3.1. Single particle dynamics

After fabricating the composite, prolate particles, we observed the sedimentation of single, isolated particles to better understand their dynamics and to extract the terms in the mobility matrix (1.6). The response of a single particle to an external force or torque informs its effective interactions with neighbouring particles (Goldfriend et al. Reference Goldfriend, Diamant and Witten2017; Witten & Diamant Reference Witten and Diamant2020). For example, a rod-shaped particle of uniform mass density will sediment without a change in its initial angle (Ramaswamy Reference Ramaswamy2001; Witten & Diamant Reference Witten and Diamant2020). This results in a diagonal drift. However, the mass polarity of our objects causes them to align with the external gravitational field, meaning that a mass polar object will rotate until its centre of geometry lies directly above its centre of mass ($\theta = 0$). For our experiments, mass polar particles were released from an initial angle of $\theta = {\rm \pi}$, so that they rotated a total of ${\rm \pi}$ radians throughout the sedimentation process. A trajectory for a single $\kappa = 2$ particle composed of Cu$+$St (see table 1) is shown in figure 2(a). Particles with larger values of $\chi$ rotated much more rapidly due to the larger gravitational torque applied to the geometric centre of the particle. This can be compared with an St–St particle in figure 2(b), which shows no preference for rotations since it has no mass polarity ($\chi =0$). For particles with $\chi =0$, we occasionally observed ‘fluttering’, or oscillations of angular orientation during sedimentation. This was likely due to interactions with the walls of the experimental chamber during slight rotations out of the quasi-2-D plane of the experiment (Mitchell & Spagnolie Reference Mitchell and Spagnolie2014; D'Angelo et al. Reference D'Angelo, Cachile, Hulin and Auradou2017).

Table 1.

The different types of particles used in our experiments along with their corresponding $\kappa$ and $\chi$ values. Materials used are steel (St), aluminium (Al), copper (Cu), Delrin plastic (Pl), tungsten carbide (Tc) and zirconium dioxide (ZrO$_{2}$). Values of $\chi$ are kept to two significant digits.

Figure 2.

Two representative examples of the particle trajectories in our single particle experiments. Here, $x=0$ is defined as the geometric centre of the particle at the earliest time. (a) Particle with $\chi > 0$ (Cu$+$St, see table 1). The left part of the panel is a composite image of the particle during the length of the experiment. The right graph shows the corresponding particle orientations, with the arrows pointing from the heavier sphere (Cu) to the lighter sphere (St). The colour bar represents time. Gravity points downward in all pictures. (b) Particle with $\chi = 0$ (St$+$St).

To quantitatively capture the coupling between the external force and dynamics of single particles, we applied the mobility matrix formalism (1.6). Because we are using a quasi-2-D geometry, the complexity of the problem is reduced since the particle can only rotate in the plane. However, the mobility coefficients will be different from those measured in an unbounded, 3-D fluid. With two planar walls, our experimental setup is most similar to a Hele-Shaw cell, where the mobility matrix formalism has already been successfully implemented (Bet et al. Reference Bet, Samin, Georgiev, Eral and van Roij2018) and tested (Georgiev et al. Reference Georgiev, Toscano, Uspal, Bet, Samin, van Roij and Eral2020). Because we are considering symmetric prolate particles, the mobility matrix in the body frame (indicated by superscript $b$) is reduced to

(3.1)\begin{equation} \begin{pmatrix} v_{x}^b\\ v_{y}^b\\ \omega_{z}^b \end{pmatrix} = \frac{1}{6{\rm \pi}\eta R} \begin{pmatrix} a_t & 0 & 0\\ 0 & b_t & 0\\ 0 & 0 & \dfrac{3a_r}{4R^2} \end{pmatrix} \begin{pmatrix} F_{x}^b\\ F_{y}^b\\ \tau_{z}^b \end{pmatrix},\end{equation}

where $v_{x}^b$ and $v_{y}^b$ are the translational velocities in the body frame and $\omega _{z}^b$ is the angular velocity perpendicular to the plane of motion. Here, $F_{x}^b$ and $F_y^b$ are the components of the gravitational force in the body frame, and $\tau _{z}^b$ is the external torque from gravity about the particle's centre of geometry (see figure 1). The dimensionless translational mobility coefficients $b_t$ and $a_t$ represent mobility along the major and minor axes of the particle ($b_t>a_t$). The dimensionless rotational mobility coefficient is $a_r$. These coefficients should be identical for all of our particles with the same $\kappa$ and $R$, regardless of the internal density distribution ($\chi$). They characterize the drag from the external flow, which applies stress on the surface of the particle.

