Random motion of a circle microswimmer in a random environment

For any nonzero angular noise the meandering transition is smeared and the orbiting state is no longer present. Particles can reach different obstacle clusters by random reorientations, and this will always happen provided one waits long enough. The smearing of the phase transition is illustrated in figures 2(a) and (c) in terms of the diffusivity D as the density is increased towards the ideal meandering transition. The distance to the transition is characterized by a separation parameter $\varepsilon =\left({n}^{{\ast}}-{n}_{\mathrm{m}}^{{\ast}}\left(R\right)\right)/{n}_{\mathrm{m}}^{{\ast}}\left(R\right)$. Indeed, transport occurs now at any obstacle density due to the orientational noise. Yet, by lowering the angular noise D_θ the translational diffusion D(R, D_θ, n*) approaches the case of the ideal microswimmer. Thus, the smearing of the transition is a continuous process and the results of the ideal microswimmer [6] remain valid for sufficiently small noise. The rate of convergence depends on the orbit radius. For small orbit radii [figure 2(a)] the convergence rate is slightly slower then for the larger ones [figure 2(c)]. This is due to the higher value of the obstacle density of the meandering transition for smaller orbit radii.

Zoom In Zoom Out Reset image size

It should be mentioned that if translational diffusion was included, the meandering transition would also be smeared even in the absence of rotational noise. However, for realistic microswimmers the rotational diffusion is anticipated to be the main effect and the additional smearing due to translational diffusion should be small.

Now we zoom out from the meandering transition and consider the whole density range [figures 2(b) and (d)]. For the intermediate obstacle densities the diffusivity D increases when the angular noise is increased up to its optimal value for a small orbit radius [figure 2(b)] and a large one [figure 2(c)]. At larger n*, closer to the percolation transition, transport is dominated by the obstacles, in particular, by the crowding-enhancement transport mechanism, while angular noise is of minor importance. This can be seen in detail in the insets in figures 2(b) and (d). However, we note that the maximum diffusivity for each fixed curvature almost never exceeds the maximum value of the diffusivity in an ideal system, figures 2(b) and (d).

To get a more broad view we plot isodiffusivity contours in the state diagram spanned by the obstacle density n* and the orbit curvature σ/R, as in the case of the ideal circle microswimmer [6] or the magneto-transport problem [55]. The diagrams are presented for increasing values of angular noise D_θ, see figures 3(a)–(d). In the second row, figures 3(e)–(h), the ratio of the diffusivity to its value for the idealized system (D_θ = 0) is shown. For small noise D_θ, the diffusivity in the region above the meandering transition, ${n}^{{\ast}}{ >}{n}_{\mathrm{m}}^{{\ast}}\left(R\right)$ is very similar to the ideal case [figure 3(a)]. Only very close to the percolation transition there is a sharp drop of the diffusivity since swimmers become trapped in isolated pockets. In most of the parameter space corresponding to the diffusive state of the ideal circle swimmer, the values of the diffusivity D are again similar for both models. To the left of the meandering transition line, i.e. in the orbiting state, this ratio remains undefined [figure 3(e)]. Away from it, in most of the diffusive state, the light shading of the color coding indicates that the diffusivity there is very similar to the idealized case [figure 3(e)]. Approaching the meandering transition of the idealized model the diffusivity rapidly decreases, yet remains nonzero [figure 3(a)] in stark contrast to the idealized system where no diffusion in this region occurs.

Zoom In Zoom Out Reset image size

For intermediate noise D_θ [figures 3(b) and (c)], the dependence of the diffusivity on the orbit radius is still strong even for the obstacle-free system n* = 0, as suggested by equation (9). Upon increasing the density of obstacles the diffusivity again grows, however, the growth occurs mainly in the regime of the ideal diffusive state. For obstacle densities below the ideal meandering transition the dependence on the obstacle density is rather weak, such that the isodiffusivity lines are almost horizontal. An amplification of diffusion in the ideal diffusive state is revealed in figures 3(f) and (g).

