Synthetic, structural and mechanistic studies of interactions of carbon, nitrogen, and oxygen nucleophiles with bis-pentamethylcyclopentadienyl zirconium and hafnium derivatives

The reactivity of permethylzirconocene and permethylhafnocene complexes with various nucleophiles has been investigated. Permethylzirconocene reacts with sterically hindered ketenes and allenes to afford metallacycle products. Reaction of these cummulenes with permethylzirconocene hydride complexes affords enolate and σ-allyl species, respectively. Reactions which afford enolate products are nonstereospecific, whereas reactions which afford allyl products initially give a cis-σ-allyl complex which rearranges to its trans isomer. The mechanism of these reactions is proposed to occur either by a Lewis Acid-Lewis Base interaction (ketenes) or by formation of a π-olefin intermediate (allenes).

Permethylzirconocene haloacyl complexes react with strong bases such as lithium diisopropylamide or methylene trimethylphosphorane to afford ketene compounds. Depending on the size of the alkyl ketene substituent, the hydrogenation of these compounds affords enolate-hydride products with varying degrees of stereoselectivity. The larger the substituent, the greater is the selectivity for cis hydrogenation products.

The reaction of permethylzirconocene dihydride and permethylhafnocene dihydride with methylene trimethylphosphorane affords methyl-hydride and dimethyl derivatives. Under appropriate conditions, the metallated-ylide complex 1, (η^5-C_5(CH_3)_5)_2 Zr(H)CH_2PMe_2CH_2, is also obtained and has been structurally characterized by X-ray diffraction techniques. Reaction of 1 with CO affords (η^5-C_5(CH_3)_5)_2 Zr(C,O-η^2 -(PMe_3)HC=CO)H which exists in solution as an equilibrium mixture of isomers. In one isomer (2), the η^2-acyl oxygen atom occupies a lateral equatorial coordination position about zirconium, whereas in the other isomer (3), the η-acyl oxygen atom occupies the central equatorial position. The equilibrium kinetics of the 2→3 isomerization have been studied and the structures of both complexes confirmed by X-ray diffraction methods. These studies suggest a mechanism for CO insertion into metal-carbon bonds of the early transition metals.

Permethylhafnocene dihydride and permethylzirconocene hydride complexes react with diazoalkanes to afford η^2-N, N' -hydrazonido species in which the terminal nitrogen atom of the diazoalkane molecule has inserted into a metal-hydride or metal-carbon bond. The structure of one of these compounds, Cp*_2Zr(NMeNCTol_2)OH, has been determined by X-ray diffraction techniques. Under appropriate conditions, the hydrazonido-hydride complexes react with a second equivalent of diazoalkene to afford η' -N-hydrazonido-η^2-N, N' -hydrazonido species.

Large-eddy simulations of fully developed turbulent channel and pipe flows with smooth and rough walls

Studies in turbulence often focus on two flow conditions, both of which occur frequently in real-world flows and are sought-after for their value in advancing turbulence theory. These are the high Reynolds number regime and the effect of wall surface roughness. In this dissertation, a Large-Eddy Simulation (LES) recreates both conditions over a wide range of Reynolds numbers Re_τ = O(10²)-O(10⁸) and accounts for roughness by locally modeling the statistical effects of near-wall anisotropic fine scales in a thin layer immediately above the rough surface. A subgrid, roughness-corrected wall model is introduced to dynamically transmit this modeled information from the wall to the outer LES, which uses a stretched-vortex subgrid-scale model operating in the bulk of the flow. Of primary interest is the Reynolds number and roughness dependence of these flows in terms of first and second order statistics. The LES is first applied to a fully turbulent uniformly-smooth/rough channel flow to capture the flow dynamics over smooth, transitionally rough and fully rough regimes. Results include a Moody-like diagram for the wall averaged friction factor, believed to be the first of its kind obtained from LES. Confirmation is found for experimentally observed logarithmic behavior in the normalized stream-wise turbulent intensities. Tight logarithmic collapse, scaled on the wall friction velocity, is found for smooth-wall flows when Re_τ ≥ O(10⁶) and in fully rough cases. Since the wall model operates locally and dynamically, the framework is used to investigate non-uniform roughness distribution cases in a channel, where the flow adjustments to sudden surface changes are investigated. Recovery of mean quantities and turbulent statistics after transitions are discussed qualitatively and quantitatively at various roughness and Reynolds number levels. The internal boundary layer, which is defined as the border between the flow affected by the new surface condition and the unaffected part, is computed, and a collapse of the profiles on a length scale containing the logarithm of friction Reynolds number is presented. Finally, we turn to the possibility of expanding the present framework to accommodate more general geometries. As a first step, the whole LES framework is modified for use in the curvilinear geometry of a fully-developed turbulent pipe flow, with implementation carried out in a spectral element solver capable of handling complex wall profiles. The friction factors have shown favorable agreement with the superpipe data, and the LES estimates of the Karman constant and additive constant of the log-law closely match values obtained from experiment.

Acceleration Sensing, Feedback Cooling, and Nonlinear Dynamics with Nanoscale Cavity-Optomechanical Devices

Light has long been used for the precise measurement of moving bodies, but the burgeoning field of optomechanics is concerned with the interaction of light and matter in a regime where the typically weak radiation pressure force of light is able to push back on the moving object. This field began with the realization in the late 1960's that the momentum imparted by a recoiling photon on a mirror would place fundamental limits on the smallest measurable displacement of that mirror. This coupling between the frequency of light and the motion of a mechanical object does much more than simply add noise, however. It has been used to cool objects to their quantum ground state, demonstrate electromagnetically-induced-transparency, and modify the damping and spring constant of the resonator. Amazingly, these radiation pressure effects have now been demonstrated in systems ranging 18 orders of magnitude in mass (kg to fg).

In this work we will focus on three diverse experiments in three different optomechanical devices which span the fields of inertial sensors, closed-loop feedback, and nonlinear dynamics. The mechanical elements presented cover 6 orders of magnitude in mass (ng to fg), but they all employ nano-scale photonic crystals to trap light and resonantly enhance the light-matter interaction. In the first experiment we take advantage of the sub-femtometer displacement resolution of our photonic crystals to demonstrate a sensitive chip-scale optical accelerometer with a kHz-frequency mechanical resonator. This sensor has a noise density of approximately 10 micro-g/rt-Hz over a useable bandwidth of approximately 20 kHz and we demonstrate at least 50 dB of linear dynamic sensor range. We also discuss methods to further improve performance of this device by a factor of 10.

In the second experiment, we used a closed-loop measurement and feedback system to damp and cool a room-temperature MHz-frequency mechanical oscillator from a phonon occupation of 6.5 million down to just 66. At the time of the experiment, this represented a world-record result for the laser cooling of a macroscopic mechanical element without the aid of cryogenic pre-cooling. Furthermore, this closed-loop damping yields a high-resolution force sensor with a practical bandwidth of 200 kHZ and the method has applications to other optomechanical sensors.

The final experiment contains results from a GHz-frequency mechanical resonator in a regime where the nonlinearity of the radiation-pressure interaction dominates the system dynamics. In this device we show self-oscillations of the mechanical element that are driven by multi-photon-phonon scattering. Control of the system allows us to initialize the mechanical oscillator into a stable high-amplitude attractor which would otherwise be inaccessible. To provide context, we begin this work by first presenting an intuitive overview of optomechanical systems and then providing an extended discussion of the principles underlying the design and fabrication of our optomechanical devices.