Our experimental data, however, are collected in the lab frame. Thus, we first rotate all vectors and the mobility matrix by an angle $\theta$ (figure 1) to obtain the equations of motion in the lab frame:

(3.2)$$\begin{gather} \boldsymbol{\varOmega} = \begin{pmatrix} \cos(\theta) & \sin(\theta) & 0\\ -\sin(\theta) & \cos(\theta) & 0\\ 0 & 0 & 1 \end{pmatrix}, \end{gather}$$

(3.3)$$\begin{gather}\boldsymbol{\varOmega}\boldsymbol{\cdot} \begin{pmatrix} v_{x}^b\\ v_{y}^b\\ \omega_{z}^b \end{pmatrix} = \left(\boldsymbol{\varOmega}\boldsymbol{\cdot} \frac{1}{6{\rm \pi}\eta R} \begin{pmatrix} a_t & 0 & 0\\ 0 & b_t & 0\\ 0 & 0 & \dfrac{3a_r}{4R^2} \end{pmatrix} \boldsymbol{\cdot}\boldsymbol{\varOmega}^{{-}1}\right) \boldsymbol{\varOmega}\boldsymbol{\cdot} \begin{pmatrix} F_{x}^b\\ F_{y}^b\\ \tau_{z}^b \end{pmatrix}. \end{gather}$$

After multiplying and collecting terms, we use the substitutions $2c_1 = a_t + b_t$, $2c_2 = b_t - a_t$ and $c_3 = 3a_r/4$ to write the result in the following form:

(3.4)\begin{equation} \begin{pmatrix} v_{x}\\ v_{y}\\ \omega_{z} \end{pmatrix} =\frac{1}{6{\rm \pi}\eta R} \begin{pmatrix} c_{1} - c_{2}\cos(2\theta) & c_{2}\sin(2\theta) & 0\\ c_{2}\sin(2\theta) & c_{1} + c_{2}\cos(2\theta) & 0\\ 0 & 0 & \dfrac{c_{3}}{R^2} \end{pmatrix} \begin{pmatrix} 0\\ F_{y}\\ \tau_{z} \end{pmatrix}.\end{equation}

We have chosen this parametrization out of convenience. For example, in the case of a perfect sphere, $b_t=a_t$, thus $c_1=1$, $c_2=0$ and $c_3=3/4$ (1.7). We have dropped the superscript since we are referring to the lab frame where the gravitational force only points in the $y$-direction.

The matrix multiplication above gives us the following equations of motion for our particles in the lab frame:

(3.5)$$\begin{gather} v_{x}=\dot{x} = \frac{c_{2}\sin(2\theta)}{6{\rm \pi}\eta R}F_{y}, \end{gather}$$

(3.6)$$\begin{gather}v_{y}=\dot{y} = \frac{c_{1} + c_{2}\cos(2\theta)}{6{\rm \pi}\eta R}F_{y}, \end{gather}$$

(3.7)$$\begin{gather}\omega_{z}=\dot{\theta} = \frac{c_{3}}{6{\rm \pi}\eta R^{3}}\tau_{z}. \end{gather}$$

The dotted variables denote differentiation with respect to time. Similar simplified equations for single-particle dynamics in quasi-2-D geometries have been derived by Bet et al. (Reference Bet, Samin, Georgiev, Eral and van Roij2018) and Ekiel-Jeżewska & Wajnryb (Reference Ekiel-Jeżewska and Wajnryb2009). In our experiments, the net force and torque on a particle will depend on the values of $\kappa$ and $\chi$. For $\kappa =2$ particles, the net gravitational force and torque about the centre of geometry are

