Turbulent route to two-dimensional soft crystals

TL;DR

This study explores how two-dimensional turbulent flows induce a crystalline phase in soft granular particles, revealing a sharp transition influenced by flow dynamics and inter-particle forces.

Contribution

It demonstrates the emergence of a crystalline phase in soft particles under turbulence, highlighting the role of Stokes number and inter-particle interactions in phase transition.

Findings

Sharp transition between crystalline and non-crystalline phases as a function of Stokes number

Distinct preferential concentration behavior compared to flows without repulsive interactions

Quantitative measures of phase using bond order parameter and density fluctuations

Abstract

We investigate the effects of a two-dimensional, incompressible, turbulent flow on mono-disperse soft granular particles and show the emergence of a crystalline phase due to the interplay of Stokesian drag (measured through the Stokes number) and short-range inter-particle interactions. We quantify this phase through the bond order parameter and local density fluctuations and find a sharp transition between the crystalline and non-crystalline phase as a function of the Stokes number. Furthermore, the nature of preferential concentration, as characterised by the radial distribution function and the correlation dimension $D_2$, is significantly different from that of particle-laden flows in the absence of repulsive potentials.

Figures

Equations

$$\frac{d{\text{x}}_i}{dt}={\text{v}}_i;\frac{d{\text{v}}_i}{dt}=-\frac{{\text{v}}_i-\text{u}({\text{x}}_i,t)}{{\tau }_p}-\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_p}\nabla V(r_{ij});$$

$$V_{\mathrm{soft}}(r_{ij})=\{\begin{array}{ll}\frac{ϵ}{2}(1-r_{ij}/{\sigma }_{ij}{)}^2& \text{for }{r}_{ij}<{\sigma }_{ij},\\ 0& \text{for }{r}_{ij}\ge {\sigma }_{ij};\end{array}$$

$${\partial }_t\omega +\mathbf{u}\cdot \nabla \omega =\nu {\nabla }^2\omega -\alpha \omega +\mathbf{f};$$

Full text

The self-assembly of particles in a flow Whitesides and Grzybowski (2002); Scarabelli et al. (2014), because of its ubiquity, is amongst the most studied problem in the areas of turbulent transport, soft matter, granular systems and nonequilibrium statistical mechanics. In quiescent form, most dilute assemblies are liquids, which when densified, can take a crystalline or amorphous structure depending upon the dispersity of the constituents Clark et al. (1979); Pusey and Van Megen (1986); Liu and Nagel (2010); Boyer et al. (2011). Consequently, extensive studies of the rheology of such suspensions have happened Coussot and Wagner (2009); Bonn et al. (2017), motivated by diverse applications. In typical experiments and computer simulations, the role of a carrier flow in dispersing the particulate matter is trivial: indeed if there is an underlying fluid medium, they are typically simple shearing Doi and Chen (1989); Vermant and Solomon (2005). In a variety of natural and industrial processes, however, particles are dispersed in flows with non-trivial spatio-temporal correlations which are chaotic, and in extreme cases (such as a marine system) even turbulent Wells and Goldberg (1991, 1993); Kepkay et al. (1997).

This specific question of the structural properties of particulate suspension, where the underlying flow is turbulent, has surprisingly been not investigated despite significant progress in the last two decades in the area of turbulent transport of finite-sized, heavy, inertial (colloidal) particles. In this paper, we report the emergence of macroscopic particulate structures with crystalline (hexagonal) motifs even in the presence of strong mixing because of the carrier turbulent flow.

For suspended particles with a finite diameter and density, inertial effects and dissipative dynamics become important, leading to the particles detaching from the underlying flow to form strong inhomogeneities in their spatial distribution (see Fig. 1). This phenomenon, known as preferential concentration, has been extensively studied Longmire and Eaton (1992); Monchaux et al. (2012); Gustavsson and Mehlig (2016); Bec et al. (2007); Goto and Vassilicos (2008); Chun et al. (2005) and remains critical to our explanations of problems such as rain initiation in warm clouds. Although more recent studies have addressed the issue of effects such as gravity Bec et al. (2014); Good et al. (2014); Gustavsson et al. (2014) and turbophoresis Belan et al. (2014); De Lillo et al. (2016); Mitra et al. (2018) on preferential concentration, inter-particle interactions have largely been ignored except for studies on coalescences Bec et al. (2016); Li et al. (2018) and on Vicsek ordering Choudhary et al. (2015). The only example that we are aware of, which links ideas of soft matter and turbulent transport is the use of repulsive, elastic, hard sphere inter-particle interactions, which, combined with a dissipative dynamics, lead to stickiness and aggregation Bec et al. (2013). However, for most physical systems, the elastic limit is an idealised one: The most obvious particulate exchanges, such as those mediated through a soft potential O’Hern et al. (2003) has been ignored so far.