For even higher angular noise [figure 3(d)] the dependence of the diffusivity on the orbit radius fades out. This can be explained by the fact that the particle's free motion ceases to be circular, since the quality factor M is very low. Nevertheless, the diffusion coefficient increases with the density of obstacles except in the close vicinity of the percolation transition. In comparison to the ideal case, figure 3(h) shows that diffusion is suppressed in large areas of the parameter space.

The qualitative difference between the low and high noise behavior becomes immediately apparent upon inspecting typical trajectories, see figure 4. At low noise [figure 4(a)] the microswimmer trajectory in free space is composed of distorted circles, while for intermediate noise [figure 4(b)], the trajectories get significantly randomized and the noise promotes diffusion in free space.

Zoom In Zoom Out Reset image size

In summary, low values of angular noise promote transport by allowing for the possibility to reach obstacle clusters which were inaccessible in the ideal case due to a limited orbit range. At intermediate noise values, the already efficient free-space transport is enhanced by the wall-following mechanism. At very large D_θ the circular motion is so rapidly randomized that transport approaches the case of an active Brownian tracer in a disordered environment, yet the wall-following mechanism enhances transport at high obstacle densities.

It is instructive to study directly the interplay of quenched disorder and dynamic noise. Therefore we redraw the isodiffusivity lines in a new state diagram spanned by the obstacle density n* and the rotational diffusion coefficient D_θ for fixed orbit radius R, see figures 5(a)–(c). We highlight regions of amplification and suppression of diffusion by relating the computed contours to the diffusion coefficient at a small value of the rotational noise, figures 5(d)–(f). We choose a system with some rotational noise over the ideal system to have a more complete state diagram, without void regions of undefined value.

Zoom In Zoom Out Reset image size

We can see that the diffusivity maps are qualitatively similar for three representative orbit radii, figures 5(a)–(c). The diffusivity continuously increases when the obstacle density is increased up to n* ≃ 0.3. The line of constant angular noise ${D}_{\theta }={D}_{\theta }^{\text{opt}}$ separates two regions. When the angular noise strength is increased up to the optimal value ${D}_{\theta }^{\text{opt}}$, the diffusivity grows with it. However, as the angular noise is increased further the diffusivity starts to decrease for any value of the obstacle density. We note that for smaller radius [figure 5(a)] the optimal density of obstacles is very close to ${n}_{\mathrm{c}}^{{\ast}}$, while for large radius [figures 5(b) and (c)] the optimal density value is closer to n* ≈ 0.3.

Next, we consider amplification or suppression of the diffusivity with respect to its value at low rotational noise

$$\begin{equation}{D}^{\left(0.01\right)}\left(R,{n}^{{\ast}}\right){:=}D\left(R,{D}_{\theta }=0.01 {\sigma }^{-1}v,{n}^{{\ast}}\right),\end{equation} \tag{ 12 }$$

plotted in figures 5(d)–(f). In all three panels at n* = 0 we see an enhancement of diffusion D(D_θ)/D^(0.01) > 1 (indicated by the red color), which is solely due to the increase of angular noise, D_θ. When n* > 0, an increase of rotational diffusion enhances the diffusivity less and less for increasing density of obstacles. For the smallest radius R = 0.5σ, diffusion is almost always amplified [figure 5(d)]. For the intermediate radius R = 1.5σ, a region emerges at high obstacle densities where diffusion becomes suppressed when the angular noise is increased [figure 5(e)]. For the largest orbit radius R = 2.0σ, the size of this region increases [figure 5(f)]. From this we draw the conclusion that the area in parameter space covered by the suppression region increases with orbit radius.

While in general including angular noise and obstacles into the environment amplifies particle diffusion, for large values of both perturbations transport becomes hindered. The exact position of the boundary depends on the orbit radius R/σ, with smaller values of noise strength or obstacle density needed to cause a suppression of diffusion for larger radii.

