Microswimmers are typical nonequilibrium systems that are driven away from equilibrium due to their ability to utilize energy obtained from the surroundings and convert it into self-propelled motion [1, 2]. Examples of microswimmers include flagellated bacteria, eukaryotic cells, sperm, ciliates, synthetic microswimmers and so on [3–7]. Although the propulsion of microswimmers has been widely explored, most of the literature disregards the influence of memory of the medium in which they propel. In a practical scenario, microswimmers are mostly exposed to non-Newtonian fluids such as polymer solution [8–10] or crystalline medium [11, 12] or biological environments such as mucus [13], tissues etc., where memory plays a significant role in their dynamics [14–16]. Due to the presence of memory in the environment, an enhancement in the attraction and orientational ordering of the swimmers [17], an increase in rotational diffusion of Janus particles with self-propulsion velocity [18], an enhancement of swimming speed of Escherichia coli bacteria with reduced tumbling [19], etc. are observed.
Moreover, rotation plays a crucial role in the dynamics of microswimmers [20–22]. It enables the microswimmers to avoid obstacles, locate food and other resources, and interact with their surroundings. In the case of chiral microswimmers with shape asymmetry, a circular trajectory is a typical feature of their motion [23, 24]. The chiral active colloid, which is a kind of microswimmer, undergoes a velocity-dependent viscous torque that yields a circular trajectory [22, 23]. However, chirality is not the only key factor for the occurrence of rotational motion of microswimmers. Achiral microswimmers can also make a transition from random self-propulsion to persistent circular motion, purely induced by memory of the medium [25]. Since the microswimmers move in a medium with a very low Reynolds number, the influence of inertia on their dynamics is often ignored [25–28]. However, inertia is found to have an unavoidable impact on the motion of microswimmers [29–34]. Inertia can enhance (or hinder) the swimming speed of a microswimmer depending on whether the swimmer generates thrust at the back or in front of its body [35]. Inertia can also influence the mean speed of microswimmers based on their rotation–translation coupling [31].
In this paper we theoretically demonstrate that, even in the absence of any external torque or confinement, an inertial achiral microswimmer can make an unexpected transition from random self-propulsion to rotational (circular or elliptical) motion by adjusting both the inertial time and persistent duration of memory in the medium. Unlike chiral microswimmers, here the rotational motion undergoes spontaneous local directional reversals with a steady-state angular diffusion and with no specific direction of the rotational motion. Such an intriguing behaviour is represented by a minimal two-dimensional (2D) generalized Langevin model of non-Markovian dynamics of an inertial active Ornstein–Uhlenbeck particle. Moreover, a swimmer with sufficient inertia and with the presence of memory in the environment becomes confined or trapped with increase in the duration of activity even in the absence of any external confinement.
In our model, we consider the 2D motion of an inertial microswimmer in a non-Newtonian fluid with a finite memory. The equation of motion of such a microswimmer is described by the generalized Langevin model of non-Markovian dynamics of an inertial active Ornstein–Uhlenbeck particle [25, 36, 37]1 ) is the viscoelastic drag with $g(t-t^{\prime} )$ as the memory kernel or viscous kernel, which is given by1 ) is stochastic, and effectively $\sqrt{D}{\boldsymbol{\xi }}(t)$ can act as the induced active force or self-propulsion force that follows the Ornstein–Uhlenbeck process [40, 41]
