Escape kinetics of self-propelled particles from a circular cavity

Figure 2(a) depicts variation of $\tau_{ex}$ with $l_{\theta}$ for different thermal lengths. Here, variations of thermal length amount to variations of damping strength with a fixed thermal velocity. It is apparent from simulation results that for large damping, such that, $l_{th}$ is much smaller than the pore width $\delta\phi R$, $\tau_{ex}$ is insensitive to the $l_{th}$ for the entire range of self-propulsion length as long as $D_{0}<<D_{s}$. This feature can be treated as an identifying mark for the overdamped limit.

Simulation results of highly damped active particles [see Fig. 2(a, b)] show that $\tau_{ex}$ is inversely proportional to the self-propulsion length for $l_{\theta}<<R$. The opposite limit, $l_{\theta}>>R$, witnesses a linear growth of escape time with $l_{\theta}$. In between these two limits, a minimum is observed in $\tau_{ex}$ versus $l_{\theta}$. Thus, it appears that different exit mechanisms are involved in the three regimes of $l_{\theta}$ or rotational dynamics. The distinct exit mechanisms can further be anticipated by looking at particle trajectories inside the cavities (see Fig. 1(c)). For $l_{\theta}>>\{R,\;l_{th}\}$ and $l_{th}<<R$, our simulations results [see Fig. 1(d-e)] show that particles get accumulated and diffuse along 2D cavity circumference keeping $\mathbf{v_{0}}$ aligned to the position vector $\mathbf{r}$. Such type of accumulation of overdamped active particles near boundaries or obstructions is a manifestation of their persistent motions. This feature triggers interesting phenomenologies reported in the earlier studies VolpeP ; jcp1 ; JP1 ; JP2 ; R8 ; R9 ; R10 .

Slow rotational relaxation — To realize the escape mechanism out of a single-opening circular cavity for $l_{\theta}>>R$, we first consider the situation where the particles are initially positioned antipodal to the opening window and self-propulsion is anti-parallel to the pore direction. When the thermal velocity, $v_{th}<v_{0}$, the particle diffuses on the cavity surface and $\mathbf{v_{0}}$ remains radially directed. Thus, $\tau_{ex}$ here is basically time for rotational diffusion of an angle $\pi-\delta\phi/2$ [see Eq.(19) in appendix A],

Let us consider different initial conditions, at $t=0$ particles are placed at the middle of the cavity and $\mathbf{v_{0}}$ orientation has uniform distribution within the range $[0-2\pi]$. Particles exit from the cavity can be considered as a two-step process. First, particles drift to the cavity wall with velocity $\sim\mathbf{v_{0}}$, then rotational diffusion by an appropriate angle takes the particle at the exit window. Further to note, a little fraction ($\delta\phi/2\pi$) of trajectories directly hit the opening window. Based on this escape mechanism, our analytic calculation gives [see Eqs. (20-22) in appendix A],

Predictions based on this equation well corroborate simulation results (shown in Fig. 2). The second term in Eq. (7) accounts for the approximate time to reach the wall starting from the cavity centre. Further, the first term is much larger than the second one as $l_{\theta}>>R$. Thus, $\tau_{ex}$ becomes almost insensitive to the position where the particles are initially placed. Moreover, for a very narrow opening Eq. (7) can be simplified as, $\tau_{ex}\sim\pi^{2}l_{\theta}/3v_{0}$. This result is consistent with previous studies R1 ; R2 on escape dynamics from circular cavities.

Intermediate regime of rotational relaxation — With decreasing $l_{\theta}$ the mean exit time keeps decreasing until it reaches its minimum. The random search of the opening window becomes most effective when $\mathbf{v_{0}}$ changes its direction right after travelling a length equal to the cavity diameter. Thus, the minima in $\tau_{ex}\;vs.\;l_{\theta}$ plots are located at,

The minima positions based on this estimate are marked by vertical arrows in Fig. 2(a, b). Further, minima positions are independent of initial conditions. As anticipated, simulation results show that the exit time at minima $\tau_{ex}^{m}$ is directly proportional to hitting frequency at the wall, thus, $\tau_{ex}^{m}\propto R/v_{0}$. Further, both $\tau_{ex}^{m}$, as well as $(l_{\theta})_{min}$, are insensitive to the width of opening windows. This can be understood based on the following reasoning. Inside the cavity, particles [see the trajectories in Fig. 1(c)] fly from one point to another wall point after a short stay at the cavity surface. During their sojourn on the wall, they diffuse on the cavity surface over a length ($\Delta$) much larger than the pore arch length. Consequently, when particles hit the wall within a distance $\Delta$ from the pore, they most likely exit without flying to another point on the wall. This exit mechanism makes $\tau_{ex}$ insensitive to pore widths as long as $\Delta$ appreciably larger than $\delta\phi R$.

