Complex responses

Phenomena aswall slip and non uniform flow (i.e. inhomogeneities or shear banding) are typical of yield stress suspensions. This kind of artefacts affect the rheological measurements, invalidating the mechanical and flow characterization (i.e. yield stress, viscosity...). A general overview about yield stress fluids and their peculiarities i.e. artefacts are made by Coussot [16], Bonn et al. [10] and by Cloitre and Bonnecaze [14]. The velocity profiles across the gap (Couette geometry) corresponding to these artefacts are shown in Figure 1.19.

Figure 1.19 –Sketches of velocity profiles in a Couette geometry. Both homogeneous and inhomoge-neous profiles (i.e. wall slip and shear banding) are showed along the gap.

Dashed line represents the velocity profile corresponding to a homogeneous flow. The blue line shows a homogeneous flow with slip at both walls and the green one, shear banding and slip at the stator wall. Both phenomena can occur at the same time. These two phenomena are detailed in the next sections. To detect and follow these inhomogeneities there are several techniques developed re-cently [10]. In particular Diffusion Wave Spectroscopy, (DWS) and Ultrasonic Speckle Velocimetry, (USV) are detailed in the next Materials and Methods chapter.

Wall slip

In Figure 1.19 we observe that the velocities at the walls (only the right wall for the shear banding profile) for the two solid lines differ from the case of homogeneous flow (dashed line). The no-slip boundary condition is not satisfied. As described by Bingham in 1922, this is mainly due to "a lack of adhesion between the material and the shearing surface. The result is that there is a layer of liquid, between the shearing surface and the main body of the suspension" [8]. In this thin lubricating layer between the suspension and the wall, the overall deformation is localized [14]. The physical origin of this wall slip phenomenon, for both rigid and soft particles, is well detailed by Cloitre and Bonnecaze [14]. The main mechanism proposed for rigid particles (Brownian or non-Brownian suspensions) are steric depletion—

rigid particles cannot penetrate the wall, creating a thin layer (≈particle radius) with a different volume fraction comparing to the bulk— and particle migration due to the shear rate variation with the radial position in the Couette geometry.

1.3. RHEOLOGY OF COMPLEX FLUIDS:A MACROSCOPIC APPROACH 41

For soft (deformable) particles, wall slip depends on roughness or surface chemistry. Both hy-drophobic and hydrophilic surfaces determine the interaction at the solid boundaries. For hyhy-drophobic surfaces, the weak attractive interaction results in a finite slip yield stress. Hydrophilic surfaces cause a repulsive interaction and a thin layer of water wet the surface, the system slips as soon as it is sheared.

The signature of wall slip on the flow curve for a microgel suspension is showed in Figure 1.20.

Figure 1.20 – Reported by [14, 63]. Generic signatures of wall slip represented for a concentrated microgel suspension in parallel plate geometry. a) Variations of the shear stress versus the true shear rate (no slip) or apparent shear rate (slip). When sheared with rough surfaces (•), the suspension exhibits a true yield stress and the flow curve is well described by the Herschel-Bulkley equation (-). With smooth polymer (♦) and glass surfaces (), which are hydrophobic or hydrophilic respectively, the flow curves exhibits three regimes of flow. ˙γa^∗ marks the onset of full slip; the equations of the dashed lines are of the form τ =τs+kγ˙a^m withτs=0.03 and 0.6, m=0.92 and 0.50 for the glass and polymer surfaces respectively. The local rheology data corrected from the effect of slip coincide with the flow curve in the absence of slip (◦).

Three curves with different surfaces roughness and chemistry are compared. In particular, for high shear rate, all the flow curves coincide. Then below a certain value ˙γ_a^∗, smooth surfaces lead to wall slip showing a kink in the flow curve and a slip yield stressτs<τy. At the contrary the flow curve obtained with the rough surface (full circles) follow the Herschel-Bulkley equation [14].

Another way to track the signature of wall slip is to verify the rheological dependency with the gap size. As shown in Figure 1.21 for the calcite colloidal suspension studied in our work, both smooth and rough surfaces show a gap-dependency of the elastic modulus (in the linear regime): G_lin[47]. Increas-ing the gap size, the gap-dependency of the elastic modulus decreases, reachIncreas-ing a plateau. These results are in good agreement with the calculation made in the early work by Yoshimura and Prud’homme [72], reported for a parallel plates geometry in Equation (1.26). Upon increasing the gap width,h, the contri-bution of wall slip (slip velocityvs) to the apparent deformation is reduced and consequently the apparent shear rate, ˙γapp, approaches the effective one, ˙γeff.

