Direct imaging of three-dimensional structure and topology of colloidal gels

We present novel measurements of the structure of colloidal gels. Using confocal microscopy, we obtain the precise three-dimensional positions of a large number of particles. We develop quantitative descriptions of the topology of the gel, including the number of bonds per particle, the chemical or bond fractal dimension, the number of flexible pivot points and other topological parameters that describe the chainlike structure. We investigate the dependence of these parameters on the particle volume fraction and the strength of the attraction that holds the particles together. While all samples have approximately the same fractal and chemical dimensions, we find that gels formed with stronger attraction or larger volume fraction have fewer bonds per particle, 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 are explained 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 a wide range of phenomena. Their size is ideal for light scattering, and they have long served as 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 the methods for probing them by scattering. They have also served as a model for the study of phase 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 diffusive motion 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 of materials. When the particles interact exclusively through volume exclusion, they behave as hard 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 7581 7582 A D Dinsmore and D A Weitz Upon addition of a high concentration of much smaller particles, an attractive interaction is induced between the particles, and they form a colloidal gel. One of the true leaders in the study of colloidal particles, and in the development of the light scattering techniques which has made their study feasible, is Professor Peter Pusey. He has been a pioneer in the development of the light scattering techniques that have been so important to the study of colloid physics, and he has introduced the essential concepts for the study of both colloidal glasses, and colloidal gels made by depletion interactions. Much of our understanding about the properties of colloidal particles comes from light scattering studies. This is particularly true for colloidal gels. The randomness of the structure makes the ensemble averaging of light scattering techniques a very valuable probe. The true hallmark of a gel is the existence of a modulus at low frequency, and thus another valuable probe for colloidal gels has been rheology. Great progress has been made in developing an understanding of the elastic behaviour from the scattering data, thereby relating the bulk rheological properties of a gel to the microscopic structure. Unfortunately, however, while light scattering measurements do provide an excellent probe of the ensemble average properties of a colloidal gel, scattering does not provide information about the detailed local structure. A complete understanding of gel rheology, moreover, depends on much more than the average structure; the topology of connections among particles also plays a critical role in the rheogical behaviour of the network. The goal of this paper is to develop a more detailed picture of the topology of a colloidal gel, and to explore the relationship between topology and the elastic properties of the network. Topological information is not accessible from light scattering data or from rheology. Computer simulations of aggregation have categorized a number of these topological characteristics, including the chemical or bond dimension, which describes the scaling of the contour length with separation, the spectral dimension and some information about structural weak points in the gel [1, 2]. Although these parameters serve as the key inputs to phenomenological models of 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 colloidal gels. We use a confocal microscope to measure the precise three-dimensional positions of thousands of individual particles in the gel. In addition to presenting our data, we describe in detail our methods of quantifying structure and topology. It is useful to describe a gel as an entangled network of chains; the topology of individual chains and the manner of their interconnections together dictate the bulk elasticity. Here we focus on the topology and elasticity of individual chains, each consisting of a sequence of particles bonded to one another. We report on the contour length of chains, their cross-sectional area, their radius of gyration and the number of particles that can freely pivot in all directions without locally stretching bonds. We show how these properties vary in samples with different particle concentrations and strengths of attraction. We also show how these properties dictate the elasticity of individual chains. 