Underexcitation prevents crystallization of granular assemblies subjected to high-frequency vibration

Significance

The non-Brownian vibration-induced crystallization of an assembly of monodisperse spheres increases the packing fraction from the dense random value to nearly the theoretical maximum of 0.74. Increasing the rate of the crystallization helps speed up manufacturing processes in a wide range of powder technology applications, but the level of crystallization reduces above an optimal vibration frequency. Here, we show that the accepted view that high-frequency vibration reduces crystallization due to overexcitation, akin to formation of an amorphous liquid at high temperatures is erroneous. In fact, rather counterintuitively, the opposite is true with the granular assembly underexcited at high frequencies. An understanding of the mechanisms of underexcitation has enabled us to devise schemes to allow crystallization at high frequencies.

Abstract

Crystallization of dry particle assemblies via imposed vibrations is a scalable route to assemble micro/macro crystals. It is well understood that there exists an optimal frequency to maximize crystallization with broad acceptance that this optimal frequency emerges because high-frequency vibration results in overexcitation of the assembly. Using measurements that include interrupted X-ray computed tomography and high-speed photography combined with discrete-element simulations we show that, rather counterintuitively, high-frequency vibration underexcites the assembly. The large accelerations imposed by high-frequency vibrations create a fluidized boundary layer that prevents momentum transfer into the bulk of the granular assembly. This results in particle underexcitation which inhibits the rearrangements required for crystallization. This clear understanding of the mechanisms has allowed the development of a simple concept to inhibit fluidization which thereby allows crystallization under high-frequency vibrations.

Self-assembly is a powerful tool for the fabrication of molecular materials and nanomaterials. For example, in the nucleation of crystals, atoms can self-assemble as a result of thermal vibration. Similarly, the self-assembly of block copolymers is widely employed in the manufacture of nanomaterials (1). However, when considering macroscale granular particles, the kinetic energy required to facilitate particle movement far exceeds that which can be provided by thermal energy, preventing self-assembly. Berg et al. (2) were the first to show that a randomly packed assembly of hard macrosized spheres can be crystallized into an ordered crystal packing under imposed vibration. Since then, there has been a substantial amount of literature (3–11) on understanding the transition of a dry assembly of spheres from the dense random packing (DRP), with a packing fraction of $\sim 0.64$ to ordered FCC and HCP packings, with the packing fractions approaching the theoretical limit of $0.74$ for monodisperse spheres. The primary interest in this topic has been driven by applications related to powder technology, particularly in the context of enhancing packing fractions and reducing voids. More recently, this self-assembly route has also been recognized as a promising scalable approach to fabricate architected cellular materials with applications spanning structural materials (12) to battery electrodes (13, 14).

Given the technological significance and scientific interest of the problem, there exists a vast literature (3–11) on the conditions required to optimize granular crystallization. The majority of studies (3, 4, 7, 9, 10, 15) consider one-dimensional (1D) vibration at a frequency $f$ where the level of crystallization has been shown to scale with the vibration intensity $\mathrm{\Gamma }\equiv a{\omega }^2/g$ . Here, $\omega \equiv 2\pi f$ is the imposed angular frequency of vibration with amplitude $a$ and $g$ the acceleration due to gravity. Nevertheless, the understanding of the physical mechanisms of the non-Brownian (10) crystallization of the granular assemblies, where interparticle interactions are inherently dissipative, remains largely unclear. In a recent study, Grega et al. (8) developed a mechanistic understanding of the mechanisms of crystallization of dry monodisperse spheres for low to intermediate values of $\mathrm{\Gamma }$ . Specifically, they demonstrated that for $a{\omega }^2<g$ , the vibration-induced accelerations were insufficient to overcome the acceleration due to gravity, and hence the spheres could not ride over each other to rearrange and crystallize. With increasing $\mathrm{\Gamma }$ , the level of crystallization increased, but their study was limited to a regime where the level of crystallization appeared to plateau out with increasing $\mathrm{\Gamma }$.

Increasing the rate of crystallization is of practical interest to speed up manufacturing processes aiming to crystallize granular assemblies or enhance packing densities. High-frequency vibration studies using both discrete element modeling (DEM) and experiments have shown that while crystallization does indeed scale with $\mathrm{\Gamma }$ , there exists an optimum value of $\mathrm{\Gamma }$ above which the level of crystallization decreases with increasing $\mathrm{\Gamma }$ . There exists no experimental or numerical evidence that provides a mechanistic understanding of the loss of crystallinity at higher vibration intensities. The literature (4, 7, 15) ubiquitously attributes the existence of the optimum $\mathrm{\Gamma }$ to “overexcitation” that inhibits self-assembly. However, overexcitation is not clearly defined in the literature with the notion presumably being similar to thermal excitation that occurs at high temperatures and results in amorphous liquids phases rather than the formation of crystal structures—it is unclear whether this analogy is applicable to non-Brownian granular systems.

