Robust microstructure of self-aligning particles in a simple shear flow

A self-aligning particle (SAP) attains a permanent orientation without application of external torques in a low Reynolds number simple shear flow in contrast with the continuous rotation exhibited by most rigid bodies including thin fibers and disks. SAPs align close to the fluid lamellae, and we characterize the robustness of this flow alignment to secondary disturbances such as perturbation to the shear flow, Brownian motion, interparticle interactions, or the presence of a wall using dynamic simulations of representative SAP geometries. The robustness of the flow alignment of SAPs at all concentrations generates highly aligned microstructures inaccessible to traditional particle suspensions.

Figure 1

Coordinate systems, schematic of a $\mathrm{T}$ ring, and the stable ($\pm {\mathbf{p}}_S$) (solid ring) and unstable ($\pm {\mathbf{p}}_U$) (translucent ring) nodes of the SAP. All distances are in nondimensional units.

Figure 2

Orientational trajectories of an isolated (a) Equivalent torus and (b) SAP of aspect ratio $A=55$. (c) Orientation trajectories of an isolated SAP near the stable ${\mathbf{p}}_S$ and unstable ${\mathbf{p}}_U$ nodes.

Figure 3

State of the orientational dynamics of rings in the $(A,C_{\alpha }/C_{\lambda })$ space. Color represents $T/T_t$ for tumbling rings and the equilibrium angle $\delta$ for SAPs. Symbols with the same value of $C_{\alpha }/C_{\lambda }$ correspond to rings with the same cross-sectional shape. Cross sections of the rings are also shown with transparent gray background. These cross sections are illustrated such that the axis of symmetry of each ring is on the left side of the cross section.

Figure 4

Isolated particle properties (a) $T$ for tori and $\mathrm{T}$ rings ($A<A^{*}$) and $\delta$ for $\mathrm{T}$ rings ($A\ge A^{*}$) as a function of $A$. (b) $k_0$ as a function of $A$ for T rings, tori, and fibers. All order of magnitude scalings are derived from the SBT treatment of rings.

Figure 5

Effect of perturbations to the SSF. Contours of $k_{\varepsilon }/{\delta }^2$ as a function of varying amplitude and frequency of the perturbation for different modes: (a) $({\varepsilon }_{\omega 3},{\zeta }_{\omega 3})$, (b) $({\varepsilon }_{\omega 1},{\zeta }_{\omega 1})$, (c) $({\varepsilon }_{E13},{\zeta }_{E13})$, and (d) $({\varepsilon }_{\omega 1},{\zeta }_{\omega 1})$ with $({\varepsilon }_{\omega 3},{\zeta }_{\omega 3},d_{\omega 3})=({\delta }^2,0.1{\delta }^2,0.1\delta )$.

Figure 6

Orientational trajectory (solid curve) of a $A=55$ SAP in the $\left(\varphi \text{−}\theta \right)$ plane for ${\varepsilon }_{\omega 3}=1.05\sqrt{2{\delta }^2[1+{(\pi {\zeta }_{\omega 3}/\delta )}^2]}\approx 1.8\times {10}^{-3}$, ${\zeta }_{\omega 3}={10}^{-2}\delta$ and $d_{\omega 3}=0.0$ at times (a) $t=0.05{\zeta }_{\omega 3}^{-1}$, (b) $t=0.25{\zeta }_{\omega 3}^{-1}$, and (c) $t=0.8{\zeta }_{\omega 3}^{-1}$. Vectors represent the orientational velocity field $(\dot{\varphi },\dot{\theta }),$ and symbols represent: stable node ($\circ$), unstable node ($\square$), and position at the given time (x).

Figure 7

Brownian motion effects. (a) $P$, the probability distribution function of $\varphi$, for an $A=55$ SAP at various Pe using theory (thin line) and BD simulations (thick translucent line); inset shows the peak value of $P$, $P_{max}$ as a function of Pe. (b) Flow-alignment parameter, $k_0$ as a function of Pe for SAPs and ETs with $A=55$. (c) $P$ for a SAP and an ET of $A=55$ at $\mathrm{Pe}={10}^4$ and $\mathrm{Pe}={10}^5$. (d) Tumbling period as a function of Pe for SAPs and ETs with $A=55$.

Figure 8

Pairwise interactions (PIs). (a) Collision frequency, $\langle {\beta }_c\rangle$, and (b) effect of PIs on the orientational dispersion $k_1$, as a function of $A$. (c) $\varphi$ and $\theta$ of a SAP as a function of $\mathrm{\Delta }{r}_1$ during an interaction with another SAP initially separated with $\mathrm{\Delta }{r}_1={10}^3$, $\mathrm{\Delta }{r}_2=0.15,\mathrm{\Delta }{r}_3=-0.13$. (d) Schematic of hydrodynamic interactions of two SAPs in shear flow explaining the shape of the $\varphi$ trajectory in (c).

Figure 9

Impact of tumbling impurities and wall. (a) $k$ in a SAP-ET mixture as a function of $\chi$ at $n=0.1$ for $A=40$ particles. (b) $(\varphi +\delta )/\delta$ as a function of $h$.

Figure 10

Multiparticle interactions (MIs) (a) Variation of ${\langle k\rangle }_t$ with time. (b) $k$ for SAPs using MI ($\square$) and PI (-) simulations; and $k$ for ETs from PIs (- - dashed) and MIs (o) as a function of $n$. Shaded region and errorbars represent 95% confidence intervals. (c) $P\left(\varphi \right)$ at various $n$ for $A=40$ SAPs.

Figure 11

Orientational microstructure of $A=40$ SAPs and ETs viewed in the flow-gradient plane.

Figure 12

Conductivity of a composite made of $A=40$ SAPs or ETs. (a) Conductivity in the flow vorticity plane (in-plane) and conductivity in the gradient direction (out-of-plane) as a function of the particle volume fraction for SAPs and ETs. (b) Ratio of in-plane and out-of-plane conductivities for a composite of SAPs and ETs. Lines use the average values of the orientational moments from PI simulations, and symbols use the values from MI simulations.