The vibrating cantilever is an ideal candidate for microrheometry measurements, as the size is easily controlled during the fabrication procedure, the response of the cantilever may be varied depending on the type of material used in its construction and the fluid in which it is immersed. For optimum performance, a microcantilever’s stiffness may be matched to
its application. To determine a fluid’s viscosity and elasticity, both the phase between the driving force and microcantilever and the amplitude (or deflection) of the microcantilever should be known. Typical rheometers utilize both amplitude and phase to determine the vis- cous and elastic responses of the fluid. The portion of the actuator’s response in-phase with the driving force corresponds to the elastic component of the fluid, and the portion of the actuator’s response out-of-phase with the driving force by π/2 corresponds to the viscous
component of the fluid (Barnes, 2000). Additionally, actuator response (both amplitude and phase) in air can be measured and compared to actuator response in a specified fluid to determine dynamic moduli. In 2010, Christopher et al. utilized this method in various linear viscoelastic materials with a MEMS microrheometer which essentially mimicked a macroscale cone and plate rheometer. The microrheometer was modeled mechanically as a mass suspended between two springs, oscillating in a viscoelastic fluid treated as a spring and dashpot in parallel with a single relaxation time (Christopher et al., 2010). (The Maxwell model is a spring and dashpot in series.)
For core-shell microactuators, the measurement of amplitude as a function of frequency is done using brightfield microscopy; however, there is currently no method in place for the
measurement of phase. Future project instrumentation could include a synced magnetics element and video capture system such that the rotation of the magnetics is triggered at the start of the video and is always known at a particular instant, allowing for the calculation of the phase delay. Additionally, with a synced magnetics and video system, utilizing an electromagnet to drive core-shell cilia would allow for more control over the sinusoidal in- put signal. For current analysis of the deflection in a purely viscous fluid, we consider only the amplitude of the rod and assume the rod is in-phase with the driving force. Following Brucker’s treatment, the amplitude of a microcantilever exhibiting small deflection is given as (Br¨ucker et al., 2007) A(ω)= F0 �60 11 EI L3 �2 + � 52 33 πηL ln(L/2D) �2 ω2 −1/2 (7.1) Recall thatEis the elastic modulus,Iis the second moment of inertia,ηis the fluid viscos-
ity, andLandDare the length and diameter of the cantilever. Note that the model considers only small deflections, and the amplitude of core-shell cilia is not small, but on the order of half the actuator’s length (several microns).
For core-shell cilia, 10 µm long and 0.55 µm in diameter, the decay of amplitude as
a function of frequency in PBS (η0 = 1 cP) and 2.5 M sucrose (η0 = 100 cP) is shown
in Figure 7.1. The data is indicated by open (PBS) and filled (sucrose) circles. Fits were performed on each data set utilizing the model above with fitting parameters F0, E, and η. The fit of the model to PBS data appears to follow the trend fairly well, and lies within
experimental error. Fitting parameters for the PBS fit are F=0.32 nN, E= 2.0 MPa, and
either that the cilium is more damped than the model suggests or that the experimentally obtained amplitude is underestimated and should be larger than was measured. If the cilia are more damped than the model suggests, an additional term may be needed to account for an internal damping or potential effects on the rod from the no-slip boundary condition
at the floor. When performing these experiments, efforts are made to drive the microrods
in an upright conical beat, though often the microactuator tips may drop closer to the floor during part of a beat. The amplitude is measured experimentally with videos captured at 120 frames per second (8 ms between frames). Cilia tips spend a very short period of time at maximum amplitude (the period of motion at 16 Hz is∼60 ms and 7 frames are captured for each beat), and the motion at higher frequencies can be blurred, making it difficult
to approximate the precise location of a cilium tip. For 2.5 M sucrose, the fit appears to match the data very well. However, the fit parameter η=36.2 cP, a large difference from
the expected 100 cP measured on a cone and plate rheometer. Also measured on the cone and plate rheometer was a mixture of 75% 2.5 M sucrose - 25% PBS. The viscosity of this fluid mixture was∼24 cP. When exchanging fluids, efforts are made to entirely replace
the previous fluid; however, fluid mixtures do occur during fluid exchange as the fluid cell size surrounding the array of microactuators is less than 50µL. Thus, it is reasonable
to assume the fluid surrounding the cilia array was a mixture of PBS and sucrose. The factor of ten increase in fluid viscosity is not seen in the fit of the model to data taken in 2.5 M sucrose, though the elastic modulus does increase compared to buffer by 1/4,
implying the rod is stiffer in sucrose than in buffer. Note that the model I fit to both buffer
including that of a small amplitude deflection, a sinusoidally-varying driving force, and a uniform, homogeneous microactuator. Thus, the fits I receive based on these assumptions are reasonable.
Figure 7.1: Data for core-shell rod amplitude dependence on frequency in two fluids of differing viscosities, PBS (open circles), η0 = 1.05 cP, and 2.5 M sucrose (filled circles), η0 =100 cP. The rod geometry is approximately 10µm long by 0.55µm diameter, and the
experimentally applied magnetic field ranged from 13-18 mT. The solid and dashed lines on the plot are a fit of Equation 7.1 to the data with fit parameters magnetic force, elastic modulus, and fluid viscosity. The parameters for the PBS fit areF=0.32 nN, E=2.0 MPa,
andη=10.3 cP. The parameters for the fit to sucrose data areF=0.38 nN,E=2.5 MPa, and
η=36.2 cP.
In buffer, the need for a larger viscosity parameter to fit cilia amplitude in the model
implies the model considers the core-shell rods more damped than they are or experimental measurements underestimate rod amplitude. There is still much work to be done in creating a better model, one that includes the contributions of elasticity as well, before core-shell rods can be used as a robust microrheometry technique. The application of core-shell cilia
to the measurement of coagulation times is unrestricted by the need for future work, as measurements across blood samples may initially be compared relative to one another.