$\begin{eqnarray}m\ddot{{\boldsymbol{r}}}(t)=-{\int }_{-\infty }^{t}g(t-t^{\prime} )\dot{{\boldsymbol{r}}}(t^{\prime} ){\rm{d}}t^{\prime} +\sqrt{D}{\boldsymbol{\xi }}(t).\end{eqnarray}$
Here, ${\boldsymbol{r}}(t)=x(t)\hat{i}+y(t)\hat{j}$ represents the position vector of the microswimmer, m represents its mass, γ is the viscous coefficient of the medium and D is the strength of the noise. The first term on the right-hand side of equation ( $\begin{eqnarray}g(t-t^{\prime} )=\left\{\begin{array}{ll}\displaystyle \frac{\gamma }{{t}_{c}^{{\prime} }}{{\rm{e}}}^{-\tfrac{(t-t^{\prime} )}{{t}_{c}^{{\prime} }}}; & t\geqslant t^{\prime} \\ 0; & t\lt t^{\prime} .\end{array}\right.\end{eqnarray}$
The memory kernel $g(t-t^{\prime} )$ decays exponentially with time constant ${t}_{c}^{{\prime} }$, where, ${t}_{c}^{{\prime} }$ is the time that quantifies the persistent duration of memory in the medium or environment. The memory kernel often characterizes the viscoelastic nature of a non-Newtonian environment with a transient elasticity [38]. Here, the memory kernel follows the Maxwellian viscoelastic formalism by which the elastic force of the fluid completely diminishes for a sufficiently long interval of time [39]. In the vanishing limit of ${t}_{c}^{{\prime} }$, the memory kernel becomes a delta function [$g(t-t^{\prime} )=\delta (t-t^{\prime} )$]; as a result the suspension turns entirely into a viscous medium. The term ξ(t) in equation ( $\begin{eqnarray}{t}_{c}\dot{{\boldsymbol{\xi }}}(t)=-{\boldsymbol{\xi }}(t)+{\boldsymbol{\eta }}(t),\end{eqnarray}$
with the statistical properties $\begin{eqnarray}\langle {\xi }_{i}(t)\rangle =0\quad \mathrm{and}\quad \langle {\xi }_{i}(t){\xi }_{j}(t^{\prime} )\rangle =\displaystyle \frac{{\delta }_{{ij}}}{2{t}_{c}}\exp \left(-\displaystyle \frac{| t-t^{\prime} | }{{t}_{c}}\right).\end{eqnarray}$
Here, tc represents the self-propulsion time or activity time, i.e. the time up to which the microswimmer continues to self-propel along a specific direction despite making random collisions with the surrounding fluid particles. Hence, for a finite tc the system is driven away from equilibrium with an effective temperature different from the actual temperature of the bath. When ${t}_{c}={t}_{c}^{{\prime} }$, the generalized fluctuation dissipation relation is satisfied [37, 42] with an effective temperature related to the noise strength D. For ${t}_{c}={t}_{c}^{{\prime} }$ together with D = 2γkBT, the system approaches thermal equilibrium with T as the bath temperature and kB as the Boltzmann constant. η(t) is the delta-correlated Gaussian white noise and (i, j) ∈ (x, y).The mean displacement (MD) and mean-square displacement (MSD) of the microswimmer can be defined as
$\begin{eqnarray}\langle {\rm{\Delta }}{\boldsymbol{r}}(t)\rangle =\langle {\boldsymbol{r}}(t)-{\boldsymbol{r}}(0)\rangle ;\ \langle {\rm{\Delta }}{{\boldsymbol{r}}}^{2}(t)\rangle =\langle {\left({\boldsymbol{r}}(t)-{\boldsymbol{r}}(0)\right)}^{2}\rangle ,\end{eqnarray}$
with r(0) representing the initial position of the swimmer. The orientation of the swimmer can be determined by the direction of its instantaneous velocity. In order to describe the orientation of the swimmer, we introduce a parameter θ such that $\theta ={\tan }^{-1}\left(\tfrac{{v}_{y}}{{v}_{x}}\right)$ for all $\theta \in \left[-\pi ,\pi \right]$, with vx and vy being the x and y components of the instantaneous velocity v of the swimmer, respectively. Here, θ is the angle that the instantaneous velocity of the swimmer makes with the x-axis, and the time evolution of θ quantifies the instantaneous angular velocity. Hence, the