In the results section it has been elaborated that the amplification–suppression patterns are different for small (R = 0.5σ) [figure 5(d)] and large (R = 1.5σ, 2.0σ) radii [figures 5(e) and (f)]. For a small radius R/σ = 0.5 an amplification of transport with an increase of angular noise is observed at any obstacle density [figure 5(d)]. It qualitatively follows the diffusivity dependence on angular noise, given by the equation (9), where two values of angular noise, a low and high one, yield the same diffusivity value, causing similar amplification. However, for R/σ = 2.0 at intermediate-to-high densities the angular noise causes little amplification and even leads to suppression of diffusion at high noise values [figure 5(f)]. The long-time diffusivity given by equation (9) is not enough to explain this behavior.

To provide an intuitive explanation for such a drastic difference we consider the mean-squared displacement of an active circle Brownian swimmer in free space [48, 49, 53]

$$\begin{equation}\delta {r}^{2}\left(t\right)=\frac{2{v}_{0}^{2}{D}_{\theta }^{2}}{{\left({D}_{\theta }^{2}+{{\Omega}}^{2}\right)}^{2}}\left[{D}_{\theta }t-1+\frac{{{\Omega}}^{2}}{{D}_{\theta }^{2}}\left({D}_{\theta }t+1\right)+\left(1-\frac{{{\Omega}}^{2}}{{D}_{\theta }^{2}}\right)\mathrm{cos}\left({\Omega} t\right){\mathrm{e}}^{-{D}_{\theta }t}-2\frac{{\Omega}}{{D}_{\theta }}\mathrm{sin}\left({\Omega} t\right){\mathrm{e}}^{-{D}_{\theta }t}\right].\end{equation} \tag{ 13 }$$

The mean-squared displacement typically evolves through three different regimes: directed motion (at short times), oscillations (at intermediate times), and diffusion (at long times). The duration of each of the regimes depends on the time scales Ω⁻¹ and ${D}_{\theta }^{-1}$.

To explain the difference in diffusivities at different D_θ for fixed n* we consider the time needed to cover the characteristic distance between obstacles ℓ := σ/n*, termed the mean-free path length [56]. In some cases the particle can cover a distance ℓ already in the regime of directed motion (large R, large n*, low D_θ). In other cases (high D_θ, low n*, low R) the diffusive regime has been entered before the length ℓ is reached by the microswimmer. The difference can be quantified by a characteristic time τ, which can be directly inferred from the plot [figures 6(a) and (b)], as the first intersection of the mean-squared displacement curve with the line of constant ℓ². For small radius (R = 0.5σ) the mean-squared displacements given by equation (13) display oscillations for low angular noise [figure 6(a)]. Their amplitude is small in comparison to the mean-free path ℓ for all obstacle densities considered, and the intersection always occurs in the diffusive regime. Thus, the time τ can be computed using the long-time asymptote for the mean-squared displacement, δr²(t) = 4Dt, as t → ∞, together with equation (9),

$$\begin{equation}\tau =\frac{{\ell }^{2}}{2{v}_{0}^{2}{D}_{\theta }}\left({D}_{\theta }^{2}+\frac{{v}_{0}^{2}}{{R}^{2}}\right).\end{equation} \tag{ 14 }$$

Zoom In Zoom Out Reset image size

In this case, equation (9) can be used to describe the variation in the diffusivity. In stark contrast, for large orbit radius (R = 2.0) [figure 6(b)] the microswimmer can access regions of distance ℓ while displaying circular motion. Thus, equation (9) alone is not enough to explain the results.

The inverse of the characteristic time τ is displayed in figures 6(c) and (d) as a function of D_θ. For R = 0.5σ, a non-monotonic behavior with a pronounced maximum for 1/τ emerges at all densities [figure 6(c)], according to equation (14). Yet, for R = 2.0σ beyond a density n* ≳ 0.25 the dependence becomes monotonically decreasing [figure 6(d)] and solutions of the full equation (graphical solutions) deviate from the approximation, equation (14), at low values of D_θ. In both cases the dependence of the diffusivity D on the angular noise D_θ correlates with 1/τ. For R = 0.5σ the result is not surprising, as D vs D_θ is also a non-monotonic function with the maximum at the corresponding position [figure 6(e)]. From the idealized model we know that a higher obstacle density provides a larger diffusion [6], thus the amplitude of the amplification decreases with n*. Most importantly, the diffusivity never becomes suppressed [figure 6(e) inset].

