Compressible flows at small Reynolds numbers

The problem of the slow viscous flow of a gas past a sphere is considered. The fluid cannot be treated incompressible in the limit when the Reynolds number Re, and the Mach number M, tend to zero in such a way that Re ~ o(M^2 ). In this case, the lowest order approximation to the steady Navier-Stokes equations of motion leads to a paradox discovered by Lagerstrom and Chester. This paradox is resolved within the framework of continuum mechanics using the classical slip condition and an iteration scheme that takes into account certain terms in the full Navier-Stokes equations that drop out in the approximation used by the above authors. It is found however that the drag predicted by the theory does not agree with R. A. Millikan's classic experiments on sphere drag.

The whole question of the applicability of the Navier-Stokes theory when the Knudsen number M/Re is not small is examined. A new slip condition is proposed. The idea that the Navier-Stokes equations coupled with this condition may adequately describe small Reynolds number flows when the Knudsen number is not too large is looked at in some detail. First, a general discussion of asymptotic solutions of the equations for all such flows is given. The theory is then applied to several concrete problems of fluid motion. The deductions from this theory appear to interpret and summarize the results of Millikan over a much wider range of Knudsen numbers (almost up to the free molecular or kinetic limit) than hitherto Believed possible by a purely continuum theory. Further experimental tests are suggested and certain interesting applications to the theory of dilute suspensions in gases are noted. Some of the questions raised in the main body of the work are explored further in the appendices.

Non-linear dispersive waves with a small dissipation

The general theory of Whitham for slowly-varying non-linear wavetrains is extended to the case where some of the defining partial differential equations cannot be put into conservation form. Typical examples are considered in plasma dynamics and water waves in which the lack of a conservation form is due to dissipation; an additional non-conservative element, the presence of an external force, is treated for the plasma dynamics example. Certain numerical solutions of the water waves problem (the Korteweg-de Vries equation with dissipation) are considered and compared with perturbation expansions about the linearized solution; it is found that the first correction term in the perturbation expansion is an excellent qualitative indicator of the deviation of the dissipative decay rate from linearity.

A method for deriving necessary and sufficient conditions for the existence of a general uniform wavetrain solution is presented and illustrated in the plasma dynamics problem. Peaking of the plasma wave is demonstrated, and it is shown that the necessary and sufficient existence conditions are essentially equivalent to the statement that no wave may have an amplitude larger than the peaked wave.

A new type of fully non-linear stability criterion is developed for the plasma uniform wavetrain. It is shown explicitly that this wavetrain is stable in the near-linear limit. The nature of this new type of stability is discussed.

Steady shock solutions are also considered. By a quite general method, it is demonstrated that the plasma equations studied here have no steady shock solutions whatsoever. A special type of steady shock is proposed, in which a uniform wavetrain joins across a jump discontinuity to a constant state. Such shocks may indeed exist for the Korteweg-de Vries equation, but are barred from the plasma problem because entropy would decrease across the shock front.

Finally, a way of including the Landau damping mechanism in the plasma equations is given. It involves putting in a dissipation term of convolution integral form, and parallels a similar approach of Whitham in water wave theory. An important application of this would be towards resolving long-standing difficulties about the "collisionless" shock.

Physical investigations of small particles : (I) aerosol particle charging and flux enhancement and (II) whispering gallery mode sensing

Part I

Particles are a key feature of planetary atmospheres. On Earth they represent the greatest source of uncertainty in the global energy budget. This uncertainty can be addressed by making more measurement, by improving the theoretical analysis of measurements, and by better modeling basic particle nucleation and initial particle growth within an atmosphere. This work will focus on the latter two methods of improvement.

Uncertainty in measurements is largely due to particle charging. Accurate descriptions of particle charging are challenging because one deals with particles in a gas as opposed to a vacuum, so different length scales come into play. Previous studies have considered the effects of transition between the continuum and kinetic regime and the effects of two and three body interactions within the kinetic regime. These studies, however, use questionable assumptions about the charging process which resulted in skewed observations, and bias in the proposed dynamics of aerosol particles. These assumptions affect both the ions and particles in the system. Ions are assumed to be point monopoles that have a single characteristic speed rather than follow a distribution. Particles are assumed to be perfect conductors that have up to five elementary charges on them. The effects of three body interaction, ion-molecule-particle, are also overestimated. By revising this theory so that the basic physical attributes of both ions and particles and their interactions are better represented, we are able to make more accurate predictions of particle charging in both the kinetic and continuum regimes.