Fast rotational relaxation — When $l_{\theta}<<\delta\phi R$, as well as, $\tau_{\theta}$ is much shorter than any other relevant time scale of the system, it can be assumed that self-propellers exhibit almost uncorrelated diffusive motion inside the cavity. The effective diffusion constant of the active particle, $D_{eff}=D_{0}+v_{0}^{2}/2D_{\theta}$, carries the contribution from both thermal motion as well as the self-propulsion. In this limit, the success rate in random search of the opening of the diffusive particle is directly proportional to its diffusivity. Following holcman-review the exit time can be expressed as,

where, $\chi$ is a constant and its value depends on the initial conditions. For exit from the antipodal point, the cavity center and uniformly distributed initial positions within the cavity, the approximate values of $\chi$ are $2\ln 2$, $\ln 2+1/4$ and $\ln 2+1/8$, respectively holcman-review . As expected, the exit time is insensitive to the orientation of self-propulsion velocity at the starting point. Further to note, for $D_{0}\rightarrow 0$, Eq. (8) collaborates finding in Refs. R1 ; R2 .

Simulation results presented in Fig. 2 well accord with Eq. (8). It appears from both simulation data, and analytic estimations, $\tau_{ex}$ has $\gamma$ dependence through thermal diffusion constant $D_{0}$. When self-propulsion dominates over thermal diffusion, $D_{eff}\approx v_{0}l_{\theta}/2$, the exit time $\tau_{ex}$ becomes independent of the damping strength. Thus, the overdamped limit for fast rotation diffusion limit can be recognized by the inverse relation between $\tau_{ex}$ and $l_{\theta}$. Deviation from this relation in the fast rotational diffusion limit could be considered as inertial impact.

We conclude our discussion on escape kinetics in the diffusive limit with a remark on the regime of $l_{\theta}$, $R\gg l_{\theta}\geq\delta\phi R$. Under this restriction, active particles still exhibit diffusive motion inside the cavity. Nevertheless, after hitting the boundary wall, the particles diffuse on the cavity surface over a length comparable to the pore width. This diffusion on the surface leads to modification of the effective pore width. Thus, deviation from Eq. (8) gets noticeable as soon as self-propulsion length grows larger than the pore width.

We now study the impacts of an additional exit window on the narrow escape dynamics. Figure 1(b) shows that two pore centres are separated by an angle $\phi$. Thus, the chordal distance between pore centers is, $\lambda=2R\sin(\phi/2)$. Figure 3(b) shows the variation of $\tau_{ex}$ as a function of $\phi$ for different persistence lengths of self-propulsion.

As discussed earlier, for a very large self-propulsion length, $l_{\theta}>>R$, the exit process rests on diffusion on the cavity surface as long as the $F_{0}$ dominates over the inherent thermal fluctuations. If the starting point of a particle trajectory is the cavity center, it hits the wall within the time $\sim R/v_{0}$ and the subsequent diffusion on the wall surface takes the particle to the exit window. Further, active particle dynamics on the cavity wall is solely governed by its rotational diffusion. To estimate $\tau_{ex}$ analytically, particle trajectories are divided into the following three categories based on their hitting points on the wall. (i) As little as $\delta\phi/\pi$ fraction of particles directly hit exit windows and they escape without diffusing on the wall. (ii) $(\phi-\delta\phi)/2\pi$ fraction of particles arrive on the wall within the polar angle $\delta\phi/2\;{\rm to}\;\phi-\delta\phi/2$ [see Fig. 1(b)], and they need to diffuse over the angle $0\;{\rm to}\;(\phi-\delta\phi)/2$ to arrive the nearest exit point. (iii) The remaining particles find the opening window by diffusing over the angle, $0\;{\rm to}\;(2\pi-\phi-\delta\phi)/2$, right after arriving at the cavity surface. Based on this exit mechanism, we derive an analytic expression for $\tau_{ex}$ [see Eqs. (23-24) in appendix A],

This equation well corroborates simulation results presented in Fig. 3. However, it loses its validity when the self-propulsion persistence length is reduced to the order of cavity dimension. In this limit, as long as the diffusion length $\Delta$ on the wall is larger than the pore separations, $\tau_{ex}$ remains almost insensitive to $\phi$. Variations of $\tau_{ex}$ with $\phi$ become noticeable when $\lambda$ goes beyond the diffusion length $\Delta$.

