Analysing the motion of scallop-like swimmers in a noisy environment

Abstract

A scallop-like swimmer going back-and-forth (reciprocal motion) does not produce any net motility. We discuss a similar artificial microswimmer that is powered by magnetic fields. In the presence of thermal noise, the helical swimmer exhibits enhanced diffusivity during reciprocal actuation. The external magnetic drive can be further modified to break the reciprocity. Equipped with only the information on swimmer trajectories and orientations, we discuss quantitative methods to estimate the degree of reciprocity and non-reciprocity in such scenarios. The paper proposes a quantitative measure and validates the same with numerical simulations, further supported by experiments.

1. Introduction

In a seminal paper in 1977, Purcell [1] describes the motion of microorganisms at low Reynolds numbers, defined as the ratio of inertial to viscous forces. For micron-sized entities like bacteria swimming in the water, the Reynolds number, Re << 1, such that the acceleration terms in the Navier-Stokes equation can be ignored. A consequence of this environment is reciprocal motion, which is described by Purcell using the example of a Scallop [1]. “...I change my body into a certain shape and then I go back to the original shape by going through the sequence in reverse. At low Reynolds number, everything reverses just fine...So, if the animal tries to swim by a reciprocal motion, it can’t go anywhere.” To summarize, the swimming pattern of a scallop is indistinguishable when observed forward and backward in time, and this makes it a reciprocal swimmer. Clearly, this kind of motion is not conducive for microorganisms and their survival in a Low Reynolds number environment. Interestingly, there are many ways to work around this limitation and break the reciprocal symmetry (using non-reciprocal strokes) to achieve net motility. Some bacteria swim by rotating chiral (helical) flagella which leads to translation [2] or use a flexible arm like the flagellum in a mammalian sperm [3] which pushes the cell forward due to its whip-like motion. Similar strategies have been used to even make synthetic swimmers at small scales and this is important [4] as they allow study of active matter [5–7] and have potential applications in biomedicine [8–15] and microfludics [16, 17].

There are various ways to energise an artificial system of swimmers; chemical means [18, 19], electric fields [20], light sources [21], and magnetic fields [22–25]. In this paper, we will focus on magnetically powered helical swimmers that propel along a certain direction due to an external rotating magnetic field [22, 23]. It is essential to recognize that while the helical structures can be propelled in any desired direction with a rotating magnetic field, they are not self-propelled as both their speed and direction are determined by an external field. As we discuss this issue in greater detail later, please note, this problem can be solved partially by using oscillating fields [26] to have reciprocal motion, to make the swimmers more diffusive [26, 27], which increases further near a surface [28]. There are ways to break the reciprocal symmetry using magnetic drive [29], and Brownian ratchet principle [30] and also employing non-Newtonian properties, e.g. shear thinning [31] or viscoelastic nature of the surrounding medium [32, 33].

Considering the experimental system shows scalloplike motion with reciprocal strokes and occasionally broken reciprocal symmetry, we ask the question; can we analyze this motion in the presence of thermal noise and estimate the degree of reciprocal motion? To this end, we describe a technique for analyzing the trajectories of the swimmers in a noisy ambiance. We have further validated this method with experimental observations. The methodology devised in this work is not exclusive to the helical swimmers but generally applicable to any swimmer that performs a back-and-forth motion (irrespective of rectification) synchronized to an external drive in a noisy environment.

2. Results

The helical swimmers are fabricated using the Glancing Angle Deposition technique. In this process, wafer-scale micron-sized columnar structures are grown by a physical vapor deposition method, where the substrate is held at an extreme angle [34]. The method works on the principle of ’ballistic shadowing’, where the first nucleation site shadows the region behind it. The helical structures are about 3 μm in length and made of silica (Fig. 1A) with magnetic elements like iron and cobalt embedded into the structures. These structures are further magnetized along the short axis of the swimmer with a large (> 50 mT) DC field. The wafer is subjected to ultra-sonication to release the helices into de-ionized water and a few micro-liters of the liquid are placed in a microfluidic chamber. A tri-axial Helmholtz coil (Fig. 1A) which produces a uniform magnetic field at the centre is used to drive these helical structures.