(3.8)$$\begin{gather} F_{y} ={-}\tfrac{4}{3}{\rm \pi} R^{3}(\rho_{h} + \rho_{l} - 2\rho_{f})g, \end{gather}$$

(3.9)$$\begin{gather}\tau_{z} ={-}\tfrac{4}{3}{\rm \pi} R^{4}(\rho_{h} - \rho_{l})g\sin{\theta}. \end{gather}$$

Equations (3.5)–(3.7) are coupled through $\theta$, and can be solved analytically. However, the solution can be generalized by making the equations dimensionless. We used the sphere radius $R$ for a characteristic length scale and $\tau =R/U_T$ for the characteristic time scale, where $U_T$ is the terminal velocity of the lighter sphere (1.1). This non-dimensionalization results in the following equations of motion, where all variables are considered dimensionless for clarity of notation:

(3.10)$$\begin{gather} \dot{x} ={-}K_1c_{2}\sin(2\theta), \end{gather}$$

(3.11)$$\begin{gather}\dot{y} ={-}K_1(c_{1} + c_{2}\cos(2\theta)), \end{gather}$$

(3.12)$$\begin{gather}\dot{\theta} ={-}K_{2}c_{3}\sin(\theta), \end{gather}$$

(3.13)$$\begin{gather}K_1 = \frac{\rho_{h} + \rho_{l} - 2\rho_{f}}{\rho_{l} - \rho_{f}}, \end{gather}$$

(3.14)$$\begin{gather}K_{2} = \frac{\rho_{h} - \rho_{l}}{\rho_{l} - \rho_{f}}. \end{gather}$$

Equation (3.12) can be immediately solved since it is independent of the other equations. The result is

(3.15)\begin{equation} \cot\left(\frac{\theta(t)}{2}\right)= \cot\left(\frac{\theta_{0}}{2}\right) {\rm e}^{K_2c_{3}t},\end{equation}

where $\theta _0$ is the initial value of $\theta$ at $t=0$. Plugging this back into (3.10) and (3.11) and simplifying algebraically, we get

(3.16)$$\begin{gather} x(t) = x_0 + \frac{4c_{2}FK_1\cot(\theta_0/2)}{c_{3}K_{2} - c_{3}F^2 K_{2} \cot(\theta_0/2)^2} -\frac{2c_{2}K_1\sin(\theta_0)}{c_{3}K_{2}}, \end{gather}$$

(3.17)$$\begin{gather}y(t) = y_0 - \frac{K_1}{K_2 c_3}\left((c_1 + c_2)c_3K_2t + 2c_2\left(\cos(\theta_0)+\frac{1-F^2\cot^2 \left(\dfrac{\theta_0}{2}\right)}{1+F^2\cot^2 \left(\dfrac{\theta_0}{2}\right)} \right)\right), \end{gather}$$

where $F = {\rm e}^{K_{2}c_{3}t}$ is a function of time, and used here for compactness. In the limit of particles with uniform mass density ($K_2\rightarrow 0$, $\chi \rightarrow 0$), these functional forms simplify to

(3.18)$$\begin{gather} \theta(t)=\theta_0, \end{gather}$$

(3.19)$$\begin{gather}x(t) = x_0 - c_2 K_1 t \sin(2\theta_0), \end{gather}$$

(3.20)$$\begin{gather}y(t) = y_0 - K_1 t (c_1+c_2\cos(2\theta_0)). \end{gather}$$

Equations (3.18) and (3.19) verify the prediction that for polar particles of uniform density, the angle of inclination does not change, and the particle drifts laterally in the $x$-direction (Ramaswamy Reference Ramaswamy2001).