The same revised theory that was used above to model ion charging can also be applied to the flux of neutral vapor phase molecules to a particle or initial cluster. Using these results we can model the vapor flux to a neutral or charged particle due to diffusion and electromagnetic interactions. In many classical theories currently applied to these models, the finite size of the molecule and the electromagnetic interaction between the molecule and particle, especially for the neutral particle case, are completely ignored, or, as is often the case for a permanent dipole vapor species, strongly underestimated. Comparing our model to these classical models we determine an “enhancement factor” to characterize how important the addition of these physical parameters and processes is to the understanding of particle nucleation and growth.

Part II

Whispering gallery mode (WGM) optical biosensors are capable of extraordinarily sensitive specific and non-specific detection of species suspended in a gas or fluid. Recent experimental results suggest that these devices may attain single-molecule sensitivity to protein solutions in the form of stepwise shifts in their resonance wavelength, \lambda_{R}, but present sensor models predict much smaller steps than were reported. This study examines the physical interaction between a WGM sensor and a molecule adsorbed to its surface, exploring assumptions made in previous efforts to model WGM sensor behavior, and describing computational schemes that model the experiments for which single protein sensitivity was reported. The resulting model is used to simulate sensor performance, within constraints imposed by the limited material property data. On this basis, we conclude that nonlinear optical effects would be needed to attain the reported sensitivity, and that, in the experiments for which extreme sensitivity was reported, a bound protein experiences optical energy fluxes too high for such effects to be ignored.

Reactions of small molecules facilitated by metal-acceptor interactions

The role of metal-acceptor interactions arising from M–BR3 and M–PR3 bonding is discussed with respect to reactions between first-row transition metals and N2, H2, and CO. Thermally robust, S = 1/2 (TPB)Co(H2) and (TPB)Co(N2) complexes (TPB = tris(2- (diisopropylphosphino)phenyl)borane) are described and the energetics of N2 and H2 binding are measured. The H2 and N2 ligands are bound more weakly in the (TPB)Co complexes than in related (SiP3)M(L) complexes (SiP3 = tris(2- (diisopropylphosphino)phenyl)silyl). Comparisons within and between these two ligand platforms allow for the factors that affect N2 (and H2) binding and activation to be delineated. The characterization and reactivity of (DPB)Fe complexes (DPB = bis(2- (diisopropylphosphino)phenyl)phenylborane) in the context of N2 functionalization and E–H bond addition (E = H, C, N, Si) are described. This platform allows for the one-pot transformation of free N2 to an Fe hydrazido(-) complex via an Fe aminoimide intermediate. The principles learned from the N2 chemistry using (DPB)Fe are applied to CO reduction on the same system. The preparation of (DPB)Fe(CO)2 is described as well as its reductive functionalization to generate an unprecedented Fe dicarbyne. The bonding in this highly covalent complex is discussed in detail. Initial studies of the reactivity of the Fe dicarbyne reveal that a CO-derived olefin is released upon hydrogenation. Alternative approaches to uncovering unusual reactivity using metal- acceptor interactions are described in Chapters 5 and 6, including initial studies on a new π-accepting tridentate diphosphinosulfinyl ligand and strategies for designing ligands that undergo site-selective metallation to generate heterobimetallic complexes.

Towards conditional RNAi : shape and sequence transduction with small conditional RNAs

RNA interference (RNAi) is a powerful biological pathway allowing for sequence-specific knockdown of any gene of interest. While RNAi is a proven tool for probing gene function in biological circuits, it is limited by being constitutively ON and executes the logical operation: silence gene Y. To provide greater control over post-transcriptional gene silencing, we propose engineering a biological logic gate to implement “conditional RNAi.” Such a logic gate would silence gene Y only upon the expression of gene X, a completely unrelated gene, executing the logic: if gene X is transcribed, silence independent gene Y. Silencing of gene Y could be confined to a specific time and/or tissue by appropriately selecting gene X.