average angular velocity ⟨ω⟩ can be defined as $\left\langle \tfrac{{\rm{d}}\theta }{{\rm{d}}t}\right\rangle $ [25].By performing a Laplace transform on equation (1 ), the exact solution of equation (1 ) with initial conditions ${\boldsymbol{r}}(0)={{\boldsymbol{r}}}_{0}[=({x}_{0}\hat{i}+{y}_{0}\hat{j})]$ and $\dot{{\boldsymbol{r}}}(0)={{\boldsymbol{v}}}_{0}[=({\dot{x}}_{0}\hat{i}+{\dot{y}}_{0}\hat{j})]$ can be obtained as6 )), we establish a phase diagram in the inertia-memory (${t}_{c}^{{\prime} }$, Γ) parameter space (see figure 1(a)) separating the oscillatory and non-oscillatory behaviour of motion of the swimmer. The oscillatory and non-oscillatory motions of the swimmer are separated by a phase boundary ${t}_{c}^{{\prime} }{\rm{\Gamma }}=\tfrac{1}{4}$. The microswimmer experiences oscillations in its motion for the phase with ${t}_{c}^{{\prime} }{\rm{\Gamma }}\gt \tfrac{1}{4}$ and no oscillation in its motion for the phase with ${t}_{c}^{{\prime} }{\rm{\Gamma }}\lt \tfrac{1}{4}$ in the ${t}_{c}^{{\prime} }-{\rm{\Gamma }}$ parameter space. The exact calculation of frequency of oscillation shows its dependence on Γ and ${t}_{c}^{{\prime} }$. It decreases with increase in ${t}_{c}^{{\prime} }$, increases with increase in Γ and remains same along the diagonal line, i.e. for ${\rm{\Gamma }}={t}_{c}^{{\prime} }$ (see the oscillatory regime of figure 1(a)). This is the same as for the deterministic dynamics, i.e. in the absence of stochastic force [ξ(t)]. The deterministic part of equation (1 )is simply a form of delay differential equation [43–45]. In order to distinguish the oscillatory and non-oscillatory phases, in figure 1(b) we have show the time evolution of the MSD corresponding to all the points along the diagonal line AB of the ${t}_{c}^{{\prime} }-{\rm{\Gamma }}$ phase diagram (in figure 1(a)), by plotting ⟨Δr2(t)⟩ versus t. Using equation (5 ), the MD at any instant of time is calculated as
$\begin{eqnarray}\begin{array}{rcl}{\boldsymbol{r}}(t) & = & {{\boldsymbol{r}}}_{0}-\displaystyle \sum _{i=0}^{2}{b}_{i}{{\boldsymbol{u}}}_{0}{{\rm{e}}}^{{s}_{i}t}\\ & & +\displaystyle \sum _{i=0}^{2}{a}_{i}\left[{{\boldsymbol{v}}}_{0}{{\rm{e}}}^{{s}_{i}t}+\displaystyle \frac{\sqrt{D}}{m}{\displaystyle \int }_{0}^{t}{{\rm{e}}}^{{s}_{i}(t-t^{\prime} )}{\boldsymbol{\xi }}(t^{\prime} )\,{\rm{d}}t^{\prime} \right],\end{array}\end{eqnarray}$
with u0 as $\begin{eqnarray}{{\boldsymbol{u}}}_{0}={\int }_{-\infty }^{0}\displaystyle \frac{\gamma }{{{mt}}_{c}^{{\prime} }}{{\rm{e}}}^{\tfrac{t^{\prime} }{{t}_{c}^{{\prime} }}}\,\dot{{\boldsymbol{r}}}(t^{\prime} ){\rm{d}}t^{\prime} .\end{eqnarray}$
Here, si are given by the expression $\begin{eqnarray}{s}_{0}\,=\,0\quad \mathrm{and}\quad {s}_{\mathrm{1,2}}=-\displaystyle \frac{1}{2}\left(\displaystyle \frac{1}{{t}_{c}^{{\prime} }}\pm \displaystyle \frac{\sqrt{1-4{t}_{c}^{{\prime} }{\rm{\Gamma }}}}{{t}_{c}^{{\prime} }}\right).\end{eqnarray}$
The coefficients ai and bi are given by $\begin{eqnarray}{a}_{0}=\displaystyle \frac{1}{{\rm{\Gamma }}},\quad {a}_{\mathrm{1,2}}=\displaystyle \frac{\pm {t}_{c}^{{\prime} }-{t}_{c}^{{\prime} }\sqrt{1-4{\rm{\Gamma }}{t}_{c}^{{\prime} }}}{\pm \left(1-4{\rm{\Gamma }}{t}_{c}^{{\prime} }\right)+\sqrt{1-4{\rm{\Gamma }}{t}_{c}^{{\prime} }}},\end{eqnarray}$
$\begin{eqnarray}{b}_{0}=\displaystyle \frac{{t}_{c}^{{\prime} }}{{\rm{\Gamma }}},\mathrm{and}\quad {b}_{\mathrm{1,2}}=\displaystyle \frac{\pm 2{\left({t}_{c}^{{\prime} }\right)}^{2}}{\pm \left(1-4{\rm{\Gamma }}{t}_{c}^{{\prime} }\right)+\sqrt{1-4{\rm{\Gamma }}{t}_{c}^{{\prime} }}},\end{eqnarray}$