After taking the inverse cotangent of (3.15) and using standard least-squares nonlinear regression, we can fit these analytic forms to the experimental data with very good agreement. Figure 3 shows five identical experiments and their corresponding fits. For $\theta (t)$, there are only two fitting parameters, $c_3$ and $\theta _0$. Once they are determined by the fit, then $x(t)$ can be fit for the parameters $c_2$ and $x_0$. Finally, $y(t)$ can then be fit for $c_1$ and $y_0$. The curves are compared to each other by assigning $t=0$ when the particles are completely horizontal, i.e. $\theta ={\rm \pi} /2$. We also moved the $x$ and $y$ origin to correspond to $t=0$. Open symbols represent data and curves are the fits to (3.15)–(3.17). The fits for the $x$-position show more systematic deviation from the data, yet the overall displacement is also much smaller. For example, as shown in the inset in $y$ versus $t$, the residuals of these fits are comparable to the variability in $x$ versus $t$, which is a fraction of a particle radius in displacement. Although the source of the systematic asymmetry is unclear, we suspect that when particles are released from the gating mechanism, they are not perfectly parallel with the walls of the quasi-2-D chamber. If a particle's alignment varies during the rotation from $\theta = {\rm \pi}$ to $\theta = 0$, we would expect variations in the mobility coefficients (i.e. $c_2$) due to wall effects (Brenner Reference Brenner1999; Mitchell & Spagnolie Reference Mitchell and Spagnolie2014), resulting in an asymmetry in $x(t)$ about $\theta = {\rm \pi}/2$. Additionally, we do not expect errors in particle tracking to lead to systematic asymmetry even though the $x$-motion is of the order of the particle size. Tracking errors would manifest more as random noise rather than systematic deviations from theory. The data for $\theta$, $x$ and $y$ can also be fit simultaneously using a global least squares regression for all parameters, since parameters appear in multiple equations. We found less than 5 % difference in the fitted parameter values using this method, so we have only chosen to report the results of the sequential fitting. Similar analytic solutions and quality of fits were recently found in the alignment of mirror-symmetric particles in a microfluidic device (Bet et al. Reference Bet, Samin, Georgiev, Eral and van Roij2018; Georgiev et al. Reference Georgiev, Toscano, Uspal, Bet, Samin, van Roij and Eral2020).

Figure 3.

Data for five experiments with a single Al$+$Pl particle (table 1). Only 5 % of points are plotted for clarity. Initially, the heavy aluminium sphere begins above the lighter Delrin sphere. Open symbols represent data, and curves are model fits from (3.15)–(3.17). Different symbols and colours are separate experiments. Inset shows the residual difference between the model fit $y_m$ and the data $y$ for the vertical position of the particle.

One of the major assumptions of our model was that all coefficients are independent of $\chi$, and only depend on the shape of the composite, prolate particles. This is evident from (3.1), since $a_t$, $b_t$ and $a_r$ are dimensionless coefficients that only depend on the particle shape, not the density distribution. We confirmed this prediction using all fits of single particle experiments with $\kappa = 2$, as shown in figure 4(a–c). The coefficients $c_1$, $c_2$ and $c_3$ are computed directly from nonlinear least-squares regression of the data (3.10)–(3.12). For particles with $\chi = 0$ (uniform density), we used (3.18) and (3.19) to fit the data. In this form, there is no torque from gravity, so $c_3$ cannot be determined and is not shown. However, $c_1$ and $c_2$ can be determined, but are not very reliable because of experimental artefacts that affect the angle (and thus translational velocity) during sedimentation. These artefacts include small differences in the distribution of glue used between the particles, rotations out of the quasi-2-D plane and other 2-D confinement effects such as ‘fluttering’ (Brenner Reference Brenner1999; Mitchell & Spagnolie Reference Mitchell and Spagnolie2014; D'Angelo et al. Reference D'Angelo, Cachile, Hulin and Auradou2017). For finite $\chi$, the particles rotate significantly due to gravitational torque, and $c_1$, $c_2$ and $c_3$ can be determined reliably. There appears to be some small systematic trend in $c_1$, but the overall variation is small and the data for all parameters is consistent with a constant value over the range $0<\chi <0.25$.

Figure 4.