To implement the logic of conditional RNAi, we present the design and experimental validation of three nucleic acid self-assembly mechanisms which detect a sub-sequence of mRNA X and produce a Dicer substrate specific to gene Y. We introduce small conditional RNAs (scRNAs) to execute the signal transduction under isothermal conditions. scRNAs are small RNAs which change conformation, leading to both shape and sequence signal transduction, in response to hybridization to an input nucleic acid target. While all three conditional RNAi mechanisms execute the same logical operation, they explore various design alternatives for nucleic acid self-assembly pathways, including the use of duplex and monomer scRNAs, stable versus metastable reactants, multiple methods of nucleation, and 3-way and 4-way branch migration.

We demonstrate the isothermal execution of the conditional RNAi mechanisms in a test tube with recombinant Dicer. These mechanisms execute the logic: if mRNA X is detected, produce a Dicer substrate targeting independent mRNA Y. Only the final Dicer substrate, not the scRNA reactants or intermediates, is efficiently processed by Dicer. Additional work in human whole-cell extracts and a model tissue-culture system delves into both the promise and challenge of implementing conditional RNAi in vivo.

Selective hydroxylation of small alkanes by the particulate methane monooxygenase

The particulate methane monooxygenase (pMMO) catalyzes the oxidation of methane to methanol under ambient temperatures and pressures. Other small alkanes and alkenes are also substrates of this enzyme. We measured and compared the initial rate constants of oxidation of small alkanes (C1 to C5) catalyzed by pMMO. Both primary and secondary alcohols were formed from oxidation of n-butane and n-pentane. The alcohols produced from alkane oxidation can be further oxidized, probably by pMMO, to aldehydes and ketones. The apparent regioselectivity for n-butane and n-pentane is 100% 2-alcohols because the formation of primary alcohols is slower than further oxidation of these alcohols. The hydroxylation at the secondary carbons is highly stereoselective: (R)-alcohols are preferentially formed. The enantiomeric excess increases slightly with decreasing reaction temperature. The steric course of hydroxylation on primary carbons was also studied by using isotopically substituted ethane: (S)- or (R)-CH_3-CHDT, and (S)- or (R)-CD_3- CHDT and the reactions were found to proceed with 100% retention of configuration. A primary isotopic effect of k_H/k_D=5.0 was observed in these experiments.

Theory of viscous and thermal attenuation of sound by small spheres

The problem is to calculate the attenuation of plane sound waves passing through a viscous, heat-conducting fluid containing small spherical inhomogeneities. The attenuation is calculated by evaluating the rate of increase of entropy caused by two irreversible processes: (1) the mechanical work done by the viscous stresses in the presence of velocity gradients, and (2) the flow of heat down the thermal gradients. The method is first applied to a homogeneous fluid with no spheres and shown to give the classical Stokes-Kirchhoff expressions. The method is then used to calculate the additional viscous and thermal attenuation when small spheres are present. The viscous attenuation agrees with Epstein's result obtained in 1941 for a non-heat-conducting fluid. The thermal attenuation is found to be similar in form to the viscous attenuation and, for gases, of comparable magnitude. The general results are applied to the case of water drops in air and air bubbles in water.

For water drops in air the viscous and thermal attenuations are camparable; the thermal losses occur almost entirely in the air, the thermal dissipation in the water being negligible. The theoretical values are compared with Knudsen's experimental data for fogs and found to agree in order of magnitude and dependence on frequency. For air bubbles in water the viscous losses are negligible and the calculated attenuation is almost completely due to thermal losses occurring in the air inside the bubbles, the thermal dissipation in the water being relatively small. (These results apply only to non-resonant bubbles whose radius changes but slightly during the acoustic cycle.)

A small perturbation theory for cycloidal propellers

A hydromechanical theory is developed for cycloidal propellers for two limiting modes of operation wherein U » ΩR and U « ΩR, with U the rectilinear propeller speed (speed of advance) and ΩR the rotational blade speed. A first order theory is developed from the basic principles of the kinematics and dynamics of fluid motion and proceeds from the point of view of unsteady hydrofoil theory.

Explicit expressions for the instantaneous forces and moments produced by blade motions are presented. On the basis of these results an optimization procedure is carried out which minimizes the energy loss under the constraint of specified mean thrust. Under optimal conditions the propeller is found to possess high Froude efficiencies in both the high and low speed modes of propulsion. This efficiency is defined as the ratio of the average useful work obtained during one cycle of propeller operation to the average power input required to sustain the motion of the propeller during the cycle.

