In the previous section, we studied the local rheology of two systems — pure calcite atφ=11% and calcite with sodium hydroxide atφ=10%— which are characterized by identical macroscopic rheology (flow curve and storage modulus). We observed two different scenarios. At first, for pure calcite samples, the velocity profiles are homogeneous along the gap for all studied shear rates. Slip is present at both rotor and stator, especially at low shear rates whose macroscopic signature is the decrease in the flow curve of the shear stress at low shear rates. Then, for calcite with sodium hydroxide, the velocity profiles exhibit shear-bands for ˙γ≤50s−1and become homogeneous only at higher shear rates. As for the pure sample, slip is present at both rotor and stator and becomes more important at lower ˙γ. We also verified that the shear banding phenomenon does not depend on the volume concentration down to 8 %.
The main difference between the two systems lies in the interaction between the calcite particles at the microscopic scale. Indeed both the Zeta potential and Debye length are smaller in the paste with sodium hydroxide than for pure calcite, such that the DLVO interaction in calcite with NaOH is purely attractive (only Van der Waals attraction), whereas the colloids in pure calcite interact through a potential with a small repulsive barrier, comparable to thermal energy. Both systems are attractive but the attraction is stronger with addition of NaOH.
In our experiments, the existence of the shear-bands seems therefore to be correlated to a strong attraction between colloidal particles, allowing us to make a link between the microscopic colloidal interaction (i.e. DLVO barrier) and the macroscopic properties (i.e. velocity profiles).
This link between attraction and shear banding phenomena has been previously shown in the work of Bécu et al.(2006) [1]. This study focused on concentrated emulsions. By changing the concentration of surfactants, the authors could tune short range attractive forces. In particular, for a low surfactant concentration the resulting system is nonadhesive, for a higher quantity on the contrary, the depletion forces increase and the system becomes adhesive. For these two systems (adhesive and non adhesive), velocity profiles are recorded using USV and are shown in Figure 5.10.
Figure 5.10 (i) displays the results for a nonadhesive emulsion. For low/intermediate wall velocity v0, (a)-(c), the profiles show a total wall slip. Increasingv0, the velocity profiles become homogeneous (d). The adhesive emulsion in Figure 5.10 (ii) instead shows total wall slip only for very low wall velocity (a). Shear bands appear for low and intermediate values ofv0. At high wall velocity, the profiles also
5.3. DISCUSSION 133
(i) (ii)
Figure 5.10 –Adapted from Bécu et al.(2006) [1]. (i) Velocity profiles in the nonadhesive emulsion for (a)v0= 0.98, (b)v0= 1.47, (c)v0= 1.96 and (d)v0=2.94 (◦), 4.90 (•) and 9.79 mm s−1(). (ii) Velocity profiles in the adhesive emulsion for (a)v0= 0.49, (b)v0= 0.98 (◦), 1.17 (•), (c)v0= 1.47 (◦), 1.96 (•) and (d)v0=4.78 (◦), 9.78 (•) and 19.5 mm s−1(). Arrows indicate the wall velocityv0.
become homogeneous. To summarize, the nonadhesive system is homogeneous throughout the yielding transition whereas the adhesive one shows shear banding.
The common features between the Bécu et al. study and ours are that both systems - concentrated emulsions and calcite suspensions- are yield stress fluids and that shear bands occur once the attraction is increased. But it is important to point out that these two systems are intrinsically different in nature.
Our calcite paste is a colloidal gel with a fractal structure while the system studied by Bécu et al. is a jammed concentrated system, i.e. a soft paste.
Following this distinction, we can find in the literature two relevant theoretical works that address the influence of attraction on velocity profiles, in jammed systems and in fractal ones. These studies are respectively the ones from Chaudhuri et al.(2012) [2] and Irani et al.(2014) [4].
In the article of Chaudhuri et al. [2], molecular dynamics simulations of soft particles jammed sys-tems are performed to describe the flow of yield stress fluids such as foams, emulsions or polymer microgels. In this model they tune the particle interactions (from repulsive to attractive system) and study how it affects the velocity profiles.
For a non-attractive system (Figure 5.11), they find that the velocity profiles strongly fluctuate both in space and time. However these fluctuations vanish as the velocity profiles are averaged on large deformations. Shear inhomogeneities in the case of a non-attractive system are not therefore an intrinsic feature of the system, but is only a transient phenomenon.
For the attractive system the inhomogeneities of the velocity profiles are more pronounced (Fig-ure 5.12) possibly longer than the durations of the simulations or the meas(Fig-urements (but not permanent).
This could explain the observation of shear bands in the concentrated attractive emulsions of Bécu et al. [1], although these heterogeneities are not necessarly associated with a non-monotonic flow curve τ(γ)˙ , usually invoked to explain shear bands [7].
In the article of Irani et al. (2014) [4], the authors study the dynamics of assemblies of non-Brownian soft particles with a fractal structure in the vicinity of jamming. They also tuned particle interactions and measured the impact of a weak attraction on rheology.
The authors demonstrate the existence of non monotonic flow curves once attraction between
parti-Figure 5.11 – Reproduced from Chaudhuri et al. [2]. A series of representative velocity profiles for repulsive particles, averaged during a strain window ofΔγ=0.10, and sampled during steady-state flow at an imposed ˙γ =2.5x10−5s−1. These profiles reveal strong deviations from linear profiles, which strongly fluctuate both in space and time.
Figure 5.12 – Reproduced from Chaudhuri et al. [2]. A series of representative velocity profiles for attractive particles, averaged during a strain window of Δγ =0.10 (top), and Δγ =4.0 (bottom) and sampled during steady-state flow at an imposed ˙γ=10−4s−1. These profiles reveal strong deviations from linear profiles, which strongly fluctuate both in space and time. Compared to the repulsive system, these profiles remain strongly non linear at large deformationΔγ=4.0, suggesting that a linear velocity profile is not stable in the presence of strong particle adhesion.
cles is added into the system, as shown in Figure 5.13. Decreasing stressτ vs ˙γ are instable and led to persistent shear banding over a range of shear rates that does not depend on the system size [7]. Irani et al. also demonstrate that these inhomogeneities are linked to the properties of the contacting network and disappear as soon as the attraction is suppressed. This theoretical study considers fractal suspensions whose microstructure and properties are similar to our fractal gel. The only difference is that our calcite particles are Brownian whereas the soft particles of Irani et al. are not. It would be interesting to study the role of thermal energy on the micro-structure and on its dynamics.