Best-fit parameters versus $\chi$ from (3.15)–(3.17) for single particle sedimentation experiments with $\kappa =2$. The material combinations used were circle, St$+$St; diamond, Cu$+$St; plus, St$+$ZrO$_{2}$; right triangle, Cu$+$ZrO$_{2}$; square, Al$+$Pl; star, Al$+$St (see table 1). Each data point is the weighted mean of five different trials with error bars representing the standard error of the mean. (a–c) Parameters $c_1$–$c_3$ directly computed from the nonlinear regression of data in the lab frame. (d–f) Body frame coefficients: $a_t$, $b_t$ and $a_r$. Here, St$+$St is missing from $c_3$ and $a_r$ because of the limiting form of $\theta (t)$ when $\chi = 0$ (3.18), which has no dependence on $c_3$.

Using $2c_1 = a_t + b_t$, $2c_2 = b_t - a_t$ and $c_3 = 3a_r/4$, we computed the shape-dependent drag coefficients of our symmetric particles in the body frame, as shown in figure 4(d,e). Here again, $a_r$ cannot be determined from the $\chi = 0$ data, and data with finite $\chi$ are most reliable. The average values of these mobility coefficients with $\chi >0$ are shown by the dashed lines: $\bar {a}_t=0.328\pm 0.018$, $\bar {b}_t=0.404\pm 0.012$ and $\bar {a}_r=0.221\pm 0.008$. We suggest that these experimental values for the mobility coefficients can be compared directly to simulations of particles composed of spheres (Garcia de la Torre & Carrasco Reference Garcia de la Torre and Carrasco2002). In comparison to sedimentation in an unbounded, 3-D fluid, we expect our measured values of $a_t$, $b_t$ and $a_r$ to be somewhat smaller since the particles experience a larger drag due to the confining walls.

3.2. Sedimenting particle pairs

Goldfriend et al. (Reference Goldfriend, Diamant and Witten2017) theoretically examined a sedimenting suspension of mass polar spheroids using a continuum linear stability analysis. To briefly summarize their results, they considered a suspension of particles settling due to an external body force $F$ in the direction of gravity in a fluid of viscosity $\eta$. A sinusoidal concentration perturbation was applied in a direction perpendicular to the force with amplitude $c(x)$ and a characteristic wavelength $\lambda$. These fluctuations in the concentration create velocity fluctuations, $U(x)$. Balancing the change in gravitational force of the suspension versus the change in the drag force gives $c\lambda F \approx U \eta /\lambda$. By solving for the amplitude $U$, we see that $U \sim c\lambda ^2 F/\eta$. The indefinite scaling of $U$ with $\lambda$ is a demonstration of the Caflisch–Luke paradox described in the introduction. These slabs of particles will also experience vorticity of the magnitude $U/\lambda \sim c\lambda F/\eta$. For uniform spheres, this will cause a rotation of the sphere, but no drift. However, self-aligning objects will be tilted away from their preferred alignment. This causes a drift in the $x$-direction with velocity $\sim \gamma R c F/\eta$, where $\gamma =\gamma (\kappa,\chi )$ is a proportionality constant determined by the shape and mass distribution of an individual particle. For positive $\gamma$, which requires $\kappa >1$ (Goldfriend et al. Reference Goldfriend, Diamant and Witten2017), the relative velocity of the particles is positive, meaning they drift away from each other. This is the screening mechanism that stabilizes the suspension. For negative $\gamma$, which requires $\kappa <1$, they drift towards each other, leading to unbounded growth of the instability.

