A random walk description of the heterogeneous glassy dynamics of attracting colloids

We study the heterogeneous dynamics of attractive colloidal particles close to the gel transition using confocal microscopy experiments combined with a theoretical statistical analysis. We focus on single particle dynamics and show that the self-part of the van Hove distribution function is not the Gaussian expected for a Fickian process, but that it reflects instead the existence, at any given time, of colloids with widely different mobilities. Our confocal microscopy measurements can be described well by a simple analytical model based on a conventional continuous time random walk picture, as already found for several other glassy materials. In particular, the theory successfully accounts for the presence of broad tails in the van Hove distributions that exhibit exponential, rather than Gaussian, decay at large distance. 1. Dynamic heterogeneity in colloidal gels There are many systems in nature whose dynamics become slow in some part of their phase diagram, because they undergo a transition from a fluid to a disordered solid phase— like in a sol–gel transition, a glass transition, or a jamming transition. These systems are generically called ‘glassy materials’, examples of which are simple or polymeric liquids, colloidal particles with soft-core or hard-core interactions, grains, etc. As physicists, we would like to have a microscopic understanding of the slow dynamics of these materials and would like to answer, in particular, an apparently very simple question: How do particles move in a glassy material close to the fluid–solid transition? To answer this question directly, one needs to resolve the dynamics of individual particles. In experiments, this is a particularly hard task for molecular liquids, although some techniques are now available [1, 2] but becomes much easier in the colloidal and granular worlds, where direct visualization is possible [3–10]. Of course, resolving single particle dynamics is trivial in computer simulations where, for each particle in the system, the equations of motion are directly integrated. Hence, single particle dynamics have now been well documented, both numerically and experimentally, in a wide variety of materials. A most striking feature emerging from these studies is the existence of dynamic heterogeneity [11]. In terms of single particle trajectories, dynamic heterogeneity implies the existence of relatively broad distributions of mobilities inside the system. It is therefore an important task to suggest a framework to describe and interpret those data, and hopefully understand the physical content carried by single particle displacements. In this work, we study an assembly of moderately attractive colloidal particles (attraction depth U ≈ 3kBT , where kBT is the thermal energy) that undergo dynamic arrest at an ‘intermediate’ volume fraction, φc ∼ 0.44 [9]. The system is in fact intermediate between fractal gels made of very strongly attractive particles (U kBT ) at very low volume fraction, and hard sphere glasses obtained with no attraction (U ≈ 0) at a much higher volume fraction, φ ≈ 0.6. Although experiments clearly detect the presence of an amorphous phase with arrested dynamics, the nature of the transition towards this ‘dense gel’ (or low density glass!) remains unclear [12]. The transition seems too far from the so-called ‘attractive glass’ obtained at higher volume fraction in colloids with very short0953-8984/08/244126+07$30.00 © 2008 IOP Publishing Ltd Printed in the UK 1 J. Phys.: Condens. Matter 20 (2008) 244126 P Chaudhuri et al range attraction (sticky particles), so that other phenomena are usually invoked. A popular hypothesis is that gelation is in fact a non-equilibrium phenomenon due to a kinetically arrested phase separation [12, 13]. Dynamic heterogeneity in such systems has been analyzed before in just a few systems, both experimentally [9, 10] and numerically [14, 15]. In this paper, we analyze single particle dynamics on the approach to the glassy phase and show that the self-part of the van Hove distribution function is not the Gaussian expected for a Fickian process, but that it reflects instead the existence, at any given time, of colloids with widely different mobilities: Our system is dynamically heterogeneous. We then show that the simple analytical model proposed in [16] to describe data in a variety of systems close to glass and jamming transitions also describes our experimental data in a satisfactory manner. This paper is organized as follows. In section 2 we describe the system, experimental techniques, and the results obtained for the van Hove function. In section 3 we describe the model used to fit the experimental data and discuss the results. We conclude the paper in section 4. 2. Measuring single particle dynamics using confocal microscopy 2.1. Experimental system and techniques The experimental system under study is a suspension of colloidal particles interacting through a hard-core repulsion and a softer attractive interaction, induced by depletion by adding polymers. Details of the system have been presented in [9]. The dynamics of this system is observed using confocal fluorescence microscopy. The strength of the inter-particle attractive interaction, U , is determined by the concentration of polymers in the suspension. We present data for a sample at a moderate interaction strength of U ≈ 2.86kBT . We work at constant temperature T , so that our control parameter is the volume fraction of the particles, φ. We find that the system becomes a gel when φ is increased, with a transition close to φc ≈ 0.442 [9]. Measurement of different relevant statistical quantities are carried out at different φ < φc. Our procedure to vary slowly the volume fraction uses particle sedimentation. The relative buoyancy of the colloids is ρ = 0.011 g cm−3, corresponding to a gravitational height of h = kBT/( 3πa ρg) ≈ 40 particle radii a, where g is the acceleration due to gravity. Therefore, the gravitational field is small enough that it induces a very slow densification of the system. The densification is slow enough that microscopic dynamics of the colloids remains controlled by the interplay between attraction and steric hindrance, rather than by sedimentation itself. Moreover, the large asymmetry between polymer coil diffusion time, ≈0.3 s, and particle sedimentation time over one particle, ≈260 s, ensures that polymers are uniformly distributed, maintaining the interaction strength U constant in the course of the experiment. The colloidal particles are polymethyl-methacrylate (PMMA) spheres of diameter 1.33 μm, sterically stabilized by chemically grafted poly-12-hydroxystearic acid, dyed with the electrically neutral fluorophore 4-chloro-7-nitrobenzo-2 Figure 1. Three-dimensional confocal microscopy rendered image of a typical particle configuration at volume fraction φ = 0.429. oxa-1,3-diazole (NBD), and suspended in a solvent mixture of decahydronaphthalene (decalin), tetrahydronaphthalene (tetralin), and cyclohexyl bromide (CXB) that allows for independent control of the refractive index and buoyancy matching with the particles. Polystyrene polymers (molecular weight 11.4×106 g mol−1) are added at 1.177 mg ml−1 to induce a depletion attraction at a range estimated by = 2Rg = 0.28a, where Rg is the polymer radius of gyration. Using confocal microscopy, we collect stacks of images at fixed time intervals ranging from 12 to 1500 s at different φ to access short-and long-time dynamics during the approach to gelation. From the stacks of images we extract the particle positions of 103 particles in three dimensions and track their positions at better than 10 nm resolution over time. A threedimensional rendering of a typical particle configuration from a stack of images at φ = 0.429 is illustrated in figure 1. 2.2. Non-Gaussian distributions of single particle displacements In an earlier work [9], we analyzed some structural and dynamical properties of the system for different volume fractions. In particular, we analyzed in some detail the distinct part of the van Hove function, finding dynamic signatures typical of gel systems. We only presented briefly some preliminary results concerning the self-part of the van Hove function. It is the latter that we investigate in more detail here. It is defined by