Propulsion and controlled steering of magnetic nanohelices

TL;DR

This paper presents a simulation-based approach to design and control magnetic nanohelices for targeted propulsion and steering in fluid environments, overcoming thermal fluctuations at the nanoscale.

Contribution

It introduces a quantitative simulation methodology for designing magnetic nanohelices that can be accurately propelled and steered despite Brownian motion.

Findings

Successful propulsion of 30 nm magnetic nanohelices demonstrated

Protocols for controlled steering at biological temperatures developed

Fast transport achieved with external magnetic field manipulation

Abstract

Externally controlled motion of micro and nanomotors in a fluid environment constitutes a promising tool in biosensing, targeted delivery and environmental remediation. In particular, recent experiments have demonstrated that fuel-free propulsion can be achieved through the application of external magnetic fields on magnetic helically shaped structures. The magnetic interaction between helices and the rotating field induces a torque that rotates and propels them via the coupled rotational-translational motion. Recent works have shown that there exist certain optimal geometries of helical shapes for propulsion. However, experiments show that controlled motion remains a challenge at the nanoscale due to Brownian motion that interferes with the deterministic motion and makes it difficult to achieve controlled steering. In the present work we employ quantitatively accurate simulation methodology to design a setup for which magnetic nanohelices of 30 nm in radius, with and without cargo, can be accurately propelled and steered in the presence of thermal fluctuations. In particular, we demonstrate fast transport of such nanomotors and devise protocols in manipulating external fields to achieve directionally controlled steering at biologically relevant temperatures.

Equations

$$\left(\begin{array}{c}\mathbf{F}\\ \tau \end{array}\right)=\left(\begin{array}{cc}{\xi }^{\text{TT}}& {\xi }^{\text{TR}}\\ {\xi }^{\text{RT}}& {\xi }^{\text{RR}}\end{array}\right)\left(\begin{array}{c}\mathbf{v}\\ \omega \end{array}\right),$$

Full text

Introduction

The development of artificial nano and micromotors that can be controlled accurately and precisely in spatial and temporal scales has attracted rapidly increasing interest. They are typically catalytically driven [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] or magnetically propelled [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. One of the key issues for such nanomotors to be useful for practical purposes is remote motion control and maneuverability. This has been demonstrated where thermal fluctuations are not important such as in the transport, guidance, and release of cells [20, 21, 22], single-cell targeted delivery [16] and oil droplet capture [25, 26]. Catalytically driven micromotors rely on chemical gradients for self-propulsion, and motion control is made possible through externally manipulated magnetic fields [10, 11, 12, 13]. Catalysis-propelled particles require fuel such as hydrogen peroxide, acidic or alkaline solutions which poses a challenge in their biocompatibility for in vivo applications [27]. This challenge can be alleviated by creating catalytically driven micromotors that require low concentrations of fuel without compromising their propulsion speeds [28] or using micromotors that propel in biocompatible solutions [29, 30]. Magnetic propellers, on the other hand, are suitable candidates for in vivo applications since they do not require fuel and most soft biological tissues have very low magnetic susceptibilities [31]. Such propellers have been shown to be driven and elaborately maneuvered, and can perform complex tasks such as cargo fetching and delivery [15, 17, 18, 21]. Their propulsion speeds can be remotely controlled by tuning the field frequencies.

Scaling magnetic propellers down to the nanosize presents a major challenge for controlled motion and maneuverability because of thermal fluctuations[6, 13, 18]; thermal Brownian motion can severely alter the direction of motion and interfere with the propulsion. Schamel et al. [18] circumvented the challenge in nanoscale steering by driving the nanopropellers at low frequencies in gels and highly viscous media to damp out thermal fluctuations. In water where biology occurs, however, no propulsion was observed as Brownian effects dominated the propulsion [18].

