Journal of Thermodynamics

Volume 2015 (2015), Article ID 153854, 10 pages

http://dx.doi.org/10.1155/2015/153854

## Shear Thickening in Concentrated Soft Sphere Colloidal Suspensions: A Shear Induced Phase Transition

^{1}Technische Universität Berlin, Königin-Luise-Strasse 22, 14195 Berlin, Germany^{2}Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, Netherlands

Received 26 September 2014; Accepted 17 December 2014

Academic Editor: Brian J. Edwards

Copyright © 2015 Joachim Kaldasch et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A model of shear thickening in dense suspensions of Brownian soft sphere colloidal particles is established. It suggests that shear thickening in soft sphere suspensions can be interpreted as a shear induced phase transition. Based on a Landau model of the coagulation transition of stabilized colloidal particles, taking the coupling between order parameter fluctuations and the local strain-field into account, the model suggests the occurrence of clusters of coagulated particles (subcritical bubbles) by applying a continuous shear perturbation. The critical shear stress of shear thickening in soft sphere suspensions is derived while reversible shear thickening and irreversible shear thickening have the same origin. The comparison of the theory with an experimental investigation of electrically stabilized colloidal suspensions confirms the presented approach.

#### 1. Introduction

Concentrated colloidal dispersions are of technological relevance for various industrial products such as paints, pharmaceuticals, cosmetics, lubricants, and food. They are often processed at high shear rates and stresses. In strong viscous flows, colloidal dispersions exhibit a unique transition related to an increase of the shear viscosity, termed shear thickening [1]. This effect makes it harder to pump suspensions and can cause equipment damage and failure in the production processes (flow blockage). Next to reversible shear thickening, also the irreversible aggregation of the dispersion after shear thickening has been reported known as irreversible shear thickening [2]. A fundamental understanding of the relation between particle and flow properties of dense colloidal suspensions is required in order to provide clues minimizing undesired effects or exploiting useful applications of shear thickening [3].

It has long been an issue to understand the underlying mechanisms involved in shear thickening. A number of theoretical attempts were made explaining shear thickening in sheared suspensions of Brownian particles where inertial effects can be neglected (non-Brownian particles [4] and deformable spheres like emulsions, etc. are not considered here). We have to distinguish between hard sphere suspensions with an interaction potential confined to the bare particle (Born-) repulsion and soft sphere suspensions with a combination of van der Waals attraction and long-range electrostatic or steric repulsion in addition to the Born-repulsion.

The dynamics of sheared hard sphere suspensions can be studied in simulations. In dense suspensions, the particle interaction is essentially determined by lubrication forces. When particles are pushed together along the compression axis of a sheared suspension, they must overcome the viscous drag forces between neighbouring particles in order to move away from each other. Above a critical shear rate, particles stick together generating the formation of shear induced clusters (hydroclusters) [5–8]. At high volume fractions, these transient touched particles induce jamming and an increase of the viscosity at a critical shear stress [9]. An alternative concept explaining shear thickening is related to an order-disorder transition. In this approach, an ordered, layered structure of colloidal particles in a sheared suspension becomes unstable above a critical shear rate [10–13]. Shear thickening occurs when lubrication forces between neighbouring particles cause particles to rotate out of alignment of the sheared structure, destabilizing the flow. A more sophisticated approach is based on an ad hoc mode-coupling model. It describes the instability as a stress-induced transition into a jammed state [14–17]. Also suggested is that dilation of confined suspensions may cause a rapid increase of the viscosity termed discontinuous shear thickening [18].

The presented theory is confined to the study of Brownian soft sphere suspensions. As long as the attractive interaction can be neglected, soft spheres can be approximated as so-called effective hard spheres and the abovementioned mechanisms for shear thickening in hard sphere suspensions may apply. However, soft sphere particles have a much more complex interaction than hard spheres. With increasing volume fraction van der Waals attraction causes a coagulation of colloidal particles, in particular when the repulsive stabilization of the particles is small. For soft sphere suspensions, the hard sphere approach to shear thickening has to be extended. An approach aiming at understanding shear thickening in soft sphere suspensions is an activation model [19–21]. It takes advantage from the complex interaction potential and suggests that colloidal particles arranged along the compression axis of a sheared suspension may overcome the mutual repulsion at a critical shear stress. As a result, the viscosity increases when clusters of coagulated particles are formed. For a sufficiently strong attraction between particles, these clusters cannot be disrupted by the applied shear leading to irreversible shear thickening. Otherwise, shear thickening is reversible.

While the activation model is a microscopic approach, this paper presents an alternative mesoscopic approach to shear thickening in dense soft sphere suspensions. It suggests that shear thickening is related to a shear induced phase transition. The transition is formulated in terms of a thermodynamic standard concept known as the Landau model [22, 23]. The Landau model is a mean field theory originally developed to understand symmetry breaking phase transitions. It is a widely accepted theory utilized in particular to model the dynamics of fluids and polymers [24]. Here the theory is used to describe the equilibrium coagulation transition of soft spheres. Taking the coupling of density fluctuations to the viscoelastic medium into account, it can be shown that sheared dense suspensions induce clusters of coagulated particles in equivalence to the microscopic activation model.

The paper is organized as follows. First, shear thickening in suspensions of hard and soft spheres is discussed. After establishing a hydrodynamic model for a dense suspension and deriving a Landau model for the coagulation transition, the models are combined in a subcritical bubble theory to predict the occurrence of coagulated particle clusters. The application of the subcritical bubble approach to sheared dense suspensions allows an estimation of the critical stress for shear thickening. After comparing the model with an experimental investigation, the paper ends with concluding remarks on the rheology of concentrated soft sphere suspensions in relation to their equilibrium phase diagram.

#### 2. The Model

##### 2.1. The Shear Thickening in Hard Sphere Suspensions

Hard spheres have an interaction potential that is zero when particles do not overlap and infinite otherwise (Born-repulsion). The phase diagram depends on the volume fraction of the bare particles determined by where is the particle radius and the number of particles in the volume . As shown by simulations, monodisperse hard sphere suspensions form a liquid phase and a (face-centred cubic) solid phase for volume fractions in the absence of flow. However, even for a polydispersity of the particles >5%, this transition is suppressed. At a packing fraction , the relaxation time becomes large compared to typical experimental time scales. The system does not relax anymore. This jammed state is called a colloidal glass [25, 26]. Approaching the jamming volume fraction , the apparent shear viscosity of a hard sphere suspension diverges with while is the critical exponent for low shear rates [27]. The mode-coupling theory predicts that a dynamic glass transition occurs already at . It suggests that approaching the jamming volume fraction large density fluctuations with glass-like dynamics occur in concentrated suspensions. They determine the internal relaxation time of a concentrated suspension and are related to the divergence of the viscosity approaching the jamming transition by

The flow properties of sheared dense suspensions are essentially governed by . If the applied shear rate is much smaller than the inverse relaxation time , density fluctuations disappear before they can be perturbed. For the case , however, they are deformed by the convective shear flow. As a consequence, density fluctuations are compressed along the compression axis and stretched along the elongation axis of the sheared suspension while rotating in time as schematically displayed in Figure 1. This effect causes a decrease of the apparent viscosity of hard sphere suspensions termed shear thinning [28].