Site-Specific Isotopes in Small Organic Molecules

Stable isotope geochemistry is a valuable toolkit for addressing a broad range of problems in the geosciences. Recent technical advances provide information that was previously unattainable or provide unprecedented precision and accuracy. Two such techniques are site-specific stable isotope mass spectrometry and clumped isotope thermometry. In this thesis, I use site-specific isotope and clumped isotope data to explore natural gas development and carbonate reaction kinetics. In the first chapter, I develop an equilibrium thermodynamics model to calculate equilibrium constants for isotope exchange reactions in small organic molecules. This equilibrium data provides a framework for interpreting the more complex data in the later chapters. In the second chapter, I demonstrate a method for measuring site-specific carbon isotopes in propane using high-resolution gas source mass spectrometry. This method relies on the characteristic fragments created during electron ionization, in which I measure the relative isotopic enrichment of separate parts of the molecule. My technique will be applied to a range of organic compounds in the future. For the third chapter, I use this technique to explore diffusion, mixing, and other natural processes in natural gas basins. As time progresses and the mixture matures, different components like kerogen and oil contribute to the propane in a natural gas sample. Each component imparts a distinct fingerprint on the site-specific isotope distribution within propane that I can observe to understand the source composition and maturation of the basin. Finally, in Chapter Four, I study the reaction kinetics of clumped isotopes in aragonite. Despite its frequent use as a clumped isotope thermometer, the aragonite blocking temperature is not known. Using laboratory heating experiments, I determine that the aragonite clumped isotope thermometer has a blocking temperature of 50-100°C. I compare this result to natural samples from the San Juan Islands that exhibit a maximum clumped isotope temperature that matches this blocking temperature. This thesis presents a framework for measuring site-specific carbon isotopes in organic molecules and new constraints on aragonite reaction kinetics. This study represents the foundation of a future generation of geochemical tools for the study of complex geologic systems.

Solar System Small-Body Demographics with the Palomar Transient Factory Survey

Observational studies of our solar system's small-body populations (asteroids and comets) offer insight into the history of our planetary system, as these minor planets represent the left-over building blocks from its formation. The Palomar Transient Factory (PTF) survey began in 2009 as the latest wide-field sky-survey program to be conducted on the 1.2-meter Samuel Oschin telescope at Palomar Observatory. Though its main science program has been the discovery of high-energy extragalactic sources (such as supernovae), during its first five years PTF has collected nearly five million observations of over half a million unique solar system small bodies. This thesis begins to analyze this vast data set to address key population-level science topics, including: the detection rates of rare main-belt comets and small near-Earth asteroids, the spin and shape properties of asteroids as inferred from their lightcurves, the applicability of this visible light data to the interpretation of ultraviolet asteroid observations, and a comparison of the physical properties of main-belt and Jovian Trojan asteroids. Future sky-surveys would benefit from application of the analytical techniques presented herein, which include novel modeling methods and unique applications of machine-learning classification. The PTF asteroid small-body data produced in the course of this thesis work should remain a fertile source of solar system science and discovery for years to come.

I. Thermochemistry and reaction kinetics of disolvated protons by ion cyclotron resonance spectroscopy. II. Thermochemical studies of small fluorocarbons by photoionization mass spectrometry

The disolvated proton, H(OH₂)₂⁺ is employed as a chemical reagent in low pressure (˂ 10^-5 torr) investigations by ion cyclotron resonance spectroscopy. Since termolecular reactions are absent at low pressure, disolvated protons are not generally observed. However H(OH₂)₂⁺ is produced in a sequence of bimolecular reactions in mixtures containing H₂O and one of a small number of organohalide precursors. Then a series of hydrated Lewis bases is produced by H₃O⁺ transfer from H(OH₂)₂⁺. In Chapter II, the relative stability of hydrated bases containing heteroatoms of both first and second row elements is determined from the preferred direction of H₃O⁺ transfer between BH(OH₂)⁺ complexes. S and P containing bases are shown to bind H₃O⁺ more weakly than O and N bases with comparable proton affinities. A simple model of hydrogen bonding is proposed to account for these observations.

