Saturation and coercivity limit the velocity of rotating active magnetic microparticles

One class of active particles that is especially promising for biomedical microrobotic applications is rigid magnetic microswimmers propelled via rotation by a magnetic field. For these particles there is a maximum rotational frequency, hence velocity, determined by the maximum torque exerted by the field on the particle magnetization. It has been expected that velocity can always be increased by increasing the field magnitude to increase torque. This expectation holds if magnetization is constant or responds linearly to applied field, but all real materials actually respond nonlinearly, since as field strength increases the magnetization saturates and is coerced to point along the field direction. Here we show that this saturation and coercivity limit the maximum velocity of these microparticles. These effects are particularly important for soft magnetic materials. Although soft magnetic materials are used in many microswimmers, propulsion models incorporating their magnetic response are lacking. Our results are consistent with experimental observations and we predict that the limiting behavior occurs for common magnetic materials at typical rotational frequencies and field strengths, hence is relevant for current microswimmer design.

Figure 1

Schematic of rigid magnetic microswimmer and its velocity-frequency response. (a) Magnetic field $\mathit{h}$ is rotated with an angular velocity $\mathit{\omega }$ resulting in rotation of the swimmer with angular velocity $\mathit{\Omega }$ and average translational velocity ${\mathit{v}}_a$. (b) Schematic of typical velocity-frequency response for helical magnetic microswimmers. Up to the step-out frequency (${\mathrm{\Omega }}_s$), the average velocity is proportional to $\omega$.

Figure 2

Types of magnetic materials. Magnetization ($m$) response to an applied field $h$ for paramagnetic (green dashed line), hard magnetic (blue dot-dashed line), and soft magnetic (red solid line) material. Saturation magnetization ($m_s$), remanence ($m_r$), and coercive field ($h_c$) are indicated for hard and soft magnets.

Figure 3

(a) Schematic of microswimmer with a prolate ellipsoidal head. The volume of magnetic material ($V$) is not shown, and the head and tail are not to scale. Polar and azimuthal angles $\theta$ and $\varphi$ are used to describe vectors in the swimmer's body frame. (b) Demonstration of plateau behavior due to saturation and coercivity for microswimmer with head aspect ratio 2. Nondimensional step-out frequency (${\mathrm{\Omega }}_s$; left axis) plateaus as field magnitude ($h$) increases, because the magnetization approaches the field direction, as shown by the angle between magnetization and applied field (right axis). Dashed line shows $h^{-1}$ behavior of this angle. Insets show how step-out frequencies are determined from the largest nondimensional average velocity ($\widetilde{v_a}=\frac{v_a\eta L^2}{{\mu }_0{m}_s^{2}V}$) and nondimensional frequency ($\tilde{\omega }=\frac{\omega \eta L^3}{{\mu }_0{m}_s^{2}V}$) for which a solution exists, where $\eta$ is the dynamic viscosity of the fluid.

Figure 4

Plateau behavior of experimental microswimmer with square head. (a) Geometry of microswimmer with a thin square plate head [27]. Polar and azimuthal angles $\theta$ and $\varphi$ are used to describe vectors in the swimmer's body frame. (b) Experimental data for velocity versus frequency in three different field magnitudes, with uncertainty in estimation of step-out frequencies indicated. Taken from [27] and modified with permission. (c) Calculated step-out frequencies versus field magnitude for various parameter values as discussed in the main text. Table 2 shows values of $c_{s1}$, $c_{s2}$, $c_{s3}$, $m_s$, and $E$ corresponding to each plot. The inset shows more clearly the estimated experimental values of ${\omega }_{1.5}^e$ and ${\omega }_2^e$ and their uncertainties relative to our predictions.

Figure 5

Effect of material and geometry on plateau behavior. Dependence of step-out frequency (${\mathrm{\Omega }}_s$) on applied field magnitude ($h$) for labeled values of (a) saturation magnetization $m_s$, (b) volume of magnetic material $V$, and (c) head aspect ratio $R_h$. $h_0$, $m_s^0$, $V^0$, and $R_h^0$ are the values for the baseline case (blue curves) described in the text.

Figure 6

Surface discretization of microswimmer with ellipsoidal head. To scale, 2091 total discretization points.

Figure 7

Example of determination of magnetization with saturation condition. Dimensionless magnetic energy ($\tilde{e}=e/{\mu }_{0}V{m}_s^2$) for a square for different magnetization directions in an applied field with ${\theta }_h={30}^{\circ }$ and ${\varphi }_h=0$. Square points show energy computed with $\left|\mathit{m}\right|=m^{\prime }$ from Eq. (C4), and the circles show energy computed with $\left|\mathit{m}\right|=m_s$. The actual energy values (solid line) are computed with $\left|\mathit{m}\right|=\tilde{m}$; see text. (a) Field magnitude is $\left|\mathit{h}\right|/m_s=0.05$ and energy is minimized at ${\theta }_m={86}^{\circ }$ (dashed line). (b) Field magnitude is $\left|\mathit{h}\right|/m_s=0.4$ and energy is minimized at ${\theta }_m={72}^{\circ }$ (dashed line).

Figure 8

Surface discretization of microswimmer with thin square plate head. To scale, 894 total discretization points.