The aim of this investigation is to understand the inhibition of crystallization at high vibration intensities in the non-Brownian context. It will be shown, rather counterintuitively and contrary to inferences in the literature, that crystallization at high vibration intensities is inhibited not by overexcitation, but rather is a consequence of underexcitation of the assembly. This finding is made via a combination of experiments that include interrupted X-ray computed tomography (XCT) observations of the movement of particles within the assembly, high-speed photographic tracking of surface particles and discrete element simulations of the experiments over a wide range of values of $\mathrm{\Gamma }$ . We show that in a non-Brownian granular system vibrated at high vibration intensities, fluidization of the particles in a boundary layer inhibits energy/momentum transfer from the vibrating container into the bulk of the assembly, and this results in underexcitation of the granular assembly. A mechanistic understanding of the underexcitation process has then enabled us to design a setup that prevents fluidization and induces crystallization at high $\mathrm{\Gamma }$.

Results and Discussion

Vibration Intensity Sets the Level of Crystallization.

We first set out to measure and observe the crystallization of the granular assembly over a variety of vibration conditions. Approximately one hundred thousand spherical cellulose acetate balls of nominal diameter $d=2 \text{mm}$ were randomly assembled within a square-based Perspex box (Fig. 1A), and the box was subjected to one-dimensional (1D) horizontal vibration along the $x-$direction (Materials and Methods and Fig. 1A). Experiments were performed with a vibration amplitude $a=1.5 \text{mm}$ and $2 \text{mm}$ over a frequency range $8 \text{Hz}\le ƒ\le 100 \text{Hz}$ . The vibration was periodically interrupted to visualize the assembly and identify particle coordinates via XCT. An example of the reconstructed three-dimensional (3D) structure of the balls prior to vibration and after $N=3,000$ cycles (Fig. 1B and Movie S1) shows that such a vibration induced a polycrystalline assembly interspersed with amorphous regions (8). Numerous metrics can be evaluated from this XCT data, but the primary metric of interest here is the ratio $\phi$ of the number of balls within a crystal structure to the total number of balls in the assembly (16). This ratio evolves with the number of cycles and following (8), we have confirmed that it attains a steady-state value ${\phi }_{\mathit{ss}}$ for $N\ge 3,000$ cycles for all values of $a$ and $f$ investigated here. Measurements of ${\phi }_{\mathit{ss}}$ as a function of the vibration intensity $\mathrm{\Gamma }\equiv a{\omega }^2/g$ , are included in Fig. 1C. Two critical observations emerge: i) Over the entire range of frequencies and vibration amplitudes investigated here, the level of crystallization is governed by $\mathrm{\Gamma }$ in line with observations reported in ref. 8 for low values of $\mathrm{\Gamma }$ . This suggests that the ratio of the vibration acceleration $a{\omega }^2$ to the acceleration due to gravity $g$ defines the physics governing crystallization. ii) There exists an optimal $\mathrm{\Gamma }$ that maximizes ${\phi }_{\mathit{ss}}$.

Fig. 1.

(A) Schematic of the setup used to vibrate the assembly of monodisperse balls. The global coordinate system aligned with the box edges is also indicated. Gravity acts in the negative $z-$ direction, and the box is vibrated along the $x-$ direction. (B) X-ray Computed Tomography (XCT) reconstruction of the initial random assembly of the balls. The dimensions of the box and the initial random assembly are indicated on the image. The crystallized assembly after vibrating for $N=3,000$ cycles with an amplitude $a=2.0\text{mm}$ and frequency $f=20\text{Hz}$ is also included. The crystallized assembly is colored to indicate the crystal structures and thereby also the polycrystalline nature of the crystallized assembly. (C) Measurements and discrete-element predictions of the steady-state crystallinity fraction ${\phi }_{\mathit{ss}}$ as a function of the nondimensional vibration intensity $\mathrm{\Gamma }\equiv a{\omega }^2/g$ for two amplitudes $a$ . (D) Sketch illustrating the mechanisms for balls to rearrange within the assembly resulting in crystallization of the assembly. The balls need to ride-up on each other with balls in the lower layers displacing balls that lie in layers further-up in the assembly.

The crystallization response in Fig. 1C can approximately be divided into three regions: I) At low $\mathrm{\Gamma }$ , there is essentially no crystallization. For crystallization to occur, the balls need to rearrange, which involves them riding over each other as illustrated in Fig. 1D. Thus, the balls need to overcome the acceleration due to gravity. The imposed vibration acceleration $a{\omega }^2$ is insufficient at low $\mathrm{\Gamma }$ to overcome the acceleration due to gravity. Therefore, there is little or no crystallization in this regime. II) Increasing $\mathrm{\Gamma }$ results in an increase in ${\phi }_{\mathit{ss}}$ . This regime of intermediate frequencies has been extensively investigated in ref. 8 where they showed that crystallization commences at the bottom of the container and progresses upward through the box with increasing number of cycles. Levels of vibration acceleration greater than $g$ are required to enhance crystallization as balls near the bottom need to lift up balls lying on top of them as shown in Fig. 1D. At an optimum vibration intensity $\mathrm{\Gamma }={\mathrm{\Gamma }}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}\approx 3.2$ , the level of crystallization attains a maximum of ${\phi }_{\mathit{ss}}={\phi }_{\mathit{ss}}^{\mathrm{m}\mathrm{a}\mathrm{x}}\approx 0.7$ . III) For $\mathrm{\Gamma }>{\mathrm{\Gamma }}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$ , the level of crystallization decreases with increasing $\mathrm{\Gamma }$ with very low levels of crystallization occurring at $\mathrm{\Gamma }\sim 60$ , which is the maximum vibration intensity investigated here. It is at these high values of $\mathrm{\Gamma }$ that the literature (3–6, 15, 17) has attributed the reduction in crystallization to “overexcitation.” Qualitatively, the argument used to invoke the concept of overexcitation is as follows. At high vibrational frequencies, high levels of vibrational energy are input into the assembly. The high energy excites the balls out of low-energy states with the consequence that the assembly does not crystallize. However, there is neither experimental nor numerical evidence in the literature to confirm or refute this hypothesis.