with ${\rm{\Gamma }}=\tfrac{\gamma }{m}$ being the inverse inertial time scale of the swimmer. The mass of the particle is considered to be unity (m = 1) throughout the paper. From the exact solution of the dynamics (equation ( $\begin{eqnarray}\langle {\rm{\Delta }}{\boldsymbol{r}}(t)\rangle =\langle {{\boldsymbol{v}}}_{0}\rangle \left(\displaystyle \frac{1}{{\rm{\Gamma }}}+\displaystyle \sum _{i=1}^{2}{a}_{i}{{\rm{e}}}^{{s}_{i}t}\right)-\langle {{\boldsymbol{u}}}_{0}\rangle \left(\displaystyle \frac{{t}_{c}^{{\prime} }}{{\rm{\Gamma }}}+\displaystyle \sum _{i=1}^{2}{b}_{i}{{\rm{e}}}^{{s}_{i}t}\right).\end{eqnarray}$
The steady-state MD is found to be $\langle {\rm{\Delta }}{\boldsymbol{r}}{\rangle }_{{st}}={\mathrm{lim}}_{t\to \infty }\langle {\rm{\Delta }}{\boldsymbol{r}}(t)\rangle =\tfrac{\langle {{\boldsymbol{v}}}_{0}\rangle }{{\rm{\Gamma }}}$. Similarly, we have exactly calculated MSD: it can be found in the supplementary text (see the Supplementary Material (SM)2). Passing from the non-oscillatory to the oscillatory phase along the line AB in the ${t}_{c}^{{\prime} }-{\rm{\Gamma }}$ phase diagram, if the chosen parameters $({t}_{c}^{{\prime} },{\rm{\Gamma }})$ lie above the point $O(\tfrac{1}{2},\tfrac{1}{2}$) the time evolution of the corresponding MSD is oscillatory, and if these points lie below the point $O(\tfrac{1}{2},\tfrac{1}{2}$) the time evolution of the corresponding MSD is non-oscillatory (see figure 1(b)). For any point near the boundary, the amplitude of oscillation is very small and hence not visible in the MSD curves (for ${\rm{\Gamma }}={t}_{c}^{{\prime} }=0.6$).
Figure 1. (a) Phase diagram separating the oscillatory and non-oscillatory regimes in the ${t}_{c}^{{\prime} }-{\rm{\Gamma }}$ parameter space (equation ( |
The simulation of the dynamics (equation (1 )) is carried out using the second-order modified Euler method [46] and the Fox algorithm [47]. The simulation is run up to 103 time steps and the numerical integration is performed with a time step of 10−3. Averages of the physical quantities are measured over 104 realizations. By analysing the instantaneous 2D trajectories of the microswimmer in the xy-plane (see figure 2), it is interestingly observed that by changing the persistent duration of memory or (and) inertial time in the oscillatory regime of the ${t}_{c}^{{\prime} }-{\rm{\Gamma }}$ parameter space, the swimmer performs rotational (circular or elliptical) motion even in the absence of any external torque or confinement. The emergence of this behaviour is only due to the presence of inertial motion and memory in the medium or elastic nature of the fluid. Figures 2(a)–(e) present the instantaneous 2D trajectories of a microswimmer for different values of ${t}_{c}^{{\prime} }$, keeping Γ fixed. In the insets of figures 2(a)–(e), we also present the data corresponding to the time evolution of orientation (θ) of the swimmer. For ${t}_{c}^{{\prime} }=0.1$, the instantaneous trajectory (figure 2(a)) exhibits almost random behaviour, reflecting random self-propulsion of the swimmer, and the time evolution of the corresponding θ (inset of figure 2(a)) on average shows a random pattern. However, with an increase in the value of ${t}_{c}^{{\prime} }$ (i.e. at around ${t}_{c}^{{\prime} }=10$), the swimmer is influenced by the elastic nature of the suspension and hence tries to make a smooth and curvy type trajectory (figure 2(b)). With further increase in ${t}_{c}^{{\prime} }$, at around ${t}_{c}^{{\prime} }=25$ (see figure 2(c)), the swimmer likely starts to form a circular (or elliptical) trajectory. For higher values of ${t}_{c}^{{\prime} }$, the trajectories adopt an almost elliptical shape, as shown in figures 2(d) and (e) for ${t}_{c}^{{\prime} }=100$ and ${t}_{c}^{{\prime} }=200$, respectively. Hence, the increase in ${t}_{c}^{{\prime} }$ leads the swimmer to make an unforeseen transition from random self-propulsion to a different