Stability diagram for dense suspensions of model colloidal ${\mathrm{Al}}_2{\mathrm{O}}_3$ particles in shear flow

In ${\mathrm{Al}}_2{\mathrm{O}}_3$ suspensions, depending on the experimental conditions, very different microstructures can be found, comprising fluidlike suspensions, a repulsive structure, and a clustered microstructure. For technical processing in ceramics, the knowledge of the microstructure is of importance, since it essentially determines the stability of a workpiece to be produced. To enlighten this topic, we investigate these suspensions under shear by means of simulations. We observe cluster formation on two different length scales: the distance of nearest neighbors and on the length scale of the system size. We find that the clustering behavior does not depend on the length scale of observation. If interparticle interactions are not attractive the particles form layers in the shear flow. The results are summarized in a stability diagram.

Figure 1

Dependence of the particle correlation function on the $p\mathrm{H}$ value, $I=3\mathrm{mmol}$, $\dot{\gamma }=0∕\mathrm{s}$, $\Phi =35\%$. The plots for four different $p\mathrm{H}$ values are shifted against each other for better visibility by a factor of 3. For $p\mathrm{H}=4$ the particles are not clustered. Hence the structure at $\frac{r}{d}=2$ is less sharp than in the other three curves of the plot and the nearest-neighbor peak (at $\frac{r}{d}=1$) is broad. For $p\mathrm{H}=6.5$ slight clustering starts, the structures become sharper. For $p\mathrm{H}=7.7$ strong cluster formation is reflected in very sharp structures. For $p\mathrm{H}=8.5$ electrostatic repulsion nearly disappears so that no barrier between primary and secondary minimum exists anymore. The particles cannot rearrange anymore, and therefore the structures labeled by the arrows become smoothened compared to the case of $p\mathrm{H}=7.7$.

Figure 2

Correlation function, $\dot{\gamma }=100∕\mathrm{s}$, $p\mathrm{H}=7$, $I=3\mathrm{mmol}∕\mathrm{l}$, $\Phi =35\%$: one finds clustering in the primary and the secondary minimum. Note: The vertical axis is logarithmic, and a constant offset has been added before plotting. Thereby, the coincidence of the minima in the potential with the maxima in the correlation function becomes more apparent. The dotted lines denote the position of the base line before shifting.

Figure 3

Correlation function for $\dot{\gamma }=500∕\mathrm{s}$, $p\mathrm{H}=6$, $I=0.3\mathrm{mmol}∕\mathrm{l}$, $\Phi =32\%$: depending on the simulation time the peaks indicated by the arrows from below split into two (indicated by the arrows from above), and long range correlations occur (for $\frac{r}{d}>4$). This reflects the process of layer formation and appearance of a regular (hexagonal) order in the layers. From bottom to top several states for increasing time are shown. The plots are shifted vertically by a factor of 2 for better visibility.

Figure 4

Nearest-neighbor peak (primary and secondary minimum of the potential) of the correlation function $I=3\mathrm{mmol}∕\mathrm{l}$, $\Phi =35\%$: For low $p\mathrm{H}$ values clustering is prevented by the electrostatic repulsion. For high $p\mathrm{H}$ values the particles form clusters, which is reflected by an increased nearest-neighbor peak. First, shear prevents clustering, then depending on the shear rate, cluster formation takes place. Low shear rates even support cluster formation at high $p\mathrm{H}$ values.