In our previous work [32], we studied the shape dependence of rotation and propulsion of helically shaped magnetic nanoparticles at constant torque. We found that the maximum propulsion is achieved by balancing competing requirements for rotational stability and minimizing viscous drag. Of various helical shapes, we found that the helix with a circular cross-section and number of turns $N=1.25$ and length to radius ratio $L/R\approx 6$ exhibits maximal propulsion velocity (cf. Fig. 1 and Fig. LABEL:SI-Fig1). Furthermore, for well-defined propulsion and chiral separation in the presence of Brownian motion, the Péclet number must be larger than about $50$. In the present work, we extend our previous work [32] to the experimentally relevant case where the helix has a fixed magnetic moment and it is driven by an external magnetic field with varying frequency $f$. In particular, we show how steered propulsion can be optimized as a function of the magnetic interaction and $f$. We also consider the case where the helix functions as a cargo-carrying nanomotor, and present a protocol for changing the field for controlled directional steering of the nanopropellers. We use the recently developed fluctuating lattice-Boltzmann method [33, 34, 35, 36, 37] to quantitatively model the fluid environment and the nanopropellers. Details about the method are provided in the Supporting Information (SI).

Coupled translational and rotational motion by magnetic field

At the scales of micron and less, micro-organisms navigate in water based fluids at very low Reynolds numbers ${\rm Re}\ll 1$. They have developed, through billions of years of evolution, helically shaped appendages or flagella with locomotion as the primary function [39]. Artifical micropropellers have a similar shape as the flagella such that as the propeller rotates in the fluid, it pushes the fluid along the direction of its easy axis of propulsion. At low Re, this motion is described by the generalized resistance tensor:

[TABLE]

where $\mathbf{F},\mathbf{\tau},\mathbf{v},\mathbf{\omega}$ and $\mathbf{\xi}^{\alpha\beta}$ are the applied force, applied torque, translational velocity, angular velocity, and the friction matrix of the particle, respectively [40, 41]. Bodies with spherical symmetry have zero non-diagonal elements in their resistance tensor. Screw-shaped bodies have helicoidal symmetry and non-zero off-diagonal terms in Eq. 1. This means that the translational and the rotational motion are coupled i.e. rotational motion will result in translational motion and vice versa. Rotation of a magnetic helix can be induced by a torque that is generated by the interaction of its magnetic moment $\mathbf{M}$ with an external magnetic field $\mathbf{B}$. As indicated in Fig. 1, here we set $\mathbf{M}$ to be uniform and perpendicular to the helix’s easy axis of rotation. An external field is applied and the torque on the helix is given by $\mathbf{\tau}=\mathbf{M}\times\mathbf{B}$ (see Fig. 1). The field frequencies used in the simulations are $f=10-10000$ kHz. The state-of-the-art nanopropeller’s magnetic moment, which were driven in highly viscous media, is estimated to be $2\times 10^{-14}$ emu [18]. For $MB=(1.5-2.5)\times 10^{-18}$ Nm in the simulations and using the experimental value of the magnetic moment, the magnetic field strengths required would be $75-125$ mT. The fastest micropropellers (of 100-150 nm radius) had a propulsion speed of $\sim 40$ $\mu$m/s in water at 150 Hz and 5 mT [14].

Results

Dependence of Propulsion on Frequency

In the limit where Brownian motion can be neglected (${\rm Pe}\gg 1$), the total net forces and torques on the helix are due to its magnetic interaction with the external field and the hydrodynamic drag. For propulsion, an external magnetic field of constant amplitude is rotated along a plane with frequency $f$. The direction of propulsion is perpendicular to the plane of the rotating field (cf. Fig. 1). Both the magnitude of the applied torque and the hydrodynamic drag increase from zero up to $\kappa_{\parallel}f$ for frequencies that are less than $MB/\kappa_{\parallel}$, where $\kappa_{\parallel}$ is the viscous rotational drag coefficient; Fig. LABEL:SI-Fig3a shows the response of the hydrodynamic drag on the rotation of the helix. Once the helix has achieved an angular frequency of rotation that is equal to that of the magnetic field, the difference between the applied torque and the magnitude of the hydrodynamic drag goes to zero, and the phase difference between the magnetic moment and the magnetic field becomes constant (Fig. LABEL:SI-Fig3b).

