1 TÍTULO: PROGRAMA ERGONÓMICO

While the previous section demonstrated a novel bending mechanism, the magnetic field geometry used for that experiment is poorly suited for most practical circum- stances. Gradient bending requires very large gradients, which in general can only be found very close to a magnet edge. Thus, with permanent magnets it is difficult to produce stable, predictable beat shapes with a gradient. On a related note, the field bending produced by this same magnet is also non-ideal. For field bending, the field energy term in equation4.5is a function of the magnetic moment of the cilium (which is a function of field strength) and of the relative orientation of the field direction and the long axis of the cilium. As this orientational relationship is a cosine, the field-bending energy is minimized when the cilium aligns with the field. Inversely, then, the maxi- mum force is applied when the two orientations are perpendicular. Thus, to maximize deflections a general design principle is that the field strength should be large enough to saturate the magnetization of the cilium while simultaneously the field direction should point perpendicular to the cilia (or parallel to the sample floor).

The magnetic field direction near a magnet’s pole is essentially normal to the mag- net’s surface, and so in general to get the pole of a magnet very close to the array (large field strengths) means that the force generated will be zero, because the field direction is parallel to the undeflected cilium. Thus, I typically configure actuating magnets such that the dipole is oriented parallel to the sample floor in order to maximize cilia bending.

With this in mind, I outline the strategies I have developed for mimicking the two main classes of motile cilia. I remind the reader that airway epithelia cilia, and most other motile, 9 + 2 cilia, are believed to oscillate largely within a single plane and therefore execute a planar beat. Nodal cilia, in contrast, perform a rotational beat around a tilted axis.

The planar nature of the airway cilia beat is easily duplicated. While one way to generate this beat is with a simple one-dimensional oscillation of a magnet, in principle this is slightly more difficult to generate than a rotational motion, which is easily generated by connecting magnets to the pivot of a variable frequency motor. To get a planar beat, then, a magnet must be placed onto a radial arm which extends out from the axis of rotation of the motor. When this is the case, even though the magnet path is circular, in the vicinity of the cilia sample (which is only about 1mm on a side) the magnet path looks approximately linear. By placing a set of magnets onto radial arms a significant cilia beat frequency can be generated with a relatively slow motor speed. In addition, this type of actuation can also produce what Ben Evans referred to as a ‘snap-beat’. In this configuration, even for a constant rate of rotation of the magnet, the cilia beat cycle is two phased and consists of a fast stroke (the snap-beat) and a slow stroke. To some degree this biphasic behavior mimics the effective and recovery strokes of biological cilia. However, as I have mentioned the two phase beat of airway cilia is accompanied by significant differences in the contour of the cilium, as well as a velocity difference.

Furthermore, as I discussed in Chapter 2, any variations in the rate of motion of a biomimetic cilium over its beat cycle will have no effect on transport at low Reynolds number because of the lack of inertia. The fast stroke moves the fluid faster, but it does not move it any farther than the slower recovery stroke moves the fluid in the opposite direction. Thus, even though this beat is an approximation of the airway beat, it will not produce fluid transport, only oscillations of the fluid. However, this may not be true in a viscoelastic fluid. The elasticity of such a complex fluid introduces characteristic time scales to the problem which are not present in a Newtonian fluid. Thus, future work with biomimetic cilia could focus on whether a biphasic beat which is spatially symmetric can produce fluid flow when the elasticity of the fluid plays a role.

Figure 4.9: Click to Animate. A comparison of the beat of nodal cilia (at bottom, from Okada et al. 2005) with the tilted conical beat of biomimetic cilia (at top). The tilt away from vertical is evidenced by the fact that each cilium’s tip moves in an ellipse which is offset from the base of the cilium.

With this in mind, the question becomes: how can I generate fluid transport with a cilium which cannot change its shape over a beat cycle? The answer, of course, is in the beat of embryonic nodal cilia. As I have explained, these cilia are able to generate a net transport of fluid with a rotational motion which does not require shape changes. More specifically, the nodal cilium takes advantage of the presence of the lower boundary in order to generate an asymmetric flow with a spatially symmetric motion. A rotational beat of the biomimetic cilia is relatively easy to generate, as it can be done simply by placing a magnet, in the configuration I have specified (where the dipole is parallel to the surface), directly above the cilia array and rotating the

Magnet Offset Cilia array Objective N S S _Magnet N S Magnet N B B Offset a b Flow cell

Figure 4.10: In order to mimic the tilted conical beat of nodal cilia I rotate a magnet

above the sample. A rotation of the magnet around an axis which is directly above a cilium generates rotational actuation around a vertical axis, while a small offset of the magnet’s rotational axis causes the cilium to tilt away from vertical.

magnet. If the rotational axis of the magnet were perfectly aligned with a cilium, all other things being constant the cilium will perform a rotational beat which precesses around a vertical axis. As Cartwright demonstrated, this beat will only produce vortical motion and no transport (Cartwright et al., 2004). However, a tilt of the rotational axis can be generated in my biomimetic samples by a subtle lateral offset of the rotational axis of the motor from the cilium. If the offset is too large the cilia will not execute a clean rotational beat, but if the offset is small enough the beat remains rotational but is tilted away from vertical. In Chapter 5 I will address the effect of this tilt on fluid transport.