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Understanding the diffusion of a colloid in a polymer solution Wim Maeyaert Nicolas Vander Stichele Thesis voorgedragen tot het behalen van de graad van Master in de ingenieurswetenschappen: chemische technologie, optie kunststofverwerking en productontwerp Promotoren: Prof. dr. ir. P. Moldenaers Prof. dr. ir. J. Vermant Academiejaar 2011 – 2012 Master in de ingenieurswetenschappen: chemische technologie Understanding the diffusion of a colloid in a polymer solution Wim Maeyaert Nicolas Vander Stichele Thesis voorgedragen tot het behalen van de graad van Master in de ingenieurswetenschappen: chemische technologie, optie kunststofverwerking en productontwerp Promotoren: Prof. dr. ir. P. Moldenaers Prof. dr. ir. J. Vermant Assessoren: Prof. dr. ir. B. Van der Bruggen Ir. M. Vallerio Begeleider: Dr. ir. N. Reddy Academiejaar 2011 – 2012 c Copyright K.U.Leuven � Without written permission of the thesis supervisors and the authors it is forbidden to reproduce or adapt in any form or by any means any part of this publication. Requests for obtaining the right to reproduce or utilize parts of this publication should be addressed to Faculteit Ingenieurswetenschappen, Kasteelpark Arenberg 1 bus 2200, B-3001 Heverlee, +32-16-321350. A written permission of the thesis supervisors is also required to use the methods, products, schematics and programs described in this work for industrial or commercial use, and for submitting this publication in scientific contests. Zonder voorafgaande schriftelijke toestemming van zowel de promotoren als de auteurs is overnemen, kopiëren, gebruiken of realiseren van deze uitgave of gedeelten ervan verboden. Voor aanvragen tot of informatie i.v.m. het overnemen en/of gebruik en/of realisatie van gedeelten uit deze publicatie, wend u tot Faculteit Ingenieurswetenschappen, Kasteelpark Arenberg 1 bus 2200, B-3001 Heverlee, +3216-321350. Voorafgaande schriftelijke toestemming van de promotoren is eveneens vereist voor het aanwenden van de in deze masterproef beschreven (originele) methoden, producten, schakelingen en programma’s voor industrieel of commercieel nut en voor de inzending van deze publicatie ter deelname aan wetenschappelijke prijzen of wedstrijden. Preface A master thesis is a group assignment and we would like to use this page to thank all the people who helped and supported us during the realization of this scientific work. “ Genius does what it must, and talent does what it can” Bulwer Thanks to our promotors, Prof. Dr. Ir. P. Moldenaers and Prof. Dr. Ir. J. Vermant, for providing us the opportunity to conduct our master thesis under their supervision. Their constructive comments and support stimulated us to convert our interest in chemical engineering into scientific thinking. We would like to thank them for their help during our entire eduction in chemical engineering by providing a unique research and educational environment, together with all members of the CIT department including our fellow students. “ Habe einen guten gedanken, man borgt dir zwanzig” Ebner-Eschenbach We would like to express our in-depth gratitude to Dr. Ir. Naveen, our mentor during this work. Not only as a researcher but also as a friend Naveen influenced greatly our final year at CIT. We would like to thank him for all his guidance, eﬀort and time in making this master thesis better. His explanations about dynamic light scattering, the wonderful data processing software, as well as his patience in explaining all kinds of problems we encountered, were invaluable. i PREFACE We would also like to thank Ruth Cardinaels and Jeroen De Wolf for their support using the rheometer device, and Anja Vananroye for making the lab the safest place on earth. Special thanks to Denis Rodriguez Fernandez for providing ellipsoids coated with gold nanorods used for dynamic light scattering experiments in this master thesis and Ward Vanheeswijck for his help and support with our special Latex compiler and cozy times at the copy room. “La science se fait non seulement avec l’esprit, mais aussi avec le coeur” L. Pasteur We greatly appreciate the endless support of our parents during our engineering studies, and the opportunity they gave us to graduate as a chemical engineer. During those five years they stressed (many times) the importance of a good education, but more importantly they gave us the right personal values to develop into young responsible adults. Also special reference to our brothers and sister for their support, happiness and help during our time at the university and with this master thesis as a highlight. A final mention goes to our girlfriends, Elien and Alexandra, for their endless support and encouraging comments during the performance of this thesis. Even though they didn’t always follow us in our elaborations, they kept believing in our decisions. Wim Maeyaert Nicolas Vander Stichele ii Table of contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Samenvatting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of abbreviations and symbols . . . . . . . . . . . . . . . . . . xiv 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Aim and structure . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Literature study 5 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.1 Elastic materials . . . . . . . . . . . . . . . . . . . . 8 2.2.2 Viscous materials . . . . . . . . . . . . . . . . . . . . 8 2.2.3 Viscoelastic materials . . . . . . . . . . . . . . . . . 8 2.3 Bulk rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Microrheology . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4.1 Colloids . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4.1.1 Van der Waals forces . . . . . . . . . . . . . 12 2.4.1.2 Electrical double layer . . . . . . . . . . . . 12 2.4.1.3 DLVO . . . . . . . . . . . . . . . . . . . . . 13 2.4.1.4 Eﬀect of polymers on colloidal stability . . 2.4.2 Generalized Stokes-Einstein relationship . . . . . . . 14 15 2.4.2.1 Translational diﬀusion . . . . . . . . . . . . 16 2.4.2.2 Rotational diﬀusion . . . . . . . . . . . . . 18 2.4.3 Active microrheological techniques . . . . . . . . . . 19 2.4.4 Passive microrheological techniques . . . . . . . . . 19 2.4.4.1 General dynamic light scattering (DLS) . . 20 iii TABLE OF CONTENTS 2.4.4.2 Diﬀusive wave spectroscopy . . . . . . . . . 20 2.4.4.3 Video based particle tracking . . . . . . . . 20 2.5 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Materials and methods 23 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2.1 Polymers: Polyetheleneglycol or Polyetheleneoxide . 24 3.2.2 Beads . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2.2.1 Spheres . . . . . . . . . . . . . . . . . . . . 25 3.2.2.2 Ellipsoids . . . . . . . . . . . . . . . . . . . 25 3.2.3 Extra materials . . . . . . . . . . . . . . . . . . . . . 28 3.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3.1 Sample preparation . . . . . . . . . . . . . . . . . . 29 3.3.2 Rheometer . . . . . . . . . . . . . . . . . . . . . . . 31 3.3.2.1 Device . . . . . . . . . . . . . . . . . . . . . 31 3.3.2.2 Maxwell model . . . . . . . . . . . . . . . . 32 3.3.3 Dynmaic light scattering . . . . . . . . . . . . . . . . 33 3.3.3.1 General dynamic light scattering . . . . . . 33 3.3.3.2 Depolarized dynamic light scattering . . . . 36 3.3.3.3 DLS device . . . . . . . . . . . . . . . . . . 39 3.3.4 Comparison between bulk and microrheology . . . . 40 4 Experimental results and discussion 43 4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2 Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3 Dynamic Light Scattering of spherical particles and polymers 45 4.3.1 Silica particles in water . . . . . . . . . . . . . . . . 47 4.3.2 Latex particles in water . . . . . . . . . . . . . . . . 48 4.3.3 Polymers in water . . . . . . . . . . . . . . . . . . . 50 4.3.4 Latex particles in polymers . . . . . . . . . . . . . . 54 4.3.5 Comparison between DLS and rheometer . . . . . . 57 4.4 Ellipsoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Ellipsoids in water . . . . . . . . . . . . . . . . . . . 59 59 4.4.2 Ellipsoids in polymer . . . . . . . . . . . . . . . . . . 61 4.4.3 Comparison between DLS and rheometer . . . . . . 63 5 Conclusion and future research 65 A Specifications of the geometry 68 iv TABLE OF CONTENTS B Results PEG 35 kDa 70 C Matlab code for data processing 74 D Safety and hazard analysis 77 Bibliography 79 v Abstract In this scientific work, research is conducted on the use of microrheological techniques to characterize viscoelastic materials. In traditional rheological experiments, properties of complex fluids such as polymers are investigated using conventional rheometers. Recently however a new technology, called microrheology, has been developed in this field. In microrheology, rheological properties of complex fluids are investigated by tracking the movement of microparticles embedded in them. Microrheology has numerous advantages over bulk rheology and is becoming more important ever since its discovery. When adding particles to a fluid, a colloidal solution is formed. Although most colloids in nature are disklike and rodlike, current technology is based on spherical colloids. Due to the complex characteristics and the lack of suitable computational models, only minor research was done on the use of non-spherical particles in microrheology. Therefore, in this master thesis the use of nonsphercial, ellipsoidal particles will be investigated. While the spherical particles only yield a translational diﬀusion, ellipsoids also show rotational diﬀusion due to their shape anisotropy. By tracking this rotational diﬀusion more information about the viscoelastic behavior of the material can be obtained. Spherical particles in polyethylene glycol (PEG) are used to accustom to microrheology and prove the current state of the art in this field. To prove � �� the suitability of microrheology, the storage and loss modulus (G and G ) of the samples are investigated with both dynamic light scattering (DLS) and a traditional bulk rheometer. A modified, algebraical form of the general Stokes Einstein relation enables a comparison of both methods. As expected from literature, the use of microrheology with spherical particles was proven an excellent equivalent for bulk rheology. vii ABSTRACT After proving the value of microrheology for spherical particles, the second part of the thesis focuses on the possibilities of using ellipsoidal beads in this experimental technique. Regular light scattering measurements, used with spherical particles, include both translational and rotational diﬀusion in one value. But by placing a polarizer before the detection optics, scattering of the translational diﬀusion is suppressed and only rotational diﬀusion is observed. The aim was to calculate storage and loss modulus from this rotational diﬀusion and compare this result with bulk rheology, in order to prove the suitability of rotational diﬀusion in microrheology. But, as the ellipsoids used in this last part are stretched spheres in a prolate way, they are anisotropic in shape but still have isotropic scattering properties. Consequently they gave disappointing results with the polarizer, as the rotational diﬀusion was not recognized. To overcome this problem, the ellipsoids were covered with gold nanorods. The gold nanorods have excellent scattering properties and as they are attached to the ellipsoids, they scatter the rotational diﬀusion of the ellipsoids. Surprisingly however, it was discovered that not the size of the ellipsoids, but that of the gold rods instead, has to be used in calculations to obtain equivalent results between bulk and microrheology. Observing this results, this would imply for the first time that the probe size of the bead would diﬀer from the scattering size of the particle. One other explanation is that the nanorods detached from the ellipsoids due to the polymer solution and were moving freely in the polymer solution. Either way, for the first time the use of ellipsoids is shown in determining microrheology properties. Although further investigation is needed, this idea could considerably enlarge the application scope of microrheology. viii Samenvatting In dit wetenschappelijk werk wordt onderzoek gedaan naar microreologische technieken om viscoelastische materialen te karakteriseren. In traditionele reologische experimenten worden de eigenschappen van complexe vloeistoﬀen, zoals polymeren, onderzocht met conventionele reometers. De laatste jaren werden binnen dit domein echter een aantal nieuwe technieken ontwikkeld, waaronder microreologie. Hierin worden reologische eigenschappen van complexe vloeistoffen onderzocht via het nauwgezet opvolgen van de beweging van micropartikels die aan de onderzochte vloeistof werden toegevoegd. Omwille van zijn vele voordelen t.o.v. bulk reologie, blijft microreologie, vooral de laatste jaren, aan belang winnen. Door het toevoegen van micropartikels aan een vloeistof, ontstaat een colloïdale oplossing. Hoewel de meeste colloïdale deeltjes in de natuur voorkomen als schijfjes of staafjes, is de huidige technologie gebaseerd op sferische partikels. Door de complexe karakteristieken en het ontbreken van gepaste rekenkundige modellen, is er voorlopig slechts in beperkte mate onderzoek uitgevoerd naar het gebruik van niet-sferische partikels in microreologie. Desondanks zullen in deze master thesis de mogelijkheden van ellipsoïde deeltjes in microreologische technieken onderzocht worden. Waar bij sferische partikels enkel hun translationele diﬀusie kan gemeten worden, laten ellipsoïden, door hun anisotrope vorm, het ook toe hun rotationele diﬀusie te karakteriseren. Door ook deze rotationele diﬀusie te observeren, kan er meer informatie over het viscoelastisch gedrag van het materiaal achterhaald worden. Sferische deeltjes in polyethyleenglycol (PEG) worden gebruikt om de basisprincipes van microreologie te leren kennen en de huidige ontwikkelingen in dit domein te toetsen. Om de compatibiliteit tussen bulk en microreologie te bewi- ix SAMENVATTING jzen, worden de opslag en verlies modulus (G’ en G”) van verschillende samples onderzocht met zowel dynamische licht verstrooiing (DLS) als met een traditionele reometer. Een aangepaste, algebraïsche vorm van de algemene Stokes Einstein relatie laat toe om de beide resultaten met elkaar te vergelijken. Zoals beschreven in de literatuur, werd aangetoond dat microreologie met sferische partikels een geldig alternatief vormt voor bulkreologie. De ellipsoïden die vervolgens in de initiële experimenten gebruikt werden, waren uitgerekte sferen. Hierdoor hadden ze, hoewel anisotroop in vorm, isotrope verstrooiings eigenschappen. Op deze manier waren de resultaten met de polarisator, die de vertstrooiing van de rotationele diﬀusie niet herkende, niet zoals gehoopt. Als oplossing voor dit probleem werden de originele ellipsoïden bedekt met kleinere nanostaafjes uit goud. Deze hebben wel uitstekende verstrooiings eigenschappen, en aangezien ze bevestigd zijn op de ellipsoïden laten ze toe de rotationele diﬀusie van de ellipsoïden waar te nemen. Verrassend genoeg bleek uit deze laatste experimenten dat niet de lengte van de ellipsoïden gebruikt moest worden in de berekeningen om de compatibiliteit tussen bulk en microreologie aan te tonen. Het is waarlijk door de lengte van de gouden nanostaafjes mee te nemen in de wiskundige berekeningen, dat deze compatibiliteit wel werd aangetoond. Een andere mogelijkheid bestaat erin dat de goud nanostaafjes loskomen van de ellipsoïden in de polymeeroplossing. Uitgebreider onderzoek is zeker noodzakelijk, maar hoe dan ook wordt in dit werk voor de eerste keer microreologie gebruik makend van ellipsvormige partikels toegepast. Toekomstige experimenten zullen moeten uitwijzen indien het toepassingsgebied van microreologie hierdoor aanzienlijk vergroot kan worden. x List of Figures 1 Response of a material to a shear strain. . . . . . . . . . . . . . . 