Fig. 1.

A The schematic of a triaxial Helmholtz coil and an SEM image of the swimmer. B The swimmer responds to the external oscillating field in the Z direction. C Considering the probabilities of turning (CW or CCW), four sequences are discussed. D Plane of pq parameters encompassing the reciprocal and non-reciprocal swimming

As a helical nanostructure is magnetised along its short axis, upon the action of a rotating magnetic field, the object turns about its long axis; hence, bringing about a corkscrew motion that enables it to propel forward. The magnetic moment of the swimmer can be calculated by observing its dynamics [35, 36] under a rotating field with the moment, m = ω₂γ_s/B where γ_s is the rotational friction co-efficient about the short axis, B is the magnetic field amplitude and ω₂ is the upper limit of frequency of the stable phase-locked regime. Within this regime, the swimmer’s magnetic moment follows the applied field with a constant phase difference [35, 36]. However, this system still does not qualify as a self-propelled system because the direction of motion are coupled with the sense of rotation of the external field. To realize a system which consumes energy from external sources, yet having direction independent of the external drive, we applied an oscillating field, perpendicular to the 2D sample plane (also the imaging plane) (see Fig. 1A). In the absence of thermal noise, the arbitrarily oriented swimmer performs forward-backwards strokes which can be imagined as a consecutive series of clockwise (CW) and counter-clockwise (CCW) rotations (see Fig. 1B), and therefore move along in the direction of their long axis by an amount of p_h/2 where p_h is the hydrodynamic pitch of the swimmer. The dynamics which cause such a motion are dependent on two counteracting torques on the swimmer, one is due to the weight and the other due to the magnetic moment and is extensively discussed in the previous works [26, 29]. We define the strokes as forward (FW) (during a clockwise turn, CW) and backward (BW) (during a counterclockwise turn, CCW) and create a parameter space to understand it better. (Fig. 1C, D).

1. Reciprocal swimmer

In the absence of thermal noise, the swimmer shows a perfect sequence of alternate CW and CCW rotations. The probability that CW turn becomes CCW is zero and is given by p= probability (CCW → CW = 0) and for the reverse case, q = probability (CW → CCW = 0).

2. Reciprocal swimmer (time-averaged)

In the presence of noise in the system, the perfect sequence is no longer feasible, and hence the probabilities of p and q are no longer 0, yet they are equal when averaged over a long duration.

In the presence of thermal noise, diffusion of a reciprocal swimmer is enhanced [26, 27] compared to the Brownian motion. Thermal noise plays a crucial role here which is responsible for both the orientational diffusivity in the XY plane and an imperfect reciprocal sequence (Fig. 1C). Despite this, we still have an equal number of CW-CCW on average, described as case 2 above.

To overcome this limitation, we can break the reciprocity in a few ways to obtain a self-propelled swimmer showing non-reciprocal dynamics, described as case 3 later.. One such method was recently reported [29] where we used the principle of a Brownian ratchet to break the time reciprocity. This principle allows the particle to achieve net motility in a non-equilibrium system with the aid of fluctuations and an asymmetric potential [37, 38]. This is implemented in our system with an asymmetric design of the magnetic field where each oscillating cycle has two timescales and probabilities of forward–backward strokes are tied to them [29]. Alternately, the presence of a surface in close proximity to the reciprocal swimmer can result in an anomalous rise of diffusion. This is due to the difference in interaction during forward and backward strokes, where the swimmers achieve a much higher diffusivity [28] and sometimes even directed motion [39].

3. Non-reciprocal swimmer

We are faced with a third possibility, which can happen both in the presence or absence of noise where p ≠ q ≠ 0. This results in more CW (or CCW) than CCW (or CW) rotations, and a broken CW-CCW symmetry. When p=1 and q=0 or p=0 and q=1, the case corresponds to a continuous rotation, observed during the rotating magnetic field.

