INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTERJ. Phys.: Condens. Matter 14 (2002) 7581–7597 PII: S0953-8984(02)38648-XDirect imaging of three-dimensional structure andtopology of colloidal gels ADDinsmore1,2and DAWeitz11Department of Physics and DEAS, Harvard University Cambridge, MA 02138, USA2University of Massachusetts Physics Department, Amherst, MA 01003, USAReceived 25 June 2002Published 9 August 2002Onlineatstacks.iop.org/JPhysCM/14/7581AbstractWe present novel measurements of the structure of colloidal gels. Usingconfocal microscopy, we obtain the precise three-dimensional positions ofalarge number of particles. We develop quantitative descriptions of thetopology of the gel, including the number of bonds per particle, the chemicalor bond fractal dimension, the number of flexible pivot points and othertopological parameters that describe the chainlike structure. We investigatethe dependence of these parameters on the particle volume fraction and thestrength of the attraction that holds the particles together. While all sampleshave approximately the same fractal and chemical dimensions, we find that gelsformed with stronger attraction or larger volume fraction have fewer bonds perparticle, more filamentous chains and a greater number of flexible pivot points.Finally, we discuss the topological results in the context of the gel’s elasticity.Measurements of the elastic constants of individual chainlike segments areexplained with a simple model. The distribution of elastic constants, however,has a general form that is not understood.(Some figures in this article are in colour only in the electronic version)Monodisperse colloidal particles have served as a fascinating model system for the study of awide range of phenomena. Their size is ideal for light scattering, and they have long servedas a model system for the development of new scattering techniques, both dynamic and static.They have been important models in the understanding of hydrodynamic interactions and themethods for probing them by scattering. They have also served as a model for the study ofphase behaviour of atomic systems, with each colloidal particle playing the role of an atom.The larger size of the colloidal particles compared with atoms and resultant slower diffusivemotion make it feasible to use optical scattering techniques to study their behaviour. Thus,they provide a fascinating statistical model system with which to probe the phase behaviour ofmaterials. When the particles interact exclusively through volume exclusion, they behave ashard spheres and exhibit both crystalline and glassy states as their volume fraction is varied.0953-8984/02/337581+17$30.00 © 2002 IOP Publishing Ltd Printed in the UK 75817582 ADDinsmoreand D A WeitzUpon addition of a high concentration of much smaller particles, an attractive interaction isinduced between the particles, and they form a colloidal gel. One of the true leaders in the studyof colloidal particles, and in the development of the light scattering techniques which has madetheir study feasible, is Professor Peter Pusey. He has been a pioneer in the development of thelight scattering techniques that have been so important to the study of colloid physics, and hehas introduced the essential concepts for the study of both colloidal glasses, and colloidal gelsmade by depletion interactions.Much of our understanding about the properties of colloidal particles comes from lightscattering studies. This is particularly true for colloidal gels. The randomness of the structuremakes the ensemble averaging of light scattering techniques a very valuable probe. The truehallmark of a gel is the existence of a modulus at low frequency, and thus another valuableprobe for colloidal gels has been rheology. Great progress has been made in developingan understanding of the elastic behaviour from the scattering data, thereby relating the bulkrheological properties of a gel to the microscopic structure. Unfortunately, however, while lightscattering measurements do provide an excellent probe of the ensemble average properties ofacolloidal gel, scattering does not provide information about the detailed local structure. Acomplete understanding of gel rheology, moreover, depends on much more than the averagestructure; the topology of connections among particles also plays a critical role in the rheogicalbehaviour of the network.The goal of this paper is to develop a more detailed picture of the topology of a colloidalgel, and to explore the relationship between topology and the elastic properties of the network.Topological information is not accessible from light scatteringdata or from rheology. Computersimulations of aggregation have categorized a number of these topological characteristics,including the chemical or bond dimension, which describes the scaling of the contour lengthwith separation, the spectral dimension and some information about structural weak points inthegel [1, 2]. Although these parameters serve as the key inputs to phenomenological modelsof gel elasticity [3–6], they have not been directly measured to date.In this paper, we describe detailed measurements of the structure and topology of colloidalgels. We use a confocal microscope to measure the precise three-dimensional positions ofthousands of individual particlesinthe gel. In addition to presenting our data, we describe indetail our methods of quantifying structure and topology. It is useful to describe a gel as anentangled network of chains; the topology of individual chains and the manner of their inter-connections together dictate the bulk elasticity. Here we focus on the topology and elasticity ofindividual chains, each consisting of a sequenceofparticles bonded to one another. We reporton the contour length of chains, their cross-sectional area, their radius of gyration and thenumber of particles that can freely pivot in all directions without locally stretching bonds. Weshow how these properties vary in samples with different particle concentrations and strengthsof attraction. We also show how these propertiesdictate the elasticity of individual chains.1. Samples and experimental proceduresWe use poly(methylmethacrylate) (PMMA) particles, dyed with fluorescent rhodamine andsuspended in a mixture of decalin and cyclohexylbromide that matches both the density andrefractive index of the particles [7, 8]. The particles did not settle to any observable extentafter centrifuging at 6000 g
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