Asymptotic analysis of thin plates under normal load and horizontal edge thrust

We consider the radially symmetric nonlinear von Kármán plate equations for circular or annular plates in the limit of small thickness. The loads on the plate consist of a radially symmetric pressure load and a uniform edge load. The dependence of the steady states on the edge load and thickness is studied using asymptotics as well as numerical calculations. The von Kármán plate equations are a singular perturbation of the Fӧppl membrane equation in the asymptotic limit of small thickness. We study the role of compressive membrane solutions in the small thickness asymptotic behavior of the plate solutions.

We give evidence for the existence of a singular compressive solution for the circular membrane and show by a singular perturbation expansion that the nonsingular compressive solution approach this singular solution as the radial stress at the center of the plate vanishes. In this limit, an infinite number of folds occur with respect to the edge load. Similar behavior is observed for the annular membrane with zero edge load at the inner radius in the limit as the circumferential stress vanishes.

We develop multiscale expansions, which are asymptotic to members of this family for plates with edges that are elastically supported against rotation. At some thicknesses this approximation breaks down and a boundary layer appears at the center of the plate. In the limit of small normal load, the points of breakdown approach the bifurcation points corresponding to buckling of the nondeflected state. A uniform asymptotic expansion for small thickness combining the boundary layer with a multiscale approximation of the outer solution is developed for this case. These approximations complement the well known boundary layer expansions based on tensile membrane solutions in describing the bending and stretching of thin plates. The approximation becomes inconsistent as the clamped state is approached by increasing the resistance against rotation at the edge. We prove that such an expansion for the clamped circular plate cannot exist unless the pressure load is self-equilibrating.

Quantum simulation of enzyme catalysis

Separating the dynamics of variables that evolve on different timescales is a common assumption in exploring complex systems, and a great deal of progress has been made in understanding chemical systems by treating independently the fast processes of an activated chemical species from the slower processes that proceed activation. Protein motion underlies all biocatalytic reactions, and understanding the nature of this motion is central to understanding how enzymes catalyze reactions with such specificity and such rate enhancement. This understanding is challenged by evidence of breakdowns in the separability of timescales of dynamics in the active site form motions of the solvating protein. Quantum simulation methods that bridge these timescales by simultaneously evolving quantum and classical degrees of freedom provide an important method on which to explore this breakdown. In the following dissertation, three problems of enzyme catalysis are explored through quantum simulation.

Nonlinear Effects in Granular Crystals with Broken Periodicity

When studying physical systems, it is common to make approximations: the contact interaction is linear, the crystal is periodic, the variations occurs slowly, the mass of a particle is constant with velocity, or the position of a particle is exactly known are just a few examples. These approximations help us simplify complex systems to make them more comprehensible while still demonstrating interesting physics. But what happens when these assumptions break down? This question becomes particularly interesting in the materials science community in designing new materials structures with exotic properties In this thesis, we study the mechanical response and dynamics in granular crystals, in which the approximation of linearity and infinite size break down. The system is inherently finite, and contact interaction can be tuned to access different nonlinear regimes. When the assumptions of linearity and perfect periodicity are no longer valid, a host of interesting physical phenomena presents itself. The advantage of using a granular crystal is in its experimental feasibility and its similarity to many other materials systems. This allows us to both leverage past experience in the condensed matter physics and materials science communities while also presenting results with implications beyond the narrower granular physics community. In addition, we bring tools from the nonlinear systems community to study the dynamics in finite lattices, where there are inherently more degrees of freedom. This approach leads to the major contributions of this thesis in broken periodic systems. We demonstrate the first defect mode whose spatial profile can be tuned from highly localized to completely delocalized by simply tuning an external parameter. Using the sensitive dynamics near bifurcation points, we present a completely new approach to modifying the incremental stiffness of a lattice to arbitrary values. We show how using nonlinear defect modes, the incremental stiffness can be tuned to anywhere in the force-displacement relation. Other contributions include demonstrating nonlinear breakdown of mechanical filters as a result of finite size, and the presents of frequency attenuation bands in essentially nonlinear materials. We finish by presenting two new energy harvesting systems based on our experience with instabilities in weakly nonlinear systems.

Battery-Powered RF Pre-Ionization System for the Caltech Magnetohydrodynamically-Driven Jet Experiment: RF Discharge Properties and MHD-Driven Jet Dynamics

This thesis describes investigations of two classes of laboratory plasmas with rather different properties: partially ionized low pressure radiofrequency (RF) discharges, and fully ionized high density magnetohydrodynamically (MHD)-driven jets. An RF pre-ionization system was developed to enable neutral gas breakdown at lower pressures and create hotter, faster jets in the Caltech MHD-Driven Jet Experiment. The RF plasma source used a custom pulsed 3 kW 13.56 MHz RF power amplifier that was powered by AA batteries, allowing it to safely float at 4-6 kV with the cathode of the jet experiment. The argon RF discharge equilibrium and transport properties were analyzed, and novel jet dynamics were observed.

