How Confinement-Induced Structures Alter the Contribution of Hydrodynamic and Short-Ranged Repulsion Forces to the Viscosity of Colloidal Suspensions

Confined systems ranging from the atomic to the granular are ubiquitous in nature. Experiments and simulations of such atomic and granular systems have shown a complex relationship between the microstructural arrangements under confinement, the short-ranged particle stresses, and flow fields. Understanding the same correlation between structure and rheology in the colloidal regime is important due to the significance of such suspensions in industrial applications. Moreover, colloidal suspensions exhibit a wide range of structures under confinement that could considerably modify such force balances and the resulting viscosity. Here, we use a combination of experiments and simulations to elucidate how confinement-induced structures alter the relative contributions of hydrodynamic and short-range repulsive forces to produce up to a tenfold change in the viscosity. In the experiments we use a custom-built confocal rheoscope to image the particle configurations of a colloidal suspension while simultaneously measuring its stress response. We find that as the gap decreases below 15 particle diameters, the viscosity first decreases from its bulk value, shows fluctuations with the gap, and then sharply increases for gaps below 3 particle diameters. These trends in the viscosity are shown to strongly correlate with the suspension microstructure. Further, we compare our experimental results to those from two different simulations techniques, which enables us to determine the relative contributions of hydrodynamic and short-range repulsive stresses to the suspension rheology. The first method uses the lubrication approximation to find the hydrodynamic stress and includes a short-range repulsive force between the particles while the second is a Stokesian dynamics simulation that calculates the full hydrodynamic stress in the suspension. We find that the decrease in the viscosity at moderate confinements has a significant contribution from both the hydrodynamic and short-range repulsive forces whereas the increase in viscosities at gaps less than 3 particle diameters arises primarily from short-range repulsive forces. These results provide important insights into the rheological behavior of confined suspensions and further enable us to tune the viscosity of confined suspensions by changing properties such as the gap, polydispersity, and the volume fraction.

Popular Summary

Complex fluids consisting of particles suspended in an ambient fluid—such as toothpaste, lubricants, and gels—are ubiquitous in everyday life. Understanding how these complex fluids flow through confined spaces, along with the underlying forces at play, are important; flows of confined suspensions have implications in areas ranging from blood flow in thin capillaries to the design of lubricants for micromachines. Studying these confined fluids is challenging, however, as traditional flow devices do not achieve such small gaps. Here, we use a combination of experimental and simulation techniques to determine the flow behavior under extreme confinement and understand the forces that give rise to the resistance to flow, or viscosity.

We show that at small gaps, the suspension flow response is strongly coupled to the arrangements of the particles in the fluid. When a suspension is confined to a space between 6 and 15 times the particle diameter, the particles arrange themselves into layers, which gives the suspension a low viscosity. At gaps between three and six particle diameters, geometric effects such as the commensurability of the gap with the particle diameter become important, giving rise to fluctuations in the viscosity. Finally, at extreme confinements with gaps less than three particle diameters, the contact forces between particles become increasingly important, and the particles form bridges between the plates that increase the viscosity by over an order of magnitude.

With this understanding, we are able to tune the flow behavior by altering the particle arrangements using dopants or different particle concentrations.

Figure 1

The experimental apparatus and sample measurements. (a) A schematic of the shear cell, focusing on the shear zone and the force measurement device. (b) Confocal images in the shear gradient plane at three different extents of confinement. (c) The applied strain generated by the piezoelectric stage is depicted by the blue curve. A typical stress measurement that is obtained from the force measurement device is depicted by the red curve. Importantly, to extract the viscosity magnitude we use imaging to back out the effective strain amplitude. (d) The magnitude of the complex viscosity as a function of the effective strain rate for the three gaps shown in (b). The vertical gray dashed line indicates the effective strain rate that is used for the remainder of the experiments.

