Tabulación de datos luego de la aplicación de encuestas

To perform the tilted conical beat, a permanent magnet is affixed to the shaft of a rotat-

ing variable speed motor. The magnet is attached such that its poles are oriented parallel to the sample plane, as shown in Figure 5.10A. This orientation causes the magnetic field at or near the sample to also be parallel to the sample plane. As the motor and magnet rotate from 0◦ _{to 360}◦_{, the magnetic field rotates from 0}◦ _{to 360}◦_{, and a cilium which follows the}

magnetic field will follow this circular rotation. As illustrated in Figure 5.9C, when the magnetic field is nearly 90◦ _{with respect to the vertical, the cilium’s bend anglewill be as}

large as possible, restrained by its material parameters. This bend angle is analogous to the tilt angle plus half cone angle depicted in Figure 5.10B. Since the permanent magnet is fixed in space, the angle between the magnetic field and the vertical axis does not change. The permanent magnet does, however, rotate in space, and in spherical coordinates the an- gle swept out isφ, where 0 _≤φ <2π. As the permanent magnet and thus the magnetic field

N sample motor objective magnet Θ Ψ Z X

A

B

C

Figure 5.10: (A) Magnet position with respect to the sample is characterized by the mag- net’s height above the sample, and the magnet’s x-offset from the sample center. For ref-

erence, the microscope stage and objective are included in the diagram. (B) The x-offset

controls the direction of the magnetic field experienced by the cilia array and thus the tilt angleθof a cilium as further detailed in Figure 5.11. The distance from the magnet to the

sample controls the strength of the magnetic field and thus the half cone angleψ. (C) Min-

imum intensity projection of a top down view of single rod performing the tilted conical beat.

The distance from the permanent magnet to the cilia sample affects the strength of the

applied magnetic field. A shorter distance between magnet and sample will increase the field strength, thereby increasing the amplitude and half cone angle ψ (as illustrated in

Figure 5.10B) of the cilium. In addition, offsetting the center of the permanent magnet

laterally with respect to the cilia array affects the axis around which cilia rotate. Figure

5.10C is a top down view of a time lapse minimum intensity projection of a single core- shell rod performing this tilted conical beat shape. Each dark line denotes a position of the rod, the darkness of which is caused by the Ni tube. When the x-offset is zero, the magnet

is directly above the cilia array such that the rotational axis of the magnet is aligned with the cilium’s axis. As shown in Figure 5.11A, a cilium responds to this magnet orientation with an upright, conical beat shape. The tilt angleθ= 0◦.

�

B

single

cilium

magnet

cilia array

�

B

cilia array

x-offset

A

B

Figure 5.11: (A) When the axis of the permanent magnet is aligned with the cilium’s ro- tational axis,θ = 0◦. The cilium will rotate as the magnetic field rotates, and its half cone

angleψ will be determined by the magnetic field strength. (B) When the axis of the per-

manent magnet is offset by some x, the cilium’s rotational axis attempts to align itself with

the magnet’s axis producing a nonzeroθ. The cilium will then rotate around its tilted axis

as the magnetic field rotates.

With the introduction of an x-offset between the magnet center and cilia array, a cilium’s

rotational axis will point toward the magnet’s center rather than vertically, tilted at an angle

θ. As a laterally offset magnet rotates, a cilium will beat around its newly tilted axis, and

the half cone angle ψ is measured with respect to θ, as shown in Figure 5.11B. A larger

x-offset induces a larger tilt angle. The asymmetry in beat shape produced by the offset of

the magnet is integral to achieving fluid flow at the microscale, as theoretically detailed by Smith et al. (2008). Fluid flow at the microscale will be discussed in Chapter 6.

A cilium’s tilt angle, bend angle, and beat amplitude are measured using brightfield videos. To determine the tilt angleθand half cone angleψas shown in Figure 5.10B, the

straight line displacement from the rod’s base to its tip’s nearest and farthest extent (d1and

d2) in the direction of the tilt are measured over one beat cycle, as shown in Figure 5.12.

d1

d2

Figure 5.12: Minimum intensity projection of the cilium’s path over the course of several beats. The tilt angleθis calculated by first measuring the nearest and farthest extent of the

cilium tip with respect to its base and then utilizing Equation 5.3. The bend angle is found usingd1 and the rod’s known height.

The tilt angleθis then

θ=1/2�sin−1(d1/L)+sin−1(d2/L)

�

, (5.3)

and the half cone angleψis

ψ=θ₋�sin−1(d1/L)�. (5.4)

The bend angle is determined by measuring the farthest straight line displacement of the rod’s tip from its base, orθ+ψwhich is equivalent tod1in Figure 5.12, and using the rod’s

known height of L = 10 µm. In order to measure the amplitude of the cilium, the path

the ellipse’s major axis. The amplitude is measured in this way to control for varying tilt angles across biomimetic cilia specimens.

This tilted conical beat shape is the shape utilized for all experiments discussed in Chap- ter 6. I have not yet been successful in replicating the asymmetrical planar beat employed by biological airway cilia. However, with the tilted conical beat, I am able to compare my flow results in aqueous fluids directly to the biological cilia-driven flow in vertebrate embryonic nodes, the ultimate illustration of the utility of a system such as mine.