Two coils (\({Z}_{1}\)and \({Z}_{2}\)) are arranged parallelly to generate the magnetic field in one direction (the vertical or Z-direction in the laboratory frame), whereas a single-coil (Y-coil) with a soft iron core generates the field in the orthogonal direction. The soft iron core boosts the magnetic field and compensates for the lack of a complimentary coil. The inner diameter of each coil aligned in the Z-direction is chosen so that one objective lens can be placed inside it. A schematic diagram of the tree-coil assembly is shown in Fig. 2(a). A constant phase difference of \(90^\circ\) is maintained between the current applied to the Z pair of coils and the Y coil for generating a rotating magnetic field in the YZ plane, given by \({B}_{z}=A\text{c}\text{o}\text{s}\left(\omega t\right),\) and \({B}_{y}=A\text{s}\text{i}\text{n}\left(\omega t\right)\), where \(A\)is the amplitude and \(\omega\) is the frequency of the rotating magnetic field. The instantaneous magnetic field vectors for one cycle are shown in Fig. 2(b-f) at different values of \(\omega t\).
The magnetic flux density vector is included in the colourmap. A video demonstration of the same is also available in the supporting information (movie M.1).
Table I. The variation in the b/a ratio of the rotating magnetic field at different frequencies and field strength. The simulation results obtained from finite element ("COMSOL MULTIPHYSICS") calculations are given in the last column.
The length of the helical magnetic nanorobot is around \(3.5 {\mu }\text{m},\) and the thickness is around \(200-400 \text{n}\text{m}\). It is prepared by a shadow-based evaporation technique called the glancing angle deposition (GLAD) [24] method. The helix is made up of \(\text{S}\text{i}{\text{O}}_{2}\) with a small amount of iron added during evaporation to impart magnetic properties to it. Scanning Electron Microscope (SEM) image of the helical magnetic nanorobot is given in Fig. 3(a). We have measured the speed of a nanorobot moving in a water-glycerol (1:5 v/v) mixture (Newtonian media of viscosity approximately 100 cP) as a function of the frequency of the rotating magnetic field generated by the three-coil system for different field strengths. The trajectory of a nanorobot subjected to a rotating magnetic field (5 mT, 3 Hz) in the Newtonian media is shown in Fig. 3(b). Movie M.2 shows the motion of the helical magnetic nanorobots subjected to the rotating magnetic field (more details of the movie can be found in the supporting information). The helical magnetic nanorobot follows the rotating magnetic field up to a certain frequency, beyond which phase slip motion occurs. This frequency is called cut-off or the step-out frequency [25], [26]. The cut-off frequency is directly proportional to the applied magnetic field. In Fig. 3(c), we show this proportionality by increasing the magnetic field by a factor of three. As expected, the cut-off frequency increases three folds, and the maximum speed achieved by the helical magnetic nanorobots in the same medium increases substantially compared to the traditional triaxial Helmholtz coil reported till now. Also, note that the cut-off frequency is inversely proportional to the viscosity of a Newtonian medium [27]. Hence, a higher field (approx. 15 mT) will help the helical magnetic nanorobots to propel through the complex biological environment, which can have viscosity ranges more than 1000 cP [28] [29].
Due to the non-uniformities in amplitudes and phase differences between the Y and Z components of the magnetic field, the BY vs. BZ plot is elliptical in nature. The ellipticity, defined by the ratio between the lengths of the semi-minor (b) and the semi-major (a) axes, provides a measure of the uniformity of the rotating field. A perfectly rotating field has an ellipticity of one. Formation of the rotating magnetic field has been explained mathematically in the supporting information.