On further reduction of $l_{\theta}$, particles dynamics inside the cavity become completely diffusive. Recall that this regime is characterized by the restriction, $l_{\theta}\ll\delta\phi R$. For well separated opening windows, the exit process through one window is not affected by the other adjacent pores. Thus, total exit rate can be expressed as a sum of the exit rate through individual windows. This leads to, $\tau_{ex}=\tilde{\tau}_{ex}/2$. Where, $\tilde{\tau}_{ex}$ is the mean exit time through one opening when the other windows are closed. On the other hand, when the exit points are not well separated, a logarithmic dependence of $\tau_{ex}$ on the chordal distance between pore centers is observed holcman-review

where, $\tilde{\chi}$ is a constant which depends on the pore size as well as initial conditions. For very narrow exit windows and when the particle is placed at the center of the cavity at the starting point, $\tilde{\chi}$ approaches to $\ln[2(2\pi-\delta\phi)/\delta\phi]+O(1)$. We compare estimations based on the Eq. (10) with our simulation results. They fairly corroborate each other.

It is known that the mean escape time of passive Brownian particles from a cavity with well-separated exit pores is inversely proportional to the number of pores holcman-review ; JPCC-1 . However, our simulation results show that for active particles, $\tau_{ex}\sim 1/n^{\alpha}$. The exponent $\alpha$ varies in the range from 1 to 2 depending upon the persistence length of self-propulsion. For $l_{\theta}\gg R$, $\alpha$ assumes its maximum value $2$. In the opposite limit $l_{\theta}\ll R$, $\alpha=1$.

Recalling the exit mechanisms of active particles for very large $l_{\theta}$, in the present case, the particle trajectories can be divided into two categories. Some trajectories directly hit the opening, thus taking approximately $R/v_{0}$ time to exit the cavity. Another type of trajectories require rotational diffusion over the angle $0\;{\rm to}\;\pi/n-\delta\phi$ after hitting the wall, to reach the nearby exit window. Based on the same line of reasoning as Eqs. (9, 20) we deduce the following expression for the mean exit time,

Here, the first term contributes in exit time much larger than the second one as $l_{\theta}>>R$. Further, for a very narrow opening window, simplification of Eq. (11) produces, $\tau_{ex}\sim\pi^{2}l_{\theta}/3v_{0}n^{2}$. Thus, this result supports well fitted power law $\tau_{ex}\sim 1/n^{2}$ in the inset of Fig. 3(a) for large $l_{\theta}$. Moreover, assessments based on Eq. (11) represented with dashed lines in Fig. 3(a) accord with simulations.

Simulation results show that in the intermediate regime of $l_{\theta}$, the exponent $\alpha$ is still larger than 1. However, the position of the minimum in $\tau_{ex}\;vs.\;l_{\theta}$ is insensitive to the number of pore $n$. Moreover, $\tau_{ex}^{m}$ becomes independent of $n$ as soon as the spacing between pores approaches the diffusion length on the wall.

In the very fast rotational dynamics of the active particle, the self-propulsion adds up to uncorrelated thermal fluctuations. Thus, followingholcman-review one can find the expression of mean exit time as,

Simplification of this equation for very narrow and well separated pores yields, $\tau_{ex}\sim\tilde{\tau}_{ex}/n$. As mentioned earlier, $\tilde{\tau}_{ex}$ is the mean exit time when only one opening window is there. Thus, when pores are well separated, the exit through one pore is not affected by the presence of other windows.

Fast rotational relaxation – As discussed earlier, when $l_{\theta}<<R$ and $\tau_{\theta}$ is shorter than any relevant time scale of the system, the effect of self-propulsion amounts to the augmentation of the effective temperature of the system. Thus, self-propellers can be pondered as passive particles diffusing in a viscous medium at the effective temperature,

In the very low damping regimes, when the particle travels a length within the viscous relaxation time is larger than the cavity diameter, the particle moves back and forth inside the cavity [see Fig. 1(c)] and occasionally exits through the narrow opening with a probability $\delta\phi/2\pi$. Furthermore, for Maxwellian distribution of velocity, the particle drifts to the wall with the average velocity, $\sqrt{2k_{B}T_{eff}/m\pi}$. Therefore, an apprehension of collision frequency and their success probability steer the mean exit time as  JPCC-1 :

We note that the validity of this equation is strictly restricted for effective thermal persistence length, $\tau_{\gamma}\sqrt{k_{B}T_{eff}/m}>2R$ in addition to $l_{\theta}<<R$. The appraisals based on the Eq. (14) [shown by solid black lines in Fig. 4(a)] fairly accord with simulation results.