Increasing the frequency of the external field enhances propulsion due to the translational-rotational coupling (Eq. 1), as shown in Fig. 2. The propulsion is linear with the frequency since the torque attains a constant value at steady state. However, when the frequency is greater than a threshold step-out value $f_{\rm SO}=MB/\kappa_{\parallel}$, the helix eventually lags the magnetic field ((Fig. LABEL:SI-Fig4)). For frequencies less than the step-out frequency, the phase difference reaches a constant value below $\pi/2$ and the resulting external torque also attains a constant value (Fig. LABEL:SI-Fig5). For field frequencies greater than the step-out frequency, the phase difference “steps-out” of $\pi/2$ and consequently varies periodically as the process of the magnetic moment lagging the magnetic field and the magnetic field catching up to the magnetic moment repeats itself. Accordingly, the external torque varies periodically for $f>f_{\rm SO}$ as shown in Fig. LABEL:SI-Fig5. This is reflected in the strong reduction in propulsion beyond $f_{\rm SO}$ (Fig. 2). We find that the step-out frequency of the helix with an attached cargo (Fig. LABEL:SI-Fig5b) (a prolate spheroid which minimizes the Stokes drag coefficient) is smaller by about a factor of two than the step-out frequency of the helix without the cargo (Fig. LABEL:SI-Fig5a). This is consistent with the fact that the helix with an attached cargo has a greater viscous drag coefficient due to the additional surface area of the cargo than a helix without an attached cargo (the drag coefficients are given in Table LABEL:SI-Table2).

Péclet numbers

The importance of Brownian motion to propulsion can be quantified by considering the Péclet number, which is the ratio between the diffusive and the advective time scales. A Péclet number greater than one means that the advective motion dominates over thermal effects. For translational motion, the Péclet number is given by $\text{Pe}_{\rm T}=vL/D_{\rm T}$ and for rotational motion $\text{Pe}_{\rm R}=\Omega/D_{\rm R}$, where $D$, $v$ and $\Omega$, and $L$ refer to the diffusion coefficient, the propulsion velocity, angular velocity, and length of the helix, respectively [18]. In our previous study we found that nanohelices could be reliably propelled for Pe${}_{\rm T}\gtrsim 50$ [32]. For the $30$ nm helices here, the minimum frequencies for $\text{Pe}_{\rm R}>1$ are at $(1558\pm 5)$ and $(3095\pm 60)$ Hz with and without cargo, respectively (cf. Table LABEL:SI-Table1). The frequencies used in the simulations thus result in Péclet numbers that are much larger than unity for both with and without cargo as shown in Fig. 3, where we plot the propulsion velocities and the corresponding translational Péclet numbers for two different values of $MB$ with Brownian motion at $T=300$ K. We note that the highest propulsion speed of the nanohelices that were driven at $MB=2.0\times 10^{-18}$ Nm and $f=149\times 10^{4}$ Hz is $\sim 3.5\times 10^{-3}$ m/s which is $\sim 90$ times greater than the propulsion of the fastest experimental micropropellers that were driven at $150$ Hz ($\sim 40\times 10^{-6}$ m/s) [14, 42].

Directional steering of nanohelices

In addition to steering propellers along a straight path, directional maneuverability is a crucial aspect for the application of helices as nanomotors and cargo carriers. For the magnetic propellers, reversing the direction of motion is a matter of switching the direction of the rotation of the external magnetic field. Alterations in the direction of motion may be done by changing the direction of the torques or changing the perpendicular plane of the rotating field. Here we demonstrate steering both with and without thermal fluctuations and at high Péclet numbers (${\rm Pe_{T}}=370$ and 526 for 700 kHz and 1000 kHz, respectively) such that the external driving is significantly greater than Brownian motion.

Two types of steering were tested corresponding to high and low values of $MB$. The aim at high $MB$ steering is to avoid the scenario where the magnetic moment of the helix steps out of the $\pi/2$ phase with the magnetic field. High $MB$ steering subjects the helix to high external torques during the turn. Low $MB$ turning, on the other hand, results to turns for which the helix steps out of the $\pi/2$ phase for a short time and simultaneously rotates the helix in the opposite direction of the field rotation until it synchronizes back into the rotation of the magnetic field.

