On the settling speed of dilute arrays of spheres

A method is developed to calculate the settling speed of dilute arrays of spheres for the three cases of: I, a random array of freely moving particles; II, a random array of rigidly held particles; and III, a cubic array of particles. The basic idea of the technique is to give a formal representation for the solution and then manipulate this representation in a straightforward manner to obtain the result. For infinite arrays of spheres, our results agree with the results previously found by other authors, and the analysis here appears to be simpler. This method is able to obtain more terms in the answer than was possible by Saffman's unified treatment for point particles. Some results for arbitrary two sphere distributions are presented, and an analysis of the wall effect for particles settling in a tube is given. It is expected that the method presented here can be generalized to solve other types of problems.

Dynamics of cell–matrix mechanical interactions in three dimensions

The forces cells apply to their surroundings control biological processes such as growth, adhesion, development, and migration. In the past 20 years, a number of experimental techniques have been developed to measure such cell tractions. These approaches have primarily measured the tractions applied by cells to synthetic two-dimensional substrates, which do not mimic in vivo conditions for most cell types. Many cell types live in a fibrous three-dimensional (3D) matrix environment. While studying cell behavior in such 3D matrices will provide valuable insights for the mechanobiology and tissue engineering communities, no experimental approaches have yet measured cell tractions in a fibrous 3D matrix.

This thesis describes the development and application of an experimental technique for quantifying cellular forces in a natural 3D matrix. Cells and their surrounding matrix are imaged in three dimensions with high speed confocal microscopy. The cell-induced matrix displacements are computed from the 3D image volumes using digital volume correlation. The strain tensor in the 3D matrix is computed by differentiating the displacements, and the stress tensor is computed by applying a constitutive law. Finally, tractions applied by the cells to the matrix are computed directly from the stress tensor.

The 3D traction measurement approach is used to investigate how cells mechanically interact with the matrix in biologically relevant processes such as division and invasion. During division, a single mother cell undergoes a drastic morphological change to split into two daughter cells. In a 3D matrix, dividing cells apply tensile force to the matrix through thin, persistent extensions that in turn direct the orientation and location of the daughter cells. Cell invasion into a 3D matrix is the first step required for cell migration in three dimensions. During invasion, cells initially apply minimal tractions to the matrix as they extend thin protrusions into the matrix fiber network. The invading cells anchor themselves to the matrix using these protrusions, and subsequently pull on the matrix to propel themselves forward.

Lastly, this thesis describes a constitutive model for the 3D fibrous matrix that uses a finite element (FE) approach. The FE model simulates the fibrous microstructure of the matrix and matches the cell-induced matrix displacements observed experimentally using digital volume correlation. The model is applied to predict how cells mechanically sense one another in a 3D matrix. It is found that cell-induced matrix displacements localize along linear paths. These linear paths propagate over a long range through the fibrous matrix, and provide a mechanism for cell-cell signaling and mechanosensing. The FE model developed here has the potential to reveal the effects of matrix density, inhomogeneity, and anisotropy in signaling cell behavior through mechanotransduction.

Flight dynamics in Drosophila through a dynamically-scaled robotic approach

Flies are particularly adept at balancing the competing demands of delay tolerance, performance, and robustness during flight, which invites thoughtful examination of their multimodal feedback architecture. This dissertation examines stabilization requirements for inner-loop feedback strategies in the flapping flight of Drosophila, the fruit fly, against the backdrop of sensorimotor transformations present in the animal. Flies have evolved multiple specializations to reduce sensorimotor latency, but sensory delay during flight is still significant on the timescale of body dynamics. I explored the effect of sensor delay on flight stability and performance for yaw turns using a dynamically-scaled robot equipped with a real-time feedback system that performed active turns in response to measured yaw torque. The results show a fundamental tradeoff between sensor delay and permissible feedback gain, and suggest that fast mechanosensory feedback provides a source of active damping that compliments that contributed by passive effects. Presented in the context of these findings, a control architecture whereby a haltere-mediated inner-loop proportional controller provides damping for slower visually-mediated feedback is consistent with tethered-flight measurements, free-flight observations, and engineering design principles. Additionally, I investigated how flies adjust stroke features to regulate and stabilize level forward flight. The results suggest that few changes to hovering kinematics are actually required to meet steady-state lift and thrust requirements at different flight speeds, and the primary driver of equilibrium velocity is the aerodynamic pitch moment. This finding is consistent with prior hypotheses and observations regarding the relationship between body pitch and flight speed in fruit flies. The results also show that the dynamics may be stabilized with additional pitch damping, but the magnitude of required damping increases with flight speed. I posit that differences in stroke deviation between the upstroke and downstroke might play a critical role in this stabilization. Fast mechanosensory feedback of the pitch rate could enable active damping, which would inherently exhibit gain scheduling with flight speed if pitch torque is regulated by adjusting stroke deviation. Such a control scheme would provide an elegant solution for flight stabilization across a wide range of flight speeds.