In this work, we therefore address two important and related issues, namely what is the effect of soft particle interactions on clustering of particles in a turbulent flow and can such realistic interactions, contrary to naïve expectations, lead to the growth of stable crystalline structures in an ensemble of particles interacting with each other as well as an ambient turbulent fluid.

We consider an assembly of $N_{p}$ particles seeded in a two-dimensional, statistically stationary, turbulent velocity field u See Supplemental Material at [URL will be inserted by publisher] for additional details . Since we consider particles smaller than the relevant length scales of the flow, the dynamics of the $i$-th particle (characterised by its Stokes or particle response time $\tau_{p}=\frac{2\rho_{p}a^{2}}{9\rho_{f}\nu}$, where $a$ is the particle radius, $\nu$ the kinematic viscosity of the flow, and $\rho_{p}$ and $\rho_{f}$ are the particle and fluid densities, respectively) defined through it position $\textbf{x}_{i}$ and velocity $\textbf{v}_{i}$ is given by the linear Stokes drag model along with the inter-particle interaction potential $V(r_{ij})$:

[TABLE]

where the interacting short-ranged repulsive potential Durian (1995); O’Hern et al. (2003), commonly used for modelling emulsions and other soft granular suspensions, is given by:

[TABLE]

where, $r_{ij}$ is the inter-particle separation, $\sigma_{ij}$ is the sum of the radii of particles $i$ and $j$, $\epsilon=1$ sets the energy scale for particle interactions. Thus, such an interaction takes into account the energy cost of deformation, only when two particles are in contact. For our work, we mostly consider a mono-disperse assembly, where all particles have the same diameter $\sigma$. For the mono disperse suspensions, which is our focus, all particles have the same radius $a$; thence $\sigma=2a$. In the Supplemental Material See Supplemental Material at [URL will be inserted by publisher] for additional details we show additional results for bi-disperse suspension where particles have different radii.

The advecting turbulent velocity u is obtained (by using a standard pseudo-spectral method) as a solution of the two-dimensional, incompressible Navier-Stokes equation, written in the vorticity ($\omega=\nabla\times{\bf u}$) formulation as

[TABLE]

with $\nu$ the kinematic viscosity and $\alpha$ Ekman friction coefficient. The flow is solved on a 2$\pi$ periodic grid with $N=512^{2}$ collocation points, and a deterministic large scale forcing $f$, ensuring a direct enstrophy cascade regime, maintains a non-equilibrium steady state Ray et al. (2011). The flow is conveniently described by its characteristic length $l_{\nu}=\sqrt{\langle\Omega\rangle/\langle P\rangle}=0.06$ and time $\tau_{f}=\sqrt{1/\langle\Omega\rangle}=0.4$ scales, where $\Omega=\int k^{2}E(k)dk$ is the enstrophy and $P=\int k^{4}E(k)dk$ the palinstrophy Boffetta et al. (2012). We set the forcing length scale $l_{f}=1.71$ yielding a secondary time scale $l_{f}/u_{\rm rms}=1.60$, $u_{\rm rms}=\sqrt{\int E(k)dk}$. By using the flow time-scale, we define the non-dimensional Stokes number $St=\tau_{p}/\tau_{f}$ and ensure that both the grid spacing and particle diameter are much smaller than $l_{\nu}$. Further, we vary the Stokes number by using different values of $\tau_{p}$.

The effect of such inter-particle interactions is striking. In Fig. 1 we show representative snapshots of particle positions, superimposed on the background vorticity field, for both interacting (red) and non-interacting $V(r_{ij})=0$ (black) particles. In the absence of inertia ($St=0$), unsurprisingly, the difference between the two ensembles is minute. However, for finite values of $St$ (panels (a) and (b)), the particle distributions are strongly influenced by their interactions. In particular, the nature of small-scale clustering is significantly altered and, especially for $St>1$ (panel (b)), interacting particles appear to be more homogeneously distributed than the ones which are non-interacting.

