Shear viscosity of claylike colloids in computer simulations and experiments

Dense suspensions of small strongly interacting particles are complex systems that are rarely understood on the microscopic level. We investigate properties of dense suspensions and sediments of small spherical ${\mathrm{Al}}_2{\mathrm{O}}_3$ particles in a shear cell by means of a combined molecular-dynamics and stochastic rotation dynamics simulation. We study structuring effects and the dependence of the suspension’s viscosity on the shear rate and shear thinning for systems of varying salt concentration and $p\mathrm{H}$ value. To show the agreement of our results with experimental data, the relation between the bulk $p\mathrm{H}$ value and surface charge of spherical colloidal particles is modeled by Debye-Hückel theory in conjunction with a $2\mathrm{pK}$ charge regulation model.

Figure 1

Schematic stability diagram for volume fraction $\Phi =35\%$ in terms of $p$H-value and ionic strength involving three different microstructures: A clustering regime due to van der Waals attraction, stable suspensions where the charge of the colloidal particles prevents clustering, and a repulsive structure for further increased electrostatic repulsion. This work concentrates on state $A$ $(p\mathrm{H}=6,I=3\mathrm{mmol}∕\mathrm{l})$ in the suspended region, state $B$ $(p\mathrm{H}=6,I=7\mathrm{mmol}∕\mathrm{l})$ close to the border but already in the clustered region, and state $C$ $(p\mathrm{H}=6,I=25\mathrm{mmol}∕\mathrm{l})$ in the clustered region. The borders are not sharp transitions, but notable in a change of the shear viscosity.

Figure 2

(Color online) Images of four different cases. For better visibility we have chosen smaller systems than we usually use for the calculation of the viscosity. The colors denote velocities: Dark particles are slow, bright ones move fast. The potentials do not correspond exactly to the cases $A$–$C$ in Fig. 1, but they show qualitatively the differences between the different states: (a) Suspension like in state $A$, at low shear rates. (b) Layer formation, which occurs in the suspension (state $A$) at high shear rates and in the repulsive regime already at moderate shear rates. (c) Strong clustering, like in state $C$, so that the single cluster in the simulation is formed. (d) Weak clustering close to the border like in state $B$, where the cluster can be broken into pieces, which follow the flow of the fluid (plug flow).

Figure 3

Total energy depending on the shear rate $\dot{\gamma }$ for the states $A$ $(I=3\mathrm{mmol}∕\mathrm{l})$ and $C$ $(I=25\mathrm{mmol}∕\mathrm{l})$ of Fig. 1. In state $A$ the system is a stable suspension; in state $C$ cluster formation reduces the total energy at low shear rates. At $\dot{\gamma }=500∕\mathrm{s}$ $(\mathrm{Pe}=15)$, the cluster can be broken up into two parts moving in opposite directions. The two solid bodies have a larger kinetic energy than the suspension with a linear velocity profile. This explains the crossover of the two curves. For even higher $\dot{\gamma }$, the clusters are broken up in more pieces, leading ultimately to the same structure as for the suspended state $A$. The energy axis has been scaled by the total number of particles (fluid particles plus colloidal particles) and plotted in units of $k_{B}T$. The solid line is the analytical solution [Eq. (18)] for a linear velocity profile; the dashed lines are a guide to the eye.

Figure 4

Profiles of tangential velocity component $\left(v_x\right)$ in normal direction $\left(z\right)$: (a) Linear profile in the suspended regime, state $A$ of Fig. 1 $(I=3\mathrm{mmol}∕\mathrm{l})$ at $\dot{\gamma }=500∕\mathrm{s}$ $(\mathrm{Pe}=15)$. (b) Cluster formation in state $C$ $(I=25\mathrm{mmol}∕\mathrm{l})$ at $\dot{\gamma }=100∕\mathrm{s}$ $(\mathrm{Pe}=2.9)$. In principle, one could determine the viscosity of one single cluster from the central plateau, but this is not the viscosity found in experiments. There, one measures the viscosity of a paste consisting of many of these clusters. (c) Same as case (b) but with higher shear rate ($\dot{\gamma }=500∕\mathrm{s}$ $\mathrm{Pe}=15$). Hydrodynamic forces are large enough to break the cluster into two pieces. The velocity axis is scaled with the shear velocity $v_S$ for better comparability.

Figure 5

Density profiles: (a) Suspended case: State $A$ in Fig. 1 $(I=3\mathrm{mmol}∕\mathrm{l})$, at low shear rates [$\dot{\gamma }=50∕\mathrm{s}$ $(\mathrm{Pe}=1.5)$]. The density distribution is homogeneous. (b) Shear induced layer formation: This is state $A$ as in graph (a) of this figure, but for a high shear rate [$\dot{\gamma }=1000∕\mathrm{s}$ $(\mathrm{Pe}=29)$]. (c) Strong attractive forces in state $C$ $(I=25\mathrm{mmol}∕\mathrm{l})$: For low shear rates [$\dot{\gamma }=50∕\mathrm{s}$ $(\mathrm{Pe}=1.5)$] only one central cluster is formed, which is deformed slowly.

Figure 6

Comparison between simulation and experiment: viscosity in dependence of the shear rate for the states $A$ $(I=3\mathrm{mmol}∕\mathrm{l})$ and $B$ $(I=7\mathrm{mmol}∕\mathrm{l})$ of Fig. 1. Note: shear thinning is more pronounced for the slightly attractive interactions in state $B$ than for the suspended state $A$. Lines denote experimental data [17], points are results from our simulations.

Figure 7

Discrepancy between simulated viscosity and measurement for states $A$ and $B$ of Fig. 1 for different system sizes at low shear rates. The plot shows squared relative differences against $z$ extension of the simulation volume. The lines are a guide to the eye.

Figure 8

Viscosity vs volume fraction for the repulsive region ($p\mathrm{H}=6$ and $I=0.3\mathrm{mmol}∕\mathrm{l}$). The shear rate was $\dot{\gamma }=100∕\mathrm{s}$ $(\mathrm{Pe}=2.9)$. The points are simulation results, the line is a guide to the eye.