2 Typical frequency ranges for diﬀerent measurement devices [8]. . 3 Schematic representation of the rheometer. [8] 4 Stress/strain response of a material to a strain/stress application . . . . . . . . . . [11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 7 10 10 Schematic representation of the electro-static potential near a solid surface in a solution containing ions [13]. . . . . . . . . . . . 13 6 Eﬀect of adding polymers to colloidal solution: steric stabilization 7 and depletion[14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Eﬀect of adding polymers to a colloidal solution: eﬀect of bridging. 15 8 Evolution of the mean square displacement (MSD) of the particles as a function of the lag time τ [8] . . . . . . . . . . . . . . . . . . 17 9 Monomer polyethylene glycol (PEG) . . . . . . . . . . . . . . . . 24 10 Scanning electron micrograph (SEM) of silica particles [24] . . . 25 11 Rotation of ellipsoids in a fluid.(a) axisymetric rotation. (b) nonaxisymmetric rotation. . . . . . . . . . . . . . . . . . . . . . . . 26 12 Oblate and prolate ellipsoid. . . . . . . . . . . . . . . . . . . . . . 27 13 Transmission electron micrograph of polystyrene ellipsoids. . . . 27 14 Transmission electron micrograph of of gold nanorods. . . . . . . 27 15 Schematic representation of prolate ellipsoids covered with gold nanorods. [Courtesy: Sylvie Van Loon] . . . . . . . . . . . . . . . 16 28 Schematic representation of the critical overlap concentration C∗ [29] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 17 Schematic representation of the radius of gyration. [29] 30 18 Anton-Paar 501 MCR stress-controlled rheometer . . . . . . . . 32 19 Schematic representation of the cone and plate geometry [12] . . 32 20 Schematic representation of the Maxwell model . . . . . . . . . . 32 � . . . . . �� 21 Evolution of G and G described by the Maxwell model [31] . . 33 22 Schematic representation of dynamic light scattering (DLS) . . . 34 xi LIST OF FIGURES 23 Scattering oﬀ a small particle in an ideal solution by incident light [32] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Intensity measurement and autocorrelation function measured by dynamic light scattering [12] . . . . . . . . . . . . . . . . . . . . . 25 36 Schematic representation of the general dynamic light scattering setup (VV-mode). [35] . . . . . . . . . . . . . . . . . . . . . . . . 26 34 38 Schematic representation of the depolarized dynamic light scattering setup (VH-mode). [35] . . . . . . . . . . . . . . . . . . . . 39 27 The ALV/CGS-3 compact goniometer system. . . . . . . . . . . . 40 28 Bulk rheology: Storage G” and Loss modulus G’ for PEO 1000 kDa at 20 C∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Bulk rheology: Storage modulus G” for PEO 1000 kDa at diﬀerent concentrations. . . . . . . . . . . . . . . . . . . . . . . . . . . 30 44 44 Bulk rheology: Loss modulus G’ for PEO 1000 kDa at diﬀerent concentrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 31 DLS: Silica particles in water . . . . . . . . . . . . . . . . . . . . 47 32 Diﬀusion coeﬃcient: Silica particles in water . . . . . . . . . . . 48 33 DLS: Latex particles in water . . . . . . . . . . . . . . . . . . . . 49 34 Diﬀusion coeﬃcient: Latex particles in water . . . . . . . . . . . 49 35 DLS: Overview diﬀerent molecular weights PEG at 0.5 C∗ . No particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 36 DLS: PEO 1000 kDa at 0.5 C . No particles . . . . . . . . . . . . 51 37 ∗ Diﬀusion coeﬃcient: Overview PEG diﬀerent molecular weights at 0.5 C∗ . No Particles. . . . . . . . . . . . . . . . . . . . . . . . 52 38 Diﬀusion coeﬃcient: PEO 1000 kDa at 0.5C . No particles. . . . 53 39 DLS: PEO 1000 kDa at 20C . No particles . . . . . . . . . . . . 54 40 DLS: PEO 1000 kDa at 5C and 10C . Latex Particles . . . . . . 55 41 DLS: PEO 1000 kDa at 20C∗ . Latex Particles . . . . . . . . . . . 55 42 DLS: Overview PEO 1000 kDa at diﬀerent concentrations. Latex ∗ ∗ ∗ ∗ Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 43 MSD: PEO 1000 kDa at 20C . Latex Particles . . . . . . . . . . 56 44 DLS-Rheo: PEO 1000 kDa at 20C∗ . Latex Particles . . . . . . . 58 45 DLS-Rheo: PEO 1000 kDa at 10C . Latex Particles . . . . . . . 58 46 DLS: Ellipsoids in water. No polarizer . . . . . . . . . . . . . . . 59 47 Diﬀusion coeﬃcient: Ellipsoids in water. No polarizer . . . . . . 60 48 DLS: Ellipsoids in water. Polarizer . . . . . . . . . . . . . . . . . 60 49 DLS: PEG 1000 kDa at 20C Ellipsoids. No polarizer . . . . . . 62 50 DLS: PEG 1000 kDa at 20C Ellipsoids. Polarizer . . . . . . . . 62 51 DLS-Rheo: PEG 1000 kDa at 20C∗ . Ellipsoids . . . . . . . . . . 64 52 NFPA 704 chloroform and toluene . . . . . . . . . . . . . . . . . 78 ∗ ∗ ∗ ∗ xii List of Tables 1 Classification of colloidal solutions depending on the continuous medium and the dispersed phase. . . . . . . . . . . . . . . . . . . 2 11 Overview of Rg , C∗ and mesh size at diﬀerent concentrations for diﬀerent molecular weights Mw . . . . . . . . . . . . . . . . . . . . 31 3 Overview of radius of spheres in water. . . . . . . . . . . . . . . . 50 4 Overview radius of gyration for diﬀerent molecular weights. . . . 53 5 Overview of length of ellipsoids in water. . . . . . . . . . . . . . . 61 xiii List of abbreviations and symbols List of abbreviations AFM Atomic Force Microscopy DDLS Depolarized Dynamic Light Scattering DLS Dynamic Light Scattering DLVO Derjaguin, Landau, Verwey & Overbeek DWS Diﬀusive Wave Spectroscopy GSER Generalized Stokes Einstein Relationship MCR Modular Compact Rheometer MSD Mean Square Displacement PEG Polyethylene Glycol PEO Polyehtylene Oxide VDW Van der Waals VV Vertical - vertical mode VH Verical-horizontal mode xiv LIST OF ABBREVIATIONS AND SYMBOLS List of Greek symbols α Diﬀusive coeﬃcient [-] β Coherence factor [-] γ Strain γ˙ Shear rate [s−1 ] Γ Decay rate [s−1 ] ΓV V Decay rate vertical vertical mode [s−1 ] ΓV H Decay rate vertical horizontal mode [s−1 ] δ Phase angle [rad] ε Dielectric constant [Fm−1 ] ε0 Dielectric constant vacuum [Fm−1 ] ξ Mesh size [m] φ Electric potential [V] κ−1 Debye length [m] ψ0 Surface potential [V] λ Wavelength [m] σ Stress [Pa] σ0 Stress amplitude [Pa] τ Relaxation time θ Angle [rad] ∆θ Mean square angular displacement [rad] ν Refraction index ω Frequency [%] [s] [-] [rad/s] xv LIST OF ABBREVIATIONS AND SYMBOLS List of symbols a Ah c∗ D Dt Dr e E(t) E G∗ Radius of the particle Hamaker constant Critical overlap concentration Diﬀusivity coeﬃcient Diﬀusivity coeﬃcient translational Diﬀusivity coeﬃcient rotation Electron charge = 1.60217646.10−19 Electric field Intensity Elasticity modulus Complex shear modulus [m] [J] [g/L] [m2 .s−1 ] [m2 .s−1 ] [m2 .s−1 ] [C] [-] [Pa] [Pa] � Storage modulus [Pa] �� Loss modulus [Pa] Laplace transformation of complex shear modulus Intensity auto correlation function (IACF) (Electric) field auto correlation function (FACF) FACF in the vertical-vertical mode FACF in the vertical-horizontal mode Intensity field gradient of the scattered light Intensity field gradient of the incoming light Ionic strength Boltzman number=1.3806488.10−23 Eﬀective spring constant Weight average molecular weight Avogadro’s number = 6.02214179.1023 Scattering vector Radius of gyration Mean square translational displacement Temperature [Pa] G G ˜ G g2 (q, t) g1 (q, t) g1V V (q, t) g1V H (q, t) I I0 [I] kB Ks Mw NA q Rg ∆r T xvi [-] [-] [-] [-] [cd] [cd] [mol.dm−3 ] [m2 kg.s.K −1 ] [-] [g/mol] [mol−1 ] [m−1 ] [m] [m] [K] Chapter 1 Introduction 1.1 Motivation Panta rhei (π αντ ´ α ��˜ι) or ’everything flows’ are the famous words of the Greek philosopher Heraclitus and also form the etymological origin of the word rheology. Rheology is the study of the flow of matter. Many industrially important substances such as paints, beverages, chocolates, polymers and even flowing metals during processes have very complex flow behavior. The major concern of rheology is to measure, describe and establish predictions about the flow behavior of these materials. Rheology is used in everyday life, not only through the use of consumer goods, e.g. the good mouth-feeling of beverages and chocolates or the ease of application of paint, but also during the processing steps of many consumer goods. Although the formal introduction of the word rheology was established in 1929, considerable rheological research has been carried out before. During these early measurements, many devices were developed to measure the rheological properties of matter. Traditionally, the characterization of complex materials is performed using bulk rheology devices such as a rheometer. A rheometer imposes a specific stress or strain on the material and measures the resulting strain or stress in the fluid. At present the rheometer is a widely used device to investigate the rheology of complex materials. However, more recently, new rheological measurement techniques have been developed [1, 2]. One method of particular interest is microrheology. In microrheology, micro/nano-scale particles are added to a small sample of the material under investigation. The goal of microrheology is to derive the rheological properties of the material by tracking the motion of these particles embedded in them. Microrheology overcomes certain limitations of bulk rheology such as 1 CHAPTER 1. INTRODUCTION a smaller sample size and a larger range of frequencies and moduli that can be probed. Additionally, the strains exerted by microrheology are much smaller than those in bulk rheology, which makes the method useful for fragile materials. Important advantages also include the possibility of investigating heterogeneity and a reduced cost [3]. These advantages of microrheology have made this method very popular over the past decade. The use of particle trackers of diﬀerent size, shape and matter or the investigation of diﬀerent materials have raised a number of challenges within this field. A further understanding of microrheology makes it possible to investigate more materials and enables us to extract information such as the microrheological properties of the material which supplements bulk rheology. A major drawbacks of the method however, is that the material under investigation should be transparent when using dynamic light scattering[3]. 1.2 Context Over the past decades, scientists have investigated the use of spherical particles in microrheology. For many materials the use of microrheology has been proven as an equivalent of bulk rheology. Although the concept was originally proven on synthetic materials, microrheology soon started playing an important role in the investigation of biological matter. These materials are not available in large amounts which makes traditional bulk rheology, which requires larger sample sizes, inappropriate. More recently, the use of non-spherical particles in microrheological methods gained particular interest. By using non-spherical particles the translational movement of the particles, as well as the rotation of the beads can be investigated. With the spherical particles used originally, this rotational diﬀusion was not measurable. A new point of interest has emerged on the use of ellipsoidal particles. For simple, incompressible, isotropic viscoelastic materials, measurements using these particles provide redundant information which can be used to check the self-consistency of the measurements by comparing translational and rotational results for the same material property [4]. Additionally, ellipsoids can give information about the nature of depletion zones formed in certain materials as explained in section 2.6. 2 CHAPTER 1. INTRODUCTION 1.3 Aim and structure The purpose of this research is twofold. First, the equivalence of bulk rheology and microrheology with spherical particles will be investigated for polyethylene glycol solutions. In the second part, the use of ellipsoidal particles in microrheology will be investigated. In this section the major concern is to obtain an equivalence with bulk rheology concerning the rotational diﬀusion of the ellipsoidal particles. After an extensive literature study in chapter two, the used materials and methods are further explained in chapter three. The experimental work in chapter four is divided in two parts. In a first section the results from bulk rheology experiments are shown. Since bulk rheology is not the main goal of this scientific work only few experimental work was done in this part. In the second experimental part, microrheological experiments were carried out with dynamic light scattering, of which the results are given in section 4.2. As a way of acquaintance with the device, experiments on silica particles in water were performed. Later, the beads used for microrheology in this master thesis were tested in water to fully characterize them. After testing the particles in a polymer solution the results of microrheology are compared with bulk rheology for spherical particles. In the last experimental part the use of ellipsoidal particles in microrheology is investigated. Firstly translational diﬀusion of the ellipsoids is tracked with a general dynamic light scattering device. Secondly, to obtain only the scattering of the rotational diﬀusion of the particles a depolarized dynamic light scattering (DDLS) device was used. 3 Chapter 2 Literature study 2.1 Introduction The term rheology was inspired from the Greek ’panta rei’ meaning ’everyting flows’. It was coined by E.G. Bingham, one of the founding fathers of rheology. In rheology the deformation and flow of materials in response to applied stress or strain is investigated. Rheology focuses on materials with a complex molecular structure such as polymer solutions, suspensions, emulsions, which exhibit a complicated flow behavior. A key diﬀerence between a solid matter and a liquid material is their contrasting response to an applied shear strain, which is the amount of deformation perpendicular to a given plane (figure 1). When applying a force, simple Newtonian liquids dissipate the provided energy through viscous flow, while pure solids store energy and show an elastic response [5, 6]. Squishy materials on the other hand both store and dissipate energy and are called viscoelastic materials. Rheology reveals both their solid-like and fluid-like behavior, depending on the time scale used to probe the material. Figure 1: Response of a material to a shear strain. 