dynamical regime characterized by persistent rotation. The plots of corresponding to θ versus t (insets of figures 2(d) and (e)) also show slanted stripes, which is a consequence of rotational motion, either clockwise (π to −π) or counter-clockwise (−π to π). This observation reflects that during time evolution such rotational orbits spontaneously reverse their orientations. The frequency (ν) of such orientation reversals is stochastic and it depends on the parameters of the model. The simulated curve for ν as a function of ${t}_{c}^{{\prime} }$ is shown in figure 2(f), and it is found to decrease with increase in ${t}_{c}^{{\prime} }$. This is due to the fact that a swimmer with longer persistence of memory covers a larger distance and takes a longer time to reverse its orientation. Remarkably, from the trajectory plots it can be noticed that the occurrence of these orientation reversals becomes very uncommon as ${t}_{c}^{{\prime} }$ increases. At the same time, the elliptical trajectories become more stable and are clearly identified, as seen from figure 2(e). The time evolution of corresponding θ on average becomes linear over a long interval of time (see inset of figure 2(e)). Similarly, keeping ${t}_{c}^{{\prime} }$ fixed and varying Γ as well as varying both ${t}_{c}^{{\prime} }$ and Γ simultaneously in the ${t}_{c}^{{\prime} }-{\rm{\Gamma }}$ parameter space, we observe the same attributes and our assessments based on the trajectory plots are supported by the MSD calculation (see the SM and footnote 2).
Figure 2. Simulated trajectories of a microswimmer for different values of ${t}_{c}^{{\prime} }$ are plotted in (a) for ${t}_{c}^{{\prime} }=0.1$, (b) for ${t}_{c}^{{\prime} }=2$, (c) for ${t}_{c}^{{\prime} }=25$, (d) for ${t}_{c}^{{\prime} }=100$ and (e) for ${t}_{c}^{{\prime} }=200$. The insets in (a)–(e) show the evolution of the corresponding orientation θ with t. The frequency of directional reversal ν is plotted as a function of ${t}_{c}^{{\prime} }$ in (f). The other fixed parameters for all the plots are Γ = D = tc = 1 and ${{\boldsymbol{r}}}_{0}=0\hat{i}+0\hat{j}$. The colour map shows the time evolution of the trajectories, and the arrows in (e) represent the instantaneous orientation of the swimmer. A schematic diagram for the instantaneous orientation (θ) of the swimmer is depicted in the inset to (a). |
The rotational motion of the swimmer is characterized by computing the mean-square angular displacement (MSAD) ⟨Δθ2⟩ from the simulated data for θ (with Δθ = θ − ⟨θ⟩); it is plotted as a function of t for different values of ${t}_{c}^{{\prime} }$ in figure 3(a). For all ${t}_{c}^{{\prime} }$ values, the long time regimes are found to be diffusive, whereas the initial transient regimes are neither ballistic nor diffusive. The deviation from ballistic and diffusive behaviours is found to be significant for higher values of ${t}_{c}^{{\prime} }$ (see inset to figure 3(a)). In order to characterize different dynamical regimes of MSAD (⟨Δθ2⟩), we introduce the parameter α as the instantaneous slope of the MSAD curves from figure 3(a), defined by the logarithmic derivative $\alpha (t)=\tfrac{\partial \mathrm{log}\langle {\rm{\Delta }}{\theta }^{2}\rangle }{\partial \mathrm{log}t}$, and plot it in figure 3(b) for different values of ${t}_{c}^{\prime} $. If MSAD is a function that takes the power-law form, its logarithmic derivative [α(t)] is a constant over several decades in time and represents the power-law exponent [48]. The exponents α = 2 and α = 1 correspond to ballistic and diffusive behaviour of motion, respectively. In the present model, for a given ${t}_{c}^{{\prime} }$ value, the long time behaviour is always diffusive as MSAD is a linear function on a logarithmic scale