To correlate the probability parameters pq with the experimental observations, we consider a parameter plane pq (see Fig. 1D). Point (0,0) would represent the perfectly reciprocal swimmer and (0.5,0.5) represents the highest degree of anomalous strokes in a reciprocal swimmer. While (0,1) and (1,0) represent the perfect non-reciprocal swimmer, it is possible under a rotating magnetic field. Our current experimental methods adhere to oscillating fields only and it allows us to attain a maximum of 0.5 probability of anomalous strokes. Hence, (0.5,0) and (0,0.5) represent the highest possible non-reciprocal swimming strokes in our current discussion.

However, directly measuring these experimental parameters is quite challenging. One way could be to physically count the number of CW and CCW turns which requires highly involved imaging techniques. As an easier method, we consider the position and orientation information of a swimmer in the XY plane, which are easily extractible. The position information can also be used to estimate the enhanced diffusivity of the active swimmers.

We first perform a numerical simulation, where the helical swimmer is imagined to be in an XY plane with its center of mass located at r(t) = [x(t), y(t)] in the lab frame as shown in Fig. 2A. The helical structure makes an angle θ with the X-axis. The time step between the displacements is 1/1000s. A stochastic equation is numerically solved to derive the positions of the swimmer at a series of time steps. We use the Euler’s numerical method to arrive at

$$X\left(t+\text{d}t\right)=X\left(t\right)+\sqrt{2D}\xi \left(t\right)\text{d}t-\frac{D}{k_{B}T}\text{d}t\nabla U$$ (1)

where ξ(t) is a delta-correlated Gaussian process with zero-mean, such that < ξ(t)ξ(t′) >= δ(t – t′). D = k_BT /γ is the diffusion constant of the colloidal particle and includes the fluid viscosity. In the case of a helical swimmer, we approximate it to a cylinder for simplification [36, 40, 41]. The terms owing to thermal fluctuations in the Langevin equation are not straight-forward as in the case of a spherical particle, and hence we write the noise part as follows [42]

$$\left[\begin{array}{c}\dot{x}\\ \dot{y}\end{array}\right]=\left[\begin{array}{cc}\sqrt{2D_L}co{s}^2\theta +\sqrt{2D_s}si{n}^2\theta & \left(\sqrt{2D_L}-\sqrt{2D_s}\right)cos\theta sin\theta \\ \left(\sqrt{2D_L}-\sqrt{2D_s}\right)cos\theta sin\theta & \sqrt{2D_L}si{n}^2\theta +\sqrt{2D_s}co{s}^2\theta \end{array}\right]\left[\begin{array}{c}{\xi }_x\\ {\xi }_y\end{array}\right]$$ (2)

$$\dot{\theta }=\sqrt{2D_{\theta }}{\xi }_{\theta }$$ (3)

Fig. 2.

A The swimmer in the lab frame projected on its body frame, calculated using a rotation matrix. B The graph plotted shows how the swimmer displacement should have been in an ideal case in the absence of any noise and under an oscillating frequency of 5 Hz, whereas the graph at the bottom portrays the actual displacement data obtained. C The Fourier transform of the noisy data gives a peak at 5 Hz which corresponds to the driving frequency. D The Fourier peak derived from the experimental data might vary from the simulated data due to the errors in the rate of image acquisition. (Inset) The frames of a video are acquired at 90 fps (time period 0.01 s) but there is a variation in the rate of acquisition as shown in the histogram

The positions of the particle in XY plane are updated as follows:

$$\begin{array}{l}x\left(t+\text{d}t\right)=x\left(t\right)+\dot{x}\text{d}t+v_x\text{d}t\\ y\left(t+\text{d}t\right)=y\left(t\right)+\dot{y}\text{d}t+v_y\text{d}t\\ \theta \left(t+\text{d}t\right)=\theta \left(t\right)+\dot{\theta }\text{d}t\end{array}$$

Here, $\dot{x},\dot{y}\text{and}\dot{\theta }$ are the diffusive terms and v_x and v_y are the lab-frame velocity components of the swimmer. D_L and D_s are the translational diffusion coefficients of the rod along the long and short axis, respectively and D_θ is the rotational diffusion coefficient about the short axis. The values of diffusion coefficients are as follows:

$$D_L=k_{B}T/{\gamma }_1,D_s=k_{B}T/{\gamma }_2,D_{\theta }=k_{B}T/{\gamma }_3$$ (4)