High-Frequency Vibration Results in Underexcitation.

To investigate the hypothesis that overexcitation at high frequencies causes low levels of crystallization, we first resort to modeling to get a more detailed mechanistic understanding. We use the soft contact discrete element approach (Fig. 2A and Materials and Methods) to model the vibration response of the assembly. This modeling framework has been extensively employed for modeling both the deformation of granular media (18, 19) as well as for the vibration response of assemblies of balls (8). The simulations were set up to replicate the experiments as closely as possible (Materials and Methods). To understand/visualize the level of excitation, we include predictions of the spatial distribution of the magnitudes $|v|$ of the velocities of the balls on a $z-x$ plane located at $y=0$ . This surface is indicated in Fig. 2B with the velocity predictions included in Fig. 2C. These predictions show an average particle velocity magnitude $〈\left|v\right|〉$ over 30 cycles after the level of crystallization $\phi$ has attained a steady state (these results are independent of the number of cycles over which the velocities are averaged for $\ge 10$ cycles since the assembly has attained a steady state; see SI Appendix, Fig. S1). Further we restrict the predictions to only the height of the original stationary assembly of the balls for vibration at 90 Hz as tracking the ball velocities above this height in the measurements discussed subsequently is prone to errors. Two cases are included in Fig. 2C for a vibration amplitude $ɑ=2 \text{mm}$ : i) a frequency corresponding to approximately the maximum level of crystallization in Fig. 1C, viz. $f=20 \text{Hz}$ and ii) a high frequency of $f=90 \text{Hz}$ where crystallization is very low. Recalling that the vibration direction is aligned with the $x-$ direction, we clearly see that for $f=20 \text{Hz}$ a high-velocity boundary layer develops at walls that transmit the vibration velocity $a\omega$ into the granular assembly. This high velocity is then transmitted into the assembly although the transmission is more successful into the bottom layers of the assembly. Nevertheless, there is sufficient velocity that is transmitted to allow for the balls to rearrange and the assembly to crystallize over the entire height as seen in Fig. 1B.

Fig. 2.

(A) Schematic of the soft-particle contact model used in the discrete-element calculations. Contact between each particle is characterized by normal and tangential springs and dashpots along with friction acting in the tangential direction. (B) Sketch indicating the surface of the transparent Perspex box that is tracked via high-speed photography. (C) High-speed photography measurements and corresponding predictions of the average magnitude $〈\left|v\right|〉$ of the velocities of balls over 30 cycles on the $z-x$ plane located at the $y=0$ surface of the box. These velocities are averaged over 30 cycles after steady-state crystallization has been attained for vibration at an amplitude $ɑ=2 \text{mm}$ and two frequencies ƒ = 20 Hz and 90 Hz. The predictions and measurements are restricted to the height of the original stationary assembly for 90 Hz. (D) Predictions of the effective strain rate ${\dot{\epsilon }}_e$ within the 3D granular assembly. The predictions are shown on the on the $z-x$ plane located at $y=L/2=6\mathrm{c}\mathrm{m}$ for vibration with an amplitude $ɑ=2 \text{mm}$ at ƒ = 20 Hz and 90 Hz. The temporal evolution of the strain rate is shown by plotting the spatial distributions of ${\dot{\epsilon }}_e$ averaged over five cycles at selected stages prior to the assembly attaining steady-state crystallization. Alongside, we also show the discrete particle structure with the particles colored to indicate the crystal structure they are deemed to lie within.

The situation is very different in the $f=90 \text{Hz}$ case. Even though the vibration velocity $a\omega$ is 4.5 times higher, $〈\left|v\right|〉$ is lower over the whole $z-x$ surface. This seems contrary to the hypothesis that high-frequency vibration causes overexcitation, and thus, it is worth checking the fidelity of the predictions. We do this here using two metrics: i) Predictions of the steady-state crystallization fraction ${\phi }_{\mathit{ss}}$ as a function of $\mathrm{\Gamma }$ are included in Fig. 1C and show remarkable agreement with measurements (additional predictions discussing the effect of the number of balls or the height of the assembly are discussed in SI Appendix); ii) the velocities of the balls on the $y=0$ box surface can also be directly visualized via high-speed photography (Fig. 2B) and the spatial distribution of $〈\left|v\right|〉$ measured; see Materials and Methods for details of the algorithm used to track the balls. These measurements are included in Fig. 2C, and within the anticipated statistical variation they are in excellent agreement with the predictions (the minor discrepancies are down to the inherent statistical variations that are inherent in a multibody interaction problem; see SI Appendix, Fig. S2). Thus, both the measurements and predictions suggest the counterintuitive fact that the velocities of the balls are lower at higher imposed vibrational intensities (or frequencies).