for several decades and α = 1. However, the initial transient regime is neither ballistic nor diffusive, as evident from the effective power-law exponent associated with the MSAD curves in figure 3(b) [48]. For a very low value of ${t}_{c^{\prime} }$ (${t}_{c}^{{\prime} }=0.1$), the angular motion is subdiffusive (α < 1) at short time scales. However, for higher values of ${t}_{c}^{{\prime} }$, the initial transient or short time regime is the complex interplay of subdiffusive (α < 1) and superdiffusive (α > 1) motions. With increase in ${t}_{c}^{{\prime} }$, we observe that the initial subdiffusive motion is followed by the emergence of superdiffusive behaviours, where the rotational trajectories become visible. Further increase in ${t}_{c}^{{\prime} }$ results in enhancement of both subdiffusive and superdiffusive regimes. The subdiffusive behaviour implies an induced transient spatial confinement or caging effect and the superdiffusive behaviour indicate the breaking of this confinement [48]. These observations clearly suggest that in the initial transient regime there is a transition from random self-propulsion to rotational motion due to the induced transient confinement. Moreover, the long time diffusive behaviour of MSAD is due to the disappearance of the induced confinement which persists for a short interval of time and diffuses away in the asymptotic time limit.
Figure 3. (a) Log–log representation of the MSAD versus t of the swimmer for different values of ${t}_{c}^{{\prime} }$. The downward arrow points to increasing values of ${t}_{c}^{{\prime} }$. The expanded view of the transient regimes of MSAD is shown in the inset to (a). (b) α versus t for different values of ${t}_{c}^{{\prime} }$. (c) ⟨ω⟩ versus t for different values of ${t}_{c}^{{\prime} }$. The other common parameters are D = tc = Γ = 1 and ${{\boldsymbol{r}}}_{0}=0\hat{i}+0\hat{j}$. |
With increase in ${t}_{c}^{{\prime} }$, although the swimmer performs a circular or elliptical trajectory with a specific orientation due to the stochastic change of direction of rotation there is on average no preferential orientation of the swimmer. That is why the mean angular velocity ⟨ω⟩ of the rotational orbits is always found to be zero (see figure 3(c)). In figure 3(c), we plot ⟨ω⟩ as a function of t for different values of ${t}_{c}^{{\prime} }$, and it is found to be zero for all ${t}_{c}^{\prime} $ values. Besides, fixing ${t}_{c}^{{\prime} }=200$, at which the elliptical trajectories are stabilized, if we further increase the value of tc the elliptical orbits become compressed (see figures 4(a)–(c)). This is a clear indication of effective spatial trapping or confinement of the swimmer (with non-zero finite inertia) in a non-Newtonian environment, even in the absence of any external confinement. However, the frequency of orientation reversal (ν) decreases with increase in tc and saturates to a constant value for very large tc values (figure 4(d)). This is because with increase in tc, the swimmer gets slower as the self-propulsion velocity varies inversely with tc. At the same time, it gets more directed, which results in a decrease in ν. The initial transient subdiffusive and superdiffusive angular motions (figure 3(b)) of the swimmer predict the presence of an induced transient spatial confinement, which might be of an effective harmonic type. As a consequence a swimmer with finite inertia performs a rotational trajectory like an inertial active walker in harmonic confinement [49, 50]. The shape of the trajectory is found to depend on both inertia and the memory time scale. Further, compression of the trajectory or spatial effective trapping of the swimmer with increase in tc complements the above features, as in [50, 51].