1. Samples and experimental procedures We use poly(methylmethacrylate) (PMMA) particles, dyed with fluorescent rhodamine and suspended in a mixture of decalin and cyclohexylbromide that matches both the density and refractive index of the particles [7, 8]. The particles did not settle to any observable extent after centrifuging at 6000 g for 15 h, confirming the buoyancy match. We report results from ten samples with a variety of particle sizes and concentrations, as detailed in table 1. The mean particle radius, a, was measured with an uncertainty of approximately 0.05 μm from the lattice parameter of close-packed colloidal crystals and from the contact value of the pair distribution function of gelled samples. The measured radius agreed with the value Direct imaging of three-dimensional structure and topology of colloidal gels 7583 Table 1. Samples reported here, along with the labels used in the text. ξ = Rg/a. ‘Fluid-clust’ refers to samples that form a fluid of aggregates (see figure 1(a)). ‘Gel’ refers to samples that form an aggregate of macroscopic size (see figure 1(b)). Label cp φ a/Rg (nm) ξ Description A 5.1 0.03 350/38 0.11 Fluid-clust B 5.5 0.03 350/38 0.11 Fluid-clust C 9.1 0.03 350/38 0.11 Gel D 12.2 0.03 350/38 0.11 Gel E 9.3 0.06 350/38 0.11 Gel F 9.8 0.10 350/38 0.11 Gel G 6.7 0.05 350/93 0.27 Gel H 22.3 0.04 750/8.4 0.01 Gel I 7.2 0.04 750/38 0.05 Gel J 4.6 0.05 750/93 0.12 Gel determined by light scattering from particles suspended in dodecane. We determine the particle volume fraction, φ, by measuring the number density of particles, and knowing their radius, a. We induce a depletion attraction of controlled strength through the addition of polystyrene (PS, Polymer Laboratories) [9–13]. Three different molecular weights are used to vary the range of the potential, with molecular weights and estimated radii of gyration, Rg , of 96 000 (Rg = 8.4 nm), 1.95 × 106 (Rg = 38 nm) and 11.6 × 106 (Rg = 93 nm). This allows us to vary the ratio of size of the polymer and the size of the spheres, ξ ≡ Rg/a. We define the concentration of polymer, cp, as the mass of polymer, in milligrams, divided by the volume of the solvent, in millilitres (the total volume multiplied by 1 − φ). The maximum strength of the depletion attraction, Ud0, is numerically approximately equal to cp, when measured in kB T , in samples with a = 350 nm (samples A–G) and approximately twice cp in samples with a = 750 nm (samples H–J). The process of incorporating the rhodamine dye leaves a charge on the particles, resulting in a long-range repulsion. The magnitude of the electrostatic repulsion is estimated by measuring the gelation time and comparing it with the DLCA model [14]. In samples with a = 750 nm (G–J), the estimated charge magnitude lead to a repulsive interaction of UC(r) = 0.3kB T/(r − ac), where ac is the radius of the sphere corrected for the thickness of the steric layer, whose thickness is about 10 nm [14]. Sample cells consist of a #1 1 2 cover slip glued to a standard glass microscope slide. Glass spacers are used to construct a gap with 1 mm thickness; the sample area is approximately 10 mm × 30 mm. Adhesive cured with ultraviolet light (Norland NOA) is used to construct the empty cell. The colloid is injected into the cell through a 3 mm hole, which is subsequently sealed using 5 min epoxy. To avoid contact between the colloid and the wet epoxy, the cell is only filled half-way with colloid. After the epoxy has hardened for at least a day, sample cells are shaken, then placed on the microscope stage within 30 s to verify complete breakup of any aggregates. To avoid disturbance of the sample by motion of the air bubble, samples are left affixed to the sample stage during the observation period, which is typically several days. Real-space images are collected with a confocal microscope with a scanning slit (Noran Oz). Images are acquired at 15 or 30 frames per second, with pixel sizes of either 0.086 or 0.22 μm in the xand y-directions. Image stacks typically contain approximately 100 images, each offset by 0.2 μm in the z-direction. Fluorescence is excited using with the 488 nm line of an Ar+ ion gas laser; the fluorescence is filtered with a 500 nm long-pass filter, and is detected with a photomultiplier tube [8]. Images are analysed using IDL routines to filter the images and locate particle positions in three dimensions [7, 15]. Images with the computer-determined particle positions overlaid are 7584 A D Dinsmore and D A Weitz Figure 1. (Top) Confocal microscope image of a thin slice through a sample in the fluid-cluster phase (sample A). (Bottom) Image of a sample that has formed a gel (sample D; 21 h). All of the particles in this image are part of the same cluster. The scale bars represent 10 μm. routinely checked to ensure accuracy. Each three-dimensional field of view contains between 5000 and 15 000 particles. We estimate that less than ∼0.1% of particles are not detected by the software. The uncertainty in the particle positions is estimated to be 50 nm [8]. 