Figure 5

Structure factor for some selected examples, with $p\mathrm{H}=6$ fixed for all plots: $\dot{\gamma }=500∕\mathrm{s}$, $I=0.3\mathrm{mmol}∕\mathrm{l}$: (a) $\Phi =20\%$ and (b) $\Phi =35\%$; $\dot{\gamma }=0$, $I=25\mathrm{mmol}∕\mathrm{l}$: (c) $\Phi =40\%$ and (d) $\Phi =10\%$. The curves are shifted vertically for better visibility. In case (a) ten layers can be identified in the system, resulting in the strong peak close to 5. But, since the particles in the layers can still move freely, there is no second-order peak. In case (b) layers are formed, but particles are moving from one layer to the other, disturbing the flow. As a result the nearest-neighbor peak is much broader. Due to the structure in the layers, a second-order peak appears. In case (c) the interaction is strongly attractive, hence the particles approach each other and the nearest-neighbor peak is shifted to higher $\mathbf{k}$ vectors. In case (d) the volume fraction is much less. The slope of the low-$\mathbf{k}$ peak is much flatter, which depicts that the cluster is fractal.

Figure 6

(Color online) Snapshots of the systems analyzed in Fig. 5: In case (a) one can see the layers resulting in the sharp peak in the structure factor. In case (b) the layers are packed closer due to the higher volume fraction. Collisions between particles of neighboring layers happen more frequently. In case (c) one big cluster is formed. The particles are packed densely. In case (d) the fractal nature of the system can be seen directly.

Figure 7

Low-$\mathbf{k}$ peak for different $p\mathrm{H}$ values and different shear rates. The ionic strength $I$ is kept constant at $I=3\mathrm{mmol}∕\mathrm{l}$ and the volume fraction is always $\Phi =35\%$. For $\dot{\gamma }=0∕\mathrm{s}$ the particles tend to cluster in the secondary minimum of the potential. This clustering can easily be broken up, if shear is applied. If the $p\mathrm{H}$ value is increased, shear cannot prevent cluster formation anymore. At low shear rates $(\dot{\gamma }=100∕\mathrm{s})$ clustering is even enhanced, since the particles are brought closer to each other by the shear flow. Note that the offset of the plots reflects the finite size of our system combined with the closed boundary conditions (see the text).

Figure 8

Demixing parameter for constant $p\mathrm{H}=6$ and volume fraction $\Phi =35\%$ for three different shear rates: $\dot{\gamma }=0∕\mathrm{s}$ (a), $100∕\mathrm{s}$ (b), and $500∕\mathrm{s}$ (c). Without shear only very weak demixing takes place. For $\dot{\gamma }=100∕\mathrm{s}$ the strongest demixing can be observed for the system with the highest ionic strength $I=20\mathrm{mmol}∕\mathrm{l}$. For lower ionic strengths the effect decreases until it disappears completely at $I=1\mathrm{mmol}∕\mathrm{l}$, where the repulsive regime is reached. If the shear rate is further increased $(\dot{\gamma }=500∕\mathrm{s})$, the clusters are less stable so that the system becomes less inhomogeneous. For $I=5\mathrm{mmol}∕\mathrm{l}$ the demixing is even suppressed completely.

Figure 9

Mean squared displacement at $p\mathrm{H}=6$ for different ionic strengths, without shear. One can see a ballistic regime for short times, a central plateau and a collective long time movement which can be a movement of a whole cluster or cage escape events of single particles. Depending on the ionic strength, the central plateau is more or less pronounced. A comparison of the plateau for different simulations can be used to decide, if a certain state belongs to the repulsive region of the stability diagram. A state well in the liquid microstructure should be used as a reference for the comparison.

Figure 10

Stability diagram (plotted for $\Phi =35\%$ and without shear) depicting three regions: a clustered region (filled circles), a suspended regime (open squares), and a repulsive structure (filled squares). In the clustered region particles aggregate which leads to inhomogeneity in the system. In the suspended regime, the particles are distributed homogeneously in the system and they can move freely. In the repulsive regime the mobility of the particles is restricted by electrostatic repulsion exerted by their neighbors. As a result they arrange in a local order which maximizes nearest-neighbor distances. The borders between the regimes are not sharp. They depend on the shear rate and on the volume fraction. Therefore we have indicated the crossover regions by the shaded patterns. The lines are guides to the eye.