Figures 4(a)-4(d) demonstrate the case of high $MB$ steering. Two $45^{\circ}$ turns were performed at $1000$ kHz and $MB=10\times 10^{-18}$ Nm. The steering protocol is as follows: first, the nanohelix is propelled for $2.5\times 10^{6}$ time steps by applying an external torque in the $-x$-direction. Then the direction of the torque is instantaneously changed towards $-(\hat{x}+\hat{y})$ for the same amount of time steps (Fig. LABEL:SI-Fig6). In the final part, the helix is propelled in the $-y$-direction. Small oscillations in the displacement along the long axis are present as shown in Figs. 4(b) and 4(c) for $0<t<0.7\times 10^{-4}$ s when the helix is propelled along the $-\hat{x}$, and for $1.4\times 10^{-5}<t<2.1\times 10^{-4}$ s when the helix is propelled along $-\hat{y}$. These oscillations in the displacement are not observed in the direction of the propulsion as shown in Fig. 4(a) and in the transformed coordinates $x^{\prime}$ rotated by $45^{\circ}$ from the $x$-axis when the helix is turned by $45^{\circ}$ at $t=0.7\times 10^{-5}$ s (Fig. 4(d)). These turns were done at high $MB$ such that the magnetic moment of the helix does not step out of the $\pi/2$ phase with the magnetic field. Despite this, the large torque causes the helix to turn with a sudden motion causing a sharp jump in the displacement during the turns (at $t=0.7\times 10^{-5}$ s and at $t=1.4\times 10^{-5}$ s). These jumps are more prominent in the parametric plot of the trajectory in Fig. 4(e) where it can also be seen that the intended trajectory is not closely followed by the helix.

In contrast, for low steering at $MB=1.5\times 10^{-18}$ Nm with the steering protocol described above, the magnetic moment of the helix momentarily steps out of the $\pi/2$ phase difference. During this stage, it rotates in a direction opposite to the magnetic field until its rotational motion is synchronized with that of the field. This makes controlled directional steering difficult to achieve as demonstrated in Fig. LABEL:SI-Fig10 with abrupt, sharp $90^{\circ}$ turns.

To overcome this difficulty and to ensure that the helix is kept synchronized with the magnetic field, gradual changes in the direction of the field must be made. To this end, we have devised the following steering protocol. At the beginning the helix is first synchronized with the field (by propelling it for $15$ rotations of the magnetic field here). The actual turning begins by fixing the magnetic field in a given direction for a period of time $\delta t=T_{1}$ to allow the helix to realign with the fixed field as it prepares to be steered. To achieve a turn, the field is then rotated in small increments that add up to $\theta$. After this the field direction is fixed again for $\delta t=T_{3}$. The time interval for which the gradual changes in the field are done must be $f^{-1}<\delta t<\tau_{\text{relax}}$, where $\tau_{\text{relax}}\approx\delta R^{2}/D_{T}$ is the time scale to traverse a distance of $\delta R\leq L$ by diffusion. In this manner asynchronization is avoided and successful $\theta=45^{\circ}$ and $\theta=90^{\circ}$ turns are demonstrated for $MB=1.5\times 10^{-18}$ Nm in Figs. 4(f)-4(i). Without Brownian motion (blue trajectories in Figs. 4(g) and 4(i)) the intended trajectory is faithfully followed due to the lack of sudden jumps. In the presence of thermal fluctuations, the trajectory follows the intended path within the region of diffusive spreading which can be mitigated by optimizing the propulsion velocity. In the simulations we chose $T_{1}=4.4\times 10^{-6}$ s, $T_{2}=2.9\times 10^{-6}$ s and $T_{2}=1.5\times 10^{-6}$ s.

The shaded regions in the parametric plots (Figs. 4(g) and 4(i)) represent the regions of diffusional spreading caused by Brownian motion at $T=300$ K as obtained from the mean square displacement measurements. This inherent spreading, of width $\delta s$, that is orthogonal to the direction of propulsion is controlled by the Brownian tracer diffusion coefficient of the helix during the straight segments of the path by $\delta s\approx\sqrt{D_{\rm T}t_{s}}$, where $t_{s}$ is the duration of time from the start of the propulsion. Thus, diffusional spreading can be mitigated by increasing the propulsion velocity (for a given path length), or reducing $D_{\rm T}$ by lowering the temperature, increasing the propeller size and increasing the viscosity of the fluid [32].