Here, γ₁, γ₂, and γ₃ are the friction coefficients given by the mobility tensor [35, 36, 40]. The diffusion coefficients can be estimated theoretically, which compares well with the experimental measurements. The speeds correspond to the motion of the swimmer showing forward and backward motion along the long axis, transformed to lab frame. When the field is switched from +B to -B as shown in Fig. 1B, then the swimmer, within a duration δt does a clockwise turn and while doing so moves a distance of p_h/2 where p_h is the hydro-dynamic pitch of the swimmer. Conversely, when the field is switched from -B to +B, the swimmer does a counterclockwise turn and moves a distance of p_h/2. In the simulation, we consider a probability distribution set according to the definition of pq plane. We assign a probability to each turn and this results in a certain velocity (p_h/2δt) in either a forward or backward direction. For the swimmers considered here, p_h ranged between 400 and 600 nm, while δt depended on the frequency of magnetic actuation, and ranged between 7 ms and 11 ms. It is important to note that though the displacement is assumed to be the same (p_h/2) in simulations, the presence of the surface can modify this. Although not shown, this can be easily included in the analysis method described here.

The change in position of the swimmer after each time step is given by δr = r(t + 1) – r(t). The displacement can be decomposed in the body axis along the short and long axes of the swimmer. δr is expressed as δx′ and δy′ relative to the body frame axis (SL) [43]. Figure 2A shows how the two axes are related due to a rotation, δx′ = Rδx where $R=\left(\begin{array}{cc}cos\theta & sin\theta \\ -sin\theta & cos\theta \end{array}\right)$ Here, θ = [θ(t + 1) + θ(t)]/2, as the θ barely changes during 1/1000s. Normally, with no noise in the system, the alternate sequence of positive and negative displacements corresponding to forward–backwards motion is easy to recognise as shown in the top graph of Fig. 2B. But in the experimental data, the role of thermal noise is significant, as shown in the graph described in the lower portion of Fig. 2B. This makes it difficult to understand what is happening with the swimmer under an oscillating field.

We consider the Fourier space to investigate if any information can be extracted related to the forward–backwards strokes of the swimmer, as shown in numerical and experimental data shown in in Fig. 2C and (Fig. 2D). The physical meaning of these observations will be discussed later. In this analysis, the Fast Fourier Transform was implemented in MATLAB for a sequence of δx′ displacements for a total time T (see Eq. 5).

$$y\left(k\right)={\displaystyle \sum _{t=1}^T\delta x^{\prime }\left(t\right)e^{\left(-2\pi i\right)\left(t-1\right)\left(k-1\right)/T}}$$ (5)

Comparing the simulated and experimental data are not completely straightforward due to errors creeping into the recording of data, especially due to variation in the frame rate. To verify this issue, a DAQ is connected to the camera via LabView and frames are captured at 90 frames per second (fps) for several minutes. In Fig. 2D (inset), the histogram represents the data acquired through the DAQ, showing a distribution of acquired rate of frames. If we take this into consideration and incorporate the errors in our numerical model, we observe changes in the Fourier peak as shown in Fig. 2D. Although the peak height slightly decreases, as we will see further, this does not interfere with the validity of the technique. The peak characteristics start to differ significantly (not shown here) as the error goes up but fortunately, the error due to the camera recording (Fig. 2D (inset)) is not high enough. This suggests that even with the existing experimental error, the Fourier peak remains a reliable method to extract additional information about the nature of a swimmer.

In Fig. 3, we plot few examples of the Fourier transform of the displacement signal for certain values of pq. For reasons that will be clear later, we separately consider the time series of expected forward and backward displacements of the swimmer. This was possible because the external actuation time series was known, so for each time step, it was possible to assign an expected displacement, in either forward or backward direction, assuming a perfect reciprocal sequence.

Fig. 3.