Topics in LIGO-related physics: interferometric speed meters and tidal work

In the quest to develop viable designs for third-generation optical interferometric gravitational-wave detectors, one strategy is to monitor the relative momentum or speed of the test-mass mirrors, rather than monitoring their relative position. The most straightforward design for a speed-meter interferometer that accomplishes this is described and analyzed in Chapter 2. This design (due to Braginsky, Gorodetsky, Khalili, and Thorne) is analogous to a microwave-cavity speed meter conceived by Braginsky and Khalili. A mathematical mapping between the microwave speed meter and the optical interferometric speed meter is developed and used to show (in accord with the speed being a quantum nondemolition observable) that in principle the interferometric speed meter can beat the gravitational-wave standard quantum limit (SQL) by an arbitrarily large amount, over an arbitrarily wide range of frequencies . However, in practice, to reach or beat the SQL, this specific speed meter requires exorbitantly high input light power. The physical reason for this is explored, along with other issues such as constraints on performance due to optical dissipation.

Chapter 3 proposes a more sophisticated version of a speed meter. This new design requires only a modest input power and appears to be a fully practical candidate for third-generation LIGO. It can beat the SQL (the approximate sensitivity of second-generation LIGO interferometers) over a broad range of frequencies (~ 10 to 100 Hz in practice) by a factor h/h_SQL ~ √W^(SQL)_(circ)/W_circ. Here W_circ is the light power circulating in the interferometer arms and W_SQL ≃ 800 kW is the circulating power required to beat the SQL at 100 Hz (the LIGO-II power). If squeezed vacuum (with a power-squeeze factor e^-2R) is injected into the interferometer's output port, the SQL can be beat with a much reduced laser power: h/h_SQL ~ √W^(SQL)_(circ)/W_circe^-2R. For realistic parameters (e^-2R ≃ 10 and W_circ ≃ 800 to 2000 kW), the SQL can be beat by a factor ~ 3 to 4 from 10 to 100 Hz. [However, as the power increases in these expressions, the speed meter becomes more narrow band; additional power and re-optimization of some parameters are required to maintain the wide band.] By performing frequency-dependent homodyne detection on the output (with the aid of two kilometer-scale filter cavities), one can markedly improve the interferometer's sensitivity at frequencies above 100 Hz.

Chapters 2 and 3 are part of an ongoing effort to develop a practical variant of an interferometric speed meter and to combine the speed meter concept with other ideas to yield a promising third- generation interferometric gravitational-wave detector that entails low laser power.

Chapter 4 is a contribution to the foundations for analyzing sources of gravitational waves for LIGO. Specifically, it presents an analysis of the tidal work done on a self-gravitating body (e.g., a neutron star or black hole) in an external tidal field (e.g., that of a binary companion). The change in the mass-energy of the body as a result of the tidal work, or "tidal heating," is analyzed using the Landau-Lifshitz pseudotensor and the local asymptotic rest frame of the body. It is shown that the work done on the body is gauge invariant, while the body-tidal-field interaction energy contained within the body's local asymptotic rest frame is gauge dependent. This is analogous to Newtonian theory, where the interaction energy is shown to depend on how one localizes gravitational energy, but the work done on the body is independent of that localization. These conclusions play a role in analyses, by others, of the dynamics and stability of the inspiraling neutron-star binaries whose gravitational waves are likely to be seen and studied by LIGO.

Innovations of wide-field optical-sectioning fluorescence microscopy : toward high-speed volumetric bio-imaging with simplicity

Optical microscopy has become an indispensable tool for biological researches since its invention, mostly owing to its sub-cellular spatial resolutions, non-invasiveness, instrumental simplicity, and the intuitive observations it provides. Nonetheless, obtaining reliable, quantitative spatial information from conventional wide-field optical microscopy is not always intuitive as it appears to be. This is because in the acquired images of optical microscopy the information about out-of-focus regions is spatially blurred and mixed with in-focus information. In other words, conventional wide-field optical microscopy transforms the three-dimensional spatial information, or volumetric information about the objects into a two-dimensional form in each acquired image, and therefore distorts the spatial information about the object. Several fluorescence holography-based methods have demonstrated the ability to obtain three-dimensional information about the objects, but these methods generally rely on decomposing stereoscopic visualizations to extract volumetric information and are unable to resolve complex 3-dimensional structures such as a multi-layer sphere.