γ˙app=γ˙eff+2vs(τ)

h (1.26)

Emulsions and foams under flow show a similar gap-dependence [6]. To avoid wall slip different meth-ods can be used. Firstly as already mentioned rough surfaces with a roughness comparable to the order

Figure 1.21 –Storage modulus in the linear regimeG_linof calcite paste (φ=20%) as a function of the gap size, for different smooth and rough surfaces.

of magnitude of the microstructure can be used [10], e.g. deformable objects (droplets in emulsions..).

In this case the surface asperities avoid or destroy the lubricating film at the wall [14]. Another way is to use more specific geometries like vane or helical tools [14, 10]. Finally it is possible to play with the chemistry, tuning the surface interaction into attractive or repulsive, depending on the specific nature of the system [14]. In particular, the attraction between the surface and the complex fluid can lead to the suppression or reduction of slip.

Shear banding

Shear banding refers generally to flow inhomogeneities and occurs in disordered materials from com-plex fluids to rocks [7]. In particular, yield stress fluids (both simple and thixotropic) show stress het-erogeneities.

In the case of simple yield stress fluids, flow inhomogeneities are not intrinsic and are linked only to stress heterogeneities resulting in shear localization [10]. Flow inhomogeneities can also depend on specific geometry conditions [10]. This happens for example, in a large gap of a Couette geometry, where consequently there is a large stress variation between the walls (i.e. rotor and stator). Therefore the lower stress value can be below the τy determining a non-shearing zone. Another example is a confined geometry with a gap width close to the characteristic microstructure size of the fluid, this can produce a local structure rearrangements showing a heterogeneous flow [10].

For thixotropic yield stress on the contrary, shear bands are intrinsic shear localization and geometry independent. They are linked to the existence of a critical shear rate as thixotropic yield stress fluids [10].

In Figure 1.19, the green line divides the gap region in two areas. The first is close to the rotor (left side) and flows like a liquid. The other one close to the stator is an arrested (or solid) region. This unstable flow corresponds to a range of critical strain between 0 and ˙γc (corresponding to the sheared zone).

The origin of shear bands for Brownian suspension is interpreted as a competition between spontaneous aging and shear-induced rejuvenation. Aging is linked with attractive interactions and promotes shear bands [10]. For dense non Brownian particles the origin of shear banding is linked to the competition between sedimentation, shear and volume fraction heterogeneities [10]. More details can be found in the review of Bonn et al. [10].

1.4. CALCITE PASTE:A COLLOIDAL SUSPENSION WITH COMPLEX RHEOLOGY 43

1.4 Calcite paste: a colloidal suspension with complex rheol-ogy

Calcite dispersed in water results in attractive interactions which promote aggregation of the calcite particles [2]. Literature on calcite suspensions focuses mainly on flow measurements, quantifying the variation of the yield stressτy(or the viscosityη) changing the particle interactions by superplasticizers addition [51, 53, 61, 19, 21, 20, 11].

In particular our calcite system is well described by DLVO forces, as detailed in Chapter 4. Then we assume that the non-DLVO forces presented above are not relevant in comparison to DLVO ones for the interactions between our calcite particles.

1.4.1 Zeta potential

At the microscopic level, Zeta Potential measurements determine the electrostatic (repulsive) contribu-tion into DLVO calculacontribu-tion. The nature of interaccontribu-tion drives the final mechanical properties of suspen-sion.

In the case of calcite paste, determining the Zeta potential is a complex problem. In the litterature in fact there is no consensus on both its value and sign. It is found to depend, among others, on pH, CO2 pressure [55, 52, 27, 67], concentrations of solid calcite [65], specific surface area, concentration of other dissolved ions [61], and possible presence of impurities for natural calcite [69, 60]. An accurate and complete review of the Zeta potential on artificial and natural calcite is proposed by Al Mahrouqi et al. (2017) [1]. In Section 2.3.1 we propose a detailed explanation about the main chemical mechanisms on calcite surface, determining the value of Zeta potential.