Figure 2

Magnitude of the normalized complex viscosity versus normalized gap. A fivefold decrease in the viscosity is observed below gaps smaller than 15 particle diameters. For gaps lower than 3 particle diameters a steep increase in the viscosity is observed. Very similar trends are observed in the experiment (green circles) and simulation (red squares) data. Uncertainty in the experiments corresponds to the level of background noise. The gray dashed line indicates the viscosity trend for a Newtonian fluid.

Figure 3

Layering under confinement. Histograms of the normalized $y$ coordinate of the particle positions from experiments [green histograms, (a),(b)] and lubrication-repulsion simulations [red histograms, (c),(b)] for small (a),(c) and large (b),(d) gaps. The bottom plate corresponds to $y=0$. Collectively these histograms indicate strong layering with increasing confinement.

Figure 4

Relating viscosity to the gap-dependent order parameter. (a) The order parameter as a function of gap from both experiments and the lubrication-repulsion dynamics simulation. (b) The normalized viscosity as a function of the order parameter for the experiments and the lubrication-repulsion dynamics simulation. The decrease in the viscosity is well correlated with the increase in the order parameter with an $R$ value of 0.8031.

Figure 5

Force contributions to the total stress as determined from the two simulation techniques. (a) The total (red), hydrodynamic (yellow), and short-ranged repulsion contributions (pink) to the relative viscosity of the suspension as calculated by the lubrication-repulsion dynamics simulation. The viscosities are scaled by the total bulk viscosity in all cases. The inset shows comparable decreases in the viscosities arising from short-range repulsive and hydrodynamic interactions. (b) Comparison of the hydrodynamic viscosity from the lubrication-repulsion dynamics simulation, where a lubrication approximation is used between particle pairs, and the full hydrodynamic viscosity from Stokesian dynamics. Here, the viscosity is normalized by the suspending fluid viscosity. Both simulations show similar bulk viscosities and viscosity oscillations at low gaps. The lubrication-repulsion simulation shows up to a factor of 3 reduction in viscosity for normalized gaps less than 10.

Figure 6

Buckled phase viscosity. (a) The shear geometry and a schematic of the three-dimensional microstructure of the buckled phase and perfectly layered phase. The spheres of the same color indicate particles that move together with same velocity and displacement. (b) The magnitude of the complex viscosity as a function of the gap. The images show the $x\text{−}z$ cross section of the microstructure at the indicated values of the gap.

Figure 7

Tuning the viscosity under confinement. (a) The decrease in the viscosity under confinement for two different volume fractions. (b) The viscosity of a bidisperse system at three different number ratios of small to large particles, $r=0$, 1, and 3. Also shown are the lubrication-repulsion simulation results for $r=1$.

Figure 8

Bridge formation under confinement. Panels (a) and (b) show an example of a bridge seen in the simulations at gaps of 3.5 and 2.5 particle diameters, respectively. (c) The number of system spanning bridges as a function of gap. We find a sharp increase in the number of bridges corresponding to the increased viscosity at extreme confinement.

Figure 9

The suspension microstructure in the shear vorticity plane from simulations. Panels (a) and (b) show the structure formed in the lubrication-repulsion simulations at gaps 3.5 and 3.9 particle diameters, respectively. Panels (c) and (d) show the structure formed in the Stokesian dynamics simulations at gaps 3.5 and 4 particle diameters, respectively.

Figure 10

Rheology and microstructure of a low volume fraction ($\varphi =0.38$) monodisperse suspension. (a) The experimentally observed variations in the viscosity between gaps 2.5–4 particle diameters. We see no measurable oscillations due to incommensurability. The inset shows the viscosity variation over the full range of gaps. (b) The viscosity of the suspension measured by the lubrication-repulsion simulations. Panels (c)–(f) show the microstructure from experiments (c),(d) and simulations (e),(f). Panels (c) and (e) are at a gap of 3.5 particle diameters and panels (d) and (f) at a gap of 3.9 particle diameters. Collectively, the structural data demonstrate the absence of layering and buckled structure in the sample.