In our experiments, we examined the particle-level interactions by measuring the relative separation of pairs of prolate ($\kappa >1$) particles as they repel each other during sedimentation. We placed two particles heavy-side down in adjacent slots of the plastic gate so that their initial separation was 3.3 mm. Each experiment was conducted five times for reproducibility. Figure 5 shows a representative selection of settling trajectories for various values of $\kappa$ and $\chi$. These are composite images of the particles during sedimentation, spaced 3.33 s apart. The arrows to the right of each panel show the orientations of each particle during sedimentation and the colour represents time. First, particles with $\chi = 0$ heavily influenced each other. Their dynamics were typically characterized by one of the particles rotating or flipping completely. This particle often lagged behind the other one, which did not flip, but followed a curved trajectory. This can be seen in both figures 5(a) and 5(d). The particles did not preferentially align to gravity, and instead produced a variety of dynamics. For example, the periodic variation in separation visible in figure 5(d) is reminiscent of Kepler orbits observed in sedimenting pairs of disks (Chajwa et al. Reference Chajwa, Menon and Ramaswamy2019). In fact, a periodic variation in the relative position between adjacent, sedimenting prolate particles was theoretically predicted by Claeys & Brady (Reference Claeys and Brady1993) (see their figure 4). On average, we did not observe a net repulsion or attraction between our particles with a uniform mass distribution ($\chi =0$).

Figure 5.

Experimental trajectories of two-particle interaction experiments. In each alphabetic panel, the image shows a composite of the particles’ trajectories during sedimentation. The graph shows the orientation of the particles with arrows pointing from the heavier particle to the lighter particle(s). The colour is used to show when two arrows are at the same time during their transit. Here, $x=0$ is defined as the halfway point between the particle centres on the first frame. Panels (a–c) are $\kappa =2$ particles, panels (d–f) are $\kappa =3$ particles. Panels (a) and (d) show particles with $\chi =0$. Panels (b) and (e) show particles with the smallest $\chi$, Cu$+$St (see table 1). Panels (c) and (f) show particles with the largest $\chi$, Cu$+$Pl (see table 1).

For particles with $\chi > 0$, there was an immediate rotation and repulsion between the particles leading to a horizontal separation that grew with time. Eventually the particles would align with the external gravitational field, and the separation saturated. This is shown in figure 5(b,c) for $\kappa = 2$ and figure 5(e,f) for $\kappa = 3$. The finite width of the quasi-2-D chamber, 4$R$, introduced a length scale that could potentially set an upper limit on the range of the repulsive interaction. However, we observed that the final separation between the particles could be as much as 30$R$ (figure 5e) for smaller values of $\chi$. The repulsive effect was most prominent for particles composed of materials with closely matched densities (i.e. copper and steel). Although this may seem counter-intuitive at first, particles with $0 < \chi \ll 1$ can rotate away from vertical more easily, and thus experience a larger repulsion and horizontal drift. As $\chi \rightarrow 0$, we expect one of the particles to be able to flip entirely if they are close enough to interact strongly, leading to the periodic type of interactions observed for $\chi =0$ (figures 5a and 5d). In this limit, the eventual behaviour of the sedimenting particles should be determined both by $\chi$ and by the initial separation.

The inverse relationship between $\chi$ and the mutual repulsion was also predicted by Goldfriend et al. (Reference Goldfriend, Diamant and Witten2017). The authors found that the growth rate of the horizontal velocity fluctuations scaled as $\gamma = \kappa ^{2/3}/3\chi$ for highly prolate particles ($\kappa \gg 1$). To quantify this effect in our experiments, we chose to measure the total change in horizontal separation, ${\rm \Delta} H$, between the particles’ geometric centres in each experiment. This is plotted in figure 6(a) as a function of $\kappa$. Generally, the separation increased with $\kappa$. However, to compare between each set of experiments that corresponded to different values of $\chi$, we multiplied the final separation by $\chi ^\alpha$, where $\alpha$ was determined by simultaneous fitting of all the data to the following form:

(3.21)\begin{equation} \chi^\alpha\frac{{\rm \Delta} H}{2R}=A(\kappa-1),\end{equation}

where we have imposed the requirement that there be no repulsion for $\kappa =1$ (i.e. single spheres). The fit was performed by subtracting the left- and right-hand sides of (3.21), squaring the difference and summing over all data points. The best fit values for the parameters were $\alpha =0.39\pm 0.05$ and $A=1.28\pm 0.20$, where the errors represent one standard error. The fit shows very good agreement with the data, as plotted in figure 6(b).