H⁺ transfer from H(OH₂)₂⁺ to several Lewis bases also occurs at low pressure. In Chapter III the relative importance of H₃O⁺ transfer and H⁺ transfer from H(OH₂)₂⁺ to a series of bases is observed to be a function of base strength. Beginning with CH₃COOH, the weakest base for which H⁺ transfer is observed, the importance of H⁺ transfer increases with increasing proton affinity of the acceptor base. The nature of neutral products formed from H(OH₂)₂⁺ by loss of H⁺ is also considered.

Chapters IV and V deal with thermochemistry of small fluorocarbons determined by photoionization mass spectrometry. The enthalpy of formation of CF₂ is considered in Chapter IV. Photoionization of perfluoropropylene, perfluorocyclopropane, and trifluoromethyl benzene yield onsets for ions formed by loss of a CF₂ neutral fragment. Earlier determinations of ΔH^°_f298 (CF₂) are reinterpreted using updated thermochemical values and compared with results of this study. The heat of formation of neutral perfluorocyclopropane is also derived. Finally, the energetics of interconversion of perfluoropropylene and perfluorocyclopropane are considered for both the neutrals and their molecular ions.

In Chapter V the heats of formation of CF₃⁺ and CF₃I⁺are derived from photoionization of CF₃I. These are considered with respect to ion-molecule reactions observed in CF₃I monitored by the techniques of ion cyclotron resonance spectroscopy. Results obtained in previous experiments are also compared.

The Design and Synthesis of Transition Metal Complexes Supported by Non-innocent Ligand Scaffolds for Small Molecule Activation

This dissertation focuses on the incorporation of non-innocent or multifunctional moieties into different ligand scaffolds to support one or multiple metal centers in close proximity. Chapter 2 focuses on the initial efforts to synthesize hetero- or homometallic tri- or dinuclear metal carbonyl complexes supported by para-terphenyl diphosphine ligands. A series of [M₂M’(CO)₄]-type clusters (M = Ni, Pd; M’ = Fe, Co) could be accessed and used to relate the metal composition to the properties of the complexes. During these studies it was also found that non-innocent behavior was observed in dinuclear Fe complexes that result from changes in oxidation state of the cluster. These studies led to efforts to rationally incorporate central arene moieties capable managing both protons and electrons during small molecule activation.

Chapter 3 discusses the synthesis of metal complexes supported by a novel para-terphenyl diphosphine ligand containing a non-innocent 1,4-hydroquinone moiety as the central arene. A Pd⁰-hydroquinone complex was found to mediate the activation of a variety of small molecules to form the corresponding Pd⁰-quinone complexes in a formal two proton ⁄ two electron transformation. Mechanistic investigations of dioxygen activation revealed a metal-first activation process followed by subsequent proton and electron transfer from the ligand. These studies revealed the capacity of the central arene substituent to serve as a reservoir for a formal equivalent of dihydrogen, although the stability of the M-quinone compounds prevented access to the PdII-quinone oxidation state, thus hindering of small molecule transformations requiring more than two electrons per equivalent of metal complex.

Chapter 4 discusses the synthesis of metal complexes supported by a ligand containing a 3,5-substituted pyridine moiety as the linker separating the phenylene phosphine donors. Nickel and palladium complexes supported by this ligand were found to tolerate a wide variety of pyridine nitrogen-coordinated electrophiles which were found to alter central pyridine electronics, and therefore metal-pyridine π-system interactions, substantially. Furthermore, nickel complexes supported by this ligand were found to activate H-B and H-Si bonds and formally hydroborate and hydrosilylate the central pyridine ring. These systems highlight the potential use of pyridine π-system-coordinated metal complexes to reversibly store reducing equivalents within the ligand framework in a manner akin to the previously discussed 1,4-hydroquinone diphosphine ligand scaffold.

Chapter 5 departs from the phosphine-based chemistry and instead focuses on the incorporation of hydrogen bonding networks into the secondary coordination sphere of [Fe₄(μ₄-O)]-type clusters supported by various pyrazolate ligands. The aim of this project is to stabilize reactive oxygenic species, such as oxos, to study their spectroscopy and reactivity in the context of complicated multimetallic clusters. Herein is reported this synthesis and electrochemical and Mössbauer characterization of a series of chloride clusters have been synthesized using parent pyrazolate and a 3-aminophenyl substituted pyrazolate ligand. Efforts to rationally access hydroxo and oxo clusters from these chloride precursors represents ongoing work that will continue in the group.