5 CHAPTER 2. LITERATURE STUDY Traditional rheological experiments are performed with mechanical rheometers. However, over the past decades new techniques have been developed to measure the viscoelastic behavior of complex materials. Microrheology is one of them and uses embedded micron or nano-sized particles to locally deform a sample. The goal of microrheology is to derive the rheological properties of the material from the motion of the colloidal particles embedded within it. The particles can be either thermally excited (passive rheology) or moved using external forces (active rheology). Microrheological methods enjoy certain advantages over bulk rheological techniques [3] . • In microrheology the required sample volume is much smaller (� 1 ml), which make it possible to study rare and expensive materials, including biological materials that are diﬃcult to obtain in large quantities. • The frequency range which can be probed may be orders of magnitude greater in microrheology than in conventional bulk rheology (figure 2). • Microrheological techniques exert very low strains which is useful to measure the viscoelastic properties of fragile materials. • Microrheology allows to measure local anisotropy in inhomogeneous systems. • Microrheology is a non contact method. If the samples are toxic or hazardous they can be placed in a glass tube and measured using a dynamic light scattering setup. Besides the many advantages of microrheology there are some limitations to the method too [7]. • The most important limitation is that the samples in microrheology most be transparent to light when using dynamic light scattering device. • Microrheology is computationally intensive. The raw data coming from the device need to be transformed in order to deduce the rheological properties of the matter. • If the probe motion is slow, the time to collect suﬃcient information is large and therefore this method is also not useful for very stiﬀ or viscous materials. 6 CHAPTER 2. LITERATURE STUDY • If the polymer chains in the sample are far apart (large mesh size) very large probes are necessary to measure the viscoelastic behavior of the material. If the probes are too large, only part of the probe scatters light emitted by the laser of the dynamic light scattering device, giving erroneous results. This will be further explained in chapter 3. Figure 2: Typical frequency ranges for diﬀerent measurement devices [8]. In order to fully understand the subject the general information about viscoelastic materials will be given in section 2.2. Later bulk rheology experiments will be explained briefly. To fully understand microrheological methods a brief introduction is given on colloids. In microrheology, the full frequency dependence of the viscoelastic moduli is obtained from the mean square displacement of embedded particles using the generalized Stokes-Einstein relation, this is discussed in section 2.4.2. Next, active microrheological methods are listed up. As these methods require sophisticated instrumentation they will not be discussed in detail. In the next session the passive measurements will be discussed with special attention to dynamic light scattering. In section 2.5 the state of the art about the subject is given. 7 CHAPTER 2. LITERATURE STUDY 2.2 Viscoelasticity 2.2.1 Elastic materials The behavior of pure solids or elastic materials can be described by Hooke’s law [9]: (1) σ = Eγ where σ is the stress, E the elasticity modulus and γ is the strain. This law states that the stress is directly proportional to the strain. All the energy given to the material while loading is stored in the material. When the load is removed all the energy is released again. 2.2.2 Viscous materials For viscous materials Newton’s law describes their behavior: . (2) σ = ηγ . where σ is the stress, η the viscosity and γ the shear rate. In this case the stress is directly proportional to the shear rate. Perfectly viscous materials obey this law. When a viscous material is loaded all the energy is dissipated. These are called Newtonian fluids. 2.2.3 Viscoelastic materials Viscoelastic materials are in between the two extremes (Newtonian fluids and Hookian solids). The elastic susceptibility of a viscoelastic material is given by the complex shear modulus G∗ (ω). For an oscillatory shear strain at a frequency ω, G∗ (ω) determines the stress induced in the material. The real part of the � complex modulus Re(G∗ (ω)) = G (ω) is the in phase response of the medium to the applied strain and is called the elastic or storage modulus [6]. It is a measure of the elasticity and the storage of energy of the investigated material. �� The imaginary part of the complex shear modulus Im(G∗ (ω) = G (ω) is the out of phase response to the applied strain and is called the viscous or loss modulus. � It is related to the viscosity of the material and the dissipation of energy. G (ω) �� and G (ω) are related by the Kramers-Kronig relations [10]. 8 CHAPTER 2. LITERATURE STUDY 2.3 Bulk rheology For bulk rheological measurements a rheometer is used. There are two diﬀerent types of rheometers: drag flows and pressure driven rheometers. The former applies a shear stress or shear strain between a moving and a fixed solid surface while the latter generates a shear by a pressure diﬀerence over a closed channel. Pressure driven rheometers are less frequently used. Of both rheometers many diﬀerent types exist, which will not be discussed in detail. In a stress/strain controlled rheometer, after applying an oscillatory stress or strain (figure 4) the stress/strain response of the material is measured. For an elastic material the response in completely in phase with the applied stress or strain, while for an viscous material only an out of phase component is observed. A viscoelastic material has a situation in between (figure 4). The response will have the same frequency but will be shifted by a phase angle δ. The response will be decomposed into an in phase component and an out of phase component. If for example a sinusoidally deformation is applied to a material the strain γ is given by [11]: (3) γ = γ0 sin(ωt) where γ0 is the amplitude of the strain. The resulting stress σ in the material is then given by: (4) σ = σ0 sin(ωt + δ) where τ0 is the amplitude of the stress. If the resulting stress is decomposed into the in phase (sin) and out of phase (cos) component, the stress is given by: � �� � �� σ = σ + σ = σ 0 sin(ωt) + σ cos(ωt) (5) It can be derived that the relationship between the phase angle and the in and out of phase component is: �� σ tanδ = 0� σ0 (6) Now a modulus can be defined as the ratio of the stress in the material and the strain applied to it. The decomposition of the stress thus results in the two dynamic moduli: � σ G = 0 γ0 � 9 (7) CHAPTER 2. LITERATURE STUDY �� �� G = σ0 γ0 (8) So from equation 6 follows: �� G tanδ = � (9) G If the real and imaginary part of the modulus are combined, the complex modulus G∗ can be defined: σ0 = |G∗ | γ0 (10) � �� or G∗ is a complex number with a real part: G and an imaginary part: G . � G∗ = G + iG �� (11) During rheology measurements G∗ is measured and later decomposed into its real and imaginary part, to describe its rheological behavior. Figure 3: Schematic representation of the rheometer. [8] Figure 4: Stress/strain response of a material to a strain/stress application [11] 10 CHAPTER 2. LITERATURE STUDY 2.4 Microrheology In order to use microrheological methods, particles must be added to the investigated material. Because the embedded particles are of microscopic scale or smaller, a colloidal solution is formed. 2.4.1 Colloids The name colloids comes from the Greek ’κολλα’ which means ’to stick’. A colloidal system consists of a dispersed phase and a continuous phase. A colloidal systems satisfies two important properties. Firstly the behavior of the system is mainly determined by the thermal fluctuations of the particles due to the Brownian motion. Secondly the surface properties predominate the bulk properties of the system. A classification of colloids can be found in Table 1. In microrheology we mainly deal with suspensions [12]. ❤❤❤❤ ❤❤❤ ❤❤ dispersed phase ❤❤❤❤ ❤❤❤ ❤❤ SOLID LIQUID GAS SOLID Solid Foam Foam LIQUID Suspension Emulsion Foam GAS Aerosol Aerosol Gas continous phase Table 1: Classification of colloidal solutions depending on the continuous medium and the dispersed phase. The stability of a colloidal system is a key feature in microrheological measurements. Stability is defined as the ability of the system to remain in the current state. This means that the particles should not stick together or settle down in the polymer solution but in contrast stay dispersed in the continuous phase. There are two ways of describing the stability of colloidal systems: thermodynamical and through kinetical stability. Thermodynamical stability is the state of minimal Gibbs free energy. Normally the energy of colloidal systems is very high and they are therefore thermodynamically unstable. Kinetic stability comes when the transition to a lower energy state becomes so slow that no aggregation occurs. The tools to influence the stability of a colloidal systems are the interaction forces between the particles. There are mainly two forces, attraction and repulsion, acting on colloidal systems. In order to obtain stability a balance between them should be established. It is very important to understand and to manipulate these forces. The repulsive forces are electrostatic forces, steric repulsions, electrosteric repulsions and solvatation. Among the attractive forces depletion, hydrogen bonds, hydrophobic 11 CHAPTER 2. LITERATURE STUDY interactions and Van der Waals forces are observed. To investigate the stability of a colloidal suspension the overall eﬀect of the diﬀerent forces should be determined. Derjaguin, Landau, Verwey & Overbeek (DLVO) established a theory which bears their name. The DLVO theory describes the balance between the attractive Van der Waals forces and the repulsive electrostatic forces. 2.4.1.1 Van der Waals (VDW) forces To describe the VDW forces between macroscopic bodies one needs to add the contributions of all the atoms in the solid. This means that the calculation depends on geometry. Hamaker calculated the total interaction between two spherical particles. For R1 and R2 >> H the potential is given by: φ=− Ah R 1 R 2 ( ) 6H R1 + R2 (12) Where R1 is the radius of the first sphere, R2 the radius of the second sphere, H the distance between the spheres and Ah the Hamaker constant. The Hamaker constant H is generally a function of density and the nature of the interaction. The VDW forces depend on the mobility of the electrons and also on geometry. 2.4.1.2 Electrical double layer The repulsive electro-static forces of the particles come from the electrical double layer surrounding the particle. This double layer comes from the charged surface attracting ions of opposite charge. To describe the electrical double layer diﬀerent models exist. A frequently used model is the Gouy-Chapman model. This model describes the interaction of a charged surface with the ions in the solution and the formation of a double layer. For a positively charged surface, the anions in the solution tend to balance the positive surface charge. The counter ions are not rigidly held, but tend to diﬀuse into the liquid phase to the solid surface. This tendency decreases as more ions reach to surround the surface. The kinetic energy of the counter ions will partly aﬀect the thickness of the resulting double layer. The Gouy-Chapman model was extended by Stern, whose model takes into account the finite size of ions and hence cannot approach the surface closer than a few nanometers. The ions can be adsorbed onto the surface up to the ’slipping plane’ through a distance known as the Stern Layer (figure 5). The potential at the slipping plane is known as the zeta potential. Concerning electrostatic repulsion, the zeta potential is sometimes considered more significant than the surface potential [12, 13]. Knowledge of this potential allows to determine the electro-static forces between the particles, which have a great influence on the stability of the colloidal system. 12 CHAPTER 2. LITERATURE STUDY Figure 5: Schematic representation of the electro-static potential near a solid surface in a solution containing ions [13]. 2.4.1.3 DLVO The DLVO theory describes the force between charged surfaces interacting through a liquid medium. As already stated, it combines the eﬀects of the VDW attraction force and the electrostatic repulsion force due to the double layer. The overall potential is a function of the distance from the surface x and is given by: φ(x) = −Ah R + 2π�ψ0 exp(−κx) 12x (13) where R is the radius of the particle, Ah the Hamaker constant, ψ0 the standard surface potential, x is the distance between the particles in nm and ε the dielectric constant. Also the variable κ appears in formula (13). κ is the inverse of the Debye length or the distance over which the potential decreased with 66%. The exact formula of is given by: κ= � 2e2 NA I εε0 kB T (14) κ is given in [nm]. In formula 14, e is the charge of an electron, NA the Avogadro number, I is the ionic strength, ε is the dielectric constant, ε0 is the 13 CHAPTER 2. LITERATURE STUDY dielectric constant vacuum, k B the Boltzmann number and T is the temperature [K]. Summary: In order to determine the potential three factors need to be determined: 1. the Debye length (κ−1 ) 2. the Hamaker constant (Ah ) 3. the potential at the surface of the particles (ψ0 ) 2.4.1.4 Eﬀect of polymers on colloidal stability Adding polymers to a colloidal solution influences the stability of the system. Dissolved polymers can either have a repulsive or an attractive eﬀect on the solid particles. Repulsive forces come from the steric interaction between the polymer chains sticking on the surface of the particles. The stability of the system is determined by the adsorption and the chemical anchoring of the polymer chains. The stability is less sensitive for changes in pH and ionic strength. The steric interactions can be controlled by the length of the polymer chains, the density and rigidity of the polymer chains , and the solvent quality (figure 6). Attractive forces between the solid particles appear when the polymers do not interact with the solid surface. Two eﬀects are observed: depletion and bridging. Depletion occurs when the osmotic pressure is out of balance. When the polymer chains are located between the solid spheres, their entropy decreases and the chains tend to restore their entropy loss by leaving the space between the particles. Consequently there is an imbalance in osmotic pressure around the particles by which the particles are pushed towards each other. Bridging occurs when parts of the polymer chain adsorb on the solid surface. This results in an attractive force working on large distances (figure 7). The eﬀects of adding polymer chains to a colloidal solution depends mainly on the molecular weight, the concentration and the adsorption of polymer on the solid surface. 14 CHAPTER 2. LITERATURE STUDY Figure 6: Eﬀect of adding polymers to colloidal solution: steric stabilization and depletion[14]. Figure 7: Eﬀect of adding polymers to a colloidal solution: eﬀect of bridging. 