Noise and diffusion of a vibrated self-propelled granular particle

TL;DR

This study investigates the motion of vibrated granular particles, demonstrating that despite their unique noise-propulsion relationship, their behavior aligns with the active Brownian particle model, supporting its use in predicting collective phenomena.

Contribution

The paper provides experimental validation of the active Brownian particle model for vibrated granular particles, highlighting its applicability despite the particles' unique noise and propulsion relationship.

Findings

Granular particles exhibit features predicted by the ABP model.

The noise-propulsion relationship in granular media is unique but still fits the ABP framework.

The analysis method can be applied to other systems for model validation.

Abstract

Granular materials are an important physical realization of active matter. In vibration-fluidized granular matter, both diffusion and self-propulsion derive from the same collisional forcing, unlike many other active systems where there is a clean separation between the origin of single-particle mobility and the coupling to noise. Here we present experimental studies of single-particle motion in a vibrated granular monolayer, along with theoretical analysis that compares grain motion at short and long time scales to the assumptions and predictions, respectively, of the active Brownian particle (ABP) model. The results demonstrate that despite the unique relation between noise and propulsion, granular media do show the generic features predicted by the ABP model and indicate that this is a valid framework to predict collective phenomena. Additionally, our scheme of analysis for validating the inputs and outputs of the model can be applied to other granular and non-granular systems.

Figures

Equations

$$\dot{\vec{r}}(t)$$

$$\dot{\theta }(t)$$

Full text

Introduction

Background

Active materials (Ramaswamy, 2010; Marchetti et al., 2013) manifest striking nonequilibrium collective dynamics because they are made up of self-propelled entities. That is, in addition to diffusive motion produced by a noisy environment, their constituent particles also have propulsion along some internal, body-fixed axis. Examples include flocks of creatures on land and in the sea and sky; cells and in vitro cell extracts; and phoretic colloids and emulsions. Some of the most striking experimental demonstrations of nonequilibrium collective effects were first shown in vibrated granular matter. These include spontaneous vortex formation (Blair et al., 2003), coupling of orientability and activity to produce persistent flows and giant number fluctuations (Narayan et al., 2007), boundary migration (Deseigne et al., 2012), and modified crystallization dynamics (Briand and Dauchot, 2016).

Motivation

These fascinating collective effects can theoretically be shown to emerge from microscopic models in which there is an abstracted description of single-particle motion. A theoretical cornerstone is the active Brownian particle (ABP) model (Marchetti et al., 2016), in which a Brownian particle propels itself with a constant speed along a body-fixed axis that rotates diffusively 111A related but distinct model is the run-and-tumble model (Tailleur and Cates, 2008; Cates and Tailleur, 2013).. The two key dynamical ingredients in this model—self-propulsion and diffusion—are independent physical processes that together contribute to the single-particle motion. The assumptions of this popular model are well satisfied by some physical systems, and it has successfully predicted, among other phenomena, phase separation and an active solid phase in self-propelled particles (Fily and Marchetti, 2012; Redner et al., 2013; Stenhammar et al., 2013; Wysocki et al., 2014; Cates and Tailleur, 2015; Redner et al., 2016; Krinninger et al., 2016).

Self-propulsion in granular particles arises from different physical considerations than in other soft and living systems. Here, both the propulsion and the noise share the same non-thermal origin. In a two-dimensional granular fluid, dynamics are driven by a vibrating boundary which energizes particles via collisions in the vertical direction. These collisions act as a source of high-frequency noise that leads to diffusion in the other two dimensions (Yadav and Kudrolli, 2012). Particles with anisotropy in shape (Tsai et al., 2005; Narayan et al., 2006; Daniels et al., 2009) or other properties (Deseigne et al., 2012) will also be preferentially propelled along a body-fixed axis by the collisions. Further, collisions are locked to the deterministic vibrations of the plate. This could lead to temporal or spatial correlations of the noise, as well as a non-thermal noise spectrum (Zik et al., 1992; D’Anna et al., 2003).

Goal

In this article, we seek to quantify single-particle motion of a vibrated granular system and to test the theoretical paradigm in this setting. To do so, we measure the noise-driven short-time motion and check the assumptions of the ABP model. We then compare the mean long-time motion with the predictions of the model. We show that model parameters derived from the short- and long-time measurements are self-consistent. To test the robustness of our results, we have carried out this procedure on various combinations of particle design, containment, and vibration. More generally, we present a template for systematically relating observed short- and long-time motion to each other and to the predictions of the active Brownian particle model, not just in granular matter but in other active particle systems as well.

Method

In the simplest version of the ABP model, the center-of-mass velocity of an object is the sum of two contributions, a constant velocity $v_{0}$ directed along a body-fixed orientation $\hat{n}={\cos\theta\choose\sin\theta}$, and a translational noise $\vec{\eta}$. This is accompanied by rotation of $\hat{n}$ driven by a noise term $\xi$.

