Harnessing confinement and driving to tune active particle dynamics

Active matter is a generic class of nonequilibrium systems wherein the driving and dissipation occur at the level of the constituent individual units. Some of the earliest studies of active matter focused on the hydrodynamic description of collections of self-propelled units to explain coherent phenomenon seen in bird flocks, wildebeest herds, fish schools or insect swarms [1, 2, 3, 4, 5]. The orientational degrees of freedom of the critters are an integral part of their dynamics in these studies, with the interaction between motility and the orientational degrees of freedom leading to complex phases with rich behavior [5]. However, it is not so much the orientational degrees of freedom nor the anisotropic interactions, but rather the broken detailed balance in the dynamics of the individual units that is the crucial aspect of an active system [6]. Minimal models of self-propelled particles that incorporate this essential aspect, in addition to isotropic interactions between individuals, have been of interest in recent years [7]. Synthetic systems, such as self-propelled droplets [8, 9], colloidal systems [10, 11] and granular media [12, 13], provide experimental realisations of such scalar active matter.

Theoretical models of scalar active matter are typically of three broad types: Active Brownian particles (ABP), exemplified by the motion of self-phoretic colloids [10, 11] and driven granular particles [12, 13], move with a constant speed and their motility direction undergoes rotational diffusion. Run-and-tumble-particles (RTP), inspired by the swimming motion of bacteria [14], also move with a constant speed along a given direction for a certain “run-time”, after which they undergo an instantaneous “tumble” that randomises the direction of motion [15]. Active Ornstein-Uhlenbeck particles (AOUP) are driven by a self-propelled noise which has a finite-correlation time with a Gaussian strength [16]. The essential distinguishing feature in these three models is the characteristics of the active noise driving the self-propelled motion. The changes in propulsion velocity are governed by rotational diffusion for ABPs, by a Poisson process for RTPs, and by an exponentially correlated Gaussian noise for AOUPs. In spite of these contrasting active noise contributions, collections of interacting particles of all three kinds show motility-induced phase-separation (MIPS) [17] and scaling behavior in a single-file geometry [18], pointing to underlying universal features.

In this work, we seek to control the noise characteristics of a scalar active particle and thereby its self-propelled motion. Using an experimental system comprised of a driven granular particle, we demonstrate that laterally confining an ABP-like particle in a narrow channel leads to the emergence of an active noise reminiscent of an RTP-like motion, and that the characteristics of this noise can be controlled by our empirical parameters. To understand this dynamics, the effects of the confining channel are modeled using a simple potential for the orientation of the active particle and we then show that this maps the dynamics of the relative internal coordinate to that of a Brownian particle moving in a periodic potential subject to a constant force. Our results are a clear experimental demonstration of tuning the noise characteristics of an active particle by employing lateral confinement, suggesting new ways to control active particle dynamics.

Our experimental system is comprised of circular 3-D printed disks (particles) of diameter $d$, with an asymmetric leg on one side (Fig. 1 and Supplementary Information); when placed on a flat surface (aluminum disk) and driven using an electromagnetic shaker (Supplementary Information), the particles propel in-plane along a fixed body axis. Owing to its design, a slight shape anisotropy is evident in the 2-dimensional projected diameter of the particle along the propulsion direction (Fig. 1). In unconfined two-dimensional space, the particle is an active, motile object executing a self-propelled random walk (Supplementary Movie 1) with continuously and noisily varying position $\mathbf{r}=(x,y)$ and orientation $\varphi$ (Fig. 1, Supplementary Information). In addition to self-propulsion, the particles exhibit chiral motion characterized by an angular speed $\omega$. Altogether, the motion of the active particle in the unconfined 2-dimensional space is well characterized by an ABP model [13] for a chiral active particle (see MSD analysis in the Supplementary Information): the particle position evolves according to $\dot{\mathbf{r}}=v_{0}\;\hat{\mathbf{e}}(\varphi)+\sqrt{2D_{t}}\;\bm{\eta}(t)$ where $\hat{\mathbf{e}}(\varphi)=\cos\varphi\;\hat{\mathbf{x}}+\sin\varphi\;\hat{\mathbf{y}}$ is the local direction of motion, the over-dot indicates a time-derivative, $v_{0}$ is an active speed, $D_{t}$ is the translational diffusion coefficient, and the stochastic term $\bm{\eta}(t)$ is a Gaussian white-noise process with zero-mean and unit-variance. The angular coordinate $\varphi$ of the ABP performs a random walk governed by $\dot{\varphi}=\omega+\sqrt{2D_{r}}\;\zeta(t)$ with $\omega$ the (chiral) angular velocity, $D_{r}$ the rotational diffusion coefficient of the ABP while $\zeta(t)$ is a zero-mean and unit-variance Gaussian white-noise process. In our experimental set-up, we are able to control these various parameters of the ABP model, viz. $\omega$, $D_{r}$, $v_{0}$ and $D_{t}$, using the driving $\Gamma$ of the electro-magnetic shaker (Supplementary Information).