The concept of optical-sectioning techniques, on the other hand, is to detect only two-dimensional information about an object at each acquisition. Specifically, each image obtained by optical-sectioning techniques contains mainly the information about an optically thin layer inside the object, as if only a thin histological section is being observed at a time. Using such a methodology, obtaining undistorted volumetric information about the object simply requires taking images of the object at sequential depths.

Among existing methods of obtaining volumetric information, the practicability of optical sectioning has made it the most commonly used and most powerful one in biological science. However, when applied to imaging living biological systems, conventional single-point-scanning optical-sectioning techniques often result in certain degrees of photo-damages because of the high focal intensity at the scanning point. In order to overcome such an issue, several wide-field optical-sectioning techniques have been proposed and demonstrated, although not without introducing new limitations and compromises such as low signal-to-background ratios and reduced axial resolutions. As a result, single-point-scanning optical-sectioning techniques remain the most widely used instrumentations for volumetric imaging of living biological systems to date.

In order to develop wide-field optical-sectioning techniques that has equivalent optical performance as single-point-scanning ones, this thesis first introduces the mechanisms and limitations of existing wide-field optical-sectioning techniques, and then brings in our innovations that aim to overcome these limitations. We demonstrate, theoretically and experimentally, that our proposed wide-field optical-sectioning techniques can achieve diffraction-limited optical sectioning, low out-of-focus excitation and high-frame-rate imaging in living biological systems. In addition to such imaging capabilities, our proposed techniques can be instrumentally simple and economic, and are straightforward for implementation on conventional wide-field microscopes. These advantages together show the potential of our innovations to be widely used for high-speed, volumetric fluorescence imaging of living biological systems.

Source characteristics of recent and historic earthquakes in Central and Southern California: results from forward modeling

The long- and short-period body waves of a number of moderate earthquakes occurring in central and southern California recorded at regional (200-1400 km) and teleseismic (> 30°) distances are modeled to obtain the source parameters-focal mechanism, depth, seismic moment, and source time history. The modeling is done in the time domain using a forward modeling technique based on ray summation. A simple layer over a half space velocity model is used with additional layers being added if necessary-for example, in a basin with a low velocity lid.

The earthquakes studied fall into two geographic regions: 1) the western Transverse Ranges, and 2) the western Imperial Valley. Earthquakes in the western Transverse Ranges include the 1987 Whittier Narrows earthquake, several offshore earthquakes that occurred between 1969 and 1981, and aftershocks to the 1983 Coalinga earthquake (these actually occurred north of the Transverse Ranges but share many characteristics with those that occurred there). These earthquakes are predominantly thrust faulting events with the average strike being east-west, but with many variations. Of the six earthquakes which had sufficient short-period data to accurately determine the source time history, five were complex events. That is, they could not be modeled as a simple point source, but consisted of two or more subevents. The subevents of the Whittier Narrows earthquake had different focal mechanisms. In the other cases, the subevents appear to be the same, but small variations could not be ruled out.

The recent Imperial Valley earthquakes modeled include the two 1987 Superstition Hills earthquakes and the 1969 Coyote Mountain earthquake. All are strike-slip events, and the second 1987 earthquake is a complex event With non-identical subevents.

In all the earthquakes studied, and particularly the thrust events, constraining the source parameters required modeling several phases and distance ranges. Teleseismic P waves could provide only approximate solutions. P_(nl) waves were probably the most useful phase in determining the focal mechanism, with additional constraints supplied by the SH waves when available. Contamination of the SH waves by shear-coupled PL waves was a frequent problem. Short-period data were needed to obtain the source time function.

In addition to the earthquakes mentioned above, several historic earthquakes were also studied. Earthquakes that occurred before the existence of dense local and worldwide networks are difficult to model due to the sparse data set. It has been noticed that earthquakes that occur near each other often produce similar waveforms implying similar source parameters. By comparing recent well studied earthquakes to historic earthquakes in the same region, better constraints can be placed on the source parameters of the historic events.

The Lompoc earthquake (M=7) of 1927 is the largest offshore earthquake to occur in California this century. By direct comparison of waveforms and amplitudes with the Coalinga and Santa Lucia Banks earthquakes, the focal mechanism (thrust faulting on a northwest striking fault) and long-period seismic moment (10^(26) dyne cm) can be obtained. The S-P travel times are consistent with an offshore location, rather than one in the Hosgri fault zone.