The fidelity of the predictions allows us to use the numerical model to further investigate the differences between the crystallization mechanisms at low and high frequencies. Crystallization requires the rearrangement of the balls, i.e., relative motion between the balls. Velocity is not a direct indicator of this relative motion, and so we define an effective strain rate within the assembly based on the relative positions and velocities of the ball centers. This is accomplished by first calculating the spatial gradients of the velocities of the centers of the balls and then defining the strain rate tensor ${\dot{\epsilon }}_{\mathit{ij}}$ as the symmetric component of the velocity gradient tensor; see Materials and Methods and SI Appendix for further details. For visualization purposes, we define an effective strain rate ${\dot{\epsilon }}_e\equiv {\left({\sum }_I{\dot{\epsilon }}_I^2\right)}^{\frac{1}{2}}$ , where ${\dot{\epsilon }}_I$ is a principal strain rate and the sum is over the three principal components of ${\dot{\epsilon }}_{\mathit{ij}}$ . Predictions of the spatial distribution of ${\dot{\epsilon }}_e$ on the $z-x$ mid-plane located at $y=\mathrm{L}/2=6 \text{cm}$ are included in Fig. 2D for the $f=20 \text{Hz}$ and 90 Hz cases at different stages during the crystallization process. In these plots we show ${\dot{\epsilon }}_e$ averaged over five cycles to smoothen-out local fluctuations. Alongside, we also show the discrete particle structure on the $y=L/2$ plane with the balls colored to show the local crystal structure they are deemed to lie in using the polyhedral template matching (PTM) (20) algorithm (see Movie S2 for a clear visualization of the changes in strain rate during vibration and the corresponding effect on crystallization). For vibration at $f=20 \text{Hz}$ crystallization commences at the bottom of the box and progresses upward. This is consistent with the spatial distributions of ${\dot{\epsilon }}_e$ which show bands of high strain rate approximately at locations where the assembly is currently crystalizing. On the other hand, for vibration at $f=90 \text{Hz}$ , the central portion of the box containing the bulk of the balls has a negligible strain rate. This confirms that the low velocities transmitted into the bulk of the box during high-frequency vibration indeed translate to low strain rates implying negligible crystallization. It thus seems clear that high-frequency vibration results in underexcitation rather than overexcitation of the assembly, and it is this underexcitation that translates to low levels of crystallization. While this is clearly contrary to the hypotheses in the literature, the mechanism for underexcitation of assemblies subjected to high-frequency vibration is not immediately evident. We emphasize that the key prediction that during high-frequency vibration the relative motion between particles (i.e., strain rate) is primarily restricted to a boundary layer is independent of the choice of the deformation rate measure. This is illustrated in SI Appendix, Fig. S3 where we include three additional choices of velocity gradient measures with no change in this major conclusion.

Fluidization of Assembly Prevents Momentum Transfer.

The walls perpendicular to the vibration direction push on the particles in the assembly and are thus primarily responsible for transmitting momentum into the assembly. The underexcitation at high frequency is clearly related to this momentum transfer mechanism. A feature consistent between the low- and high-frequency vibrations is that a boundary layer of high velocity and strain rate develops at the walls perpendicular to the vibration direction (Fig. 2 C and D). However, these results in Fig. 2 are not directly useful to help deduce the mechanism of underexcitation.

To get clues into the differences between the mechanisms of momentum transfer in the $f=20 \text{Hz}$ and 90 Hz cases, we need to visualize the motion of the balls within the assembly over the duration of the imposed vibrations. To enable this, we doped 200 balls with zinc iodide (Materials and Methods) and randomly placed them near one of the walls perpendicular to the vibration direction (Fig. 3A and Movie S3). These balls then act as tracer balls during XCT as their X-ray signature differs significantly from the undoped balls (SI Appendix, Fig. S4). The vibration experiment was then conducted as before, but a tomographic scan was taken prior to vibration ( $N=0$ ) and then after at $N=1,000$ cycles to visualize the motion of the tracers over the 1,000 cycles. These measurements for the 20 Hz and 90 Hz cases shown in Fig. 3 B and C reveal several key insights into the mechanisms of momentum transfer. For vibration at $f=20 \text{Hz}$ , the balls that were initially adjacent to the wall (at $N=0$ ) “diffuse” into the bulk of the assembly after 1,000 cycles. Something rather different is seen to occur in the $f=90 \text{Hz}$ case. The balls near the wall do not diffuse into the bulk but rather seem to be ejected along the wall they were placed near and then propagate over the top of the assembly (see Movie S4 for the temporal evolution of the tracer ball positions). Thus, the momentum imparted into the assembly by vibration is not transmitted into the bulk of the assembly, but rather is redirected into a thin boundary layer at the sides and then over the top of the assembly. This results in underexcitation of the bulk of the assembly. We also employed simulations to confirm the validity of these experimental findings. The corresponding simulation results (with 200 balls randomly labeled as the tracers) included side-by-side to the observations indeed confirm the mechanism described above. This reinforces the fact that the tracer ball observations are not an outcome of artifacts created by the inclusion of doped balls into the experimental assembly.

Fig. 3.