We now turn our attention to the behaviour of these active particles when they are confined to quasi one-dimensional channels – grooves of width $w$ that run along the periphery of the aluminum disk as concentric circles (Fig. 1). The geometric configuration of the channels is such that the particles are constrained radially to within $\leq 1\%$ of the channel radius (Supplementary Information) and can still explore all possible values of $\varphi$. When the confinement $\delta=d/w\approx\rm 0.88$ is sufficiently small (i.e., a wide channel) the orientation co-ordinate $\psi$ samples all directions uniformly ($P(\psi)$ for various confinements $\delta$ are not shown) and since this co-ordinate is coupled to the position co-ordinate $\theta$, the particle executes a one-dimensional persistent random walk along the channel (Fig. 2A, left panel; Supplementary Movie 2). The chirality of particle is evident in the steady drift of the orientation $\psi$.

A striking difference is manifest when $\delta$ is increased to $\approx\rm 0.9$: instead of evolving continuously, the orientation of the particle fluctuates noisily along one of the channel directions i.e. $\psi=\pm\pi/2$, only to stochastically and abruptly switch direction (Fig. 2A middle panel). The discrete orientation jumps $|\Delta\psi|\sim\pi$ occur either “downhill” in a direction dictated by the chirality of the particle or less frequently “uphill” (blue and red arrows, respectively in Fig. 2B). As the confinement $\delta$ is further increased to $\approx 0.92$, the switches in the particle orientation become much less frequent – in the particular run shown (Fig. 2A, right panel), they occur only “downhill”. When $\psi\sim\pm\pi/2$, the particle executes a “run” along the channel for a typical “run-time” duration $\tau$ until it stochastically switches to a “run” in the opposite direction (Fig. 2B; Supplementary Movie 3).

Taken together, the characteristics discussed above are reminiscent of a run-and-tumble particle in one-dimension, with position co-ordinate $\theta$. As such, the relevant equation describing the dynamics of $\theta$ is that of an ABP confined to move on a circle of radius $R$ (Supplementary Information), viz.

where $\nu=v_{0}/R$, $\mathcal{D}=D_{t}/R^{2}$, $\xi(t)$ is a Gaussian white-noise process with zero-mean and unit-variance, and $\sigma(t)=\sin[\psi(t)]$ is the active noise. This model is validited by first computing the empirical two point correlation $C(|t-t^{\prime}|)=\langle\sigma(t)\;\sigma(t^{\prime})\rangle$ of the active noise, assuming $\sigma(t)$ is a stationary stochastic process. The correlation function decays exponentially with a time-constant $\tau_{\theta}\sim 9s$ (Fig. 3A). For $\psi\sim\pm\pi/2$, the active noise $\sigma(t)\sim\pm 1$, and thus $\nu$ would correspond to the active speed of the RTP-like motion for $\theta$. The bounded values $\sigma(t)\in[-1,1]$ and the exponential decay of $C(|t-t^{\prime}|)$ thus justify the effective dynamics of $\theta$ in equation (1) as the position coordinate of an RTP-like particle.

We next compare the empirically measured mean-squared-displacement (MSD) $\langle[\Delta\theta(t)]^{2}\rangle$ with that predicted from equation (1), using the values of $v_{0}$ and $D_{t}$ inferred from the two-dimensional experiments for the ABP model:

An excellent agreement is found between the empirical MSD and that given by equation (2) (Fig. 3B). Of note in the MSD are the crossovers from an initial diffusive regime (governed by passive translational diffusion), to a super-diffusive regime (governed by the self-propelled motion of the particle), and then, eventually, to a diffusive regime for time-scales $t\gg\tau_{\theta}~{}(\sim{\rm 9}s)$ corresponding to the decorrelation of the persistent motion driven by the active noise $\sigma(t)$. The effective diffusion coefficient in this asymptotic regime has an active contribution $\sim\nu^{2}\,\tau_{\theta}$ in addition to the (angular) translational diffusion coefficient $\mathcal{D}$ [19, 20]. This additional contribution to the effective diffusivity could dramatically increase with increasing confinement as the particle becomes more RTP-like. Indeed, the effective diffusion coefficient does increase with confinement, and in fact for the highest confinements, a crossover to the eventual diffusive regime is not seen on the timescale of our experiments (Fig. 3B, inset). These results conclusively suggest that equation (1) is indeed a good descriptor for the dynamics of $\theta$, and also that the parameters describing the motion in confinement remain reasonably similar to the particle motion in two-dimensions (more details in the Supplementary Information).

The empirically measured dwell time distribution of $\psi$ (equivalently, the run time distribution of $\theta$) distribution of $\psi$ is well fit by a bi-exponential function

where $k_{\rm f/b}$ are the forward (“downhill”) and backward (“uphill”) rates of the switching between the states $\psi\sim\pm\pi/2$ (Fig. 3C). It is clear from the above expression that asymptotically, i.e., for $k_{\rm f}\tau\gg 1$ and $k_{\rm f}\gg k_{\rm b}$, the dwell-time distribution $P(\tau)\sim k_{\rm b}\,e^{-k_{\rm b}\tau}$, decaying exponentially with the rate $k_{\rm b}$. For the same driving strength $\Gamma$, the time-scales that govern the decay of the two-point correlation $C(|t-t^{\prime}|)$ and the crossover of the MSD $\langle[\Delta\theta(t)]^{2}\rangle$ from the super-diffusive to the diffusive regime are comparable to the backward (“uphill”) switching rate i.e. $1/k_{\rm b}\sim\tau_{\theta}$.

Several remarks are in order. First, it should be noted that unlike the standard RTP model with an exponentially distributed run time between tumbles, our active particle dynamics demonstrates a non-monotonic distribution $P(\tau)$ of run-times $\tau$. Second, the discrete jumps in the trajectory of $\psi$ (Fig. 2B) are reminiscent of the positions of a molecular motor stepping along a polymeric track [21, 22]. Third, the fit of equation (3) (Fig. 3C) suggests the presence of (at least) two characteristic switching rates in the RTP-like dynamics of $\theta$ driven by the active noise $\sigma(t)$. Incidentally, equation (3) is used to model the dwell time distributions of a two-state molecular motor [22]. All the above points hint at a possible similarity between the dynamics of our active particle in confinement and those of a processive molecular motor. We next show that this is indeed the case.

We conjecture that the effects of the confining channel on the particle can be captured by a periodic force of the form $F_{\rm wall}=\lambda\,\sin 2\psi$ in the equation for the orientation co-ordinate, with $\lambda$ being the strength of the wall-particle interaction. In doing this, we simplify the interactions which are quite complicated indeed – they not only depend on the particle and channel surface roughness and the resultant friction, but possibly also on the amplitude of driving $\Gamma$. However, the step-like trajectories, and the preponderance of the values of $\psi$ close to $\pm\pi/2$, motivate us to test this functional form of $F_{\rm wall}$ to describe the particle motion. Following this intuition, the dynamics of $\psi$ is modelled by the following equation (details in Supplementary Information):

where $D=D_{t}/R^{2}+D_{r}$ with $R$ the (mean) radius of the confining channel, $\overline{\xi}$ is a zero-mean unit-variance Gaussian white noise process, and the prime denotes differentiation with respect the argument of its function. The effective potential

thus governs the dynamics of $\psi$. Notice that $U(\psi)=V(\psi)+\omega\,\psi$ is a periodic function of $\psi$. Thus, the Langevin equation (4) describes the stochastic dynamics of a Brownian particle with a position coordinate $\psi$ moving in a periodic potential $U$, subject to a constant “force” $\omega$ (Fig. 4A). It is straightforward to solve the Fokker-Planck equation for $\mathcal{P}(\psi)$ in the steady-state, corresponding to this motion (Supplementary Information). This theoretical $\mathcal{P}(\psi)$ agrees well with the empirical distribution, relying on a single fitting parameter $\lambda$ (Fig. 4B, Supplementary Information). However, $\lambda$ depends on the driving $\Gamma$ (Supplementary Information) which is altogether not surprising given that it encapsulates the effective strength of the complicated interactions between the active particle and the confining channel.

