Consider a Spherical Cow. Nonspherical Colloid Transport

Nonspherical Colloid Transport. The majority of colloids in the environment are non-spherical in shape, such as clay particles, various types of bacteria (e.g. bacilli), human blood cells, and polymer molecules. Various researchers have demonstrated that nonspherical particles displayed different transport and retention behaviors compared to spheres. Unlike spheres in a force field, the forces acting on nonspherical colloids strongly depend on orientations. As such, particle rotation, which may be irrelevant for homogeneous spheres, becomes crucial in governing the transport and retention behaviors of nonspherical (here, rod-shaped) particles.

Particles in fluid rotate in response to hydrodynamic flow and Brownian motions. Jeffery predicted that without considering any external forces and diffusion effects, free rigid ellipsoidal particles rotate indefinitely over a certain period (known as Jeffrey’s orbit) in a linear shear flow field. From Jeffrey’s prediction, spheres rotated faster compared to rods, and spheres tended to follow flow streamlines despite rotation, whereas rods rotated slowly when their long axes were nearly parallel to the flow directions, and rapidly otherwise (i.e. “tumbling" motion). Moreover, for inertial rod particles, their rotation was coupled with oscillation along the direction of flow velocity gradient (Figure 8).

Figure 8. Simulated particle trajectories for both 1 µm sphere and aspect ratio 6 rod particles from parallel plate geometry, under pore water velocity 5 m d^-1, with all external forces ignored. Channel height: 62.5 µm. (a) Euler angle θ versus particle displacement for sphere. (b) Euler angle θ versus particle displacement for rod. (c) Particle center position in z-direction versus particle displacement for sphere. (d) Particle center position in z-direction versus particle displacement for rod

The magnitude of oscillation increased with increasing shear and decreased when the rod approached the wall. The presence of external forces or diffusion would disturb these so-called Jeffrey orbits. For example, under gravity or colloidal forces, ellipsoids still exhibited similar rotation (and oscillation) patterns, but their overall translation trajectories slowly drifted across flow streamlines and toward the directions of these forces. Depending on the particle size, diffusion (especially Brownian rotation) may alter or totally disrupt the Jeffrey orbits through the orientation dependency of hydrodynamic drag forces.

The interplay of rotation dynamics due to flow hydrodynamics and diffusion led to the distinct retention behaviors of rod-shaped particles from their equivalent-volume spheres, as demonstrated from Lagrangian colloid trajectory simulations under favorable conditions. For example, at a pore water velocity of 5md^-1, instead of the characteristic minimum collector efficiency corresponding to the 1-2 µm colloid size seen for spheres, rods showed maxima in that size range, with two residual minima astride the maximum; e.g., “hump" (Figure 9).

Figure 9. Simulated single collector efficiencies (ⴄ) as a function of particle size and aspect ratio from Happel model under favorable conditions under pore water velocity of 5 m d^-1. AR1-AR6 represent ellipsoids with aspect ratios 1-6, respectively

At the 5md^-1 velocity, rotation of larger size rods (>2 µm) was predominantly due to hydrodynamic shear over diffusional rotation. The rotation rendered the rods to behave like spheres prescribed by their long axes, leading to their greater retention (~ 1.5-2 fold) compared to their equivalent-volume spheres. For small size rods (<200 nm), rotation due to diffusion became dominant, allowing them to behave as “spinning bodies," so that rods showed similar or slightly less attachment compared to spheres.

However, for rod sizes from 200 nm to 2 µm, rods attached significantly more than spheres, with the maximum attachment occurring around 1 µm diameter. For this size range of colloids, rotation due to shear and rotation due to diffusion were each significant, the combination of which caused mid-size rod-shaped colloids to drift considerably across streamlines and achieve greater retention. These retention trends were true for other rod aspect ratios (ranging from 2 to 6). At a given particle size, the discrepancies in attachment between rods and spheres increased with increasing aspect ratio. As flow velocity increased (e.g. 20-200 m d^-1), the “hump" shape shifted to the small particle size region. At very low velocities of 0.1 md^-1, the “hump" disappeared, and the rods exhibited similar or slightly less retention compared to spheres. These behaviors suggested that particle shape greatly affected particle attachment in groundwater conditions, and the effect of shape on retention strongly depended on particle size and fluid velocity.

The distinct transport and retention behaviors of rod-shaped particles, when compared to spheres, originate from their distinct response to flow and/or force perturbations. In terms of attachment, large and small rod-shaped colloids (>2 µm and <200 nm) may be represented by the concept of “effective spheres," but doing so, one will lose insight on the distinct transport and rotation dynamics differentiating rod-shaped particles from spheres. More importantly, since rod-shaped particles transport farther prior to a given retention, simplifying rods as spheres will underestimate the risks that nonspherical pathogens may pose in many environmental processes.