Figure 6.

Graphs of our experimental response parameter, ${\rm \Delta} H$, versus $\kappa$. Here, ${\rm \Delta} H$ is defined as the difference between the final and initial horizontal separation. The colours and shapes represent different material combinations of the composite particles. By order of increasing $\chi$, they are Cu$+$Pl (upright triangle), Al$+$St (upside down triangle), Tc$+$St (star) and Cu$+$St (circle) (see table 1). Panel (a) is the raw data, with each data point being an average over five runs and error bars representing one standard deviation. Panel (b) is the same data collapsed using the best fit parameters obtained from (3.21).

In general, the predictions from Goldfriend et al. (Reference Goldfriend, Diamant and Witten2017) are in excellent qualitative agreement with our experiments, yet the scaling, $\alpha \sim 0.39$, is quite different than that predicted by Goldfriend et al. (Reference Goldfriend, Diamant and Witten2017): $\alpha \sim 1$. There are a few reasons that can explain this discrepancy. (1) $\gamma$ represents an instantaneous response for an initially uniform concentration of particles. Here we are using the final separation, ${\rm \Delta} H$, which is essentially an integral of the repulsion between the particles in time. (2) The quasi-2-D environment should screen the long-ranged, $1/r$ hydrodynamic interactions (Beatus et al. Reference Beatus, Bar-Ziv and Tlusty2012), so one may expect a different theoretical scaling between $\gamma$ and $\chi$ based purely on geometry. (3) Our quasi-2-D chamber may introduce other effects that depend on the thickness of the chamber, for example, it is well known that the net viscous drag force on a sedimenting particle can be dependent on the distance to a nearby wall (Brenner Reference Brenner1999; Mitchell & Spagnolie Reference Mitchell and Spagnolie2014). (4) Our particles are not perfect examples of the prolate and oblate ellipsoids discussed by Goldfriend et al. (Reference Goldfriend, Diamant and Witten2017). Despite these differences, the experimental data with different values of $\kappa$ and $\chi$ can be reasonably collapsed using the dependence listed in (3.21).

Lastly, we verified that this mutual repulsion led to more uniformly distributed suspensions of many particles. We filled our quasi-2-D chamber with 29 particles with $\kappa = 2$. The left column of figure 7 shows that for particles with $\chi =0$, there is no preferential alignment to gravity, resulting in a large spread of particle separations, both vertically and horizontally. Particles can flip very easily and often come into contact. Some of the particles experienced small rotations out of the plane as well. The right column of figure 7 illustrates that particles with $\chi =0.24$ followed a more uniform spatial distribution. All particles tended to align with gravity, resulting in a mutual repulsion. When particles are in close proximity, they tilted away from the vertical and drifted apart, similar to figure 5. Surprisingly, the particles with $\chi = 0.24$ did not spread as much in the vertical direction as $\chi = 0$, suggesting that vertical fluctuations in concentration may be suppressed for $\chi >0$. An intuitive explanation for this behaviour stems from the variations in vertical velocities of particles. For $\chi = 0$, particle rotations lead to a spread in vertical terminal velocities (3.17), whereas particles with $\chi >0$ are mostly aligned to gravity and sediment at the same rate. Although vertical fluctuations were not directly addressed by Goldfriend et al. (Reference Goldfriend, Diamant and Witten2017), we hypothesize that the mutual repulsion in mass polar particles also suppresses the ‘clumping instability’ observed in uniform suspensions (Chajwa et al. Reference Chajwa, Menon, Ramaswamy and Govindarajan2020) that leads to large vertical separations between particles.

Figure 7.

Series of three sequential images of 29 particles sedimenting in our quasi-2-D chamber. (a) ($\chi = 0$) shows St$+$St particles and (b) ($\chi = 0.24$) shows St$+$Al particles (table 1). The top row shows images at the same vertical position near the top of the experimental chamber at early times, the middle row shows the same particles later in time and the bottom row shows the particles near the end of the experiment, at the bottom of the chamber.