Inhomogeneities in the particle distribution are conveniently characterized by the correlation dimension $D_{2}$, defined through the probability of having two particles within a distance $r$, namely $P^{<}_{2}(r)\sim r^{D_{2}}$, or equivalently through the small-scale behavior of the radial distribution function $g(r)\sim r^{D_{2}-2}$. Figure 2 shows $D_{2}$ as a function of $St$ (including in the non-interacting case), for various values of the packing fraction $\phi={N_{p}\sigma^{2}}/({16\pi})$. We fix $\sigma=5.0\times 10^{-4}$, and choose $N_{p}=1\times 10^{5}$, $1.5\times 10^{5}$ and $2.0\times 10^{5}$, to obtain $\phi=0.05\%,0.075\%$ and $0.10\%$, respectively. For a given packing fraction and $St\ll 1$, the value of $D_{2}$, within error-bars, are indistinguishable from the non-interacting particles. This is because at such small values of $St$, there is hardly any small-scale clustering and therefore such dilute suspensions are only weakly affected by the short-range particulate interactions. However, as particles cluster, i.e. for $St=\mathcal{O}(1)$, soft granular repulsions dominate and, unlike the non-interacting ensemble, the interacting particles spread more — as a result of the competing interactions of vortical-ejection due to inertia and the short-range inter-grain repulsive energy cost — and with a larger value of $D_{2}$. This effect also leads to a slight shifting to the left of the value of $St$ where $D_{2}$ attains its minimum. This effect is of course accentuated with increasing packing fractions. For Stokes numbers a bit larger than 1, although the centrifugal vortical-ejection weakens, particles still tend to cluster in straining zones. However, the strength of the short-range repulsive forces ensure that interacting particles spread out more and sample the flow homogeneously, overcoming the bias due to inertia (Fig. 1(b)). We see evidence of this in our measurements, which show that for $St>1$ the correlation dimension $D_{2}$ asymptotes to the physical dimension 2 much faster for the interacting than for the non-interacting case. Furthermore, the minimum value of $D_{2}$ also increases as a function of $\phi$. Our results for packing fractions up to $\phi=50\%$ (see Fig. S1 in Ref. See Supplemental Material at [URL will be inserted by publisher] for additional details ) shows clearly that the minima in the correlation dimension becomes shallower and shallower; however it is a matter of speculation if the minima does not exist at even higher levels of packing.

Thus, as observed and discussed, inter-particle interactions in particle-laden turbulent flows clearly have an effect on the degree and nature of preferential concentration. But are such particle interactions strong enough to overcome turbulent mixing and nucleate crystalline structure? There are recent experimental studies which suggest that such aggregates may develop in homogeneous turbulence Petersen et al. (2019). To answer this question — and provide compelling evidence — it is essential to work with a much larger packing fraction and particle diameter, as is common in studies of granular systems Radin (2008); Henkes et al. (2011). (We however ensure that the diameters are still much smaller than the fluid characteristic length scale $l_{\nu}$ for our model to be valid.) We therefore choose $\sigma=5\times 10^{-3}$, and different particle numbers, namely, $N_{p}=2\times 10^{5}$ ($\phi=10\%$) and $N_{p}=10^{6}$ ($\phi=50\%$).

A useful indicator of how densely such particles are packed, due to inertia and interactions, is to look at the local packing fraction $\phi_{\rm local}={N_{\Delta}\pi\sigma^{2}}/({4\Delta^{2}})$, where $N_{\Delta}$ is the number of particles in a small square of side $\Delta=4\,\delta x$, where $\delta x$ is the width of our Eulerian grid. In Fig. 3(a) we show a representative pseudo-color snapshot of the local packing fraction for particles with $St=1$ ($\phi=50\%$). As we would expect, in a given snapshot, there are regions which are extremely dense and the local packing fraction far exceeds its average $\phi$, with some regions hyper-packed due to the softness of the particles. We now examine the structure of these densely packed regions by using the standard approach of Voronoi tessellation. In Fig. 3(b) we show the Voronoi construction, corresponding to a zoomed in region of panel (a); furthermore we color, on a greyscale, each cell by the coordination number $z$ (values shown in the adjacent colorbar) of the particle. We find, surprisingly, that these soft particles do form hexagonal lattices, as suggested by the predominance of $z=6$ (light grey) and the cells of our Voronoi tessellation, with almost perfect crystalline order.