Figure 4. Instantaneous swimmer trajectories for different values of tc are plotted for (a) tc = 1, (b) tc = 10 and (c) tc = 50 for fixed ${t}_{c}^{{\prime} }=200$. Insets: corresponding time evolution of the instantaneous orientation. The frequency of directional reversals ν is plotted as a function of tc in (d). The other common parameters are Γ = D = 1 and ${{\boldsymbol{r}}}_{0}=0\hat{i}+0\hat{j}$. |
The trapping effect with tc is supported by the probability distribution of the 2D transient position of the swimmer (see figure 5). For a finite value of tc, the distribution takes a Gaussian shape. As tc increases, the distribution becomes narrow and transforms into a sharp peak structure (see figures 5(a)–(c)). It turns into a delta function as tc becomes infinitely large, reflecting strong confinement of the swimmer for a sufficiently long duration of activity. Finally, in order to understand the impact of noise strength in the rotational motion of the swimmer, we present the 2D trajectories of a swimmer in figure 6 for different values of D. It is observed that for deterministic dynamics, i.e. in the absence of noise (D = 0), the swimmer performs oscillatory motion along the diagonal line, as expected (figure 6(a)). In the presence of noise (D ≠ 0), for any arbitrary values of memory (${t}_{c}^{{\prime} }$) and inertia (Γ) even in the oscillatory regime (${\rm{\Gamma }}{t}_{c}^{{\prime} }\gt \tfrac{1}{4}$), the swimmer performs random self-propulsion (figure 6(b)). Fixing the D value, upon tuning either ${t}_{c}^{\prime} $ or Γ or both, the swimmer performs rotational motion in the oscillatory regime (see figure 6(c)). Fixing both ${t}_{c}^{{\prime} }$ and Γ, with further increase in D value, the rotational trajectories are enhanced without altering the shape and orientation (figures 6(d)–(e)). Hence, these observations clearly suggest that for a finite noise strength an inertial swimmer in a non-Markovian environment with suitable memory makes a transition from random self-propulsion to rotational motion. Moreover, the frequency of orientation reversal is found to be almost constant with D (figure 6(f)).
Figure 5. 2D probability distribution of the swimmer for (a) tc = 1.0, (b) tc = 50 and (c) tc = 100, respectively. The other fixed parameters are Γ = D = 1 and ${t}_{c}^{{\prime} }=200$. The time of observation is taken to be of 1000 time steps. |
Figure 6. Simulated trajectories of a microswimmer for different values of D: (a) D = 0, (b) and (c) D = 0.0001, (d) D = 1 and (e) for D = 10. In (b) ${t}_{c}^{{\prime} }=1$, in (a) and (c)–(e) ${t}_{c}^{{\prime} }=200$. The other common parameters are Γ = 1 and ${{\boldsymbol{r}}}_{0}=0\hat{i}+0\hat{j}$. |
In conclusion, we have theoretically probed the self-propulsion of an inertial-activated microswimmer in a non-Newtonian environment with finite memory. We investigate the motion of the swimmer by solving the generalized Langevin dynamics of an inertial active Ornstein–Uhlenbeck particle. The obtained results surprisingly uncover a transition from random self-propulsion to rotational (circular or elliptical) motion upon tuning both the inertial time scale and persistent duration of memory in the medium. This rotational motion is followed by spontaneous local orientation reversals, such that the average angular velocity is zero and the time asymptotic angular motion is diffusive. This implies that although the swimmer performs rotational motion, on average there is no preferential orientation of the rotational motion at steady state. Moreover, these rotational orbits of the swimmer appear only in the oscillatory regime of the memory–inertia (${\rm{\Gamma }}-{t}_{c^{\prime} }$) parameter space. Unlike a synthetic microswimmer with shape asymmetry [23], here the emergence of circular or elliptical orbits is due to the presence of both inertia of the swimmer and memory in the stochastic medium. Interestingly, such a swimmer with non-zero finite inertia effectively gets spatially trapped with increase in the duration of activity even in the absence of any external confinement. In the absence of either inertia or memory, rotational motion does not occur (see SM and footnote 2). Our findings might be relevant to Paramecium-type swimmers or inertial squirmers [35, 52].