[TABLE]

The translational and rotational noise terms are Gaussian with zero mean, no spatial or temporal correlations, and variances $2D_{T}$ and $2D_{R}$, respectively. Thus the short-time dynamics are fully characterized by three parameters, $D_{R}$, $D_{T}$, and $v_{0}$, which determine the statistical behaviour of the particle over longer times. An equivalent parameterization can be given in terms of the two diffusion coefficients and a persistence length $\ell_{p}=v_{0}/D_{R}$.

Experiment

System

Introduce experiment.

Shaking apparatus to limit any non-vertical

motion.

Experiment.— The self-propelled grains in our experiment are confined in a cell with a circular aluminium base (of diameter 203 mm) and an acrylic lid, as shown in figure 1 (Walsh and Menon, 2016). The cell is vibrated vertically by an electromagnetic shaker (LDS456) through a flexible coupling to a square air bearing that constrains horizontal motion (see for example (Harris and Bush, 2015)). The vibration frequency in these experiments is held fixed at $f=50$ Hz, and the vibration amplitude is varied from 10 to $20g$.

Tracking procedure.

Particle motions are captured by video imaging at a frame rate of 120 fps $=2.4f$. Every particle is marked with a central white dot to detect position $\vec{r}$ with a precision of $0.02s$; two smaller dots at the corner define the orientation $\theta$ of the propulsion axis with a precision of 0.03 rad. The positions and orientations are tracked over time to generate individual particle trajectories.

Describe experiment – system size, particle

size.

All particles we use have a square footprint of width $s=6.28\pm 0.04$ mm, a maximum height of $3.9\pm 0.1$ mm, and are made of ABS thermosetting resin. The cell height of 4.76 mm constrains the particles to quasi-two-dimensional motion. Since our goal in this article is to study single-particle motion, we maintain a low area fraction of particles, ($\phi\sim 2\%$) so that collisions between particles are rare. Particles within $1.5s$ of the cell boundary are excluded from the analysis to eliminate any edge effects.

Data

Energy and momentum injection are determined by

particle-wall interaction.

Asymmetry in the particle design causes them to behave as self-propelled polar ‘walkers’ by rectifying the noise provided by the vertical vibration (see figure 1). Dynamical asymmetry is introduced both through the particle geometry as well as by frictional properties (Narayan et al., 2007; Kudrolli, 2010; Deseigne et al., 2012): a short length of nitrile rubber inset along one edge provides enhanced friction, while a bevel on the opposite edge produces asymmetric rocking of the particle following a collision with the floor (sup, ). Tuning the design parameters, along with the confining gap and the vibration amplitude, provides some control over the parameters of motion. In this article, we exploit this flexibility to test the quantitative predictions of the active Brownian particle model for six distinct configurations (sup, ) of walkers. We first discuss the short-time dynamics, followed by the long-time mean behavior, showing data from one of these configurations.

Noise in vibrated particles

Direct noise characterization (short-time

dynamics)

Characterization of noise.— To directly characterize the noise generated by particle-wall collisions, we measure the translational and rotational velocity at a short timescale. Recall that the ABP model assumes that the noise terms are Gaussian-distributed and uncorrelated over space and time. These velocity components are calculated from the numerical derivative of the orientation and position of the particle (sup, ). The translational velocity is resolved into a longitudinal $\dot{r}_{\parallel}$ and a transverse $\dot{r}_{\perp}$ component, parallel and perpendicular to $\hat{n}$, respectively.

Velocity distribution

Histograms for the velocities are shown in figure 2. The distributions all show a single peak, and no long tails. The moments from these distributions characterize the noise parameters: the variances provide values for the diffusion constants, and the mean of the longitudinal velocity is the short-time measurement for $v_{0}$. Higher moments show small deviations from a pure Gaussian distribution, with perhaps the largest quantitative departure being the skewness of the longitudinal velocity component (sup, ).

Time autocorrelation

The autocorrelations of the rotational and translational velocities show only very short time correlations (see figure 2). The longest is that of the rotational autocorrelation, whose decay time $\tau$ is on the order of the vibration period $1/f$. We incorporate this small but finite correlation time into the ABP model (Daniels et al., 2009) as an exponential correlation in the rotational noise $\left\langle\xi(t)\,\xi(t^{\prime})\right\rangle=(D_{R}/\tau)e^{-|t-t^{\prime}|/\tau}$ (sup, ).

Spatial correlations

Spatial noise correlations between particles could lead to more complex collective behaviors not accounted for by minimal models such as the ABP. Despite the highly correlated nature of the noise source over space as well as time, we find no significant spatial correlations between particles. We compute interparticle spatial correlations as a function of particle separation distance, which do not persist beyond steric interactions (sup, ). Thus the nonrandom and highly correlated motion of the plate does not induce correlations between particles.