2. Phase behaviour and aggregation kinetics In this section, we provide descriptions of the final state of the samples as well as the kinetics of aggregation. Most samples formed colloidal gels. The aggregation is quite slow, presumably owing to the weak, long-range electrostatic repulsion. 2.1. Phase behaviour All of the ten samples discussed in this article undergo random aggregation; we do not observe any crystalline structures. In table 1, we list the samples discussed here along with their parameters and their final state. Two samples form a fluid of aggregates of finite size, which we refer to as ‘fluid-cluster’ samples [13, 16], as shown by a typical image in figure 1(a), which is for sample A, with Direct imaging of three-dimensional structure and topology of colloidal gels 7585 Ud0 = 5.1, φ = 0.03, a = 350 nm and Rg = 38 nm. Clusters that had formed after 5 h remained in suspension for at least 21 h with no further growth in mean cluster size. Sample B (cp = 5.5, φ = 0.03) behaved in a similar way; samples with lower cp did not aggregate and samples with larger cp (Ud0 > 6, for φ = 0.03) formed a gel. The underlying origin of the fluid-cluster phase is not clear. The most obvious possibility is that the clusters are in dynamic equilibrium, with cluster formation and breakup occurring at equal rates. However, we never observed a cluster break or a particle spontaneously dissociate from an aggregate. Thus, the origin of this cluster-fluid phase remains uncertain. All other samples (C–J) formed a gel within the experimentally accessible timescale (≈50 h). An optical micrograph of the cross-section of a typical sample of such a gel is shown in figure 1(b), which is from sample D, after 29 h. We verified that the depletion-induced aggregation is reversible: clusters melted when cp was reduced by gentle dilution. To avoid shear-melting the clusters during dilution, we constructed a sample cell containing a compartment isolated by a filter membrane. We then gently added solvent (cp = 0) to one side of the membrane, which protected aggregates within the compartment from convective flow. As the polystyrene polymer diffused out of the compartment through the membrane, clusters dissociated into single particles in a matter of minutes. 2.2. Aggregation kinetics To study the aggregation kinetics, we homogenize the samples by shaking them, breaking all clusters, as verified by taking images within 30 s. We then acquire a time-lapse series of three-dimensional images to probe the evolution of aggregate structure. The interval between three-dimensional images (typically >30 s) was too large to allow tracking of individual particles from one three-dimensional image to the next. Thus, we monitor the evolution of the clusters, rather than single-particle kinetics. Early in the aggregation process, the clusters were small and numerous, so the finite size of the field of view does not significantly affect the cluster statistics. We define individual clusters topologically, by finding all particles that are connected to one another through depletion ‘bonds’. Bonded particles are identified by their separation. The probability of inter-particle separations r shows a well defined minimum at r = r∗, where r∗ is slightly larger than 2a + 2Rg . Particles with separation less than r∗ are defined as being bonded. The mean number of particles per cluster, 〈n(t)〉, increases approximately linearly with time at early times, when the clusters are small, providing a sufficient number of clusters to obtain reliable statistics. At later times, there are insufficient clusters to reliably determine the growth kinetics. Thus, we find 〈n(t)〉 = t/τ , for 30 s < t < 300 s. However, the value of τ varies from 102 to 103 s in samples A–D, which is significantly larger than the diffusion-limited aggregation time, or the average time it takes for two single particles to collide due to diffusion, τagg = R2 sphere/(6φD) ≈ 5 s [17]. Linear growth is expected for purely diffusion-limited cluster aggregation (DLCA) [18], and has been observed for DLCA of colloidal gold [19] and for polystyrene spheres [20, 21]. However, in the latter study, while the aggregation time was initially given by τagg, it then increased to approximately 3.6 τagg for t > 15τagg. In our samples the measured aggregation time was even larger, approximately 10τagg; this is most likely due to the electrostatic repulsion. By contrast, for samples undergoing reactionlimited cluster aggregation (RLCA) [22], 〈n(t)〉 is expected to grow exponentially. However, many systems behave in a manner intermediate between these two limiting regimes, and a linear growth of the clusters is often observed in this regime [23, 24]. Thus, it is most likely 7586 A D Dinsmore and D A Weitz cp = 12.2, φ = 0.03 (D) cp = 9.3, φ = 0.06 (E) cp = 9.1, φ = 0.03 (C) cp = 5.5, φ = 0.03 (B) cp = 5.1, φ = 0.03 (A) n/ n N (n ) n 2. 2 / N 0 0.1 1.0 10.0 100.0 10-4 10-3 10-2 10-1 100 101