2.4.2 Generalized Stokes-Einstein relationship (GSER) In microrheology the stochastic thermal energy of particles, embedded in the material that needs to be investigated, is used to derive the rheological properties of the material. Particles suspended in a liquid undergo both translational as well as rotational diﬀusion due to the Brownian motion. Stokes and Einstein derived a method to obtain diﬀusion of a material by tracking Brownian motion. Their results are combined in the generalized Stokes-Einstein relation (GSER). Significant attention has been given to microrheological methods using translational diﬀusion of the colloids so far. The GSER for translational and 15 CHAPTER 2. LITERATURE STUDY rotational diﬀusion are similar and will be derived further in this section. 2.4.2.1 Translational diﬀusion First the motion of a sphere in a purely viscous material is considered and afterwards this is generalized for viscoelastic materials. In a purely viscous medium the particles simply undergo Brownian motion. The dynamics of particle motions are given by the time dependent position correlation function, also known as the mean square displacement (MSD). This MSD reflects the response of the material to the stress applied to it by the thermal motion of the beads and is given by [5]: � � � � 2 ∆�x2 (τ ) = |�x(t + τ ) − �x(t)| (15) where x is the particle position, τ is the lag time and the bracelets indicate an average over all times t. From this MSD the diﬀusion coeﬃcient (D) of the particles can be determined from the diﬀusion equation [8]: �∆�x(τ )� = 6Dτ (16) The diﬀusion coeﬃcient is related to the radius of the particle ’a’ and the solvent viscosity η of the surrounding fluid via the Stokes-Einstein relationship [8]: D= kB T 6πηa (17) where kB is the Boltzmann constant, T the absolute temperature. On the other hand when a particle is embedded in a purely elastic medium its movement will be limited and the MSD will reach a maximum value, determined by the elastic modulus of the material (figure 8). This derivation can be generalized for more complex materials, exhibiting both viscous and elastic behavior. As can be seen in figure 8 for simple fluids the MSD of the particles evolve linearly with time. For more complex materials, the linear behavior of the MSD disappears and the MSD evolves diﬀerently with τ . � � ∆�x2 (τ ) ∼ τ α (18) where is α< 1 and is called the diﬀusive coeﬃcient. In the purely elastic part the particles are limited in their movement (α =0) and the MSD reach the plateau. To further derive the GSER, a complex fluid will be modeled 16 CHAPTER 2. LITERATURE STUDY as an elastic network within a simple Newtonian fluid. After a computational intensive derivation, the GSER in the Laplace domain can be found in literature as [5]: � G(s) = kB T πas �∆� r(s)� (19) This is the generalized Stokes-Einstein relationship for translational diﬀusion � � ˜ ˜ of colloids, where G(s) and ∆r(s) are the complex shear modulus and MSD in the Laplace domain respectively. This formula can be explained intuitively. By equating the thermal energy of � 2 � the particles, k B T with its elastic energy Ks ∆rmax /2 an expression of Ks is obtained. This Ks is the eﬀective spring constant and depends on the elasticity � of the matter surrounding the beads. The elastic modulus G of the surrounding material thus can be expressed as a function of this Ks . The factor that relates both, is a factor of length: the dimension of the probe. The resulting formula gives: G� ∼ kB T 2 �∆rmax �a (20) This simplified explanation shows the main idea behind equation 19. Figure 8: Evolution of the mean square displacement (MSD) of the particles as a function of the lag time τ [8] 17 CHAPTER 2. LITERATURE STUDY 2.4.2.2 Rotational diﬀusion For rotational diﬀusion the derivation is only slightly diﬀerent. The rotational motion can be described by the mean square angular displacement [15, 16]: � ∆θ2 (t) � (21) Deriving the viscoelastic modulus for rotational diﬀusion is similar to translational diﬀusion where a is replaced by a3 (or L by L3 for ellipsoids) because rotational diﬀusion depends on the volume swept by the particle. The rotational GSER for a sphere is given by: � G(s) = kB T � � � 4πa3 s ∆θ(s) (22) This equation has to be modified with a correction factor for non-spherical particles, depending on their shape [8]. This result of the generalized Stokes-Einstein relationship is noteworthy: by observing the time-evolution of the MSD of the particles the frequency dependent viscoelastic response can be obtained. It is important to note that the use of the GSER requires that the size of the bead is larger than any structural length scales of the material. For example, in a polymer network, the size of the beads should be significantly larger than the characteristic mesh size [5]. This will be explained in detail later in chapter 3. ˜ To compare this with the bulk rheology experiments, G(s) need to be transformed into the Fourier domain to obtain G∗ (ω) . Generally this can be done in two ways. The first method is by calculating the inverse unilateral Laplace transform and then taking the Fourier transform [5]. In this method the real data are transformed into the complex plane and can cause significant errors in G∗ (ω) near the frequency extremes [10]. To overcome this problem an alternative method is developed. Here the complex shear modulus is estimated algebraically by using a local power law to describe the mean square displacement of the beads in the complex fluid [10]. This is the method used in this master thesis and will be discussed in detail in chapter 3. 18 CHAPTER 2. LITERATURE STUDY 2.4.3 Active microrheological techniques In active microrheological methods the probes in the colloidal solution are agitated by an external force. In a certain way active methods are analogous to conventional bulk rheology in which an external stress is applied and the resultant strain is measured. However the way stress is applied is diﬀerent. Active microrheological methods use embedded particles to deform the material locally and measure the viscoelastic response of the material. The advantages of active microrheological methods are mainly twofold. First, they allow to probe stiﬀ materials, secondly they can measure non-equilibrium behavior, because large stresses can be applied. As only passive techniques have been used in this thesis, only the most important methods that exists are listed up [17]: • optical tweezers measurement • magnetic manipulation technique • atomic force microscopy (AFM) 2.4.4 Passive microrheological techniques In contrast with active microrheological methods in passive methods the embedded particles are not forced to move. In passive methods the thermal energy of the beads is responsible for their movement. This thermal energy is given by k B T , where k B is the Boltzman constant and T the absolute temperature. The only energy input to the beads is their Brownian motion. Thus, in order to have measurable movement of the beads, the surrounding material should be soft enough. The displacement of the particle is highly dependent on the stiﬀness of the material that surrounds it. An embedded particle will only move when this thermal energy is bigger than the energy needed to deform the sur� rounding material. For an elastic material the elastic modulus G is related to the particle size by the equation below [5]: � Gy kB T = 3 2 a a (23) where y is the particle displacement, a is the radius of the particles or the probe. From this formula it can be derived that the upper limit of the elastic modulus that can be mainly determined by the particle size of the beads, by the possibility to investigate small particle displacements (y) and the temperature. 19 CHAPTER 2. LITERATURE STUDY 2.4.4.1 General dynamic light scattering (DLS) Th main passive microrheological technique is dynamic light scattering. The details of this method will be explained in chapter 3 materials and methods. 2.4.4.2 Diﬀusive wave spectroscopy (DWS) Diﬀusive wave spectroscopy extends the technique of DLS to samples dominated by multiple scattering. The light will hit many particles and will be scattered many times. Therefore as the individual scatterer move only a small fraction of the total wavelength of the incident light, an aggregate change of the total path length by one wavelength will be observed. Therefore DWS is sensitive at shorter length scales and thus faster time scales than DLS because these measurements are made in the single scattering limit. A combination of both techniques makes it possible to probe a large range of frequencies, which is the key advantage of microrheology [8, 18]. 2.4.4.3 Video based particle tracking Video based particle tracking is a technique to track the movement of embedded solid particles in many types of systems. The technique has certain advantages but it is especially useful to probe the local structure of the material. This technique provides a direct visualization of possible inhomogeneities that can be present in the sample while at the same time it tracks about hundred beads simultaneously [10]. 2.5 State of the art First results about microrheology were published in 1995 by T.G. Mason and D.A. Weitz. They were the first to test the applicability of the GSER � �� equation and the further derivation towards G and G for several distinctly diﬀerent complex fluids. The first system investigated was a suspension of silica particles in ethylene glycol with a radius a of 0.21 µm. They measured the mean square displacement using diﬀusing wave spectroscopy. The bulk rheology was performed with a strain controlled rheometer using a sample cell with a doublewall couette geometry. Excellent agreement was found between bulk rheology and DWS. They performed a second test on a polymer solution at a suﬃcient high concentration to have an entangled network. They used polyethylene oxide with a molecular weight of 4.106 g and polystyrene latex spheres with a = 0.21 and a volume fraction of 0.02. Again excellent agreement between micro and bulk rheology was observed. A third experiment was the study of an emulsion, 20 CHAPTER 2. LITERATURE STUDY comprised of uniformly sized oil droplets with a radius of 0.53 µm and a volume fraction of 0.62. Again, very good agreement was obtained. During the following years the use of spherical particles in microrheology have been investigated in depth. For example the eﬀects on probe size was explored by Q. Lu and M.J. Solomon in 2007. Also the limits of the application of microrheology have been investigated by Todd M. Squires and T.G. Mason. The use of non-spherical particles in microrheology is a newer development. With non-spherical particles not only the translation but also the rotation of the particle can be investigated. In 2003 Z. Cheng and T.G. Mason have established the fundamental principle of rotational diﬀusion microrheology on anisotropically shaped wax (α-eicosene) microdisks in an aqueous polyethylene oxide (PEO) solution. They used light streak tracking and compared the results with those obtained with a strain mechanical rheometer with a concentric cylinder geometry. The rotational diﬀusion measurements of the viscoelastic shear modulus of the polymer solution did agree well with the experiments with the mechanical rheometer. At present, no one has studied microrheology using ellipsoidal particles (prolate) and DLS. This master thesis is the first scientific work to study this system. Details on this ellipsoids used in this master thesis can be found in chapter 3 materials and methods. 21 Chapter 3 Materials And methods 3.1 Introduction Using a traditional mechanical rheometer, the bulk viscoelastic properties can be investigated by applying oscillatory strain and measuring the stress response of the material. This provides a direct measure of the storage and loss modulus. In the last decade, other techniques have been developed and improved that also allow investigating soft materials for local viscoelastic behavior [19, 20, 21]. In this thesis, first microrheological experiments are performed using spherical particles. This is done to get familiar with preparing correct samples and the measuring methods. In a later stadium it will be investigated using nonspherical particles. Dynamic Light Scattering (DLS) is used to measure the dynamics of the probes suspended in the polymer solution. In order to investigate the viscoelastic properties of a simple, isotropic uncrosslinked flexible polymer, experiments are performed on polyethylene glycol (PEG) and polyethylene oxide (PEO) solutions in semi-dilute regime using particles of varying form. Diﬀerent molecular weights of polyethylene glycol (PEG) are used in diﬀerent concentrations above the overlap concentration C∗ to obtain many diﬀerent samples with varying viscoelastic properties. By measuring the thermal or Brownian motions of the particles, the elastic and viscous moduli of the samples can be obtained. Using the technique of dynamic light scattering, the dynamics of the probe particles in the polymer sample are determined as a function of time. The data measured with these microrheology experiments should then have a frequency overlap with data found with traditional rheology experiments. To calculate the moduli from the dynamic light scattering experiments a modified algebraic form of the generalized Stokes-Einstein equation is used as explained in section 3.4.4. The expectation is that for simple uncross linked polymeric systems, this method should show excellent similarity between bulk and micro-rheology data. 23 CHAPTER 3. MATERIALS AND METHODS 3.2 Materials 3.2.1 Polymers: Polyetheleneglycol (PEG) or Polyetheleneoxide (PEO) Polyethyleneglycol (PEG), is a hydrophilic polymer or oligomer build with chains of monomers of ethylene glycol (-CH2 -CH2 -O-) with an hydroxyl group (-OH) at both ends. PEG exists in diﬀerent chain lengths. Shorter chains are only a couple of hundreds monomers in length and are liquid at room temperature. The longer chains are up to ten thousands monomers in length and are solid at room temperature. Polyethyleneglycol with long chains and high molecular weights (over 35000) is also called Polyethyleneoxide (PEO) [22], because the role of the hydroxyl groups are negligible. In this masterthesis experiments were performed on long chains with molecular weights varying from 3350 until 1000000 g mol−1 or 3.35 to 1000 kDa. During this thesis, the molecular weight will often be mentioned in Dalton [Da]. It is defined as one twelfth of the rest mass of an unbound neutral atom of carbon-12 in its electronic and nuclear ground state. One Dalton is approximately equal to the mass of one proton or one neutron and can be used as an equivalence for 1 g mol−1 [23]. The choice for PEG was obvious for diﬀerent reasons. First of all it is a safe and easy polymer to work with. Also, it has good viscoelastic properties and it is rather easily soluble in water. Another important fact is that it was available in diﬀerent molecular weights in the lab. Applications of PEG range from being a half fabricate in the production of polyurethane to being an ingredient in suppository pills. It can also be used to conserve wood in the domain of archeology. Figure 9: Monomer polyethylene glycol (PEG) 24 CHAPTER 3. MATERIALS AND METHODS 3.2.2 Beads 3.2.2.1 Spheres In order to test the DLS, in the first range of experiments a solution of spherical silica particles in a water was used. These particles, with a theoretical diameter of 30 nm, are typically amorphous and nonporous (figure 10). Their scattering intensity is strong and overshadows any scattering from dust particles. This is an advantage because of the clean data and easy fitting. In a second phase, spherical surfactant free sulfate latex particles were used with a diameter of 210 nm. These are typically made of polystyrene. The diameter of the particles is suﬃciently small to ensure that the particles are small enough to have colloidal properties. Also important is that this diameter is big enough to ensure that the particles are larger than the mesh size of the polymer. This mesh size, which depends on the polymer concentration, is the average size of the space between the polymer coils. Particles should be bigger than this mesh size to probe the material, if not they would just diﬀuse through the material without probing the matrix . Calculations to prove this are shown in paragraph 3.3.1. Figure 10: Scanning electron micrograph (SEM) of silica particles [24] 3.2.2.2 Ellipsoids So far, very few research has been done on the use of ellipsoidal particles to compare bulk rheology with microrheology. This will be the subject of this master thesis. Ellipsoids have three main directions to move in a fluid. They can translate, rotate about its axis of symmetry or rotate about its nonaxisymmetric rotation (figure 11). The motivation for using ellipsoidal particles in microrheology for the investigation of viscoelastic materials is twofold [4]. 