Fourier transform of the displacement signal for sample values of pq. This can be done in two ways; with forward displacement series and backward displacement series

For example, (A_f)_FW corresponds to the Fourier peak amplitude for displacements expected to occur in the forward direction at frequency f.

With p=q=0, we observe a single peak at the actuating frequency (5 Hz), which is expected as the sequence of displacements would have an alternating forward–backward step at actuation frequency f. However, if we change pq to a non-zero value, we observe a peak at twice the actuating frequency (2f). This occurs due to anomalous turns where two turns of the same type are repeated, resulting in the emergence of an additional frequency at 2f. Indeed, even higher harmonics can emerge in principle but corresponding events are less probable and therefore not considered here. Also interesting to note is when a swimmer performs reciprocal strokes i.e. p = q ≠ 0, then the amplitude of a peak at frequency 2f is almost equal, implying A_2fFW = A_2fBW. On the contrary, if there are non-reciprocal strokes, i.e p ≠ q ≠ 0, then there are higher (or lower) chances of a peak at 2f in forward (or backward) displacements, i.e. A_2fFW ≠ A_2fBW.

The take away message from Fig. 3 is how the peak height at frequencies f (here 5 Hz) and 2f (here 10 Hz) vary as a function of the probability parameters p and q. With p and q as independent variables, two functions can be expressed as f_FW (p, q) = (A_2f/A_f)_FW and f_BW (p, q) = (A_2f/A_f)_BW. These two functions can be numerically simulated and plotted as a function of p and q, representing different combinations of reciprocal and non-reciprocal strokes, as shown in Fig. 4A, B. Here, the swimmer is actuated at a frequency of 5 Hz. In parallel, experimentally obtained trajectories of a swimmer can be analyzed to estimate the Fourier peak amplitudes. Subsequently, given f_FW (p, q) and f_BW (p, q), we could deduce the pq values corresponding to the experimental data.

Fig. 4.

A The pq plane where the ratios of FW displacement are plotted as a function of p and q. B Similarly, the ratios of BW displacement are plotted as a function of p and q

To validate whether mapping of the experimental data to the numerical results shown in Fig. 4 give accurate information about the experimental p, q, we consider a swimmer undergoing both reciprocal and non-reciprocal actuation at 5 Hz. This experimental data is further analysed and compared with Fig. 4A, B to arrive at the values of p=0.1, q=0.12 for reciprocal strokes and p=0.3, q=0.22 for non-reciprocal strokes (see Fig. 5A). Assuming these specific values of pq, we calculated the mean square displacement of the swimmer as a function of time, which was found to match the experimental diffusivity. Considering the motility of the swimmers was a function of pq, it is expected that the diffusivities would depend on pq as well. Figure 5 illustrates various examples of different swimmers and the corresponding pq values deduced using f_FW (p, q) and f_BW (p, q) as the lookup tables from Fig. 4.

Fig. 5.

The figure describes various swimmers performing reciprocal and non-reciprocal strokes and their corresponding pq values. It can be observed that the diffusivity also varies according to the variation of pq. A, B The experimental and simulated data for a swimmer actuated at 5 Hz where pq values are obtained from the parameter plane. C The experimental and simulated data for a swimmer actuated at 10 Hz

It is important to note that the method described here provides greater microscopic information about the swimmer, and can distinguish between various degrees of reciprocity (or non-reciprocity) of a swimmer from simple trajectory and orientation analysis. As expected, the higher the non-reciprocal strokes, higher the diffusivity of the swimmer and hence as you move away from the diagonal of the pq plane, the diffusivity of the swimmer increases.

3. Conclusion

The artificial scallop-like swimmers can exhibit both reciprocal and non-reciprocal swimming strokes. One such example is the magnetically driven helical nanoswimmers whose dynamics have been discussed in detail in previous works [26, 28, 29]. In this paper, we describe a technique to extract information from the trajectories of the swimmer and process it to identify the degree of reciprocity and non-reciprocity. This exercise can be extended to all scallop-like swimmers where the time stamp of the sequence of strokes is known.

Data availability statement

No data associated in the manuscript.