For large radius R = 2.0σ the argument is more subtle [figure 6(f)]. At low densities, n* ≈ 0.05, the behavior is the same as for small radius. For high densities n* ≳ 0.2 the diffusivity remains almost constant (similar to its value of the idealized model) as the noise is increased. It becomes slightly amplified around the optimal noise value ${D}_{\theta }^{\text{opt}}$, and for D_θ larger than ${D}_{\theta }^{\text{opt}}$ the diffusion is suppressed. Considering the diffusivity ratios [figure 6(f) inset] only the low-density n* = 0.05 is similar to the R = 0.5 case [figure 6(e) inset]. All other curves do not quite follow the 1/τ curves at low noises. While the characteristic time to cover ℓ² only increases with D_θ [figure 6(d)], the diffusivity D nevertheless is amplified around the optimal angular noise ${D}_{\theta }^{\text{opt}}$ [figure 6(f) inset]. For larger values of angular noise the 1/τ and D curves decrease in a correlated way [figures 6(d) and (f) inset]. So, the characteristic time τ is no longer sufficient, rather we anticipate that the entire distribution of times needed to cover an entire range of path lengths slightly larger than ℓ determines the transport properties.

For an ideal microswimmer the time τ to cover an arbitrary distance λ increases until the length scale becomes equal to the orbit diameter, λ = 2R. For larger values of λ > 2R the characteristic time τ does not exist, as the ideal microswimmer cannot travel farther than 2R. If a small amount of angular noise is introduced, length scales λ > 2R become accessible, but the time to cover them is significantly larger compared to the time to cover λ = 2R. The dependence of τ on an arbitrary length scale λ is rather steep at low noise values in the vicinity of λ ≈ 2R, see figure 7. For R = 0.5σ this steep increase occurs far away from the characteristic lengths defined by the largest considered densities [figure 7(a)], and thus has no impact on the amplification–suppression pattern. For R = 2.0σ the values of ℓ(n*) at n* ≳ 0.2 are comparable with the diameter 2R, for example, ℓ(n* = 0.25) = 2R = 4σ, [figure 7(b)]. The amplification of diffusion at low noises (${D}_{\theta }\ll {D}_{\theta }^{\text{opt}}$) is absent because the characteristic times to cover small distances are comparable with the ones for the ideal system τ(λ < 2R, D_θ ≪ 1) ≈ τ(λ < 2R, D_θ = 0). Yet, times to cover slightly farther distances are quite large $\tau \left(\lambda { >}2R,{D}_{\theta }\ll {D}_{\theta }^{\text{opt}}\right)\gg \tau \left(\lambda {< }2R,{D}_{\theta }\ll {D}_{\theta }^{\text{opt}}\right)$ and do not affect the transport properties [figure 7(b), red and orange curves]. If the angular noise is increased up to its optimal value, ${D}_{\theta }\approx {D}_{\theta }^{\text{opt}}$, the time to cover small distances also increases, $\tau \left(\lambda {< }2R,{D}_{\theta }\approx {D}_{\theta }^{\text{opt}}\right){ >}\tau \left(\lambda {< }2R,{D}_{\theta }=0\right)$ [figure 7(b), yellow curve]. However, the time to cover larger distances decreases drastically and the steep increase becomes smooth. The latter effect is more important and an amplification of transport is observed. With a further increase of angular noise beyond its optimal value all characteristic times also increase, both for small λ < 2R and large λ > 2R length scales [figure 7(b), aquamarine curve]. Hence, transport becomes suppressed. This picture holds for obstacle densities below n* = 0.3 while the average length scale ℓ remains comparable to 2R. As soon as the density becomes too low, for example n* = 0.05, the simple mechanism described for R = 0.5 is valid.