(A) Visualization of the placement of the tracer balls near a wall of the box that lies perpendicular to the direction of vibration. (B) XCT reconstructions of the granular assembly prior to vibration $(N=0)$ and after $N=1,000$ cycles at an amplitude $a=2 \text{mm}$ and frequencies ƒ = 20 Hz and 90 Hz. Only the tracer balls are shown for the sake of clarity. The corresponding discrete element predictions are also included with only the 200 balls deemed to be the tracers shown. (C) XCT reconstructions of the assembly shown in (B) after $N=1,000$ cycles but now with the nontracer balls shown translucent to give a visualization of the tracers within the assembly.

The tracer balls give a full 3D picture of the mechanism of momentum redirection for high-frequency vibration. However, to complete the picture an understanding of the dynamics of this process is important. High-speed photographs of the assembly surface over a single-vibration cycle (after steady state) for vibration at $f=20 \text{Hz}$ and 90 Hz and amplitude ɑ = 2 mm are included in Fig. 4A (especially in the 90 Hz case, we see a rarefied “gaseous” layer of balls above the assembly making the tracking of the ball velocities error prone above the height of the stationary assembly as mentioned in the context of Fig. 2C). The contrast between the $f=20 \text{Hz}$ and 90 Hz cases is clear. For vibration at $f=20 \text{Hz}$ the balls maintain contact with the wall perpendicular to the vibration direction and this of course means that momentum is transmitted into the assembly from the walls. By contrast at $f=90 \text{Hz}$ , the large acceleration $a{\omega }^2$ of the walls results in loss of contact between the balls and the walls as the wall pushes on the assembly of balls (see Movies S5 and S6). After the loss of contact, no further momentum can be transmitted into the assembly from the vibrating box, and thus the total momentum transmitted into the assembly is lower than what would be initially anticipated.

Fig. 4.

(A) A montage of high-speed photographs showing the balls on the $z-x$ plane located on the $y=0$ surface of the box. The photographs show images over one vibration cycle (after steady state) of the box with an amplitude $a=2 \text{mm}$ at ƒ = 20 Hz and 90 Hz. Each image is labeled by a nondimensional time $\overline{t}=tf$, where the time $t=0$ at the start of the cycle. For reference, we show the height of the stationary assembly by a black solid line and the height of the assembly during steady-state vibration by a solid red line. The increase in height $\mathrm{\Delta }H$ of the assembly is indicated in the 90 Hz case. (B) Sketch illustrating the mechanism of the formation of the gap between the walls of the box due to the fluidization of the balls for vibration at ƒ = 90 Hz. We illustrate both the transient state (prior to the assembly attaining a steady state) and the associated flow of the balls over the top surface of the assembly resulting in the increase in height $\mathrm{\Delta }H$ as well as the steady state where the gap alternate sides. The markings (I) and (II) correlate with the 90 Hz images in (A).

So, given that we have a dense random packing of the balls that to a good approximation does not change volume, we ask how is the gap between the balls and the wall accommodated? This can be understood from the high-speed images (Fig. 4A) where we have included a horizontal line to indicate the height of the stationary assembly ( $H_{N=0}=4.5 \text{cm}$ ). The height of the assembly subjected to $f=90 \text{Hz}$ vibration increases by $\Delta \approx 10 \text{mm}$ . Taken together with the observations in Fig. 3B that balls adjacent to the wall traverse over the top surface of the assembly, we can now see the mechanism for the formation of the gap that we illustrate via sketches in Fig. 4B. In the transient state (i.e., before steady state has been attained), the high imposed accelerations “fluidize” balls in a boundary layer adjacent to the wall. This fluidized layer then flows over the top of the box resulting in the increase in height $\mathrm{\Delta }H$ of the assembly observed in steady state and forms the gap between both walls perpendicular to the vibration direction and the assembly. Subsequently, when steady state is attained the vibration of the box simply results in the gap alternating sides as shown in Fig. 4B with the central assembly of balls remaining essentially stationary. By contrast no such height increase is seen at $f=20 \text{Hz}$ with no gap formation that enables momentum transfer from the box into the assembly. (We have included two horizontal solid lines, a black line to indicate the initial height of the stationary assembly and a red line indicating the steady state height in both 20 Hz and 90 Hz high-speed images in Fig. 4A.)

Fluidization of granular assemblies has been extensively investigated (21–24) in the context of granular flows but not so in the context of vibrating assemblies. Our experimental results suggest that the key nondimensional group governing the crystallization (and therefore also the fluidization response) is $\mathrm{\Gamma }\equiv a{\omega }^2/g$ (Fig. 1C). On the other hand, fluidization in granular flows is known to be set by the nondimensional ratio of the inertial stresses to the confining stresses. Specifically, the nondimensional parameter $I\equiv \dot{\gamma }d/\sqrt{P/\rho }$ (where $\dot{\gamma }$ is the shear strain rate, $P$ the pressure, $\rho$ the density of the parent material of the granular media, and $d$ the diameter of the granular particles) is known to govern the onset of fluidization (21). So how can we rationalize that $\mathrm{\Gamma }$ governs crystallization even when we know fluidization is occurring? Some interpretation allows us to see that $I$ can be reexpressed in terms of $\mathrm{\Gamma }$ in the context of granular vibration. The shear strain rate in the boundary layer is expected to scale as $\dot{\gamma }\propto a\omega /d$ , while the increase in pressure within the assembly is due to the increase in the height $\mathrm{\Delta }H$ of the assembly as shown in Fig. 4A. The gap that is formed scales with the vibration amplitude $a$ as material of that extent needs to be moved for the box to translate by the vibration amplitude $a$ . Then, volume conservation dictates $\mathrm{\Delta }H\sim aH/L$ where $H$ is the initial height of the assembly. Thus, for a given $H/L$ , $P\sim \rho g\mathrm{\Delta }H\propto \rho g\mathrm{a}$ . Substituting these expressions for $\dot{\gamma }$ and $P$ in the definition of $I$ it follows that $I\propto \sqrt{\mathrm{\Gamma }}$ . Thus, it is unsurprising that $\mathrm{\Gamma }$ governs the level of crystallization of the assembly as it also includes the condition when fluidization sets in. As a corollary it is worth mentioning that since the boundary layer is expected to be on the order $a$ , the conclusion that fluidization will prevent crystallization at high-vibration intensities is only expected to hold in the practical case when the in-plane box dimension $L\gg a$.