We measured the signals for BZ and BY using a gaussmeter at specific spatial locations for different values of the field amplitudes (3 mT, 5 mT, 10 mT, 15 mT) and three frequencies (1 Hz, 5 Hz, 10 Hz) continuously for three minutes. We construct ellipses by plotting the accumulated signals of \({\text{B}}_{\text{Z}}\) and \({\text{B}}_{\text{Y}}\) together in each case. Figure 4 shows two such ellipses. Here green arrow indicates the amplitude of \({\text{B}}_{\text{Y}}\) and the black arrow shows the amplitude of \({\text{B}}_{\text{Z}}\). Figures 4(a) and 4(b) represent 3 mT magnetic field rotating at 1 Hz frequency and 10 mT magnetic field rotating at 10 Hz frequency, respectively. We have done an elliptical fit of the experimental data points to get the values of the ellipticity, i.e., b/a ratio. The variation of this ratio at a particular point in the workspace of the coil system with the amplitude and the frequency of the rotating fields is shown in Table \(\text{I}\). The experimental values of the b/a ratio vary between .89 to .94, which is very close to 1. To verify the observed values of the b/a ratio, we have simulated our design in the 'COMSOL MULTIPHYSICS' software. The simulated value of ellipticity is .93 for all the cases, which is close to the experimental values. Hence, we can conclude that the rotating magnetic field is almost uniform irrespective of the variation of field and frequency in our experimental regime. We have checked the ellipticity for multiple cycles of field variation accumulated over three minutes and found the deviation to be negligible. The uniformity in the magnetic field did not change with time which suggests no observable effect of Joule heating. An elaborated discussion on the variation of the b/a ratio with time can be found in the supporting information.
In the next set of experiments, we examined the change in the b/a ratio at various locations in the XY plane. This allows us to quantify the spatial variation of the uniformity of the rotating field. The coil setup is fixed on a 3-axis translation stage to get a spatial resolution of 1 mm in the movement of the coil while the magnetic field is recorded. Figure 5(a) shows the XY plane, and the arrows indicate the axes. The green arrow is the XX' axis, the blue arrow is the YY' axis, and the origin is marked in red. The grid shown in the inset has an interval of 1 mm between its markings. We consider our experimental region as an imaginary square grid of sides 6 mm on the XY plane. The b/a ratio is calculated from the observed data at each point of the intersection of that grid.
The expected variation of the b/a ratio in the XY plane was simulated and represented as a colormap, as seen in Fig. 5(b). There is minimal variation (around 5%) in the b/a ratio along the XX' direction if the YY' axis position remains unchanged. The field is almost circular (i.e., the b/a ratio @ 1) at the origin. As we move away from the origin along the YY' direction, the b/a ratio decreases. The minimum value of the b/a ratio is about 0.8. It indicates that the rotating magnetic field is almost uniform within our experimental regime, with the field being almost circular at the origin. The simulation matches with the experimentally observed data, as shown in Fig. 5(c). It confirms the consistency of our experiment.
The inductance value of the pair of Z coils is 320 mH, and resistance is 79 Ω, and the Y coil has an inductance of 225 mH and a resistance of 28 Ω. Thus, the relative phase difference between the coil may change with frequency because the impedance of the pair of Z coils is different from the Y coil. We have used an oscilloscope to record the input and output signals of a coil simultaneously. The experiments are done separately for the pair of Z coils and the Y coil at frequencies 1 Hz, 5 Hz, and 10 Hz. The phase difference between the input and output signal is calculated in each case to get the actual phase difference between the magnetic field originating from two different directions.
Figure 6 represents the variation in the norm magnetic fields in the XY plane. Figures 6(a) and 6(b) indicate the variation of \({\text{B}}_{\text{Z}}\)and \({\text{B}}_{\text{Y}}\) along YY' axis for two values of XX': 0 mm, and 3 mm, respectively. \({\text{B}}_{\text{Z}}\)remains almost constant whereas \({\text{B}}_{\text{Y}}\) changes as a function of distance along YY' axis for a fixed driven current. Measurement shows that \({\text{B}}_{\text{Y}}\) is decaying from 21 mT to 11 mT along YY' axis.
The ratio of \({\text{B}}_{\text{Z}}\) and \({\text{B}}_{\text{Y}}\)which is defined as the b/a ratio is calculated by fitting ellipses at different values of YY' and XX'. This ratio has been plotted in Fig. 5(c). The colormap shown in Fig. 5(c) is consistent with the simulation results given in Fig. 5(b). Hence, from Figs. 5(c), 6(a), and 6(b), it is evident that a uniform rotating field of maximum amplitude of 15 mT can be achieved at the center of the assembly.
In our coil assembly, two coils have air cores, and one coil has an iron core. Field distribution in the coil assembly when all of them are excited with 0.1 A DC is studied in "COMSOL MULTIPHYSICS" software. Colormap of magnetic flux density norm is given in Fig. 7(a).