To quantify this degree of crystallinity, we use the standard measure of the bond order parameter $\psi_{6}$  Steinhardt et al. (1983) for a given particle, $\psi_{6}(\textbf{r}_{i})=\frac{1}{N_{b}}\left|\sum_{m=1}^{N_{b}}\exp{6\,\iota\,\theta_{mi}}\right|$ here $N_{b}$ is the number of nearest neighbors of the $i$th particle, and $\theta_{mi}$ is the angle between the $x$-axis and the bond joining the $i$th particle with the $m$th particle. Before we turn to the full statistics of the bond order parameter, it is useful to first look at a coarse-grained measure of this quantity. In analogy to our definition of a local packing fraction, we define a local bond order parameter $\psi_{6,\rm local}=({1}/{N_{\Delta}})\sum_{i=1}^{N_{\Delta}}\psi_{6}(\textbf{r}_{i})$. The map for $\psi_{6,\rm local}$, corresponding and strongly correlated to the packing shown in Fig. 3(a), is displayed in Fig. 3(c). Such a coarse-grained description shows macroscopically large regions with a very high value of $\psi_{6,\rm local}$ consistent with the visual suggestion of crystallinity in Fig. 3(b). Also, note that wherever the soft particles are hyper-packed, the crystallinity is lost, and we have a re-entrant melting scenario Zu et al. (2016) highlighted by the dashed black oval in Figs. 3(a) and (c).

Starting from an initial spatially homogeneous particle homogeneous, we have observed that the inhomogeneous structures, as illustrated in Fig. 3, emerge over a timescale $t\approx 5\tau_{f}$ after the turbulent flow is switched on; please refer to Refs. See Supplemental Movie at [URL will be inserted by publisher] for an animation showing the evolution of $\phi_{\rm local}$ and $\psi^{6}_{\rm local}$ as well as how the dynamics freezes on switching off the flow ; See https://youtu.be/uTazvh1ETD8 for an animation showing the evolution of $\phi_{\rm local}$ and $\psi^{6}_{\rm local}$ as well as how the dynamics freezes on switching off the flow for an animation of this evolution as well as what happens subsequently when the flow is switched off. In the steady state, the crystalline aggregates are seen to be continuously advected by the underlying flow, and therefore undergo breaking and coalescing at a constant rate as the straining zones evolve with time. Thus, in most cases, a typical crystalline aggregate is transient in nature. To estimate the time scale on which a particular crystalline aggregate persists, we track a subset of particles from our simulations which form a particular crystalline structure. This subset of particles form a crystalline structure at $t=47\tau_{f}$, with most of the constituents having $\psi_{6}\sim 1$, and persists for time-scales of the order of a few $\tau_{f}$ before breaking away to form new crystalline structures See Supplemental Movie at [URL will be inserted by publisher] for an animation showing the formation and breaking up of a crystalline aggregate ; See https://youtu.be/BzQPgpQPW5Y for an animation showing the formation and breaking up of a crystalline aggregate .

Further, we have also investigated what happens when in steady state conditions, the underlying turbulent flow is suddenly switched off abruptly. The suspended particles should also come to rest, and it indeed does See Supplemental Movie at [URL will be inserted by publisher] for an animation showing the evolution of $\phi_{\rm local}$ and $\psi^{6}_{\rm local}$ as well as how the dynamics freezes on switching off the flow ; See https://youtu.be/uTazvh1ETD8 for an animation showing the evolution of $\phi_{\rm local}$ and $\psi^{6}_{\rm local}$ as well as how the dynamics freezes on switching off the flow , over a timescale determined by $\tau_{p}$. The interesting finding is that the crystalline aggregation in the form of the spatially inhomogeneous structures remain intact; see Figs. S4 (a) and (b) in the Supplementary Material See Supplemental Material at [URL will be inserted by publisher] for additional details as well as Refs.See Supplemental Movie at [URL will be inserted by publisher] for an animation showing the evolution of $\phi_{\rm local}$ and $\psi^{6}_{\rm local}$ as well as how the dynamics freezes on switching off the flow ; See https://youtu.be/uTazvh1ETD8 for an animation showing the evolution of $\phi_{\rm local}$ and $\psi^{6}_{\rm local}$ as well as how the dynamics freezes on switching off the flow .

The snapshots of Fig. 3 naturally lead us to examine the behaviour of $\psi_{6}$ as a function of $St$ for a reasonably high packing fraction ($\phi\sim 10\%$). Given the non-equilibrium, spatio-temporal variation of the advecting turbulent flow, as discussed above, it is of course natural that all the particles would not arrange themselves in a hexagonal lattice. We therefore look at the mode $[\psi_{6}]$, the value of $\psi_{6}$ that occurs most frequently amongst all particles and over time. In the non-crystalline phase, the largest number of particles have a value close to 0 whereas in the crystalline phase this value tends to 1.