Historic earthquakes in the western Imperial Valley were also studied. These events include the 1942 and 1954 earthquakes. The earthquakes were relocated by comparing S-P and R-S times to recent earthquakes. It was found that only minor changes in the epicenters were required but that the Coyote Mountain earthquake may have been more severely mislocated. The waveforms as expected indicated that all the events were strike-slip. Moment estimates were obtained by comparing the amplitudes of recent and historic events at stations which recorded both. The 1942 event was smaller than the 1968 Borrego Mountain earthquake although some previous studies suggested the reverse. The 1954 and 1937 earthquakes had moments close to the expected value. An aftershock of the 1942 earthquake appears to be larger than previously thought.

Mean-speed measurements in two-dimensional, incompressible, fully-developed turbulent channel flow

Mean velocity profiles were measured in the 5” x 60” wind channel of the turbulence laboratory at the GALCIT, by the use of a hot-wire anemometer. The repeatability of results was established, and the accuracy of the instrumentation estimated. Scatter of experimental results is a little, if any, beyond this limit, although some effects might be expected to arise from variations in atmospheric humidity, no account of this factor having been taken in the present work. Also, slight unsteadiness in flow conditions will be responsible for some scatter.

Irregularities of a hot-wire in close proximity to a solid boundary at low speeds were observed, as have already been found by others.

That Kármán’s logarithmic law holds reasonably well over the main part of a fully developed turbulent flow was checked, the equation u/u_t = 6.0 + 6.25 log₁₀ y^ut/v being obtained, and, as has been previously the case, the experimental points do not quite form one straight line in the region where viscosity effects are small. The values of the constants for this law for the best over-all agreement were determined and compared with those obtained by others.

The range of Reynolds numbers used (based on half-width of channel) was from 20,000 to 60,000.

Two topics in elementary particle physics. I. Crossing as a group and elimination of exotic channels. II. Real parts of meson-nucleon forward scattering amplitudes.

I. Crossing transformations constitute a group of permutations under which the scattering amplitude is invariant. Using Mandelstem's analyticity, we decompose the amplitude into irreducible representations of this group. The usual quantum numbers, such as isospin or SU(3), are "crossing-invariant". Thus no higher symmetry is generated by crossing itself. However, elimination of certain quantum numbers in intermediate states is not crossing-invariant, and higher symmetries have to be introduced to make it possible. The current literature on exchange degeneracy is a manifestation of this statement. To exemplify application of our analysis, we show how, starting with SU(3) invariance, one can use crossing and the absence of exotic channels to derive the quark-model picture of the tensor nonet. No detailed dynamical input is used.

II. A dispersion relation calculation of the real parts of forward π^±p and K^±p scattering amplitudes is carried out under the assumption of constant total cross sections in the Serpukhov energy range. Comparison with existing experimental results as well as predictions for future high energy experiments are presented and discussed. Electromagnetic effects are found to be too small to account for the expected difference between the π^-p and π⁺p total cross sections at higher energies.

Spontaneous and stimulated light emission due to radiative recombination in forward biased lead telluride P-N junctions

Since the discovery in 1962 of laser action in semiconductor diodes made from GaAs, the study of spontaneous and stimulated light emission from semiconductors has become an exciting new field of semiconductor physics and quantum electronics combined. Included in the limited number of direct-gap semiconductor materials suitable for laser action are the members of the lead salt family, i.e . PbS, PbSe and PbTe. The material used for the experiments described herein is PbTe . The semiconductor PbTe is a narrow band- gap material (E_g = 0.19 electron volt at a temperature of 4.2°K). Therefore, the radiative recombination of electron-hole pairs between the conduction and valence bands produces photons whose wavelength is in the infrared (λ ≈ 6.5 microns in air).

The p-n junction diode is a convenient device in which the spontaneous and stimulated emission of light can be achieved via current flow in the forward-bias direction. Consequently, the experimental devices consist of a group of PbTe p-n junction diodes made from p –type single crystal bulk material. The p - n junctions were formed by an n-type vapor- phase diffusion perpendicular to the (100) plane, with a junction depth of approximately 75 microns. Opposite ends of the diode structure were cleaved to give parallel reflectors, thereby forming the Fabry-Perot cavity needed for a laser oscillator. Since the emission of light originates from the recombination of injected current carriers, the nature of the radiation depends on the injection mechanism.