Appendix A discusses attempts to access [Fe₃Ni]-type clusters as models of the enzymatic active site of [NiFe] carbon monoxide dehydrogenase. Efforts to construct tetranuclear clusters with an interstitial sulfide proved unsuccessful, although a (μ₃-S) ligand could be installed through non-oxidative routes into triiron clusters. While [Fe₃Ni(μ₄-O)]-type clusters could be assembled, accessing an open heterobimetallic edge site proved challenging, thus prohibiting efforts to study chemical transformations, such as hydroxide attack onto carbon monoxide or carbon dioxide coordination, relevant to the native enzyme. Appendix B discusses the attempts to synthesize models of the full H-cluster of [FeFe]-hydrogenase using a bioinorganic approach. A synthetic peptide containing three cysteine donors was successfully synthesized and found to chelate a preformed synthetic [Fe₄S₄] cluster. However, efforts to incorporate the diiron subsite model complex proved challenging as the planned thioester exchange reaction was found to non-selectively acetylate the peptide backbone, thus preventing the construction of the full six-iron cluster.

Small scale turbulence in high Karlovitz number premixed flames

The purpose of this thesis is to characterize the behavior of the smallest turbulent scales in high Karlovitz number (Ka) premixed flames. These scales are particularly important in the two-way coupling between turbulence and chemistry and better understanding of these scales will support future modeling efforts using large eddy simulations (LES). The smallest turbulent scales are studied by considering the vorticity vector, ω, and its transport equation.

Due to the complexity of turbulent combustion introduced by the wide range of length and time scales, the two-dimensional vortex-flame interaction is first studied as a simplified test case. Numerical and analytical techniques are used to discern the dominate transport terms and their effects on vorticity based on the initial size and strength of the vortex. This description of the effects of the flame on a vortex provides a foundation for investigating vorticity in turbulent combustion.

Subsequently, enstrophy, ω² = ω • ω, and its transport equation are investigated in premixed turbulent combustion. For this purpose, a series of direct numerical simulations (DNS) of premixed n-heptane/air flames are performed, the conditions of which span a wide range of unburnt Karlovitz numbers and turbulent Reynolds numbers. Theoretical scaling analysis along with the DNS results support that, at high Karlovitz number, enstrophy transport is controlled by the viscous dissipation and vortex stretching/production terms. As a result, vorticity scales throughout the flame with the inverse of the Kolmogorov time scale, τ_η, just as in homogeneous isotropic turbulence. As τ_η is only a function of the viscosity and dissipation rate, this supports the validity of Kolmogorov’s first similarity hypothesis for sufficiently high Ka numbers (Ka ≳ 100). These conclusions are in contrast to low Karlovitz number behavior, where dilatation and baroclinic torque have a significant impact on vorticity within the flame. Results are unaffected by the transport model, chemical model, turbulent Reynolds number, and lastly the physical configuration.

Next, the isotropy of vorticity is assessed. It is found that given a sufficiently large value of the Karlovitz number (Ka ≳ 100) the vorticity is isotropic. At lower Karlovitz numbers, anisotropy develops due to the effects of the flame on the vortex stretching/production term. In this case, the local dynamics of vorticity in the strain-rate tensor, S, eigenframe are altered by the flame. At sufficiently high Karlovitz numbers, the dynamics of vorticity in this eigenframe resemble that of homogeneous isotropic turbulence.

Combined, the results of this thesis support that both the magnitude and orientation of vorticity resemble the behavior of homogeneous isotropic turbulence, given a sufficiently high Karlovitz number (Ka ≳ 100). This supports the validity of Kolmogorov’s first similarity hypothesis and the hypothesis of local isotropy under these condition. However, dramatically different behavior is found at lower Karlovitz numbers. These conclusions provides/suggests directions for modeling high Karlovitz number premixed flames using LES. With more accurate models, the design of aircraft combustors and other combustion based devices may better mitigate the detrimental effects of combustion, from reducing CO₂ and soot production to increasing engine efficiency.