Recall that when escape kinetics is free from the inertial impact, the mean escape time is inversely proportional to the effective diffusion constant. Thus, one can identify the limit until inertia effects in the exit process make their presence felt by observing deviation from the following empirical relation holcman-review ,

Where $\Lambda$ is a constant which depends on the cavity and pore sizes in addition to the initial conditions. For a very narrow pore size, $\delta\phi/2\pi\rightarrow 0$, the expression of $\Lambda$ can be obtained from Eq. (8). For an arbitrary size and design of the opening window, the numerical value of $\Lambda$ can be extracted from simulation data using least square fitting. The critical thermal length $l_{th}^{c}$ represents the onset of an overdamped regime depicted by vertical arrows in Fig. 4(a). As expected, the inception of an overdamped regime moves to smaller values of thermal length with decreasing pore size. It is apparent from the simulation results that the impact of inertia can safely be ignored as long as the effective thermal persistence length is much shorter than the shortest length scale involved in the escape kinetics. This requires the restriction, $\tau_{\gamma}\sqrt{k_{B}T_{eff}/m}<\delta\phi R$ for the problem at hand. Based on this requisition, $l_{th}^{c}$ can be estimated as,

The predicted threshold value is close to the observed one marked by vertical arrows in Fig. 4(a).

Slow rotational relaxation — On the other hand, when self-propulsion length is much larger than the cavity size, $l_{\theta}>>R$, the impact of self-propulsion can no longer be accounted for through the effective temperature of the system. Simulation results show that for the vanishingly small damping (such that $l_{th}>>R$), irrespective of the values of self-propulsion velocity and rotational relaxation time, particles randomly oscillates back and forth inside the cavity until they exit by hitting at the opening window. The thermal translational motion largely oversees the motion inside the cavity and the motion weakly turns on the self-propulsion. The mean exit times in this limit is given by JPCC-1 ,

The theoretical prediction well corroborates simulation results for $l_{th}>>R$. With decreasing thermal length, the persistent self-propulsion motion takes control over particle dynamics inside the cavity. Here, the overdamped limit is identified by $\gamma$ independent mean exit time given by the Eq. (7). Thus, the inertial impact free regimes can easily be perceived from $\tau_{ex}\;vs.\;l_{th}$, presented in Fig. 4(c).

Intermediate regime of rotational relaxation – For a proportionate scale of the cavity size and self-propulsion length, the exit time graces minimum in the overdamped regime as reported in the foregoing section. However, in the very low damping regime, the escape kinetics for $l_{\theta}\sim R$ evinces kindred hallmarks like other two limits of rotational dynamics, $l_{\theta}>>R\;{\rm and}\;l_{\theta}<<R$. The asymptotes here too can approximately be determined using Eq. (16). Moreover, escape kinetics carries inertial impact until, $l_{th}>\delta\phi R$.

Simulation results in Fig. 2(a) show that in the underdamped limit, the positions of the minima in $\tau_{ex}\;vs.\;l_{\theta}$ have been shifted to the longer self-propulsion length. The location of the minima hinges on both the damping strength as well as the self-propulsion velocity. This trait is attributed to the emergence of another length scale, $l_{F}=v_{0}\tau_{\gamma}$. Due to its impact, the escape rate becomes maximum when the self-propulsion motion persists over a length $2l_{F}$ above the cavity diameter. The position of the minima estimated based on this reasoning,

represented by vertical arrows in Fig. 2(a). They are fairly consistent with simulation results.

