscopiche di fluidi complessi e solidi amorfi e’ insoddisfacente. Cio’ si deve soprattutto alla compresenza di fenomeni che hanno luogo a livelli e scale di lunghezza molto diversi. In particolare, nella maggior parte dei casi, non e’ detto che il livello microscopico (dominato da interazioni dirette fra i costituenti primari) coincida con il livello di descrizione dei continui o, come si suol dire, idrodinamico. Tutto questo e’ vero soprattutto a densita’ intermedie fra il limite diluito (dove teorie a due corpi sono solitamente valide) e quello concentrato (laddove di solito domina la repulsione da volume escluso). Di conseguenza, finora non e’ stato possible comprendere appieno la risposta macroscopica a un campo esterno, ne’ per sospensioni colloidali fluide ne’ per sospensioni colloidali arrestate dinamicamente. Al fine di districare codesta complessita’, in questo lavoro ci proponiamo di sviluppare un concetto semplice ma potenzialmente di grande impatto. Tale principio puo’ riassumersi nel modo seguente: il livello microscopico e quello idrodinamico vengono considerati separatamente utilizzando il concetto di “clusters” come particelle effettive o “rinormalizzate”. Percio’, in buona approssimazione, il livello delle interazioni microscopiche influenza piu’ che altro il processo di formazione di “clusters”. Quindi, la descrizione a livello dei continui o idrodinamico puo’ essere fatta applicandola direttamente a “clusters” trattati come fossero le particelle primarie costituenti il sistema e utilizzando come frazione volumica quella occupata dai “clusters”. Combinando questo principio di “coarse-graining” con metodi standard della meccanica statistica e dei continui, oltre che con originali studi sperimentali e di simulazione numerica, possiamo finalmente spiegare e in alcuni casi anche descrivere quantitativamente la risposta macroscopica a campi di taglio di sospensioni colloidali interagenti sia nel regime fluido (nei termini della viscosita’ del fluido) sia nel regime dinamicamente arrestato (nei termini del modulo elastico di taglio). Riportiamo anche osservazioni sperimentali, in sospensioni aggreganti semi-diluite, di fenomeni finora noti solo in sospensioni molto concentrate. Tali fenomeni (“shear-thickening”, “shear-induced jamming”, “yield-stress”) possono essere spiegati e interpretati solo utilizzando il concetto base di “coarsegraining” proposto e sviluppato in questa sede.
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Table of Contents 1 Introduction…………………………………………………..4 1.1 Fluid colloidal suspensions under shear flow………………………….………..4 1.2 Arrested colloidal suspensions as model amorphous solids…………………10 1.3 Overview………………………….………………………………………………...16
2 Kinetic theory of the shear-induced aggregation of Brownian particles……………………………………...……18 2.1 Introduction……………………………………………………..………………….18 2.2 Derivation, results, and discussion………………………………………………18 2.3 Overview……………………………………………………………………………31
3 Aggregation kinetics in nondilute colloidal suspensions under shear flow: theory and experiments ……………………………………..…………………………….32 3.1 Introduction………………………………………………………………………...32 3.2 Experimental………………………………………………………….……………33 3.3 Effective medium theory……………………………….……..…………………..36 3.4 Comparison with experimental data………………………………………….….39 3.5 Overview………………………………………….…………………………………43
4 Kinetics of shear-induced clustering and mesoscopic growth………………………………………………………….46 4.1 Introduction…………………………………………………………………………46 4.2 Materials and methods……………………………………………………………47 4.3 Kinetics of shear-induced clustering: an off-site small angle light scattering study……………………………………………………………………………………..49 4.4 Overview……………………………………………………………………………57
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5 The shear viscosity of aggregating colloidal suspensions………………………………………….……….59 5.1 Introduction……………………………………………………………….………..59 5.2 The viscosity of hard-sphere suspensions…………………………….………..59 5.3 The viscosity of aggregating suspensions: coarse-graining and the effective particle model………………………………………………………………................61 5.4 Comparison between experiments and model predictions…………………...63 5.5 Understanding and modelling the anti-thixotropy of colloidal suspensions…………………………………………………………………………….67 5.6 Overview……………………………………………………………………………69
6 Rheology of aggregating suspensions ………………..71 6.1 Introduction…………………………………………………………………………71 6.2 Dynamic frequency sweep on sheared suspensions after flow cessation: gelation transition and the emergence of rigidity………………………72 6.3 Shear-rate sweep on sheared suspensions after flow cessation: shear thinning and shear thickening………………………………................74 6.4 Overview……………………………………………………………………………79
7 The macroscopic linear response of amorphous solids: affine theory………………………………………….81 7.1 Introduction………………………………………………………………………...81
7.2 Continuum theory of shear elasticity in random solid states……………........82 7.3 Discussion and applications……………………………………………………...88 7.4 Overview……………………………………………………………………………90
8. The macroscopic elasticity of arrested attractive colloids: coarse-graining of structural inhomogeneity …………………………………………………………………...91 8.1 Introduction………………………………………………………………………...91 2
8.2 Model and predictions: the dense regime………………………………………92 8.3 Model and predictions: double ergodicity-breaking and the intermediate density regime………………………………………………………………………….98 8.4 Elastic phase diagram of attractive colloids…………………………………...101 8.5 Overview…………………………………………………………………………..102
10 Concluding remarks……………………………………103 Appendix A…………………………………………………..107 Appendix B…………………………………………………..108 References…………………………………………………...120 Curriculum Vitae…………………………………………....125
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1. Introduction 1.1 Fluid colloidal suspensions under shear flow 1.1.1 The Smoluchowski equation, shear-induced aggregation, and the microscopic level As opposed to polymer melts, the rheology of colloidal suspensions has remained a challenging issue for statistical mechanics till nowadays. This may be partly due to the higher sensitivity to shear of colloidal systems where even modest shear rates can result in significant perturbations. Another reason lies in the smaller length scales and geometrical environment which are relevant microscopically: since the 70’s the dynamics of polymers has been effectively described by means of a “mean-field tube” with >103 primary constituents (chains) [1], whereas the number of neighbours around a colloid particle, both in the liquid and in the solid state, is always