In Fig. 4(a), we plot $[\psi_{6}]$ (black filled circles) as a function of $St$ and see a remarkable behaviour. For extremely small or large values of the Stokes number, $[\psi_{6}]=0$ whereas for values of $St$ around 1, where, as seen in Fig. 2, particles show significant preferential concentration, $[\psi_{6}]=1$. This behaviour is remarkable as it shows a sharp transition between a crystalline and non-crystalline phase (as a function of the Stokes number). Of course such a characterisation makes sense only if there is a macroscopically large fraction of the total particles which show $[\psi_{6}]$ as reported in Fig. 4(a). We thus calculate the fraction of particles $N_{[\psi_{6}]}/N_{p}$ (along with their errorbars calculated over time) having $\psi_{6}=[\psi_{6}]$ (green square symbols). At a practical level, $\psi_{6}$ of course varies from 0 to 1; therefore it is convenient for us to bin values of $\psi_{6}$. In particular, we count all particles with values of $0.0\leq\psi_{6}\leq 0.2$ as $\psi_{6}=0$ and those in the range $0.8\leq\psi_{6}\leq 1.0$ as $\psi_{6}=1.0$. We have checked that slight variations in this range does not change our results (Fig. 4). Not surprisingly, in the homogeneously dispersed phase ($St=0$) we find close to $70\%$ of the particles are associated with $\psi_{6}=0.0$. However as $0<St\ll 1.0$, the spheres tend to weakly cluster leading to a greater variation in $\psi_{6}$. Thus for this range of Stokes numbers, while $[\psi_{6}]=0$, there is a decrease in $N_{[\psi_{6}]}/N_{p}$ as seen in Fig. 4(a). It is only when the Stokes numbers reach values where particles preferentially concentrate most strongly (see Fig. 2 for comparison), that $[\psi_{6}]=1.0$ and a consequent rise and plateauing of the value of $N_{[\psi_{6}]}/N_{p}$. Thus, in this cryastalline phase where $[\psi_{6}]=1$, we immediately see (Fig. 4(a)) that a majority of particles arrange themselves in perfect hexagonal order, indicating that indeed there is a dynamical structural transition with Stokes number. For higher Stokes numbers, the particles start to disperse more homogeneously, and the crystalline phase is lost, for reasons discussed before, leading to $[\psi_{6}]=0$. While for finite Stokes numbers, there are still great variations in the overall values of $\psi_{6}$ associated with each particle, as the Stokes number becomes larger and larger, the majority fraction $N_{[\psi_{6}]}/N_{p}$ of those with $\psi_{6}=0$ increases. The results from our simulations, shown in Fig. 4(a), are consistent with this theoretical understanding of such soft-sphere suspensions.

In the crystalline phase, the variation of the degree of crystallinity $\langle\psi_{6}\rangle$ with the packing fraction is shown in Fig. 4(b). Clearly, and as conjectured, $\langle\psi_{6}\rangle$ increases with $\phi$; remarkably even for very dilute suspension, we do see an average crystalline order with $\langle\psi_{6}\rangle$ significantly higher than the value for random packing.

While the two curves in Fig. 4(a) give quantitative evidence for the emergence of a macroscopic crystalline order in an ensemble of particles in a turbulent flow, it is useful to also examine the mean bond order parameter $\langle\psi_{6}\rangle$ as a function of the Stokes number. The non-monotonic behaviour of this measure, seen in the curve for $V_{\rm soft}$ in Fig. 4(c), is consistent with that seen for $[\psi_{6}]$. Of course the average value does not switch between 0 and 1 because of local fluctuations in the particle arrangement. For the range of Stokes numbers where crystalline structures are observed, we find good evidence for onset of crystallinity by measuring $\langle\psi_{6}\rangle$ as a function of $\phi$ (Fig. 4(c)).

