Unifying disparate rate-dependent rheological regimes in non-Brownian suspensions

A typical dense non-Brownian particulate suspension exhibits shear thinning (decreasing viscosity) at low shear rate or stress followed by a Newtonian plateau (constant viscosity) at intermediate shear rate or stress values which transitions to shear thickening (increasing viscosity) beyond a critical shear rate or stress value and finally undergoes a second shear thinning transition at extremely high shear rate or stress values. In this study, we unify and quantitatively reproduce all the disparate rate-dependent regimes and the corresponding transitions for a dense non-Brownian suspension with increasing shear rate or stress. We employ discrete particle dynamics simulations based on the proposed mechanism to elucidate its accuracy. We find that a competition between interparticle interactions of hydrodynamic and nonhydrodynamic origins and the switching in the dominant stress scale with increasing the shear rate or stress lead to each of the above transitions. Inclusion of traditional hydrodynamic interactions, attractive or repulsive Derjaguin-Landau-Verwey-Overbeek (DLVO) interactions the interparticle contact interactions, and a constant friction (or other constraint mechanism) reproduces the initial thinning as well as the shear thickening transition. However, to quantitatively capture the intermediate Newtonian plateau and the second shear thinning, an additional nonhydrodynamic interaction of non-DLVO origin and a decreasing coefficient of friction, respectively, are essential, thus providing an explanation for the presence of the intermediate Newtonian plateau along with reproducing the second shear thinning in a single framework. Expressions utilized for various interactions and friction are determined from experimental measurements and hence result in excellent quantitative agreement between the simulations and previous experiments.

Figure 1

Schematic showing the typical rheological flow curve for dense non-Brownian suspensions. This rheological behavior is commonly observed for non-Brownian suspensions [4, 5, 7, 9].

Figure 2

Schematic showing the coefficient of friction $\mu$ (thin black line) and the dimensionless normal force magnitude $|{\mathbf{F}}_n|$ (thick red line) between a close particle pair as a function of dimensionless interparticle gap $\lambda =h/h_r$. Boundary, partial elastohydrodynamic, and full film lubrication regimes in the Stribeck curve are demarcated based on the value of $\lambda$. Similarly, dominant interparticle interactions in each of these regimes are also shown in red font. The insets at the top show the various regimes in terms of separation between two close particles. The arrows in these insets are shown to qualitatively indicate the size of the interparticle gap and the range of the dominant interparticle interaction with respect to the roughness and the particle size.

Figure 3

(a) Relative viscosity as a function of dimensionless shear rate $\dot{\gamma }/{\dot{\gamma }}_0$ for two different volume fractions (PS denotes present simulations) compared against experiments (EX denotes experiments, ${\dot{\gamma }}_0=200{\text{s}}^{-1}$ for experimental data) of Chatté et al. [5]. The volume fractions are scaled with dry close packing fraction ${\varphi }_d$ for direct comparison. For the simulations ${\varphi }_d=0.66$. (b) Probability distribution function (dotted lines) of the average dimensionless interparticle gap $\langle \lambda \rangle$ with increasing $\widehat{\dot{\gamma }}= \dot{\gamma }/{\dot{\gamma }}_0$ (legend) along with the friction coefficient (solid lines) for the logarithmic decay friction model. Dotted lines are spline fits to guide the eye. Dashed lines demarcate the transition between interaction ranges as explained in Fig. 2.

Figure 4

Scheme of the physics involved in the shear thinning (I and II), Newtonian plateau (III), and shear thickening–shear thinning (IV) regimes in the rheological behavior of a typical dense non-Brownian suspension. The insets at the top show the approximate interparticle gaps in regimes I–IV. In these insets, the outermost circle represents the range of DLVO forces, the inner orange circle represents the range in which non-DLVO forces are dominant, and the innermost circle represents the particles. Thus, with increasing shear rate we observe different regimes depending on which forces are dominant in the suspension on average as depicted by the overlaps of different force zones in the insets.

Figure 5

(a) Contributions from hydrodynamic (${\eta }_r^H$), noncontact (${\eta }_r^{NC}$) (DLVO and non-DLVO), and contact (${\eta }_r^C$) interactions to the total relative viscosity of the suspension for $\varphi /{\varphi }_d\approx 0.86$. The trends in the respective contribution follow from Fig. 4. Lines are for guiding the eye. (b) Flow curve for different coefficient of friction functions for $\varphi /{\varphi }_d\approx 0.86$.

Figure 6

(a) DLVO force (repulsion and attraction) profiles as a function of dimensionless gap for two different attractive force magnitudes. (b) Effect of varying the attractive force magnitude on the first shear thinning regime.

Figure 7

(a) Effect of changing the magnitude of the non-DLVO force. The Newtonian plateau disappears in the absence of non-DLVO forces. Thus, we can reproduce the thinning-thickening-thinning transition using the proposed model as well. This is useful for suspensions which do not have a significant Newtonian plateau, e.g., silica particles [22]. (b) Order metric $Q_6$ for two different non-DLVO force magnitude. The gradual increase in $Q_6$ in the first shear thinning regime and peak in the Newtonian regime hint at the link between ordering and the initial shear thinning Newtonian plateau [22].

Figure 8

(a) Different friction laws tested for the sensitivity analysis of the model to $\mu$. Here $-aln(1-\lambda )+b$ is the logarithmic decay model derived from the experimental measurements from Ref. [5]. The black solid line ($a=\frac{1}{2}$ and $b=-0.2$), red dashed line ($a=\frac{1}{4}$ and $b=-0.2$), and pink dotted line ($a=\frac{1}{2}$ and $b=-0.5$) show how $\mu$ for the logarithmic decay model changes with dimensionless interparticle gap $\lambda =h/h_r=1+\delta /h_r$. The blue dash-dotted line shows a hypothetical exponentially decaying $\mu$. Finally, the gray solid line shows the model of Brizmer et al. [58] for $\mu$ which has been previously used in the literature to explain the shear thinning in dense non-Brownian suspensions [47]. (b) Variation of viscosity for different friction models. If we use a constant $\mu$ instead, we will not observe the second shear thinning regime. The data show that the friction model determines the viscosity jump during the shear thickening transition and the slope in the second shear thinning regime. The flow curve for suspensions which do not exhibit the second shear thinning can be obtained by choosing a constant coefficient of friction.

Figure 9

Effect of varying particle surface roughness height ${\epsilon }_r$ on the suspension viscosity. Increasing surface roughness results in a stronger initial shear thinning and increases the viscosity during and beyond the ST transition. Note that the ST transition is governed by the direct contact between the particles which is due to the breaking of the lubrication film due to the particle asperities. Here all the other parameters are the same as given in Table 1, except ${\epsilon }_r$, which is varied. Here $\varphi =52\%$. The increase in the viscosity with roughness in the thinning regimes is consistent with Ref. [52] and the increase in the viscosity during the ST jump is consistent with Ref. [72]. The viscosities in the Newtonian plateau are comparable for small change in the roughness values.

Figure 10

(a) Evolution of second normal stress difference $N_2$ with applied dimensionless shear rate; $N_2$ is always negative. (b) $N_2$ scaled by the shear stress in the suspension. This plot shows that $N_2$ mimics the stress in the system.