25 CHAPTER 3. MATERIALS AND METHODS First, for simple, incompressible, isotropic linear viscoelastic materials, investigating both translational and rotational diﬀusion gives redundant information. This information can be used as a self-consistency check between G∗ (ω) translation and rotational diﬀusion. If in contrast, the moduli measured with translational diﬀusion diﬀer from those measured with rotational diﬀusion, extra information can be extracted from the coupling of the investigated material and the probes or from the anisotropy of the material. Second, some materials are known to form depletion zones. These materials show regions of lower polymer concentration around particles. As the ellipsoid will rotate freely about its axis of symmetry, the lack of polymer macromolecules around the particle aﬀects its axisymetric rotation much more significant than its translational or non-axisymmetric diﬀusion [4]. Quantitative measurements of the axisymetric rotation compared to the translational or non-axisymetric diﬀusion will give information about the nature and extent of the depletion zone. Figure 11: Rotation of ellipsoids in a fluid.(a) axisymetric rotation. (b) nonaxisymmetric rotation. • General ellipsoids The ellipsoidal particles were made of polystyrene. They are made by stretching spheres in one direction, which makes them prolate (figure 12). The beads used, have a theoretical length of approximately 650 nm which is bigger than the spherical ones and an aspect ratio P of around 3.9. This means that the length of the ellipsoids was 3.9 times longer than the width. These dimensions proved suﬃcient to match the mesh size of the polymers (figure 13). 26 CHAPTER 3. MATERIALS AND METHODS • Ellipsoids covered with gold The general ellipsoids used are anisotropic in shape. To observe only their rotational diﬀusion, depolarized setup of the DLS was used, as will be explained in section 3.3.3. But although they are anisotropic in shape they still have isotropic scattering properties. As a result the polarization of the scattered light from the sample is the same as the polarization direction of the incoming light, giving disappointing results when using the polarizer. This is why the ellipsoids were covered with gold nanorods (figure 14), which have excellent, anisotropic scattering properties and make it possible to observe the rotational diﬀusion of the ellipsoids with the polarizer. In figure 15 a schematic representation of the ellipsoids covered with gold nanorods is given. Figure 12: Two diﬀerent kinds of stretching spheres are possible to obtain ellipsoids. The results are called oblate ellipsoids or prolate ellipsoids. In this master thesis prolates were used. [25] Figure 13: Transmission electron micrograph of polystyrene ellipsoids. Figure 14: Transmission electron micrograph of of gold nanorods. 27 CHAPTER 3. MATERIALS AND METHODS Figure 15: Schematic representation of prolate ellipsoids covered with gold nanorods. [Courtesy: Sylvie Van Loon] 3.2.3 Extra materials • Water Of course water was necessary during the complete duration of the thesis. To clean the tools for example, regular tap water was used to wash the soap oﬀ and then normal demineralised water was used to make everything clean. Even a third check was done with ultra pure water. The ultra pure water was used to make the samples and to dissolve the polymers. It is obtained out of a Sartorius Stedium Arium 611DI system and has a conductivity of 0, 055 µS/cm. • Sodium Chloride In order to obtain a constant level of ionic strength (electro-static double layer) and to reduce possible eﬀects of CO2 absorption in the samples, a 25 millimolar solution of Sodium Chloride (NaCl) in water was used to dissolve the polymers. • Chloroform & Aluminum foil To prevent bacterial growth in the samples a drop of chloroform was added in the main polymer solution. All samples were covered with alumina foil to prevent degradation from the light. 28 CHAPTER 3. MATERIALS AND METHODS 3.3 Methods 3.3.1 Sample preparation In order to get adequate and correct results, all the samples were created with the greatest care and patience. The first thing was to triple clean every tool used in the process. This was essential to avoid dust particles to interfere because dust particles also scatter light, which would result in bad experimental measurements. For the diﬀerent molecular weights of the PEG used, diﬀerent concentration samples were made ranging from 0.1 to 20 times the overlap concentration C∗ . This overlap concentration indicates the point where the solution begins to exhibit viscoelasticity due to the entanglements of the polymer coils. At this concentration the neighboring polymer coils are overlapping which each other (figure 16). The overlap concentration C∗ was calculated using the following equation [26]: C∗ = Mw 4 3 3 N a Rg (24) Where Na is Avogrado’s number and Rg the radius of gyration of the polymer. The radius of gyration is given by an empirical relation [27]: Rg = 0.21Mw(0.58±0.031) (25) The squared radius of gyration gives the average squared distance between monomer units and the center of mass of the polymer coil. About 92% of the polymer mass are within the radius of gyration (figure 17) [28]. Figure 16: Schematic representation of the critical overlap concentration C∗ [29] 29 CHAPTER 3. MATERIALS AND METHODS Figure 17: Schematic representation of the radius of gyration. [29] In first instance 4 molecular weights of PEG were used: 3.35 kDa, 8 kDa, 20 kDa and 35 kDa. Later on PEO of molecular weight 1000 kDa was used to obtain higher viscoelastic samples. All molecular weight samples were produced as follows: A 100 ml of de-ionized water was mixed with 0.145 g of N aCl to obtain a 25 millimolar N aCl − H2 O solution. This salt was added to ensure a constant ionic strength in the solution. For every molecular weight the right amount was measured to prepare a 100 ml 20 C∗ stock solution. The polymers were added iteratively into the stock solution while mixing it very gently on a magnetic stirrer at a temperature of 40° C to homogenize. After adding a drop of chloroform to prevent bacterial growth in the polymer, the stock solutions were covered in alumina foil to prevent degradation by light and it was put on a rolling shaker for a couple of days. When the polymer was completely dissolved, the 20 C∗ stock solution was used to create less concentrated samples. The right proportions of the stock solution were mixed with more saltwater to obtain 10 ml samples of diﬀerent concentrations. In these smaller samples tiny drops of the beads were added, and the sample was thoroughly shaken by a vortex mixer. It was very important to make sure that the bead diameter was bigger than the mesh size of the polymer sample as explained earlier. The mesh size is normally only a few nanometers and can be calculated using the following equation [26]: ξ = Rg (C ∗ /C)0.75 (26) For both the spherical particle of 210 nm diameter and the ellipsoidal particle of 650 nm this was satisfactory. Another important aspect was to inject a high enough bead concentration to ensure the bead scattering to be dominant over the polymer scattering. However a too high concentration would induce multiple scattering instead of the desired single scattering. A good balance between these 30 CHAPTER 3. MATERIALS AND METHODS was necessary to have a good scattering sample. The final step to get the samples ready for the DLS measurements was to add them in glass tubes of 1 cm diameter. These were carefully washed with ethanol and deionized water. For the rheometer experiments no extra procedure was necessary and the same samples were used as the ones for DLS. The beads did not eﬀect the rheology data. Mw [g/mol] 3350 8000 20000 35000 100000 Rg [m] 2.44084E-09 4.05452E-09 6.91734E-09 9.58584E-09 6.76766E-08 C∗ [g/ml] 0.091324338 0.047580685 0.023953756 0.015752088 0.001278925 ξ[m] at C = C∗ 2.44084E-09 4.05452E-09 6.91734E-09 9.58584E-09 6.76766E-08 ξ[m] C = 20 C∗ 1.29811E-10 2.15632E-10 3.67884E-10 5.09803E-10 3.59924E-09 Table 2: Overview of Rg , C∗ and mesh size at diﬀerent concentrations for different molecular weights Mw . 3.3.2 Rheometer 3.3.2.1 Device The bulk rheology experiments were performed on an Anton-Paar MCR 501 stress controlled rheometer (figure 18). A cone and plate geometry (CP50-1/Ti) with a diameter of 49.94 mm and a cone angle of 0.993° (figure 19). A solvent trap with evaporation blocker was used during all the experiments. Additionally water was put around the loaded sample to further prevent evaporation. This was done with the greatest care so as to prevent the water from mixing with the polymer sample. More information about the geometry can be found in appendix A. The instrument has a fully automatic tool recognition system and a T ruGapT M system to monitor and control the real gap, eliminating errors from thermal expansion and normal force during experiments. Temperature control is achieved by a Peltier element and all experiments were done at 25°C. The gap between the cone and plate geometry was fixed at 0.048 mm and a sample of 0.57 ml was loaded. The sample material that bulged out from in between the cone and the plate was scraped away. Measurements were controlled and analyzed using Physica RheoPlus software. All samples were pre-sheared for 100 seconds at a shear rate of 10 s−1 . Afterwards a strain sweep was performed in order to determine the linear viscoelastic region. All subsequent frequency sweep measurements are performed at strain within the linear viscoelastic regime. Experiments are repeated twice to check their reproducibility. 31 CHAPTER 3. MATERIALS AND METHODS Figure 18: Anton-Paar 501 MCR stress-controlled rheometer Figure 19: Schematic representation of the cone and plate geometry [12] 3.3.2.2 Maxwell model Maxwell derived a mathematical model to describe viscoelastic behavior of materials. The model describes the dual nature of a viscoelastic fluid starting from simple phenomenological tools. A Hookean spring represents a perfect solid material, while a dashpot will be the model for a Newtonian fluid. In the Maxwell model both elements are placed in series (figure 20). Figure 20: Schematic representation of the Maxwell model Mathematically G(t) is represented in the model as [30]: G(t) = G0 exp( −t ) τ (27) where τ is the relaxation time or the time where 66% of the stress in the material is released. The storage and loss modulus can be defined as: 32 CHAPTER 3. MATERIALS AND METHODS � G (ω) = �� G (ω) = � G0 ω 2 τ 2 ∝ ω2 1 + ω2 τ 2 (28) G0 ωτ ∝ω 1 + ω2 τ 2 (29) �� The evolution of G and G is given in figure 21. Since PEO is a viscoelastic material measured in the linear viscoelastic regime, it should look like the Maxwell model. The slopes of the moduli in figure 21 are logically explained by the proportional relation of the moduli to the frequency ω in equations 28 and 29. � �� Figure 21: Evolution of G and G described by the Maxwell model [31] 3.3.3 Dynamic light scattering (DLS) 3.3.3.1 General dynamic light scattering for spherical particles The main passive microrheological technique is dynamic light scattering. A DLS experiment is based on the elastic scattering of light by particles undergoing Brownian motion in a suspension. A laser beam hits on a sample and the light is scattered by the particles into a detector placed at a certain angle θ with respect to the incoming beam (figure 22). According to the Rayleigh theory, the incident light and its electrical field will distort the electron distribution in the molecule or particle and induce an oscillating dipole that will re-radiate light, as illustrated in Figure 23. 33 CHAPTER 3. MATERIALS AND METHODS Figure 22: Schematic representation of dynamic light scattering (DLS) Figure 23: Scattering oﬀ a small particle in an ideal solution by incident light [32] As the particles move and rearrange in the sample, the intensity of the light I(t) that reaches the detector fluctuates in time. From this constantly changing signal I(t) the autocorrelation function of the intensity of the light, g2 (q, t), is calculated. This function states how the intensity of the light at time t is correlated to the intensity of the light at time t + τ , and is given by [33, 34] : g2 (q, τ ) = �I(t)I(t + τ )� �I(t)� 2 (30) where the brackets�� indicate an average over time. This means the intensity at time t + τ is compared with the initial intensity at time t. For small τ , the particles are close to their initial position and the correlation between I(t) and I(t + τ ) is very strong. For larger τ the particles had time to move a lot and the correlation will fall down (figure 24). The autocorrelation function of the intensity is hence a direct measure of the Brownian motion of the particles [35]. 34 CHAPTER 3. MATERIALS AND METHODS Although the intensity of the light is measured, one is interested in the electric field intensity E(t), but this electric field intensity is too diﬃcult to measure. Using the Siegert relation a correlation between the electric field intensity correlation function g1 (τ ) and the correlation function for the intensity of the light is given by [5, 35]: g2 (q, τ ) = 1 + β |g1 | 2 (31) where β is the coherence factor, depending on the laser beam and instrumentation optics. For idealized conditions of perfect coherence β=1. In practice however β is slightly lower than 1. If all the particles are statistically independent, and moving randomly due to thermal impulses only, then [5, 35] g1 (q, τ ) = exp{−Γτ } (32) where Γ is the relaxation rate and τ the correlation time. For optically isotropic, monodisperse spheres Γ is related by the translational diﬀusion of the particles by [35]: Γ = Dt q 2 (33) where Dt is the translational diﬀusion coeﬃcient and q is the scattering wave vector given by: q= 4πν θ sin( ) λ 2 (34) where ν is the refraction index of the sample and λ is the wavelength of the laser in vacuum. The translational diﬀusion coeﬃcient can be related to the mean square displacement < ∆r2 (τ ) > of the particles by: Dt = < ∆r2 (τ ) > 6t (35) If formula 35 is combined with 32, the electric field autocorrelation function is given by: g1 (τ, q) = exp( −q 2 < ∆r2 (τ ) > ) 6 (36) Formula 36 couples the electric field autocorrelation function to the MSD of the spherical particles. This result is of major concern in this master thesis because from this MSD the complex shear modulus can be obtained by use of the GSER as explained in chapter 2. 