Zoom In Zoom Out Reset image size

So far we have presented our study from the perspective of increasing the angular noise strength at a constant density of obstacles. However, in an experimental setup such a strategy might be not possible. While in experiments it is difficult to control the rotational diffusion coefficient, the obstacle density may be varied. For a direct comparison with experimentally accessible control parameters we represent the data in terms of two dimensionless quantities, the quality factor M and the ratio of diffusion in a crowded system to the diffusion in an obstacle-free system. We show how the diffusivity is amplified or suppressed at different obstacle densities for a range of quality factors. We discuss D = D(R, M, n*) as a function of the quality factor M with fixed orbit radius and obstacle density, rather than as a function of the (reduced) orientational diffusivity. Therefore we define the rescaled diffusivity

$$\begin{equation}\tilde {D}=\tilde {D}\left(R,M,{n}^{{\ast}}\right){:=}D\left(R,M,{n}^{{\ast}}\right)/D\left(R,M,{n}^{{\ast}}=0\right),\end{equation} \tag{ 15 }$$

which by construction reduces at zero obstacle density to $\tilde {D}\left(R,M,{n}^{{\ast}}=0\right)=1$. Indeed, from the simulations we see that for low obstacle density $\tilde {D}\approx 1$ holds for quality factors below the optimal one, M ≲ M^opt = 1/2π [figures 8(a) and (b)]. At these low densities, increasing the quality factor beyond the optimal one leads to an increase of the rescaled diffusivity. Moreover, a larger amplification is observed for a larger R/σ ratio. This observation suggests that scattering at an isolated obstacle is more efficient in accelerating transport if the orbit radius is larger than the obstacle size. Propagation around the boundary of a small obstacle cluster, or even a single obstacle, facilitates meandering thus increasing the diffusivity. This effect becomes more pronounced for larger radii.

Zoom In Zoom Out Reset image size

For somewhat larger densities, the diffusivity shows a minimum at the optimal quality factor value [figures 8(c)–(f)]. This occurs together with another striking observation: for densities n* ≲ 0.25–0.30 there is an approximate data collapse for different radii R [figures 8(e) and (f)]. This feature is a peculiarity for microswimmers and is connected to the wall-following mechanism. We have checked that for specular scattering off the obstacles as in magneto-transport this data collapse does not occur.

In the case of the highest density n* = 0.35 close to the percolation transition ${n}_{\mathrm{c}}^{{\ast}}$ there is still a pronounced minimum in the amplification at M = M^opt [figure 8(g)], yet the dependence on the orbit radius becomes significant. In the ideal model at obstacle densities around n* = 0.35 the disordered structure ceases to amplify transport and starts to suppress it, as at ${n}_{\mathrm{c}}^{{\ast}}$ one expects the diffusivity to become zero. At this density the pockets of the void space appear connected by narrow channels. From the data one can conclude that particles with smaller orbit radii are more efficient in moving through such a structure than their counterparts with larger orbit radii. Then the order of the curves at high density [figure 8(g)] is reversed in comparison to the low density n* = 0.05, [figure 8(a)]. The same effect has been reported for ideal microswimmers where it was also seen that the diffusivity at very high obstacle densities is larger for smaller orbit radii [6], compare aquamarine points in figure 2(b) R = 0.5σ and (d) R = 2.0σ.

To summarize this part, we find that at all densities the shape of the amplification curve remains the same. At low M there is no to little amplification. At M^opt the amplification is minimal, but then quickly increases for larger values of M.

The properties described above can serve as a guideline for future experiments. The quality factor of the orbit can be easily identified for the given experimental probe particle (biological or artificial) and then one can verify the prediction for the amplification curves at different obstacle densities. The best correspondence between experiment and theory is expected for particles that are well described by our model. This is those that become trapped around obstacles and can travel large distances before departing from the obstacle boundary.