# Microstructure, local dynamics, and flow behavior of colloidal suspensions with weak attractive interactions

## Abstract

We present a comprehensive micro- and macrorheological study of the effect of weak depletion attraction (Ψdep ≈ 1–10 kBT) on dense colloidal suspensions stabilized by short-range repulsive interactions. We used aqueous polymer dispersions as model system and demonstrated the unique capabilities of multiple particle tracking (MPT) to disclose structural changes in such technically important systems exhibiting many characteristic features of hard sphere systems. Below the hard sphere freezing point ϕc, viscosity increases monotonically with increasing Ψdep due to the transition from a fluid to a fluid/crystalline and finally to a gel state. Above ϕc, increasing attraction strength first results in a viscosity reduction corresponding to the formation of large, permeable crystals and then in a viscosity increase when a network of dense, small crystals forms. The fraction of the fluid and crystal phase, particle concentration in each phase as well as the modulus of the micro-crystals are obtained, the latter decreases with Ψdep. Above the colloidal glass transition strong heterogeneities and different local particle mobility in the repulsive and attractive arrested states are found. Particles are trapped in the cage of neighboring particles rather than in an attractive potential well. The intermediate ergodic state exhibits uniform tracer diffusivity.

## Introduction

Understanding phase behavior, structure, and dynamics of colloidal dispersions is of utmost importance for two complementing reasons. First, they serve as model systems for studying fundamental physical phenomena like crystallization, gelation, and glass formation at conveniently accessible time and length scales1,2,3. Second, they are widely used in the fabrication of nanostructured materials in emerging innovative fields of application, e.g., photonic crystals, and in well-established commodity coatings and adhesives, often as precursors enabling the processing of materials after drying and sintering for final use in their solid state. A key technologic challenge is to control their flow properties to meet the many requirements of processing and application. Numerous experimental and theoretical investigations as well as numerical simulations have addressed the above-mentioned phenomena, and so-called hard sphere model systems characterized by just one physical parameter, namely the volume fraction ϕ, are used to study condensed matter physics4,5,6. It must be kept in mind, however, that the hydrodynamic interactions among suspended colloids mediated by the surrounding solvent are different than the interactions in molecular fluids. They affect diffusivity7, high frequency rheology8, and even crystallization9, thus somewhat limiting the ability to transfer the results found for colloidal suspensions to molecular fluids. Nevertheless, hard sphere colloidal model suspensions are of invaluable importance for understanding colloidal suspensions in general. In particular, phase behavior and flow properties of suspensions with short-range repulsive particle interactions can be qualitatively described using the effective volume fraction ϕ taking into account repulsive thermodynamic forces (hard sphere mapping)10.

In addition to macroscopic rheological experiments and scattering techniques, confocal scanning microscopy is a powerful tool for studying the structure and dynamics of colloidal hard sphere systems. Dynamic heterogeneities and structural relaxation phenomena near the colloidal glass transition11,12, have been investigated as well as the microscopic structure of shear thinning and shear thickening suspensions13. In these experiments, the thermal motion of several thousand particles is tracked with high spatial and temporal resolution14. The particle diameter is typically between 1 and 5 μm and polymer particles are suspended in appropriate organic solvent mixtures to provide refractive index matching, not only because of technical imaging issues, but also to mimic hard sphere properties as closely as possible. In some studies, the suspensions are density matched to avoid sedimentation and, particularly, gravitational effects on crystallization15. Technically relevant polymer dispersions, however, typically contain particles that are an order of magnitude smaller; not index-matched to the solvent; solved in water; not likely to form a sediment; and are stabilized via short-range steric, electrosteric, and electrostatic repulsion to provide a shelf life of at least several months. Such suspensions are utilized as emerging innovative materials16,17, but, more importantly, today they are used in numerous kinds of coatings or adhesive products and annual global production is on the order of 10 million tons. Accordingly, new insight into their structure and processability has a potentially huge impact on a wide range of fields. Here we present comprehensive microrheological studies on such turbid aqueous colloidal polymer dispersions covering a broad concentration range from the dilute to the glassy state. Our focus is on the effect of weak attractive interactions induced by non-adsorbing polymers dissolved in the aqueous phase. We used multi particle tracking (MPT) microrheology, i.e., we simultaneously tracked the Brownian motion of several hundred fluorescent tracer particles added to a colloidal suspension. This technique was originally developed to study cell biology systems18,19, but was later applied to study heterogeneity, e.g., in clay suspensions20,21, polymeric thickener solutions22,23, and agarose24and food gels25. Here, we applied it to highly turbid colloidal suspensions for the first time. Particle trajectories can be tracked within a focal plane ∼30 μm deep within these samples, which corresponds to more than 100 particle diameters, i.e., artifacts due to the sample surface can be excluded. One can calculate the linear viscoelastic modulus G* from the time dependence of the mean square displacement (MSD) of individual particles26, and for homogeneous fluids this agrees well with the macroscopic modulus obtained from bulk mechanical rheometry. Here we calculate the viscosity ηMPT from MSDs of freely diffusing tracers in a viscous fluid