35 CHAPTER 3. MATERIALS AND METHODS Figure 24: Intensity measurement and autocorrelation function measured by dynamic light scattering [12] 3.3.3.2 Depolarized dynamic light scattering for ellipsoidal particles In order to investigate the rotational diﬀusion of the ellipsoids an depolarized dynamic light scattering (DDLS) device is used because that polarization direction of the incoming light is diﬀerent from the polarization direction of the scattered light from the sample. This may occur for several reasons, e.g. if the particles exhibit a suﬃciently large shape and/or optical anisotropy. In standard DLS experiments the polarization of the incoming beam is normal to the plane defined by the wave vectors of the incident and the scattered beam respectively. The intensity of the scattered light is usually measured without the use of a polarizer (VV-mode), as seen in figure 25. In a DDLS experiment however, a polarizer is used that is oriented 90° with respect to the polarization of the scattered beam. (VH-mode), as given in figure 26. The correlation function contains information about both translational and rotational diﬀusion. But as the polarizer is set at 90° only the light scattered whose polarization direction has been changed is measured. Consequently the coupling between translational and rotational diﬀusion is measured. In general DLS measurements (VV-mode) the decay rate is given by [38]: ΓV V = D t q 2 (37) which is analog to equation 33. However in the VH-mode the decay mode is given by: ΓV H = Dt q 2 + 6Dr 36 (38) CHAPTER 3. MATERIALS AND METHODS As the translational diﬀusion cannot be completely blocked, the rotational diﬀusion is still present in equation 38. Dr represents the rotational diﬀusion coeﬃcient. Consequently, analog to equation 32, the electric field autocorrelation functions are given by: g1V V = exp(−Dt q 2 τ ) (39) g1V H = exp[−(Dt q 2 + 6Dr )τ ] (40) Where equation 39 is again completely analog to equation 32. As this technique was used to calculate the diﬀusivity and radius of the prolate ellipsoids, formulas to derive the diﬀusion coeﬃcient for prolate ellipsoids are given in equation 41 and 42. For thin rods with a length of L and an aspect ratio (P ) in a fluid of viscosity η and temperature T formulas where derived by Boersma and Brenner [39, 40], but for ellipsoids with small aspect ratios, as the one used in this master thesis, correction factors Cr and Ct are added. They are required to account for the relative importance of end eﬀects due to the finite rod length [41]. Dt = kB T (ln(P ) + Ct ) 3πηL (41) Dr = 3kB T (ln(P ) + Cr ) 3πηL3 (42) Empirical expressions to relate Ct and Cr to the hydrodynamic dimensions are given by [41]: Cr = 0.312 + 0.565P −1 − 0.100P −2 (43) Ct = −0.662 + 0.917P −1 − 0.050P −2 (44) In order to couple the electric field autocorrelation function to the mean square angular displacement of the probes, the electric field autocorrelation function of the VV-mode and the VH-mode are equated to obtain: g1V V exp(−Dt q 2 τ ) = = exp(6Dr τ ) V H exp[−(Dt q 2 + 6Dr )τ ] g1 (45) Additionally, equivalent to equation 35, the rotational diﬀusion coeﬃcient is given by: Dr = � ∆θ2 2 37 � (46) CHAPTER 3. MATERIALS AND METHODS Combining equation 45 and 46 and taking natural logarithm on both sides gives: ln( � � g1V V ) = 3 ∆θ2 τ V H g1 (47) From formula 47 the mean square angular displacement can be obtained from the electric field autocorrelation function. By use of the GSER for rotational diﬀusion, as explained in chapter 2, the complex shear modulus can be derived. Figure 25: Schematic representation of the general dynamic light scattering setup (VV-mode). [35] 38 CHAPTER 3. MATERIALS AND METHODS Figure 26: Schematic representation of the depolarized dynamic light scattering setup (VH-mode). [35] 3.3.3.3 DLS device The DLS set-up used in this master thesis consists of a goniometer that holds the sample, a coherent light source and detection optics that measure the intensity of the scattered light as a function of time. By placing the system on a Newport air damped table everything is stabilized and mechanical vibrations are reduced. The light source is a continuous wave (CW) laser that emits red light with an operational wavelength of 632,8 nm at a power of 35mW. This wavelength ensures enough distance from the absorption lines of the sample to prevent local heating. The compact goniometer system (ALV/CGS-3) defines the scattering geometry and also holds the sample. It consists of an arm that holds the detector and a vat with a sample holder (figure 27). The detector is placed on a movable arm that can rotate about the center of the vat from 15 to 150 degrees. The vat is filled with toluene and the sample is positioned such that the laser beam passes through the center. Toluene is used because it reduces undesirable reflections and refractions as it has a similar index of refraction as the glass of the sample. The vat is connected with a fluidsbath to control the temperature of the sample. The duration of the experiments varied depending on the concentration. The higher the polymer concentration, the longer the experiments were executed. In water the correlation function went to zero within 30 seconds per angle while for the most concentrated samples experiments took over 10000 seconds per angle. 39 CHAPTER 3. MATERIALS AND METHODS Figure 27: The ALV/CGS-3 compact goniometer system. 3.3.4 Comparison between bulk and microrheology From the raw data obtained from microrheology, the MSD was calculated as explained earlier. This MSD is plugged in the general Stokes Einstein relation (GSER) as deduced in chapter 2 to obtain G(t) for translational or rotational diﬀusion. In order to compare with the bulk rheology, these data need to be transformed from the time domain to the Fourier domain G∗ (ω) and be decou� �� pled into G and G . Generally two methods were developed to do the transformation from the time to the frequency domain. The first methods takes the inverse unilateral Laplace transformation and then afterwards takes the Fourier transform [5]. In practice however, the numerical calculation of this method can give significant errors in G∗ (ω). Therefore an alternative method was developed. In this method a local power law is used to describe the MSD of the beads in the fluid. With this local power law the complex shear modulus can be estimated algebraically. In this method no numerical transformations are used, thus avoiding the significant errors in G∗ (ω). The power law is determined from the logarithmic time derivative of the MSD [8]. For spherical particles the GSER for translational diﬀusion in the Fourier domain is given by: G∗ (ω) = kB T πaiω �∆r2 (τ )� (48) Now, an expression needs to be found for the storage and the loss modulus. � 2 � � �� If ∆r (τ ) can be written in a local power law form, G (ω) and G (ω) are 40 CHAPTER 3. MATERIALS AND METHODS given by [8]: � (49) �� (50) G (ω) = G(ω) cos(πα(ω)/2) G (ω) = G(ω) sin(πα(ω)/2) where: kB T (51) πa �∆r2 (1/ω)� Γ(1 + α(ω)) � � � � In equation 51, the local power law α(ω) is given by �(∂ln ∆r2 (τ ) /∂lnτ )�τ =1/ω G(ω) = and Γ represents the gamma function. The gamma function comes from the � � Fourier transformation of the local power law. ∆r2 (1/ω) represents the mag� � nitude of ∆r2 (τ ) evaluated at τ = 1/ω. The DLS data was converted into actual microrheology data using Matlab. The Matlab code, with the exact calculations can be found in Appendix C. For ellipsoidal particles the rotational diﬀusion is tracked, and the GSER for rotational diﬀusion needs to be used. G∗ (ω) = � kB T 4πa3 iω �∆θ2 (ω)� �� (52) The derivation of G and G is analog as in the translational case. The Matlab code, with the exact calculations can be found in Appendix C. 41 Chapter 4 Experimental results and discussion 4.1 Overview In this chapter the experimental results will be presented and analyzed. First the bulk rheology results will be shown. The focus in the rheology experiments lies � �� in the range of angular frequencies where crossover between G and G occurs. � �� Although not all samples will show the perfect crossing of G and G , the trend is definitively there. Of course a considerably part of this chapter will focus on the DLS results. In a structured way the results will handle the diﬀerent particles in water, the polymers without particles and the various particles in polymers of diﬀerent molecular weights and concentrations. An important sample that will come back throughout the whole chapter is the PEG 1000 kDa at a 20 C∗ concentration. In the last but most important part, the results from bulk rheology will be compared with microrheology from DLS. A special attention will be given to the work with the ellipsoids and the unique results obtained. 4.2 Rheology After investigating many diﬀerent concentrations and molecular weights of PEG (or PEO), the PEO sample of 1000 kDa with a concentration of 20 C∗ was chosen to compare bulk rheology with microrheology. The reason here fore was that it showed suﬃcient viscoelastic behavior. � �� In figure 28 the evolution of the storage (G ) and loss (G ) modulus as a function of angular frequency (ω), measured with the stress controlled rheome- 43 CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION ter, is shown. From this result it can be seen that at lower frequencies viscous �� � response is dominant as G � G . At higher frequencies the elastic response also � �� becomes significant and eventually there will be crossover between G and G . �� If the frequency goes further, the elastic modulus G will be more significant. As one can see, the slope of the moduli are as expected with the Maxwell model. � Figure 28: Bulk rheology: Storage modulus (G ) with open symbols and loss �� modulus (G ) with filled symbols, as a function of angular frequency for 1000 kDa PEO at 20 C∗ . � Figure 29: Bulk rheology: Storage modulus (G ) for diﬀerent concentrations from 5 C∗ to 20 C∗ , as a function of angular frequency for 1000 kDa PEO. 44 CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION �� Figure 30: Bulk rheology: Loss modulus (G ) for diﬀerent concentrations from 5 C∗ to 20 C∗ , as a function of angular frequency for 1000 kDa PEO. In order to investigate the reliability of the measurements, the results of PEO 1000 kDa with a concentration of 20C∗ are compared with samples of 15C∗ , 10C∗ and 5C∗ . Because the samples of the lower concentrations become too low viscous, the viscoelastic behavior disappears. The lower limits of the rheometer are reached and the measurements for the lower polymer concentrations become less accurate. The fact that the date of the 5C∗ and 10C∗ are almost exactly the same, proves this statement. Another derivation can be made out of figures 29 and 30. The more viscous the sample, the lower the angular frequency at which cross over happens. This is normal as this frequency is the inverse of the relaxation time of the sample. The higher the viscosity, the higher the relaxation time and thus the earlier the cross over. On the other hand the moduli of the cross over are getting higher for more viscous samples. To conclude, more viscous samples have higher moduli at cross overs and the cross over frequency is earlier. 4.3 Dynamic Light Scattering of spherical particles and polymers The first experiments were done with the purpose of getting to know the DLS system and to getting familiar with the outputs and the computational intensive data processing. To avoid complications in the beginning, the first experiments were done in water. Besides gaining experience with the methodology the second goal of the experiments was to gain an insight in the formulas to calculate 45 CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION diﬀusion, viscosity and radius. The results of the DLS measurements come in two columns. The first column represents the time τ, in milliseconds and the second contains the values of the normalized intensity auto correlation minus one, g2 − 1. In the results, shown in the following figures, these outputs were plotted and fitted using a stretched exponential. This stretched exponential or Kohlrausch-Williams-Watts (KWW) function is given as [36, 37]: g2 − 1 = A ∗ (e− ) (53) were β indicates the stretched nature of the function. It is usually between 0.9 and 1 for monodisperse particles. A represents the amplitude but is usually set to 1. Γ is the inverse of the relaxation time in seconds which denotes the average time, per angle, for which the correlation is lost. The relaxation time for each angle can be obtained from the fitting of equation 53 or in the graphs as the value on the x-axis for which the specific curve have fallen 66%. τ is the time in milliseconds coming out of the DLS. From the Γ obtained out of the fit, it is possible to calculate the diﬀusion coeﬃcient D. The relation used here is Γ = 2Dq 2 (54) By determining the diﬀusion coeﬃcient D, it is only a small step further to calculate the experimental diameter of the particles using the Stokes Einstein relation [8]: η= kB T 6πDa (55) where η is the viscosity of the sample, kB is the Boltzmann constant, T is the temperature and a is the radius of the particle. One should also realize that the radius calculated here, includes the Debye length κ−1 . As the samples used here are simple water samples the approximation form can be used [12]: 0.304 κ−1 [nm] = � I(M ) I is the ionic strength, which in this case is 0.025 mol/l. 46 (56) CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION 4.3.1 Silica particles in water As a matter of testing, the first experiments were done with spherical silica particles with a diameter of 30 nm. Silica is a wonderful material for dynamic light scattering as it scatters very good. All the experiments were done for scattering angles 30°-150°, but to have a better visualization only three angles are presented in figure 31. Figure 31: DLS: The normalized intensity auto correlation function g2 − 1 is plotted as a function of time for spherical silica particles with diameter 30 nm in water. Only 3 diﬀerent angles 50°, 90° and 150° are presented. As expected the exponential decay, which represents the correlation function, occurs smoothly as the particles are monodisperse. As the particles are almost not hindered in the water, they can move freely and the correlation function goes to zero. The experiment took only 30 seconds per angle which indicates that the correlation was lost rapidly. The relaxation times are between 0.1 and 1 millisecond. For the next calculations all the angles are taken into account. Plotting Γ values for each angle as a function of q 2 shows the expected linear relation (figure 32). By linear plotting, it was possible to obtain the slope, which gave the diﬀusion coeﬃcient. In this case D is 1.5 10−11 [m2 /s] which is high as expected. The calculations to verify the particle size are given in Table 3. 47 CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION Figure 32: Diﬀusion coeﬃcient: The inverse relaxation time Γ in inverse seconds (dots) is plotted as a function of the scattering vector for the silica particles in water sample. Using a linear fit, the slope is obtained which is 2 times the diﬀusion coeﬃcient. 4.3.2 Latex particles in water The particles used now are surfactant free white sulfate latex particles with an apparent diameter of 210 nm. It are these particles that will be put in the polymer samples in a later stage. Again only the 50°, 90° and 150° angles are represented (figure 33). As with the silica particles the curves are smooth and rapidly going down and the fit covers the data nicely. Experiments were again at 30 seconds per angle. 48 CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION Figure 33: DLS: The normalized intensity auto correlation function g2 − 1 is plotted as a function of time for spherical Latex particles with diameter 210 nm in water. Only 3 diﬀerent angles 50°, 90° and 150° are presented. As the graphic shows, the relaxation times are around 1 millisecond. This indicates that the correlation was a bit longer here than with the silica particles. This is as expected as the particles are bigger here. Again D is obtained by a linear plot of the inverse relaxation time Γ and the scattering vector q (figure 34). The diﬀusion coeﬃcient, which is 2.27 10−12 [m2 /s], is a bit smaller than with the silica particles, which is also logical. Figure 34: Diﬀusion coeﬃcient: The inverse relaxation time Γ in inverse seconds (dots) is plotted as a function of the scattering vector for the Latex particles in water sample. Using a linear fit, the slope is obtained which is 2 times the diﬀusion coeﬃcient. 49 CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION Now that all the data are given, an overview in table form shows the results. The viscosity of water is used with a value of 0.89 mPa.s. The experimental radii are calculated and when the Debye Length is subtracted from the these and the result is multiplied by two, the adjusted diameter is found. As one can see, the errors are negligible and the results are very close to the sizes provided by the manufacturers. Form + Size Diﬀusivity [m2 /s] Radius a [m] Debye Length κ−1 [m] Experimental Diameter [nm] Absolute Error [nm] Relative Error [%] Spherical Silica 30 nm 1.49E-11 1.64E-08 1.9 E-09 29.08 0.92 3.07 Spherical Latex 210 nm 2.27E-12 1.07E-07 1.9 E-09 211.90 1.90 0.90 Table 3: Overview of radius of spheres in water. 4.3.3 Polymers in water The next series of experiments were polymers in water. The first experiments were performed for the four lower molecular weights (3.35, 8, 20 and 35 kDa) for a very dilute concentration of only 0.5C∗ . The goal of this series of experiments is to characterize the polymers and to check the theoretical radius of gyration Rg , calculated in chapter 3, with the one obtained out of the measurements. Figure 35 shows the intensity autocorrelation function of tau for 4 diﬀerent Mw PEG in water at 0.5 C∗ , and figure 36 shows the same relation for PEO 1000 kDa at 0.5C∗ . Immediately one can observe that the average relaxation times for these samples are very short. With values around 10−2 ms the relaxation times are significantly lower then these of the particles in water. The polymer coils are thus very small and diﬀuse very fast. One can also see the small but certain evolution of increasing relaxation time for higher molecular weights. All the experiments were performed during 30 seconds per angle. The smaller the angle, the harder it is to obtain correct results. This is because at these angles even the tiniest dust particles scatter hard. For this reason, the 50° angle in some measurements was not of suﬃcient quality to present so the 60° angle is shown instead. 50 CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION Figure 35: DLS: The normalized intensity auto correlation function g2 − 1 is plotted as a function of time for PEO 1000 kDa for the diﬀerent molecular weights (3.35, 8, 20 and 35 kDa) at a concentration of 0.5 C∗ . Only 3 diﬀerent angles per sample are presented. No particles are present in this sample, just polymer. Figure 36: DLS: The normalized intensity auto correlation function g2 − 1 is plotted as a function of time for PEO 1000 kDa at 0.5 C∗ in water. Only 3 diﬀerent angles 50°, 90° and 150° are presented. No particles are present in this sample, just polymer. The graphs of the polymers in water are obviously less smooth than those 51 CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION of the particles in water. This is because the polymer coils scatter less than the solid particles. The coils are not in a solid predefined state and their form and measurable size is constantly changing. In order to obtain the radius of gyration of the polymer, the diﬀusivity was also calculated for these samples. Figure 37 shows the inverse relaxation time Γ as a function of the scattering angle q2 for 4 diﬀerent Mw PEG in water at 0.5 C∗ , and figure 38 shows the same relation for PEO 1000 kDa at 0.5C∗ .The values of the diﬀusivity coeﬃcients make sense as they are decreasing with higher concentration. It can be remarked that the data of the 1000 kDa sample is less good then the others. Figure 37: Diﬀusion Coeﬃcient: The inverse relaxation time Γ in inverse seconds is plotted as a function of the scattering vector for the diﬀerent molecular weights of PEG (3.35, 8, 20 and 35 kDa) at a concentration of 0.5 C∗ . Using a linear fit, the slope is obtained which is 2 times the diﬀusion coeﬃcient. All the diﬀusivity coeﬃcients are in table 4. 52 CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION Figure 38: Diﬀusion Coeﬃcient: The inverse relaxation time Γ in inverse seconds is plotted as a function of the scattering vector for PEO 1000 kDa at a concentration of 0.5 C∗ . Using a linear fit, the slope is obtained which is 2 times the diﬀusion coeﬃcient. Again an overview in table form is given, and the experimental radius of gyration, obtained out of the diﬀusivity, is compared with the theoretical Rg from chapter 3. Normally the experimental and theoretical radius of gyration should be the same, but the results show consistently lower values for the experimental values. A feasible explanation for this, is that even the 0.5 C∗ concentration was not dilute enough, meaning that the diﬀerent polymer coils were sensing each other. This oﬀ course limits the full expansion of the coil, and also explains the lower Rg as compared to the theory. Mw [kDa] Diﬀusivity [m2 /s] Experimental Rg [m] Theoretical Rg [m] 3.35 1.47E-10 1.67E-09 2.44E-09 8 9.91E-11 2.47E-09 4.05E-09 20 6.07E-11 4.04E-09 6.91E-09 35 4.39E-11 5.58E-09 9.58E-09 1000 3.41E-11 7.19E-09 6.76E-08 Table 4: Overview radius of gyration for diﬀerent molecular weights. Figure 39 shows another sample of the 1000 kDa polymer. Unlike the previous ones, this sample is at a much higher concentration of 20 C∗ . Because of the higher concentration, the coil’s movements are more diﬃcult which is shown in the elevated relaxation time of 1000 ms. This is also proven by the fact that this experiment took 10000 seconds per angle to obtain good correlation data. 53 CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION Figure 39: DLS: The normalized intensity auto correlation function g2 − 1 is plotted as a function of time for PEO 1000 kDa at 20C∗ in water for angles 60° and 90°. No particles are present in this sample, just polymer. 4.3.4 Latex particles in polymers After the previous series of experiments the characterization of the polymers and the particles was done. In order to save experimentation time and to investigate thoroughly enough, it seemed appropriate now to limit the future measurements to only 2 molecular weights. Because the rheology data showed diﬃculties with the lower molecular weights, even at higher concentrations they were not suﬃciently viscous, the 3.35, 8 and 20 kDa samples were not further investigated. Moreover, to keep this section structured and concise, the results of the PEG 35 kDa samples are shown in appendix B. In the next sessions all the attention will go to the 1000 kDa molecular weights. First the separate results are shown in figures 40 and 41 for the 5 C∗ , 10C∗ and 20 C∗ concentrations at diﬀerent angles. After, they are summarized in figure 42, which compares the diﬀerent concentrations for the angle 90°, including a 2C∗ sample. Obviously the increasing relaxation time is proof of the more diﬃcult movements of the embedded particles. 54 CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION Figure 40: DLS: The normalized intensity auto correlation function g2 − 1 is plotted as a function of time for PEO 1000 kDa at a concentration of 5C∗ and 10C∗ . In every sample 3 diﬀerent angles are plotted 50°, 90° and 150°. The particles in the sample are spherical Latex particles with 210 nm diameter. Figure 41: DLS: The normalized intensity auto correlation function g2 − 1 is plotted as a function of time for PEO 1000 kDa at a concentration of 20C∗ . Only 3 diﬀerent angles are plotted 50°, 90° and 150°. The particles in the sample are spherical Latex particles with 210 nm diameter. 55 CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION Figure 42: DLS: The normalized intensity auto correlation function g2 − 1 is plotted as a function of time for PEO 1000 kDa at diﬀerent concentrations from 2C∗ to 20C∗ . Only one angle, 90° is presented. The particles in the sample are spherical Latex particles with 210 nm diameter. Figure 43 shows the actual MSD as a function of time for the 1000 kDa 20C∗ sample. As one can see, the MSD grows slowly at lower times and faster at higher times. Figure 43: MSD: The Mean Square Displacement < ∆r(τ )2 > in [µm2 ]is plotted as a function of time in seconds for the PEG 1000 kDa sample at a concentration of 20 C∗ . 56 CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION 4.3.5 Comparison between DLS and rheometer After collecting all the necessary data in the previous parts, the main goal of the thesis is presented here: The comparison of the microrheology DLS results with the bulk rheology data. The Matlab code, created for the conversion, can be found in Appendix C. The formulas presented in paragraph 3.3.3 were used to do the conversion and are shown again for clarity. � (57) �� (58) G (ω) = G(ω) cos(πα(ω)/2) G (ω) = G(ω) sin(πα(ω)/2) where: kB T (59) πa �∆r2 (1/ω)� Γ(1 + α(ω)) � � � � In equation 59, the local power law α(ω) is given by �(∂ln ∆r2 (τ ) /∂lnτ )�τ =1/ω G(ω) = and Γ represents the gamma function (nothing to do with inverse relaxation time). The gamma function comes from the Fourier transformation of the lo� � � � cal power law. ∆r2 (1/ω) represents the magnitude of ∆r2 (τ ) evaluated at τ = 1/ω. Attention should be paid to the units of the frequency as the rheology data are in rad/s and the DLS converted data are originally in Hertz. Multiplying the DLS frequency by 2*π solves this problem. The result presented in figure 44 is for 20C∗ of the 1000 kDa polymer. For this sample good rheological data were obtained and the conversion also succeeded perfectly. The only adjustment to the original formulas presented above is a multiplication by a factor 2 in the G(ω) formula. A correction factor of 2 is also used in literature [8]. For the lower frequencies the DLS data are not shown because of too many irregularities. However, the DLS results fit the rheology data perfectly so that this result can be considered as the proof that micro-rheology is a valuable methodology. 57 CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION � �� Figure 44: DLS-Rheo: Storage modulus (G ) and loss modulus (G ) are plotted, as a function of angular frequency for 1000 kDa PEO at 20 C∗ for both the bulk rheology data and the DLS microrheology data. The particles in the sample are spherical Latex particles with 210 nm diameter. Also for the 10C∗ sample of 1000 kDa, the bulk rheology data fit the DLS microrheology data nicely. The correction factor of 2, however, was also necessary to obtain the fit presented in figure 45. � �� Figure 45: DLS-Rheo: Storage modulus (G ) and loss modulus (G ) are plotted, as a function of angular frequency for 1000 kDa PEO at 10 C∗ for both the bulk rheology data and the DLS microrheology data. The particles in the sample are spherical Latex particles with 210 nm diameter. For the PEG 35 kDa, the conversion was also done and fitted, but due to a lack of acceptable rheology data (not viscous enough) the results are presented 58 CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION in appendix B. 4.4 Ellipsoids The second part of the results focuses on the ellipsoids. In contrast to the spherical particles, the measurements with ellipsoids needed to be performed twice. Once without a polarizer to obtain translational diﬀusion, and once with a polarizer to obtain both translational and rotational diﬀusion. After proving the possibilities for microrheology for spheres, this section will show the possibilities of ellipsoids. 4.4.1 Ellipsoids in water In the first measurements, the experiments were done with ’simple’ ellipsoidal particles. However, the results of these particles in water were very disappointing, as no rotational diﬀusion was measured with the polarizer. A solution for this problem was to cover the ellipsoids (length 650 nm) with smaller gold nanorods (lenght 60 nm) as they would increase the scattering, but have the same movements as the big ellipsoids. The first experiments with these goldnanorods-covered-ellipsoids was in water and produced excellent results. The measurement without the polarizer (figure 46) was analyzed using the same methodology as spheres in water. By plotting the inverse relaxation time Γ, obtained out of the stretched exponential KWW fit, as a function of the scattering vector q2 and doing a linear fit, the slope resulted in two times the diﬀusivity coeﬃcient D. Figure 46: DLS: The normalized intensity auto correlation function g2 − 1 is plotted as a function of time for polystyrene ellipsoids with a length of 650 nm, covered with gold nanorods with a length of 60 nm, in water. Only 3 diﬀerent angles are plotted 50°, 90° and 150°. No polarizer is used. 59 CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION Figure 47: Diﬀusion coeﬃcient: The inverse relaxation time Γ in seconds (dots) is plotted in function of the scattering vector for polystyrene ellipsoids with a length of 650 nm, covered with gold nanorods with a length of 60 nm, in water. Using a linear fit, the slope is obtained which is 2 times the diﬀusion coeﬃcient. The same sample was also measured without polarizer. It was immediately obvious that this experiment had to run longer. As the polarizer blocks much of the light, the intensity measured was far below the normal conditions. However, the results are expected to be in the same range as the previous ones. Figure 48: DLS: The normalized intensity auto correlation function g2 − 1 is plotted as a function of time for polystyrene ellipsoids with a length of 650 nm, covered with gold nanorods with a length of 60 nm, in water. Only 3 diﬀerent angles are plotted 50°, 90° and 150°. Polarizer. 