Having concluded that the effective potential $V(\psi)$ is sufficient to capture the steady-state distribution of the internal coordinate $\psi$, we next show that it can also capture the dynamical properties of $\psi$. The two point correlation function $C(|t-t^{\prime}|)$ computed from simulations of the Langevin equation (4) compares well with empirical measurements (Supplementary Information). Further, the switching rates, $k_{\rm f}$ and $k_{\rm b}$, as inferred from the fits of equation (3) to the run-time distributions obtained from both the experimental and simulation data compare well. These rates and other characteristics of the RTP-like motion can be controlled by the driving amplitude $\Gamma$ (Fig. 4C, Supplementary Information). The transition rates governing the jumps of $\psi$ across successive minima of the potential can be explicitly computed in the Kramers’ approximation [23]. When $\nu\neq 0$ there are in-principle four transition rates, $k_{\rm f/b}$ and $k^{\prime}_{\rm f/b}$, across the potential minima (Fig. 4A); however, in the approximation $\nu\approx 0$ (valid for our experiments, Supplementary Information), only two transition rates $k_{\rm f/b}$ remain. The variation of the ratio $k_{\rm f}/k_{\rm b}$ from the empirical (experimental and simulation) data compares well with the analytical results obtained in the Kramers’ approximation (Fig. 4C inset). Further, in analogy with two-state dynamics of molecular motors, the chirality of our active particle makes the forward and reverse transition rates different, concomitant with the bi-exponential distribution of $P(\tau)$ (Supplementary Information). Taken together, the effective potential used to describe the confining effects of the channel is a good predictor for the dynamical properties of $\psi$.

By confining a self-propelled granular particle to a quasi one-dimensional channel, we have demonstrated emergent active noise properties, qualitatively distinct from those for the two-dimensional unconfined motion of the particle. While the two-dimensional dynamics is well described by an ABP model, RTP-like behavior emerges due to lateral confinement. This dynamics of the translational coordinate $\theta$ is driven by the internal coordinate $\psi$ of the particle, which in turn has similarities to the stepping dynamics of a molecular motor; most notably, this manifests in the non-monotonic nature of the dwell-time distribution of $\psi$ which is identical to that found in two-state processive motors. To complete the picture and capture the dynamics of $\psi$, we reduced the effects of confinement to a simple periodic force, thus mapping it to the dynamics of a Brownian particle moving in a periodic potential subject to a constant external force. Rather surprisingly, the introduction of this periodic force is sufficient to capture all features of the active particle dynamics.

We emphasize that there is no fundamental reason to expect that the approach taken here to describe the active particle dynamics in confinement – particularly, one that works for microscopic systems, in or close to equilibrium [24, 25] – should have successfully described the emergent noisy dynamics of our active particle. Our experimental system is athermal, far from equilibrium, and the various interactions between the granular surfaces are complicated, thus possibly requiring a priori, a dynamical description more complex than that given by equations (4) and (5) for the emergent statistical features of the particle motion. Therefore, the agreement we find between empirical measurements and theoretical analysis, in particular the Kramers’ transition rates is remarkable.

We conclude with a couple of comments. First, the fundamental character of an active particle is the inseperable coupling between its internal (chemical) co-ordinates and its positional degrees of freedom, while being constantly driven out of equilibrium by an energy throughput [6]. The resulting mobility in the parametric space of the internal and positional co-ordinates is non-diagonal implying that a net current along one of these co-ordinates necessarily drives a corresponding current along the other co-ordinate – our experimental trajectories (Fig. 2) are an explicit manisfestation of this picture. Second, the RTP-like discrete dynamics that we describe here is not directly imposed, for example, using explicit external means [26, 27, 28] but is rather an emergent behavior resulting from confinement. This opens up an avenue to exploit the effects of geometric constraints and non-equilibrium driving to control the noise characteristics of active particles.

We acknowledge support from the Department of Atomic Energy, Government of India, under project no. 12-R&D-TFR-5.04-0800 and 12-R&D-TFR-5.10-1100, the Simons Foundation (Grant No. 287975 to S.T.) and the Max Planck Society through a Max-Planck-Partner-Group at NCBS-TIFR (S.T.), the Department of Biotechnology, India, through a Ramalingaswami re-entry fellowship (K.V.K.) and by the Max Planck Society through a Max-Planck-Partner-Group at ICTS-TIFR (K.V.K.). We thank Pawan Nandakishore for help with the initial set-up and experiments. S.T. and K.V.K acknowledge discussions during the KITP 2020 online program on Symmetry, Thermodynamics and Topology in Active Matter.

