Microstructure and Rheology of Particle Suspension in a Yield Stress Fluid Stéphanie Deboeuf, Nicolas Lenoir, David Hautemayou, Cédric Mézière, and Guillaume Ovarlez Université Paris-Est, Laboratoire Navier (UMR 8205), CNRS, ENPC, IFSTTAR, F-77420 Marne-la-Vallée [email:
[email protected],
[email protected],
[email protected]] Keywords: Microstructure – Particles – Rheology – Suspension – X-ray microtomography – Yield stress fluid
ABSTRACT Numerous industrial and natural fluids, such as fresh concrete or debris flows, are made of coarse particles suspended in a yield stress fluid. Such suspensions exhibit complex rheological behaviours: elasto-plasticity, shear-thinning (non constant viscosity), normal stress differences, ... In the light of results about shear-induced microstructure in concentrated suspensions of spheres in Newtonian fluids, we experimentally addressed the role of the particle microstructure in this complex rheology. We use a model system made of non-Brownian spherical hard particles suspended in a concentrated emulsion. We associate rheological measurements to X-ray microtomography imaging techniques allowing for the determination of high resolution pair distribution functions in three dimensions. We characterize the rheological behavior and the microstructure for various shear histories (squeeze flow and rotational flow, transient and steady state, various low and high shear rates) and we relate it to their elastic and plastic properties. We show that the elastic and plastic properties of the suspensions depend strongly on shear history; in particular, for certain initial states, the materials display a strain hardening behavior. We relate all these complex properties to the observed shear-history-dependent microstructures. 1.
INTRODUCTION
In a concentrated suspension of particles, even in the absence of long-range interactions between particles, the suspension microstructure (spatial distribution of particles) is strongly related to stress transmission in the suspension and to its rheology (e.g., flow of the fluid is hindered by the solid particles, high stresses develop in lubrication fluid films between neighbouring particles). In the case of particle suspension in a Newtonian fluid, a simple shear flow induces an anisotropic and asymmetric microstructure in the shear-velocity plane (Parsi and Gadala-Maria 1987, Blanc et al. 2013). This anisotropy originates from direct interparticle interactions, and is at the origin of the development of normal stress differences. Moreover, any change in the spatial characteristics of flow is shown to result in a progressive change of the microstructure and, as a result, to a time-dependent macroscopic behaviour. Here, we address the characterization of the suspension microstructure in three dimensions, and the role of both the shear history and the linear and non-linear properties of the interstitial fluid on the microstructure and the rheological properties of the suspension. 2. EXPERIMENTS AND METHODS 2.1.
Experimental set-up and system
Our model suspension is made of non-Brownian spherical hard particles of crystalline polystyrene (PS) of diameter 140µm ± 15µm suspended in a concentrated emulsion: an aqueous iodine-loaded phase is dispersed by a high shear mixer in an oil phase. Such a concentrated emulsion has a yield stress: below the yield stress, it has a solid-like elastic behaviour; above the yield stress, it has a liquid-like viscoplastic (shear thinning) behavior. Using a yield stress fluid is relevant to investigate the role of microstructure on
both the linear and non linear properties of suspensions, and allows imaging the suspension microstructure since it is freezed as soon as the stress is decreased below the yield stress. The experimental set-up (Figure 1) allows viscosimetric measurements in a plate-plate geometry and Xray microtomography imaging. The 10 mm radius polymethyl methacrylate (PMMA) plates of the geometry are roughened and the gap is generally 2 mm. Shear is controlled (by torque or velocity) by a rheometer and is possible both for squeeze and rotational flows.
Figure 1: Parallel-plate geometry allowing for a simple shear flow and X-ray microtomography imaging, one radiograph and one horizontal slice of the suspension at 35% volumetric concentration. 2.2.
Three dimensional imaging facility and image processing
The X-ray CT measurements are conducted with an Ultratom scanner from RXSolutions at Laboratoire Navier. Figure 1 presents the experimental set-up and examples of a radiograph and a horizontal slice. The spatial resolution (both source spot size and voxel size) is 12µm. In practice, a desired shear history is imposed to the suspension in the rheometric cell, then stopped by controlling zero rotation and squeeze velocities as long as needed for the complete scanning (about 1 hour). The suspension microstructure is ensured to be exactly the same just before and after the interruption of shear because of the existence of the yield stress. Note that the contrast between the PS particles and the concentrated emulsion in the images is increased by the addition of iodine in water.