Mean particle behavior (long-time

correlations)

Long-time diffusion and mobility.— We turn to characterizing the longer-term motions of the particles, via three correlation functions built from the observed $\vec{r}(t)$ and $\hat{n}(t)$. These data are compared to functions obtained from integrating the active Brownian particle model over time and averaging over the noise distribution. Some of these or similar functions have been measured before (Yadav and Kudrolli, 2012; ten Hagen et al., 2011), but here we perform a sequence of fits to each one of the three pertinent ABP model parameters. We show correlation functions and fits in figure 3 for one of our experimental configurations.

We first examine the autocorrelation of the particle orientation $\left\langle\hat{n}(t)\cdot\hat{n}(0)\right\rangle$, as shown in figure 3(a). We find a nearly exponential decay with a small persistence at short time. The ABP model, including the short autocorrelation time $\tau$, predicts $e^{-D_{R}\left[t-\tau\left(1-e^{-t/\tau}\right)\right]}$. The parameter $\tau$ is held fixed in all subsequent fits at a value determined from all six particle configurations; this is a short time-scale, consistent with the autocorrelation in figure 2(c). This leaves the exponential decay rate $D_{R}$ as the only free parameter in fitting each particle configuration.

Second, as shown in figure 3(b), to characterize the self-propelled motion we measure the time-dependent correlation function $\left\langle\vec{r}(t)\cdot\hat{n}(0)\right\rangle$, which is the displacement, after lag time $t$, projected along the initial orientation. As one may anticipate, this initially grows linearly with a constant speed $v_{0}$ for a duration set by the rotational timescale $1/D_{R}$. Eventually the displacement saturates at a characteristic length scale, which defines a persistence length $\ell_{p}$. After this, the particle has lost its initial orientation and diffuses isotropically. The ABP model predicts an antisymmetric function $\ell_{p}e^{\tau D_{R}}\left(1-e^{-D_{R}|t|}\right)\operatorname{sign}(t)$, which grows linearly at short times with slope $v_{0}$ then asymptotically approaches the persistence length $\ell_{p}=\frac{v_{0}}{D_{R}}$. The model describes the data with the exception of a small but systematic asymmetry that may be due to finite system size; we therefore fit to the anti-symmetric part of our data. The fit is a single-parameter fit for the persistence length $\ell_{p}$.

Third, we compute the mean squared displacement $\langle[\vec{r}(t)-\vec{r}(0)]^{2}\rangle$, as shown in figure 3(c). The ABP prediction for mean squared displacement is $2\ell_{p}^{2}e^{\tau D_{R}}\left(D_{R}t-1+e^{-D_{R}t}\right)+4D_{T}t$. There are three distinct stages of motion: (i) at times shorter than $D_{T}/v_{0}^{2}$, translational diffusion dominates and we expect linear growth with constant $D_{T}$; (ii) later the self-propulsion dominates and displacement grows ballistically with speed $v_{0}$ until (iii) the orientation diffuses at the timescale of $1/D_{R}$ and the displacement grows linearly again with effective diffusion constant of $\ell_{p}^{2}e^{\tau D_{R}}D_{R}/2+D_{T}$. The model describes the data well over all of these stages. The earliest timescale, of translational diffusion, in our experiment is pushed to an extremely short time, i.e., $D_{T}/v_{0}^{2}\lesssim 1/f$. Therefore $D_{T}$ is not reliably determined by the fits, and we plot a two-parameter fit with $\ell_{p}$ varied as well. Alongside the total mean squared displacement, we show the longitudinal and transverse body-frame components (sup, ).

Comparing parameters between short- and long-time

measurements

Noise vs. diffusion and mobility.— Heretofore, we have extracted values for $D_{R}$, $\ell_{p}$, and $D_{T}$ via two distinct methods: from short-time noise statistics and from fitting correlation functions from the ABP model to the long-time motion of particles. We have shown this process for one experimental configuration in figures 2 and 3. We summarize the results from six configurations in figure 4 by checking for consistency between the values extracted from the two methods. We find that the values agree well, lying close to the line of slope one in each panel. The agreement is consistent using values extracted from the single-parameter sequence of fits (red symbols), and is also robust to fitting with two free parameters (blue symbols). The assumption of uncorrelated rotational noise yields a poor match between the two values of $D_{R}$ (open symbols). The weakest agreement is for $D_{T}$, which is unsurprising as the early-time diffusive motion barely appears in our experimental time-window.

Conclusion

Our results demonstrate the commonality between vibrated granular media and other active matter. Here too, the long-term motion can be constructed based on independent contributions from noise and mobility, despite their shared physical origin. The noise has only short temporal and spatial correlations, and can be reasonably described by a Gaussian distribution. These observations justify the assumptions of the ABP model. Further, fitting the model to observed dynamics can reliably extract parameters of motion. Our results validate the use of this single-particle description to study collective effects in granular systems. More generally, we suggest that the sequence of data analysis laid out in this article is applicable to other active matter, beyond vibrated granular media.

Acknowledgements

We gratefully acknowledge funding from NSF-DMR 1506750 and NSF-REU 1359191 (LW, SS, NM); and NSF-DMR 1149266, NSF-MRSEC 1420382, and IGERT-DGE 1068620 (CW, AB).