The total intensity of the light emitted from the PbTe diodes was observed over a current range of three to four orders of magnitude. At the low current levels, the light intensity data were correlated with data obtained on the electrical characteristics of the diodes. In the low current region (region A), the light intensity, current-voltage and capacitance-voltage data are consistent with the model for photon-assisted tunneling. As the current is increased, the light intensity data indicate the occurrence of a change in the current injection mechanism from photon-assisted tunneling (region A) to thermionic emission (region B). With the further increase of the injection level, the photon-field due to light emission in the diode builds up to the point where stimulated emission (oscillation) occurs. The threshold current at which oscillation begins marks the beginning of a region (region C) where the total light intensity increases very rapidly with the increase in current. This rapid increase in intensity is accompanied by an increase in the number of narrow-band oscillating modes. As the photon density in the cavity continues to increase with the injection level, the intensity gradually enters a region of linear dependence on current (region D), i.e. a region of constant (differential) quantum efficiency.

Data obtained from measurements of the stimulated-mode light-intensity profile and the far-field diffraction pattern (both in the direction perpendicular to the junction-plane) indicate that the active region of high gain (i.e. the region where a population inversion exists) extends to approximately a diffusion length on both sides of the junction. The data also indicate that the confinement of the oscillating modes within the diode cavity is due to a variation in the real part of the dielectric constant, caused by the gain in the medium. A value of τ ≈ 10^-9 second for the minority- carrier recombination lifetime (at a diode temperature of 20.4°K) is obtained from the above measurements. This value for τ is consistent with other data obtained independently for PbTe crystals.

Data on the threshold current for stimulated emission (for a diode temperature of 20. 4°K) as a function of the reciprocal cavity length were obtained. These data yield a value of J’_th = (400 ± 80) amp/cm² for the threshold current in the limit of an infinitely long diode-cavity. A value of α = (30 ± 15) cm^-1 is obtained for the total (bulk) cavity loss constant, in general agreement with independent measurements of free- carrier absorption in PbTe. In addition, the data provide a value of n_s ≈ 10% for the internal spontaneous quantum efficiency. The above value for n_s yields values of t_b ≈ τ ≈ 10^-9 second and t_s ≈ 10^-8 second for the nonradiative and the spontaneous (radiative) lifetimes, respectively.

The external quantum efficiency (n_d) for stimulated emission from diode J-2 (at 20.4° K) was calculated by using the total light intensity vs. diode current data, plus accepted values for the material parameters of the mercury- doped germanium detector used for the measurements. The resulting value is n_d ≈ 10%-20% for emission from both ends of the cavity. The corresponding radiative power output (at λ = 6.5 micron) is 120-240 milliwatts for a diode current of 6 amps.

Finite-Element Simulations of Earthquakes

This thesis discusses simulations of earthquake ground motions using prescribed ruptures and dynamic failure. Introducing sliding degrees of freedom led to an innovative technique for numerical modeling of earthquake sources. This technique allows efficient implementation of both prescribed ruptures and dynamic failure on an arbitrarily oriented fault surface. Off the fault surface the solution of the three-dimensional, dynamic elasticity equation uses well known finite-element techniques. We employ parallel processing to efficiently compute the ground motions in domains containing millions of degrees of freedom.

Using prescribed ruptures we study the sensitivity of long-period near-source ground motions to five earthquake source parameters for hypothetical events on a strike-slip fault (M_w 7.0 to 7.1) and a thrust fault (M_w 6.6 to 7.0). The directivity of the ruptures creates large displacement and velocity pulses in the ground motions in the forward direction. We found a good match between the severity of the shaking and the shape of the near-source factor from the 1997 Uniform Building Code for strike-slip faults and thrust faults with surface rupture. However, for blind thrust faults the peak displacement and velocities occur up-dip from the region with the peak near-source factor. We assert that a simple modification to the formulation of the near-source factor improves the match between the severity of the ground motion and the shape of the near-source factor.

For simulations with dynamic failure on a strike-slip fault or a thrust fault, we examine what constraints must be imposed on the coefficient of friction to produce realistic ruptures under the application of reasonable shear and normal stress distributions with depth. We found that variation of the coefficient of friction with the shear modulus and the depth produces realistic rupture behavior in both homogeneous and layered half-spaces. Furthermore, we observed a dependence of the rupture speed on the direction of propagation and fluctuations in the rupture speed and slip rate as the rupture encountered changes in the stress field. Including such behavior in prescribed ruptures would yield more realistic ground motions.