Since the results reported so far have been obtained for soft particles, whereby two particles can overlap/penetrate to mimic deformation at contact for real grains or droplets, it is important to check if this phenomenon is universal and, in particular, persists even for hard-core particles, where the repulsive forces will be stronger at contact and thereby affect local packing. In order to do so, we performed simulations where we model the hard cores particles via the Weeks-Chandler-Anderson (WCA) inter-particle potential Weeks et al. (1971) - $V_{\rm WCA}(r_{ij})=4\epsilon[(\sigma/r_{ij})^{12}-(\sigma/r_{ij})^{6}]+\epsilon$ for $r_{ij}\leq 2^{\frac{1}{6}}{\sigma}$ and [math] otherwise. For this system, we again measure $\langle\psi_{6}\rangle$ (in Fig. 4 (b), red squares), with a packing fraction twice as large as that used for soft disks, as a function of $St$. Also see Fig. S2 in Supplementary Information See Supplemental Material at [URL will be inserted by publisher] for additional details for local maps for the WCA particles, similar to what has been shown in Fig. 3 for the harmonic disks. Our measurements show that the degree of maximum crystallinity, at $St\sim 1$, is marginally smaller than that for particles with soft interactions and that $\langle\psi_{6}\rangle$ falls off faster as a function of $St$ for the assembly of hard disks as compared to soft disks. Nevertheless, the basic mechanism of the formation of crystalline aggregates seem to be unchanged even for the extreme case of the WCA potential.

Another consideration is the polydispersity of the granular particles and it’s effects on the crystalline aggregates Russo and Wilding (2017); Sampedro Ruiz et al. (2019); Sampedro Ruiz and Ni (2020). To study this we use a bi-disperse assembly of particles with the particle radius ratio of 1.4 O’hern et al. (2003). As expected, for such mixtures, the spatial extent of local crystallinity diminishes significantly when compared to the mono-component system (see Fig. S3 of the Supplementary Material See Supplemental Material at [URL will be inserted by publisher] for additional details ). Nevertheless, even for such bi-disperse suspension, dense packings leading to locally jammed structures, induced by the carrier flows, do still occur. Indeed this issue along with the role played by the energy scale $\epsilon$ needs a detailed study in future.

In conclusion, we have shown that two-dimensional particle-laden turbulent flow, lead to crystalline self-assembly, albeit spatially inhomogeneous, because of the complementary effects of drag-induced preferential concentration and inter-particle interactions A recent experiment on particles settling under gravity show signatures of crystalline phases in incompressible flows (A. Petersen, L. Baker, and F. Coletti, J. Fluid Mech. 864, 925-970 (2019). The central role played by particle inertia is apparent in the sharp transition between crystalline aggregates as a function of the Stokes number. Our work also shows that elasticity of the particles lead to a modification in the nature of preferential concentration of heavy inertial particles which lie at the heart of several natural and industrial processes.

We also demonstrate that these crystalline aggregates survive even after the turbulent flow is switched off (Fig. S4 of the Supplementary Material See Supplemental Material at [URL will be inserted by publisher] for additional details and Refs. See Supplemental Movie at [URL will be inserted by publisher] for an animation showing the evolution of $\phi_{\rm local}$ and $\psi^{6}_{\rm local}$ as well as how the dynamics freezes on switching off the flow ; See https://youtu.be/uTazvh1ETD8 for an animation showing the evolution of $\phi_{\rm local}$ and $\psi^{6}_{\rm local}$ as well as how the dynamics freezes on switching off the flow ) by setting $\textbf{u}(\textbf{x}_{i},t)=0$ in Eq. 1. Thus, this provides a route to engineer such inhomogeneous particulate assemblies, for example via drying of the suspending fluid, as it often does in nature. We further note that the chosen value of $\epsilon$ sets the energy scale such that potential energy is comparable to the kinetic energy as shown in Fig. S3 of the Supplementary Material See Supplemental Material at [URL will be inserted by publisher] for additional details .

It is also important to keep in mind that, given that there have hardly been any studies of particulate structures in a turbulent flow, our work focuses on the most simple and general framework as commonly used in turbulent transport problems. The most important simplification that we have used is to ignore the feedback of the particles on the flow, as well as lubrication forces or the effect of porosity in the packed structures. It has been shown in an earlier work  Saw et al. (2014) that a one-way coupled model for Stokesian particles is a valid assumption in turbulent flows. Furthermore, it was shown in Bec et al. (2013), while studying the problem of elastic collisions in particle-laden turbulent flows, that at least in the small Stokes limit, the effect of short-range lubrication was merely the renormalisation of the effective relaxation time. However, it should be left for future work to actually examine in detail the role of lubrication and porosity in stabilizing such crystalline structures.

Finally, we note that such inhomogeneous crystalline aggregation has recently been observed in assemblies of self-propelled particles Palacci et al. (2013). However, the morphology of the clusters in the case of the turbulence led aggregation is perhaps very different, and this needs to be explored further. However, in a broader context, the diverse non-equilibrium routes for such self-assembly opens up fascinating avenues.