60 CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION Again an analysis is made of the samples. The viscosity of water is used with a value of 0.89 mPa.s and the Debye length is again subtracted from the obtained radius. After multiplying this by 2 the wanted diameter (or length in this case) is found. The theoretical length of the ellipsoids is very close to what was measured and calculated with the DLS data and the errors are negligible. Form Name/ TEM size Diﬀusivity [m2 /s] Radius a [m] Debye Length κ−1 [m] Experimental Diameter [nm] Absolute Error [nm] Relative Error [%] Ellipsoid Polystyrene 650nm + gold 60 nm 7.43E-13 3.30E-07 1.9 E-09 656.59 6.59 1.01 Table 5: Overview of length of ellipsoids in water. 4.4.2 Ellipsoids in polymer Ellipsoids covered with gold nanorods were mixed in the polymer sample. To limit the experimentation times, for high concentrations more then 10 000 seconds per angle were necessary, ellipsoids were only added in the 20 C∗ sample of the 1000 kDa. Both experiments, with and without polarizer, were executed. Again the obtained data was good and the fit was decent. The results are shown in figures 49 and 50. 61 CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION Figure 49: DLS: The normalized intensity auto correlation function g2 − 1 is plotted as a function of time for polystyrene ellipsoids with a length of 650 nm, covered with gold nanorods with a length of 60 nm, in PEG 1000 kDa at concentration 20 C∗ . Only 3 diﬀerent angles are plotted 50°, 90° and 150°. No polarizer is used. Figure 50: DLS: The normalized intensity auto correlation function g2 − 1 is plotted as a function of time for polystyrene ellipsoids with a length of 650 nm, covered with gold nanorods with a length of 60 nm, in PEG 1000 kDa at concentration 20 C∗ . Only 3 diﬀerent angles are plotted 50°, 90° and 150°. A polarizer is used. 62 CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION 4.4.3 Comparison between DLS and rheometer In order to compare the microrheology of the ellipsoids with the obtained rheology new conversions were made. Not only translational diﬀusion, but also rotational diﬀusion had to be taken into account. For translational diﬀusion, the formulas were the same as with the spheres, but for the rotational diﬀusion new formulas were necessary to obtain the comparison. The Matlab code is presented in appendix C. A revision of the necessary equations for rotational diﬀusion, explained in paragraph 3.3.3.2, shows the way of working: In order to couple the electric field autocorrelation function to the mean square angular displacement of the probes, the electrical autocorrelation function of the VV-mode and the VH-mode, obtained by DLS, are divided to obtain: g1V V exp(−Dt q 2 τ ) = = exp(6Dr τ ) V H exp[−(Dt q 2 + 6Dr )τ ] g1 (60) The rotational diﬀusion coeﬃcient is given by: Dr = � ∆θ2 2 � (61) Combining both results and taking the natural logarithm, the following relation is obtained. : ln( � � g1V V ) = 3 ∆θ2 τ V H g1 (62) Unlike the translational diﬀusion, the rotational diﬀusion is independent of the scattering angle and analogously to formula 59 the following equation was used: G∗ (ω) = kB T 4πa3 iω �∆θ2 (ω)� (63) In figure 51 the fit is shown, both for the translational and rotational diﬀusion, with the bulk rheology data of the sample. The rotational diﬀusion was not optimally obtained but the fit is still good. As the PEO 1000 kDa sample of 20C∗ was made several months before the ellipsoids were investigated, the original rheology could not be trusted and the sample was again measured with the rheometer after the DLS experiments. As expected, the polymer degraded already and was comparable with a 15C∗ sample instead of the original 20C∗ . Initially the fitting of the DLS data on the bulk rheology was completely wrong. After thorough consideration, it was discovered that this was due to the size of the ellipsoids. Surprisingly the data should not be fitted with the length of the ellipsoids, but with the length of the gold nanorods. This is a remarkable dis- 63 CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION covery as this has never been discovered before. There are two possible ways to look at this result. The first explanation is simply that the gold nanorods have come oﬀ the original big ellipsoids and that they are responsible for the scattering. Due to a lack of time, multiple experiments were impossible to actually confirm or deny this hypothesis. A second way of looking at the result is that this might actually be a major braketrough in microrheology. What happens is that although the big ellipsoids, who satisfy the mesh size, are responsible for the movements, it is the gold nanorods that are responsible for the scattering. If this is the case, major opportunities present themselves, like measuring viscoelastic properties of gels and other materials with big mesh sizes. However, as it is unknown what actually happens in the polymer sample, no definitive conclusion can be drawn. More investigation is definitely necessary. � �� Figure 51: DLS-Rheo: Storage modulus (G ) and loss modulus (G ) are plotted, as a function of angular frequency for 1000 kDa PEO at 20C ∗ (theoretically) for both the bulk rheology data from the rheometer as the DLS microrheology data. The particles in the sample are polystyrene ellipsoids with a length of 650 nm, covered with gold nanorods with a length of 60 nm. 64 Chapter 5 Conclusion and future research In this scientific work, research was conducted on the use of microrheological techniques to characterize viscoelastic materials. Current state of the art in microrheology focuses on the use of spherical particles embedded in the fluid under investigation. However this master thesis focused on the use of ellipsoidal particles to deduce the viscoelastic behavior of the material. After performing traditional bulk rheology measurements using a rheometer device, the obtained results were compared with microrheology in order to prove their equivalence. The polymer used during the experimental work was polyethylene glycol (PEG), a hydrophilic polymer with hydroxyl groups at both ends. In the first experimental part of the thesis traditional bulk rheology experiments were performed. Although this method is well established some diﬃculties were encountered working with low viscous samples such as PEG of lower molecular weight and small concentrations. Due to the low viscoelasticity of these samples, the rheometer was pushed to its limits and the data were not reproducible. Therefore, the decision was made to look at PEG with higher molecular weights, and a standard sample was made at suﬃciently high concentration. A PEO 1000 kDa sample at 20C∗ became the standard throughout the thesis and provided steady and acceptable data with a visualization of the cross� �� over of storage (G ) and loss modulus (G ). To obtain even better data, higher concentrations were also investigated. But due to their increasing turbidity, the samples became useless in dynamic light scattering experiments. During this thesis the dynamic light scattering device was the main microrheological technique. Prior experiments with spherical silica particles embedded in water, allowed to familiarize with the concept of dynamic light scattering and the computationally intensive data processing. Afterwards, diﬀerent molecular weights and concentrations of PEG and PEO were investigated using embedded 65 CHAPTER 5. CONCLUSION AND FUTURE RESEARCH spherical particles. Most experiments were performed with a multiangle setup, covering 13 diﬀerent angles. Consequently, the duration of the experiments for more concentrated samples covered several days, as the diﬀusion coeﬃcient of the colloids diminished. The results obtained for spherical particles in polymers were of high quality and most of the results were fitted perfectly using a stretched exponential fit. Afterwards, the mean square displacement obtained from microrheology was converted into actual viscoelastic moduli, in order to compare the DLS results with the traditional bulk rheology data. In line with what literature describes, microrheology using spherical particles was confirmed as a valuable alternative for bulk rheology. After this first set of experiments the use of ellipsoids in microrheology was investigated. By using a polarizer, the scattering from translational diﬀusion was suppressed and mainly rotational diﬀusion was observed. However due to the optical isotropy of the ellipsoidal beads no light scattering was perceived. In order to overcome this problem the ellipsoids were covered with gold nanorods, which have excellent scattering properties. This led to proper results with the dynamic light scattering device. However, to obtain appropriate resemblance between bulk and microrheology for rotational diﬀusion, initial conversion, using the size of the big ellipsoids was oﬀ by several decades. Surprisingly however, using the size of the gold nanorods instead, resulted into a perfect fit with the bulk rheology data. This raised questions which can be answered with two hypotheses. A first explanation is that the gold nanorods detached from the original ellipsoids and moved freely at their own diﬀusion rate. Although the results in water do not show any indication of this, time constraints made it impossible to actually confirm or deny this hypothesis with additional experiments. A second way of looking at the result is considering this to be a new way of looking into microrheology. Although the big ellipsoids satisfy the mesh size of the polymer and probe the material, it are the gold nanorods that are responsible for the scattering and it is their length that matters in the mathematical conversion between bulk rheology and microrheology. If this is the case, major opportunities would emerge, such as measuring viscoelastic properties of gels and other materials with big mesh sizes. However, as it is currently unknown what actually happens in the polymer sample, no definitive conclusion can be drawn yet. The aforementioned conclusions make it clear that this research is far from finalized, and that in the best case only the top of the iceberg has been exposed. Further investigation will need to clarify this. 66 l Appendices Appendix A Specifications of the geometry l 68 APPENDIX A. SPECIFICATIONS OF THE GEOMETRY 69 Appendix B Latex particles in PEG 35 kDa Diﬀerent concentrations of the 35 kDa molecular weight are shown. First the results are shown for the 5 C∗ , 15C∗ and 25 C∗ at diﬀerent angles and to summarize a graph is shown that compares these concentrations for the angle of 90°, including a 0.1 C∗ . Obviously the increasing relaxation time is proof of the more diﬃcult movements of the embedded particles. The measuring time also went up to 10 000 seconds per angle during these diﬀerent measurements. The long time per angle made the various experiments very time consuming. PEG 35 kDa concentration at 5C ∗ . Latex particles. 70 APPENDIX B. Results PEG 35 kDa PEG 35 kDa concentration at15C ∗ Latex particles. PEG 35 kDa concentration at25C ∗ Latex particles. PEG 35 kDa overview diﬀerent concentrations. Latex particles 71 APPENDIX B. Results PEG 35 kDa Conversion DLS and rheometer PEG 35 kDa Because for PEG 35 kDa the rheology data were less successful, the conversion of the DLS data was done without comparing to bulk rheology. Diﬀerent concentrations are presented in increasing concentration. This way � �� one can see the evolution of the cross-over between G (ω)and G (ω) appearing at lower angular frequency and higher values for the moduli. This matches the expectations perfectly as this cross-over represents the inverse of the relaxation time. In other words, the more concentrated, the higher the relaxation time and the earlier the cross-over point. DLS-Rheo Comparison: PEG 35 kDa concentration 5C ∗ Latex particles DLS-Rheo Comparison: PEG 35 kDa concentration 15C ∗ Latex particles 72 APPENDIX B. Results PEG 35 kDa DLS-Rheo Comparison: PEG 35 kDa concentration 25C ∗ Latex particles 73 Appendix C Matlab code for data processing: Translational Diﬀusion 74 APPENDIX C. MATLAB CODE FOR DATA PROCESSING 75 APPENDIX C. MATLAB CODE FOR DATA PROCESSING Matlab Code for data processing: Rotational Diffusion 76 Appendix D Safety and hazard analysis Working in a chemical lab always entails hazardous situations which should be carefully analyzed prior to the actual experimental work. Although the hazard in this master thesis were relatively limited, the potential threats are listed up. • Chloroform In order to prevent bacterial growth, a few drops of chloroform were added to our samples. Chloroform is a colorless, dense organic compound with structured formula CHCl3 . Chloroform vapors depress the central nervous system (used as anesthetic). It is dangerous at approximately 500 ppm according to the U.S. National Institue for occupational Safety and Healt.Breathing about 900 ppm for a short time can cause dizziness, fatigue, and headache. Chronic chloroform exposure can damage the liver [42]. The NFPA 704 code fo chloroform is given in figure 52. • Laser beam of DLS The laser beam from the DLS is a continuous wave (CW) laser that emits red light with an operational wavelength of 632,8 nm at a power of 35mW. This laser beam is classified as a class 3B laser, which means the laser is hazardous if the eye is exposed directly, but diﬀuse reflections such as those from paper or other matte surfaces are not harmful. Protective eye wear is typically required where direct viewing of a class 3B laser beam may occur. Class 3B lasers must be equipped with a safety interlock. 77 APPENDIX D. SAFETY AND HAZARD ANALYSIS • Toluene The DLS samples were placed in a toluene bath. Toluene is used because it reduces undesirable reflections and refractions as it has a similar index of refraction as the glass of the sample. Toluene should not be inhaled due to its health eﬀects. Low to moderate levels can cause tiredness and confusion or loss of appetite, even hearing and color vision loss have been observed. These symptoms usually disappear when exposure is stopped. Inhaling high levels of toluene in a short time may cause nausea, or sleepiness. Additionally it can cause unconsciousness, and even death[43]. In figure 52 the NFPA code of toluene is given. As can be seen from the figure toluene has intense health hazards and high flammability risks. Figure 52: NFPA 704 of chloroform and toluene. In this fire diamond every thread is given a score from 0 to 4, ranging from no hazard to severe risk. Blue stands for health risks, red for flammability, yellow for reactivity and white for special codes for unique hazards [44]. 78 Bibliography [1] F. Gittes, B. Schnurr, P.D. Olmsted, F.C. MacKintosh and C.F. Schmidt, “Microscopic Viscoelasticity: Shear Moduli of Soft Materials Determined from Thermal Fluctuations”, Physical review letter, vol. 79, pp. 3286-3289, 1997. [2] A. Palmer, T.G. Mason, J. Xu, S.C. Kuo, D. Wirtz, “Diﬀusing Wave Spectroscopy Microrheology of Actin Filament Networks”, Biophysical Journal, vol. 76, pp. 1063-1071, 1999. [3] M.J. Solomon, Q. Lu, “Rheology and Dynamics of Particles in Viscoelastic Media”, Colloid and interface science, vol. 6, pp. 430-437, 2001. [4] T.M. 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