where kB is the Boltzmann constant, T is the temperature, a is the particle radius, and D is the diffusion coefficient related to the MSD depending linearly on lag time Δr2(τ) = 4Dτ. Tracers trapped in crystalline or glassy domains exhibit a time-independent MSD directly related to the shear modulus of these regions.

To characterize sample heterogeneities, we analyzed the distribution and slopes of the MSDs. We further calculated the van Hove correlation functions, i.e., the probability distribution of particle displacement for an ensemble of N tracked particles as:

Where N(x, τ) is the number of particles found at positions between x and x + dx along the x-coordinate. P(x, τ) is Gaussian if all tracer particles are exposed to a similar environment. Deviations from this functional form reflect the presence of spatial heterogeneities and can be characterized by the non-Gaussian parameter α.

in which x4 and x2 are the fourth and second moments of P(x, τ). For a Gaussian distribution α = 0, experiments even on homogeneous systems usually yield α values slightly larger than zero but sample heterogeneities result in much larger values α ≫ 0. Finally, we used Voronoi triangulation27 and image overlay techniques to visualize the length scale of the spatial heterogeneities. This allowed us to study the non-trivial relationship between macroscopic flow behavior and sample composition (crystal fraction, size and densities) that comes with the broadening of the fluid/crystalline co-existence regime due to the introduction of weak attractive particle interactions28. We also studied changes in heterogeneity related to the so-called re-entry phenomenon, i.e., the fluidization of colloidal systems at particle loadings beyond the hard sphere glass transition, ϕ > ϕg,HS = 0.58, induced by a weak depletion attraction29.

## Results

Here, we present and discuss the results obtained for different colloidal dispersions stabilized by short-range electrosteric repulsion. Details about the chemical composition and particle size distribution are provided in the Materials and Methods section. For comparison of the results from different systems among each other and with published data, we determined an effective volume fraction from the divergence of the zero shear viscosity and, for brevity, we use the term ϕ for this quantity in the following text.

### Fluid State

We performed steady shear viscosity and MPT measurements of several colloidal dispersion systems in the fluid state at effective volume fractions of 0.05 < ϕ < 0.5. In all cases, the Brownian particle relaxation time τB = 6πηa3/kBT was less than 0.5 s and Δr2 ∼ τ, indicating free diffusion in a viscous fluid. Representative data are provided as supplementary information in Supplementary Fig. S1. The distribution of MSDs was narrow and the α-values were always close to zero, i.e., these suspensions behaved like homogeneous viscous fluids and we calculated ηMPT according to Eq. 1 from the ensemble-averaged MSD. In the steady shear experiments, all samples exhibited a constant Newtonian viscosity value at shear rates < 1s−1 as expected according to the short Brownian relaxation times. Corresponding macroscopic zero-shear viscosities η0 and the ηMPT data are plotted in Fig. 1 versus the volume fraction ϕ. All viscosity data are normalized to the solvent viscosity ηs. Besides experimental data Fig. 1 displays two widely used models relating viscosity η0 and particle volume fraction ϕ. The empirical model proposed by Maron and Pierce30 is used here assuming that the viscosity diverges at the colloidal glass transition ϕg: