Effective aspect ratio of helices in shear flow

We report the results of simulations of rigid colloidal helices suspended in a shear flow, using dissipative particle dynamics for a coarse-grained representation of the suspending fluid, as well as deterministic trajectories of non-Brownian helices calculated from the resistance tensor derived under the slender-body approximation. The shear flow produces nonuniform rotation of the helices, similarly to other high aspect ratio particles, such that more elongated helices spend more time aligned with the fluid velocity. We introduce a geometric effective aspect ratio calculated directly from the helix geometry and a dynamical effective aspect ratio derived from the trajectories of the particles and find that the two effective aspect ratios are approximately equal over the entire parameter range tested. We also describe observed transient deflections of the helical axis into the vorticity direction that can occur when the helix is rotating through the gradient direction and that depend on the rotation of the helix about its axis.

Figure 1

Schematic of simulation domain. Fixed walls one particle diameter thick in the horizontal $(x,z)$ plane (red spheres) are moved at constant, opposite speeds in the $x$ direction, thus producing simple shear with $(x,y,-z)$ = (flow, gradient, vorticity) directions. Periodic boundaries are employed in the $x$ and $z$ directions.

Figure 2

Coordinates used for measuring helix orientation relative to shear flow (Fig. 1) and parameters used for specifying helices.

Figure 3

Geometric effective aspect ratio [$r_{\psi }$ from Eq. (9)] as a function of $a=\ell /r$ for integer $n$ from $n=1$ to $n=10$. The value of $\psi$ is indicated by color. The dependence on $\psi$ is only significant at small $n$ (large pitch) and large $a$.

Figure 4

Jeffery orbit for a non-Brownian ellipsoid in a shear flow. The top image shows successive orientation snapshots ($\tilde{u}$) for a particle of aspect ratio 4, with a small center of mass velocity from left to right for clarity, colored from white (early times) to black (late times). The graph displays angles relative to gradient ($\varphi$, solid black) and vorticity ($\theta$, dashed red) directions and square of the gradient ($y$) component of the orientation unit vector (${\tilde{u}}_y^2$, dotted blue) vs. strain.

Figure 5

Trajectories of helical filaments from DPD simulations and deterministic calculations. Top: Snapshots of $\tilde{u}$ as in Fig. 4 from deterministic trajectory of a helix with $a=5$ and $n=9$, and $\varphi , \theta$, and ${\tilde{u}}_y^2$, from the deterministic (open) and simulated (filled) trajectories. Bottom: Orientation snapshots, $\varphi , \theta$, and ${\tilde{u}}_y^2$, for a helix with $a=15, n=14$.

Figure 6

(a) Plot of the effective aspect ratio $r_e$ [Eq. (11)] from simulations versus the geometric aspect ratio $r_G$ [Eq. (10)] for a range of helix parameters. Marker size is indicative of helix length, (min,max) = (8,60), and marker color is indicative of pitch with white corresponding to $p=1$ to black (dark) with $p=30$. The solid line is $r_e=r_G$ (all three panels). (b) Plot of $r_e$ from deterministic trajectories versus $r_G$ for a range of helix parameters and different initial values of $\psi$ (c) data from (b) plotted vs. $r_{\psi }$ [Eq. (9)], calculated using the initial values of $\psi$ for that run.

Figure 7

Top: Plot of ${\tilde{u}}_y^2$ (dashed) and $u_z^2$ (solid) vs. time for one simulated (left) and one deterministic (right) trajectory ($\ell =58.8, n=4, r=3$ for both) demonstrating deflection into the vorticity direction, and subsequent recovery, as the helix axis rotates past the gradient direction. Bottom: Plot of $\psi \left(t\right)$ (dotted), ${\tau }_{\theta }$ in units of $\dot{\gamma }{\alpha }_n{\ell }_c^{2}r$ (thin) and $\theta \left(t\right)$ (thick) for the same trajectories. The deflections into the vorticity direction are well predicted by ${\tau }_{\theta }$, while the variability in size and direction of deflection arises from the behavior of $\psi \left(t\right)$.

Figure 8

Surface plot showing ${\tau }_{\theta }^0$ (in units of $\dot{\gamma }{\alpha }_n{\ell }_c^{2}r$) as defined in Eq. (12) with $\ell =58.8, r=3$ and $n=4$ as a function of $\varphi$ and $\psi$.