We start with the equations for the position $\mathbf{r}$ and the orientation $\varphi$ of an active Brownian particle (ABP) in two-dimensions

where the overdot indicates a time-derivative, $v_{0}$ is the active speed, $\hat{\mathbf{e}}(\varphi)=\cos\varphi\,\hat{\mathbf{x}}+\sin\varphi\,\hat{\mathbf{y}}$ is the instantaneous direction of the active force, $D_{t}$ the translational diffusion coefficient, $\omega$ the (chiral) angular velocity, and $D_{r}$ the rotational diffusion coefficient. The stochastic terms $\bm{\eta}(t)$ and $\zeta(t)$ are uncorrelated Gaussian white noises with zero-mean and unit-variance.

From (7), it is straightforward to compute the angular mean-squared-displacement

where $\langle\cdots\rangle$ indicates an average over the realizations of $\bm{\eta}(t)$ and $\zeta(t)$. To compute the mean-squared displacement $\langle\left[\Delta\mathbf{r}(t)\right]^{2}\rangle$ of the position $\mathbf{r}$, we proceed as follows. The formal solution of (6) is

which leads to

where $\mathcal{D}=\sqrt{\omega^{2}+D_{r}^{2}}$ and $\tan\alpha=\omega/D_{r}$. The intermediate steps for evaluating the double-integral

are detailed in the next subsection.

To develop a description for the quasi 1D motion in confinement, we first measured the radial motion of the particle in the channel. From FIG. 10, we see that the motion in the radial direction is constrained to within $<1\%$ of the channel radius for all the channel widths that we have considered. As such, we can neglect the motion in the radial direction compared to that along the channel.

We transform equations (6) and (7) above to plane-polar coordinates $(r,\theta)$ and approximate the motion of our ABP in the confining channel of mean-radius $R$ by assuming that the radial coordinate is a constant at $r=R$. With this approximation, we get an equation for $\theta$ as follows

Since $\bm{\eta}(t)$ is an uncorrelated Gaussian white noise, we rewrite

where $\nu=v_{0}/R$, $\mathcal{D}=D_{t}/R^{2}$, the relative orientation coordinate $\psi\equiv\varphi-\theta$, $\xi(t)$ is a Gaussian white noise of zero-mean and unit-variance, and $\sigma(t)=\sin[\psi(t)]$ is the active noise.

As shown in the main text, laterally confining our ABP in a quasi 1D channel leads to the emergence of a bi-modal peak in the probability distribution of $P(\psi)$ of the relative orientation coordinate. As argued in the main text, a simple way to capture the effect of this lateral confinement is to introduce an additional force $\lambda\sin 2\psi$ for the angular coordinate $\varphi$. This confinement-induced force has a strength $\lambda$ and a periodicity of $\pi$. Thus the equation for $\varphi$ changes to

Using (22) and (23), we get the following Langevin equation for $\psi$:

where $D=D_{t}/R^{2}+D_{r}$, $\overline{\xi}$ is a Gaussian white noise, and the effective potential is

With the agreement between experiments and simulations discussed above, we now compute the transition rates $k_{\rm f}$ and $k_{\rm b}$ from the time-series of $\psi$.

Supplementary Movie 1 (00Unconfined.avi):

Active self-propelled motion of disks in a 2-D arena. The particles have a diameter of 4.5 mm. Here the driving $\Gamma=6.71$. Previous positions are continuosly overlaid to indicate the overall trajectory of the particles.

Supplementary Movie 2 (01ConfinedABP.avi):

ABP-like motion in a confined channel. Here, the particle diameter is 4.5 mm, the channel confinement $\delta=\rm 0.88$ and the driving $\Gamma=6.71$. A red circle is used to highlight the particle. Insets show the time evolution of the position $\theta$ and the relative internal orientation $\psi$.

Supplementary Movie 3 (02ConfinedRTP.avi):

RTP-like motion in a confined channel. Here, the particle diameter is 4.5 mm, the channel confinement $\delta=\rm 0.9$ and the driving $\Gamma=6.71$. A red circle is used to highlight the particle. Insets show the time evolution of the position $\theta$ and the relative internal orientation $\psi$.