A Volumetric Constraint Prevents Fluidization and Inhibits the Underexcitation.

We have shown that the low level of crystallization at high frequencies is a result of underexcitation of the assembly. This in turn is due to the fluidization of the assembly near the walls perpendicular to the direction of vibration and is associated with an increase in the height of the assembly. As a corollary, preventing the height increase will inhibit fluidization, which in turn should permit crystallization to occur for high-frequency vibration. To investigate this, we covered the stationary assembly with a foam lid (Fig. 5A) that just touched the top surface of the assembly. The foam lid was a tight fit in the box so that an increase in the height of the assembly was greatly inhibited. In simulations, a fixed surface was used (Materials and Methods) but in both cases balls had enough freedom to rearrange under vibrational excitation.

Fig. 5.

(A) Sketch illustrating the foam lid used to prevent the height increase of the granular assembly. (B) Bar-chart showing a comparison between the measured steady state crystallization level ${\phi }_{\mathit{ss}}$ for vibration at an amplitude $ɑ=2 \text{mm}$ and frequency ƒ = 90 Hz with and without the foam lid. The corresponding discrete element predictions are also included as dashed lines. For comparison purposes, the maximum value of ${\phi }_{\mathit{ss}}$ from Fig. 1C (which corresponded to ƒ = 20 Hz at $a=2 \text{mm}$ ) is also included. (C) High-speed photography measurements of the magnitude $〈\left|v\right|〉$ of the average velocities of balls on the $z-x$ plane located on the $y=0$ surface of the box. These velocities are averaged over 30 cycles after steady-state crystallization has been attained for vibration at an amplitude $\text{a}=2 \text{mm}$ and ƒ = 90 Hz for the assembly capped by the foam lid. (D) Corresponding XCT measurements to show the movement of 200 tracer balls from the wall of the box into the bulk after subjecting the assembly to 1,000 vibration cycles.

Measurements for this assembly constrained with the foam lid are included in Fig. 5B for vibration at $f=90 \text{Hz}$ and an amplitude ɑ = 2 mm. The steady-state crystallization level ${\phi }_{\mathit{ss}}$ was now substantially higher compared to when no lid was present and in fact it is approximately equal to the maximum level of crystallization seen in Fig. 1C. These data for ${\phi }_{\mathit{ss}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ (corresponding to ƒ ≈ 20 Hz with ɑ = 2 mm) from Fig. 1C are also included in Fig. 5B. Thus, consistent with the physical picture, crystallization can occur at high frequencies because the lid prevents fluidization and allows momentum transfer into the assembly. To further confirm that indeed momentum is transmitted into the bulk of the assembly for high-frequency vibration when a lid is present, we repeated the measurements in Fig. 2 for the particle velocities on the surface of the assembly parallel to the vibration direction and the tracer particle experiments of Fig. 3. These observations are included in Fig. 5 C and D, respectively. The contrast with equivalent 90 Hz measurements in Figs.2B and 3B is clear. Now even with vibration at 90 Hz, the balls throughout the box acquire a high velocity (Fig. 5C). More importantly, the tracer balls placed near the walls of the box diffuse into the bulk (Fig. 5D) much like the 20 Hz case in Fig. 3B (Movie S7). This change is mechanism occurs because fluidization near the walls is prevented so that the gap between the walls and the assembly is minimized (SI Appendix, Fig. S5), which therefore means momentum is efficiently transferred from the walls to the assembly. The simulations also support these findings: the predicted crystallization level (Fig. 5B) and the spatial distributions of particle velocities and diffusion of tracer particles (SI Appendix, Fig. S6) agree well with the measurements confirming our original hypothesis that a lid on the assembly enhances momentum transfer for high-frequency vibration.

Concluding Remarks

The low levels of crystallization that occur for high-frequency vibration of granular assemblies have been ubiquitously attributed in the literature (3–6, 15, 17) to overexcitation. Here, using a combination of measurements (XCT and high-speed photography) and discrete element simulations, we demonstrate that in fact the opposite is true. The low levels of crystallization are an outcome of underexcitation of the assembly caused by the fluidization of the granular matter in contact with the walls perpendicular to the direction of vibration, i.e., the walls that transmit momentum into the assembly. This fluidization results in a reduction in the transmission of the momentum and therefore underexcitation of the bulk of the assembly. An understanding of this mechanism provides insights on methods to prevent fluidization and thereby enhance crystallization at high frequencies. We demonstrate one such strategy which involves constraining the height of the granular assembly using a lid. This points to a strategy that might be used for the scalable manufacture of crystalline granular assemblies in situations where speed of manufacture is critical.

