Modifying self-assembly and species separation in three-dimensional systems of shape-anisotropic particles

The behaviors of large, dynamic assemblies of macroscopic particles are of direct relevance to geophysical and industrial processes and may also be used as easily studied analogs to micro- or nano-scale systems, or model systems for microbiological, zoological, and even anthropological phenomena. We study vibrated mixtures of elongated particles, demonstrating that the inclusion of differing particle “species” may profoundly alter a system's dynamics and physical structure in various diverse manners. The phase behavior observed suggests that our system, despite its athermal nature, obeys a minimum free energy principle analogous to that observed for thermodynamic systems. We demonstrate that systems of exclusively spherical objects, which form the basis of numerous theoretical frameworks in many scientific disciplines, represent only a narrow region of a wide, multidimensional phase space. Thus, our results raise significant questions as to whether such models can accurately describe the behaviors of systems outside this highly specialized case.

Figure 1

Typical particle ordering in side views (b)–(h) or top views (a) and (i) for (a) any single-species region of vertically aligned cylinders; binary systems with $d_1\ne d_2$ for particles with (b) similar $p$ and (c) strongly differing $p$; (d) binary systems with $d_1=d_2$; sphere-rod mixtures with (e) $d_{\mathrm{rod}}<d_{\mathrm{sphere}}$ and (f) $d_{\mathrm{rod}}>d_{\mathrm{sphere}}$; (g) system in (f) with a lower concentration of cylinders $[{\varphi }_c=0.5$ as opposed to ${\varphi }_c=0.9$ for (e) and (f)]; (h) and (i) same system as (c) in a cylindrical container. All rod-rod systems shown comprise two vertical HCP layers of particles.

Figure 2

Two-dimensional projections of the phase diagram showing for a range of binary mixtures of spheres and/or spherocylinders, the ordered states produced for each combination of (a) relative particle diameter $\left(\frac{d_1}{d_2}\right)$ and length $\left(\frac{l_1}{l_2}\right)$ and (b) the aspect ratios $p_{1,2}$ of the differing constituents. State A: open circles; state B: open blue diamonds; state C: solid blue diamonds; state D: green diamonds; state E: crosses. The dashed and dotted lines in (a) show, respectively, the regions of phase space for which $d_1=d_2$ and $\frac{d_1}{d_2}=\frac{l_1}{l_2}$. All states are defined in the main text.

Figure 3

Above: Phase diagram showing the regions of $v^2\text{−}{\varphi }_c$ space for which systems of 4 mm spheres and $9.8\times 3$ mm cylinders form the ordered, segregated state E (blue squares) and disordered, mixed state F (red circles). The dotted line approximates the system's phase boundary. For $v^2<0.02{\mathrm{m}}^2{\mathrm{s}}^{-2},$ particle rearrangement is suppressed because the system is no longer fluidized. Below: Mean-squared displacement $M\left(t\right)$ for the cylinders in a monodisperse system (${\varphi }_c=1,v=0.27$, state E) and an equivalent cylinder and sphere system (${\varphi }_c=0.75,v=0.27$, state $F$); the system “doped” with spheres approximates Fickian diffusion, while the “pure” system demonstrates subdiffusive behavior, indicating glassy dynamics [47].

Figure 4

One-dimensional residence profiles for various binary systems. Each residence profile $F\left(z\right)$ can be considered as a histogram of the height of particles for a given species [15]. Data are shown for binary combinations of (a) $9.8\times 3$ and $7.8\times 3$ mm rods, (b) $23.8\times 2$ and $9.8\times 3$ mm rods, (c) $9.8\times 4$ mm rods and 3 mm spheres $\left({\varphi }_c=0.6\right)$, (d) $23.8\times 2$ and $7.8\times 3$ mm rods, and (e) 4 and 4.5 mm spheres. In all cases, longer spherocylinders (larger spheres) are represented by diamonds and shorter (smaller) species by squares. For all pure-sphere and pure-cylinder systems, equal volumes of each species are used $\left({\varphi }_i=0.5\right)$.