Figure 2: Opposite absorption signals I(r) of 5 particles as a function of distance to center r with 1 pixel (left) and 0.25 pixel (right) precision. The continuous line represents the particle radius, while the dotted line represents the maximal value of r used for the search of the symmetry centre and the sub-pixelisation. The 3D image is processed with 3D morphological operations, instead of as a stack of 2D images. However, one image is too large (1Go) to be loaded and processed as a whole at once, so that it is treated as parts of 3D sub-elements. First, it is smoothed with a gaussian filter of 1 pixel characteristic size. PS being less absorbent to X-rays than the iodine-loaded fluid, the particles are darker than the fluid in the absorption image. Through a particle, the signal is not homogeneous but heterogeneous, the intensity increasing with distance from its center. This property allows for a first estimation of the center particles from the local minima of the image intensity. Then we systematically compute the center of the
particle as its symmetry center: the center is chosen as the position minimizing the standard deviation of the intensity signal as a function of the distance to the center I(r) (Figure 2). By interpolating linearly the image of the particle, we achieve to the desired precision (1µm) on the position of the particle.
Figure 3: Pair distribution functions gz at 1pixel (left) and 0.25pixel (right) resolution 2.3.
Pair Distribution Function
From the particle positions, we compute the pair distribution function g(r) in three orthogonal planes, suggested by the cylindrical symmetry of the set-up and the flow: gr(∆ l ,∆z) in the (θz) azimuthal plane, gθ(∆ρ,∆z) in the (rz) radial plane and gz(∆ l ,∆ρ) in the (θr) horizontal plane, with ∆ρ=r-r0, ∆ l =r0(θ-θ0), ∆z=z-z0, the coordinates of the pairs of particles of cylindrical coordinates (r,θ,z) and (r0,θ0,z0). In the case of a rotational shear, the flow is along θ and the shear along z, whereas in the case of a squeeze flow, the flow is along r and the shear along z. See the enhanced spatial resolution of the pair distribution function gz by using sub-pixelisation for the center particle (Figure 3).
3. RESULTS AND DISCUSSION 3.1.
Rheological characterization: Elasto-plasticity
Figure 4: Quasi-static stress-strain τ/τ Y(γ) curves for different shear histories of the suspension and elastic modulus G as a function of stress τ during unload and load in the opposite direction. +
The quasi-static stress-strain curve of the suspension (for a constant and low shear rate) characterizes its elasto-plastic behavior (Figure 4). Starting from different shear histories, the stress-strain response is very different. When the material is first pre-sheared and then loaded in the same direction as the pre-shear, it
has an almost perfect elasto-plastic behaviour. When it is loaded in the direction opposite to that of the pre-shear, strain hardening is observed: plastic deformations are observed at low stresses, and the stress increases progressively with strain until a steady state is reached. In parallel, we characterize the elastic modulus of the suspension by oscillation measurements. We observe that the elastic modulus G drops down during shear reversal before increasing progressively towards steady state (Figure 4). These shear history dependent properties are expected to reflect changes in the suspension microstructure. Our aim is now to correlate the microstructure and these elasto-plastic properties. 3.2.
Microstructure characterization
As an illustration of the investigation of the correlations between the behavior and the microstructure are plotted two examples of the three-dimensional pair distribution functions g(r) (Figure 5) for different shear histories. When looking at a suspension before any shear far from the boundaries, the microstructure is isotropic in each plane and in three dimensions (Figure 5 left). By contrast, after a rotational steady shear, the shear-velocity plane shows an anisotropic and asymmetric microstructure (along the axis of shear) (Figure 5 right). Many other experiments have been performed: we are now able to show how the microstructure is developed during shear, and to observe its dependence on shear intensity.
Figure 5: Microstructure of a particle suspension in the three cylindrical (θz), (rz) and (θr) planes gr(∆ l ,∆z), gθ(∆ρ,∆z) and gz(∆ l ,∆ρ) before any shear (left) and after rotational shear (right).
4. CONCLUSIONS The association of X-ray microtomography imaging and rheological measurements is a powerful tool to study the relation between microstructure of a particle suspension and its rheology. Moreover, some preliminary tests have been conducted on density-matched suspensions in Newtonian fluids and showed promising results. 5. ACKNOWLEDGEMENTS This work was supported by grants from Région Ile-de-France and from Agence National de la Recherche (ANR 2010 JCJC 0905 01 – SUSPASEUIL) 6. REFERENCES Blanc, F., Meunier, A., Lemaire, E. and Peters, F. (2013). Microstructure in sheared non-Brownian concentrated suspensions. Journal of Rheology, 57 (1), 273-292. Parsi, F. and Gadala-Maria, F. (1987). Fore-and-aft asymmetry in a concentrated suspension of solid spheres. Journal of Rheology, 31 (8), 725–32