Materials and Methods

Materials and Apparatus.

High-precision spherical cellulose acetate balls (Precision Plastic Ball Co Ltd) with a nominal diameter d = 2 mm were used in all the experiments. These balls had a distribution of diameters with a SD of $20 \mathrm{\mu m}$ (SI Appendix, Fig. S7). The vibration setup (Fig. 1A) comprises a Perspex box, an electromagnetic shaker (Brüel & Kjær model V830-335), and a high-speed camera (Phantom v1610) that was used to track the positions of surface particles and extract surface velocities. The Perspex box was attached to the shaker via a stiff rig (SI Appendix, Fig. S8) to minimize accelerations in the $y$ and $z-$ directions. Two accelerometers were connected to the box to control and monitor the imposed vibrations. The data from the monitoring accelerometer along with the high-speed photography were used to confirm the imposed amplitude and frequency of vibration.

Vibration Protocol.

To avoid electrostatic attraction between the plastic balls and the Perspex box, we took the following precautions: i) the internal surface of the Perspex box was coated with a conductive spray (Licron Crystal ESD by Techspray); ii) a grounding wire was attached to the internal surface of the Perspex box; iii) the balls were coated with antistatic spray (Ambersil). Both the conducting and antistatic coatings eroded over time, and hence these coatings were reapplied after approximately every 50,000 cycles. The 100k balls were first poured into the box and the arrangement of balls was randomized by putting a lid on the box and then turning it topsy-turvy a few times. This resulted in a random packing of the balls inside the Perspex box with approximately a 60% by volume packing of the balls (Fig. 1B). The Perspex box was then secured to the stiff rig attached to the shaker (SI Appendix, Fig. S8). A power amplifier drove the electromagnetic shaker. For vibration frequencies less than $50 \text{Hz}$ , the vibration was initiated with no initial ramp. For higher frequencies, the shaker cannot attain the required frequency and amplitude instantaneously and so we employed a vibration ramp over 1 s where the vibration amplitude was increased linearly from 0.1 mm to the desired amplitude. This avoided an initial jerk to the assembly. The controller was programmed to stop the vibration after a fixed number of cycles and the capacitive decay of the output power from the amplifier then brought the box to rest without a jerk. The box was then carefully transferred for visualization via XCT: an example of the box with the 100k balls prior to vibration is included in Fig. 1B.

Visualization via XCT, Image Analysis and High-Speed Photography.

In this work, we followed the same XCT analysis that was previously established in ref. 8. Briefly, the X-ray scans were acquired using the Nikon XTH 225 ST CT system with X-rays produced from a Tungsten filament. A resolution of $200 \mathrm{\mu m}$ was achieved, and the acquired dataset postprocessed using the commercial software VGSudioMax 3.5. Each ball in the granular assembly was segmented from the surrounding air using a surface determination approach with a merger tolerance of 1%. A general Foam/Powder (FP) analysis was then performed on the balls to find their centers and diameters. The center coordinates and diameter of the balls were exported to the open-source software LAMMPS (25) to determine their crystal arrangements along with the crystallinity fractions using the polyhedral template matching (PTM) (20) algorithm. Finally, the open-source software Ovito was used to visualize the particles in the granular assembly.

A high-speed camera (Phantom v1610) was used to visualize the balls on the $z-x$ plane located at $y=0$ (Fig. 2A). The camera was set to take pictures at a frame rate of 1,800 fps and 8,100 fps for 20 Hz and 90 Hz loadings, respectively, such that we acquired 90 frames per cycle in each case. From these high-speed images we inferred the center coordinates of all tracked balls using the open-source software Blender 3.1. The magnitude $\left|v\right|=\sqrt{{v_z}^2+{v_x}^2}$ of velocities of all tracked balls was calculated where the velocities $v_z$ and $v_x$ in the $z$ and $x-$ directions, respectively, were determined by a finite difference scheme using the positions of the balls in consecutive high-speed image frames. The velocity magnitude $|v|$ was then interpolated onto a spatially fixed mesh with resolution $1 \text{mm} \times 1 \text{mm}$ and averaged over 30 cycles to calculate $〈\left|v\right|〉$.

Tracer Particle Experiments.

Zinc iodide has high X-ray attenuation and we soaked 200 balls in an 18% zinc iodide solution for 72 h. The balls absorb zinc iodide, and the increasing X-ray contrast of the balls as a function of the soaking time is seen in SI Appendix, Fig. S4. These 200 balls after soaking for 72 h were then randomly inserted into the assembly over a distance of 1cm from the side of the Perspex box parallel to $y-z$ plane. An initial XCT scan prior to vibration ( $N=0$ ) enabled visualizing of the initial locations of the tracer balls (Fig. 3), and then scans after $\text{N} = \mathrm{500,} \mathrm{1,000,} \text{and} \mathrm{2,000,}$ cycles were conducted by interrupting the vibration experiments to visualize the motion of the tracer balls (Fig. 3 and SI Appendix, Figs. S6, S9, and S10). The difference in X-ray contrast of the tracer and regular balls allowed us to isolate the tracer balls from the regular balls using the VGStudioMax 3.5 software and create the images in Fig. 3 and SI Appendix, Figs. S6, S9, and S10. While this method does not provide high time-resolution tracking unlike 2D X-ray methods (26), it provides high spatial resolutions with no simulation assumptions.