Finally, we demonstrate that smooth turns through curved or circular trajectories with or without thermal fluctuations are also possible as illustrated in Fig. 5. The helix is driven at $700$ kHz with $MB=1.5\times 10^{-18}$ N m and the perpendicular plane of the rotating magnetic field is rotated by $d\theta=\pi/\delta t^{\prime}$ for every time step where $\delta t^{\prime}>f^{-1}$ (in the simulations $\delta t^{\prime}=6.28f^{-1}$). Although the rotation of the perpendicular plane of the magnetic field is along the $\hat{y}$ direction, the resulting path shows a drift in the $y$ direction such that the intended circular path becomes helical. Left and right-handed helices were simulated with different combinations of clockwise and counterclockwise field rotations and directions of steering. In the deterministic simulations, we find that when the helix rotates such that it pushes the fluid inside the circular trajectory, the helix drifts in the opposite direction of the displaced fluid (cf. the diagram in Figs. 5(c) and 5(d))111A vector field plot of the fluid velocities can be found in the SI.. In the presence of thermal fluctuations, this drift is totally overcome by Brownian motion since the Péclet number $\rm{Pe_{T}}=8$ in the $z$-direction.

Summary and conclusions

In conclusion, we have shown that nanoscale helical propellers can be used for controlled motion and steering at high Péclet numbers. The propulsion velocities and Pe of the propellers can be conveniently adjusted through the magnetic field frequencies for frequencies that are less than the step-out value, which can be determined from the ratio of the magnetic interaction to drag as $f_{\rm SO}=MB/\kappa_{\parallel}$. Extending the range for higher propulsion velocities is possible by increasing both the field frequency and $MB$ or reducing drag. Compared with the fastest artificial controllable chiral micropropellers of $\sim 150$ nm in size, which have a maximum propulsion speed of $\sim 40$ $\mu$m/s [14, 42], our results show that nanoscale propellers of $\sim 30$ nm in size can be propelled much faster with $\sim 10^{-3}$ m/s with full control. With these propulsion speeds, it is possible to attain ${\rm Pe_{T}}>100$, allowing spatial and temporal control of the motion in the presence of thermal fluctuations.

Computational model

In the limit of $\rm{Re}=0$, the corresponding Stokes equations for particles of arbitrary shapes can be numerically solved and the friction matrix in Eq. 1 unraveled. In our recent work on helical particles we have, however, shown that the Stokes approach is not sufficient for quantitative accuracy in the case of externally driven nanohelices [32]. To this end, we employ the fluctuating lattice-Boltzmann - Molecular Dynamics (LBMD) method of Ref. [33]. It incorporates full Navier-Stokes hydrodynamics with consistent thermal fluctuations and a coupling of the fluid to extended MD particles of arbitrary shapes. The fluctuating LBMD has been extensively benchmarked for colloids and polymers [43, 44, 45]. In the present case, the translational (rotational) Reynolds number for a helix of radius $R$ and length $L$ can be shown [32] to be $\text{Re}_{\text{T}}=\rho vL/\eta$ ($\text{Re}_{\text{R}}=\rho\omega R^{2}/\eta$), where $\rho$, $v$, $\omega$ $\eta$ are the fluid density, propulsion velocity, angular velocity and fluid viscosity, respectively. In the simulations, the typical translational (rotational) Reynolds numbers are about $10^{-4}-10^{-5}$ ($10^{-4}-10^{-2}$). Details about the method are provided in the Supporting Information (SI).

Acknowledgements (not compulsory)

This work was supported in part by the Academy of Finland through its Centres of Excellence Programme (2012-2017) under Project No. 251748 and Aalto Energy Efficiency Research Programme. We acknowledge the computational resources provided by Aalto Science-IT project and CSC-IT. The graphical representations in the Fig. 1 were rendered using VMD [46].

Author contributions statement

M.M.A, M.K. and T.A.N. conceived the problem and analyzed the results. All authors co-wrote and reviewed the manuscript.