Predicting the maximum bend angle with Evans’ energy minimization model is an ap- proach to quantify an actuator’s static responsiveness. However, because an increasing drive frequency and strong environmental damping will cause a decrease in an overdamped oscillator’s amplitude, it is additionally integral to compare actuators of different materials
utilizing the magnetic torque which can be achieved. The magnetic torque quantifies an actuator’s ability to apply forces to the fluid regardless of actuator load or drive frequency. Torque is dependent only on the magnetic material (and thus moment) of an actuator and the direction of the applied magnetic field. I examine the magnetic torque on a rod-shaped actuator by assuming a stationary, upright rod where the magnetic moment is optimally aligned with the applied field. Magnetic torqueNon a dipole is written as (Jackson, 1998)
�
N =m� ×B� =mBsin(θmB) ˆn (3.32)
where Bis the applied field and m = MV f is the magnetic dipole moment (as described
in Section 3.3.1). M is the magnetization per unit volume (determined by SQUID mag- netometry measurements), V is the rod volume, and f is the volume fraction of magnetic material. Torque is minimized when the rod aligns itself with the applied magnetic field, as the cross product is equal to zero when the the angle between the dipole moment and applied field is zero.
Torque depends on the particular magnetic material utilized in an actuator and the vol- ume fraction of material, but does not depend on an actuator’s geometry. A larger volume
fraction does supply a higher torque; however, a larger volume fraction also leads to a higher modulus and stiffer actuator. Utilizing the combination of bend angle and torque is
best to optimize an actuator’s magnetic loading.
Experimentally, the magnetic field Bis typically on the order of 10 mT, though it may range from 1 mT−300 mT. Thus, in the calculation for torque, I letB=10 mT, andM=Msat.
To compare torque for composite materials and core-shell materials with volume fractions 0−1, the rod geometry will be identical to the geometry used for bend angle comparisons, 10µm or 25µm lengths and 0.5µm diameter. The maghemite saturation magnetization is
Msat,Mh = 3.8×105 A/m (used in FFPDMS), and for magnetite Msat,Mn = 4.6×105 A/m
(used in FFPDMS-NH2). Core-shell biomimetic cilia actuators are fabricated with nickel,
Msat,Ni = 5.2×105 A/m. Additionally for the core-shell material, I assume the optimal
nickel tube lengthLNi= 0.5L(determined in Section 3.3.1).
Figure 3.9 displays the torque for all three materials: maghemite composite (red lines), magnetite composite (blue lines), and nickel core-shell (black lines). The two lengths 10 µm and 25 µm are represented by solid and dashed lines, respectively. In Figure 3.9,
volume fractions from 0−1 are shown, but the highest volume fraction we have achieved thus far for FFPDMS is 0.04 and for FFPDMS-NH2 is 0.20, severely limiting the torque
on composite rods and potentially the actuation achievable in a more viscous environment. The highest volume fraction repeatably achieved for core-shell actuators thus far is∼0.6.
L = 25µm
L = 10µm
Figure 3.9: The torque of a rod-shaped actuator with diameter 0.5 µm as calculated by
Equation 3.32. Solid lines are 10 µm long rods; dotted lines represent 25 µm rods. Across all volume fractions, the nickel core-shell actuator (black lines) outperforms both the maghemite composite (red lines) and magnetite composite (blue lines) actuators. Ex- perimentally achieved volume fractions for maghemite, magnetite, and core-shell are 0.04, 0.20, and 0.60, respectively.
As the plot indicates, increasing f for FFPDMS, FFPDMS-NH2, and core-shell cilia
will continue to increase the torque on the cilium. However, a higher torque and therefore higher volume fraction does not guarantee larger deflection. We may look specifically at core-shell cilia to see this. Figure 3.10 plots both the bend angle curve and the relation for torque as functions of volume fraction. As we surpass 30% magnetic material by volume, the torque nearly doubles, but rod bend angle begins to decrease.
Figure 3.10: The torque and maximum bend angle as a function of volume fraction for core-shell cilia with a length of 10 µm and diameter of 500 nm. As torque on the rod
increases, bend angle increases until∼ f = 0.30, when the bend angle begins to decrease.
At this point, torque continues to increase, though the rod has become less responsive.
The design of various (FFPDMS, FFPDMS-NH2, and core-shell) actuators is strongly
dependent on the application for which they will be utilized; the sacrifice some amount of deflection to move in a higher viscosity fluid may or may not be a goal. Regardless, utiliz- ing both the theoretical bend angle and torque to optimize magnetic loading is beneficial in designing a rod with maximal static responsiveness under potentially large loads. In con- junction with designing the actuator such that it has a high response, the actuator should be modeled using the driven, damped harmonic oscillator equations in the first sections of this chapter to determine whether or not the high static responsiveness will also translate into a large dynamic responsiveness in both aqueous and high viscosity environments.