The Discrete Element Method (DEM) Simulations.

Three-dimensional simulations are performed using monodisperse spherical balls of diameter $d$ and mass $m_p$ . We employed the soft-particle contact model (Fig. 2A), introduced by Cundall and Strack (27), which accounted for both inter-ball interactions as well as the interactions of the balls with the sides of the box. The inter-ball contact law comprises:

• (i) A linear spring with spring constant $K_n$ and a linear dashpot with damping constant ${\gamma }_n$ connected in parallel, governing the contact force-displacement relation in the direction connecting the ball centers.

• (ii) A linear spring of constant $K_s$ and Coulomb friction coefficient $\mu$ connected in series, governing the tangential contact relationship.

The relevant equations, numerical solution procedure, and model parameters are detailed in SI Appendix.

The first step of the simulations was generation of the initial amorphous assembly of $M$ balls in the box. This was done by randomly pouring the particles into the box from height $80d$ above the base. This resulting assembly was amorphous in line with the initial assembly in the experiments. In the second step, the box was subjected to shaking by imposing a displacement in the $x-$direction while displacements in the $y-$ and $z-$directions were zero. In line with experiments, we employed a ramp-up period of 1 s (for frequencies $> 50 \text{Hz}$ ) during which amplitude of the sinusoid motion increased linearly from 0.1 mm to the desired amplitude $a.$ After this ramp-up period, the $x-$displacement was sinusoidal. This simulation protocol adequately captures the experimental shaking protocol. The imposed shaking was periodically stopped, and the balls allowed to settle until they came to rest. The arrangement of the balls was then analyzed using a protocol identical to that used to analyze the XCT images with PTM used to analyze the crystal structure given the centers and radii of all the spherical balls in the box. The strain rates in Fig. 2 were computed by first meshing the assembly using first-order tetrahedrons (4-noded tetrahedrons where the ball centers are the nodes of each tetrahedron) at any given instant where the ball positions and velocities are known from the DEM calculations. Then using the linear shape functions of tetrahedrons and the velocities of the ball centers, the instantaneous velocity gradient and strain rate ${\dot{\epsilon }}_{\mathit{ij}}$ for each tetrahedron were computed; see SI Appendix for detailed definitions. The effective strain rate ${\dot{\epsilon }}_e$ (and other velocity gradient measures discussed in SI Appendix) for the tetrahedron then follows, and the data were used to generate the spatial distributions of ${\dot{\epsilon }}_e$ shown in Fig. 2 and in SI Appendix, Fig. S3.

The DEM simulations with the lid were performed in a similar manner except that a fixed surface was placed touching the top of the initial random assembly of balls. From a numerical perspective, this surface was identical to the sides of the box with the balls able to collide against this surface but not pass through. The lid created a confining pressure akin to that created by the foam lid in experiments, but vibrational excitation allowed the balls to rearrange and crystallize.

Supplementary Material

Appendix 01 (PDF)

Movie S1.

Visualization of the arrangement of balls extracted from experiments for vibration at a = 2 mm and frequencies of f = 20 Hz and 90 Hz prior to vibration (N = 0 cycles) and after steady state crystallization (N = 3000 cycles). High crystallinity is observed at 20 Hz but the assembly remains amorphous after vibration at 90 Hz.

Movie S2.

Predictions of the effective strain rate within the 3D granular assembly for vibration at frequencies f = 20 Hz and 90 Hz with an amplitude of a = 2 mm. The video shows the temporal evolution of the effective strain rate on the Z − x plane located at y = L/2 = 6 cm from a stationary state at N = 0 up to N = 1500 cycles. We also show the corresponding discrete particle structures for both frequencies, with high crystallization observed at 20 Hz due to high strain rate during vibration. On the other hand, 90 Hz vibration results in low strain rates and therefore the assembly remains amorphous.

Movie S3.

XCT visualization of the Perspex box with the 100,000 balls along with the 200 tracer particles placed near one side of the box that lies perpendicular to the direction of vibration.

Movie S4.

Comparison of the movement of the tracer particles extracted from experiments and simulations for vibration at frequencies f = 20 Hz and 90 Hz with an amplitude of a = 2 mm over N = 1000 cycles.

Movie S5.

High-speed video showing the Z − x surface of the experiment for vibration at a frequency f = 20 Hz with an amplitude of a = 2 mm. The video shows the vibration after the assembly has attained a steady state level of crystallization.

Movie S6.

High-speed video showing the Z − x surface of the experiment for vibration at a frequency f = 90 Hz with an amplitude of a = 2 mm. The video shows the vibration after the assembly has attained a steady state level of crystallization. The increase in the height of the granular assembly and the gap formed between the assembly and the box sides perpendicular to the direction of vibration are clearly seen.

Movie S7.

Comparison of the movement of the tracer particles extracted from experiments with and without a lid for vibration at frequency f = 20 Hz with an amplitude of a = 2 mm over N = 1000 cycles. The tracer particles are seen to diffuse into the volume when the lid is present, which indicates that momentum is transferred into the bulk of the assembly in this case.

Data, Materials, and Software Availability

All study data are included in the article and/or supporting information.