For $l_{th}\gtrsim R$, an additional minimum emerges in $\tau_{ex}\;vs.\;l_{\theta}$ around point of transition from the diffusive to ballistic motion of self-propulsion. However, in most of the cases, it appears very subtle to be noticed or merges with the minimum described in Eq. (17). As evident in Fig. 2(a) and Fig. 4(d), the minimum at low $l_{\theta}$ becomes detectable when self-propulsion contributes to the effective temperature larger than $k_{B}T$ around the point of transition $l_{\theta}\sim R$. Parameters are chosen accordingly in Fig. 4(d) to get well separated two prominent minima. The appearance of the left minimum in $\tau_{ex}\;vs.\;l_{\theta}$ can thus be justified as follows. When $l_{\theta}<R$ and $l_{th}\gtrsim R$, the self-propulsion motion acts as uncorrelated random fluctuation, which adds up to the inherent thermal fluctuations leading to enhancement of the system effective temperature. Thus, the mean exit time decays with self-propulsion according to the Eq. (14). Once the $l_{\theta}$ length is comparable to the cavity size, the contribution of self-propulsion in the effective temperature subsides. As soon as $l_{\theta}$ grows much longer than cavity diameter, the self-propulsion motion, instead of contributing to effective temperature, affects escape kinetics by different mechanisms. Such chop and change between two self-propulsion regimes escorts to the left minimum.

Our simulation results reported in Fig. 5-6 present effects of multiple exit windows in the escape kinetics of an underdamped active particle. Three panels in Fig. 5 show variations of $\tau_{ex}$ with separation angle $\phi$ between two pores centres for various relative amplitudes of $l_{\theta}$ and $l_{th}$ with a fixed cavity radius. Here, our analysis considers three values of $l_{th}$, that correspond to the limits, $l_{th}\gg R$, $l_{th}\sim R$ and $l_{th}\sim R\delta\phi$, respectively.

For, $l_{th}\gg R$ and $l_{\theta}\ll R$, based on the arguments of the single-opening cavity, we expect $\tau_{ex}$ to be insensitive to the separation between two pores. Thus, $\tau_{ex}$ can be estimated using Eq. (14), however, doubling the pore width. Our expectations are supported by simulation results presented in Fig. 5(a). With gradually increasing self-propulsion length, the exit time remains independent of $\phi$ as long as $l_{\theta}\leq l_{th}$. As the self-propulsion length grows beyond this limit, the escape processes become sensitive to the pore separation angle $\phi$ [see Fig. 5(a)]. This feature could be understood assuming qualitative similarity of the exit mechanism with highly damped active particles. For $l_{\theta}\gg l_{th}$, most likely, due to the enhanced probability of accumulation and diffusion on walls, the narrow escape process gets expedited with increasing separation between pores.

Figure 5(b) shows that when the thermal length is comparable to cavity radius, the mean exit time is almost independent of the separation between two pores until $l_{\theta}<l_{th}$. However, in the opposite limit of self-propulsion lengths $l_{\theta}\gg l_{th}$, particles accumulate on the cavity wall and the rotational diffusion controls the particle’s escape dynamics. Thus, $\tau_{ex}$ closely follows the exit mechanism discussed in the context of Eq. (9). Simulation results here are fitted well Eq. (9) with an adjustable pre-factor.

For, $l_{th}\sim\delta\phi R$, simulation results show that the escape kinetics shares very similar features with overdamped regimes. However, the mean exit rate is not free from the impact of inertia as diffusion across the pore gets correlated due to inertial memory. This effect can be accounted for by considering a $\gamma$-dependent effective pore widths. Thus, simulation data could be fitted with overdamped Eqs. (9-10), however, one needs $\gamma$ dependent adjustable pore-width.

Figure 6 shows $\tau_{ex}\;vs.\;l_{\theta}$ with varying number of exit windows. The opening windows are uniformly distributed over the crater wall as mentioned in the context of overdamped particles. In contrast to the overdamped limit, our simulation results show that $\tau_{ex}\sim 1/n$ for $l_{th}>>R$ and irrespective of the values of self-propulsion length. This implies that the exit process through a pore is not affected by the presence of other opening windows no matter how close they are. Following Eqs. (14, 16), one can estimate the mean exit time from a circular cavity with $n$ pores as JPCC-1 ,

Note that this equation is strictly restricted for $l_{\theta}<<R\delta\phi$ in addition to very slow viscous relaxation. Further, as discussed earlier, the self-propulsion motion contributes through effective temperature given by the Eq. (13). This analytic estimations fairly corroborate the simulation results, as depicted in Figs. 5-6.

For all the three values of thermal lengths, corresponding to the panels (a, b, c) in Fig. 6, $\tau_{ex}\;vs.\;l_{\theta}$ exhibits a minimum. The position of the minimum is insensitive to the number of exit windows the cavity possesses. For $l_{th}\sim\delta\phi R$ and $l_{th}\sim R$, the minimum can be located using Eq. (17). However, for $l_{th}\gg R$, the minimum appears around the point of transition of self-propulsion motion from the diffusive to the ballistic regime.