Although the RF plasma source was conceived as a wave-heated helicon source, scaling measurements and numerical modeling showed that inductive coupling was the dominant energy input mechanism. A one-dimensional time-dependent fluid model was developed to quantitatively explain the expansion of the pre-ionized plasma into the jet experiment chamber. The plasma transitioned from an ionizing phase with depressed neutral emission to a recombining phase with enhanced emission during the course of the experiment, causing fast camera images to be a poor indicator of the density distribution. Under certain conditions, the total visible and infrared brightness and the downstream ion density both increased after the RF power was turned off. The time-dependent emission patterns were used for an indirect measurement of the neutral gas pressure.

The low-mass jets formed with the aid of the pre-ionization system were extremely narrow and collimated near the electrodes, with peak density exceeding that of jets created without pre-ionization. The initial neutral gas distribution prior to plasma breakdown was found to be critical in determining the ultimate jet structure. The visible radius of the dense central jet column was several times narrower than the axial current channel radius, suggesting that the outer portion of the jet must have been force free, with the current parallel to the magnetic field. The studies of non-equilibrium flows and plasma self-organization being carried out at Caltech are relevant to astrophysical jets and fusion energy research.

Biological Activity of Rhodium Metalloinsertors and the Design of Bifunctional Conjugates

The Barton laboratory has established that octahedral rhodium complexes bearing the sterically expansive 5,6-chrysene diimine ligand can target thermodynamically destabilized sites, such as base pair mismatches, in DNA with high affinity and selectivity. These complexes approach DNA from the minor groove, ejecting the mismatched base pairs from the duplex in a binding mode termed metalloinsertion. In recent years, we have shown that these metalloinsertor complexes also exhibit cytotoxicity preferentially in cancer cells that are deficient in the mismatch repair (MMR) machinery.

Here, we establish that a sensitive structure-activity relationship exists for rhodium metalloinsertors. We studied the relationship between the chemical structures of metalloinsertors and their effect on biological activity for ten complexes with similar DNA binding affinities, but wide variation in their lipophilicity. Drastic differences were observed in the selectivities of the complexes for MMR-deficient cells. Compounds with hydrophilic ligands were highly selective, exhibiting preferential cytotoxicity in MMR-deficient cells at low concentrations and short incubation periods, whereas complexes with lipophilic ligands displayed poor cell-selectivity. It was discovered that all of the complexes localized to the nucleus in concentrations sufficient for mismatch binding; however, highly lipophilic complexes also exhibited high mitochondrial uptake. Significantly, these results support the notion that mitochondrial DNA is not the desired target for our metalloinsertor complexes; instead, selectivity stems from targeting mismatches in genomic DNA.

We have also explored the potential for metalloinsertors to be developed into more complex structures with multiple functionalities that could either enhance their overall potency or impart mismatch selectivity onto other therapeutic cargo. We have constructed a family of bifunctional metalloinsertor conjugates incorporating cis-platinum, each unique in its chemical structure, DNA binding interactions, and biological activity. The study of these complexes in MMR-deficient cells has established that the cell-selective biological activity of rhodium metalloinsertors proceeds through a critical cellular pathway leading to necrosis.

We further explored the underlying mechanisms surrounding the biological response to mismatch recognition by metalloinsertors in the genome. Immunofluorescence assays of MMR-deficient and MMR-proficient cells revealed that a critical biomarker for DNA damage, phosphorylation of histone H2AX (γH2AX) rapidly accumulates in response to metalloinsertor treatment, signifying the induction of double strand breaks in the genome. Significantly, we have discovered that our metalloinsertor complexes selectively inhibit transcription in MMR-deficient cells, which may be a crucial checkpoint in the eventual breakdown of the cell via necrosis. Additionally, preliminary in vivo studies have revealed the capability of these compounds to traverse the complex environments of multicellular organisms and accumulate in MMR-deficient tumors. Our ever-increasing understanding of metalloinsertors, as well as the development of new generations of complexes both monofunctional and bifunctional, enables their continued progress into the clinic as promising new chemotherapeutic agents.

Spectroscopic investigations of inter- and intramolecular processes in crystalline benzene

I. Introductory Remarks

A brief discussion of the overall organization of the thesis is presented along with a discussion of the relationship between this thesis and previous work on the spectroscopic properties of benzene.

II. Radiationless Transitions and Line broadening

Radiationless rates have been calculated for the ³B_1u→¹A_1g transitions of benzene and perdeuterobenzene as well as for the ¹B_2u→¹A_1g transition of benzene. The rates were calculated using a model that considers the radiationless transition as a tunneling process between two multi-demensional potential surfaces and assuming both harmonic and anharmonic vibrational potentials. Whenever possible experimental parameters were used in the calculation. To this end we have obtained experimental values for the anharmonicities of the carbon-carbon and carbon-hydrogen vibrations and the size of the lowest triplet state of benzene. The use of the breakdown of the Born-Oppenheimer approximation in describing radiationless transitions is critically examined and it is concluded that Herzberg-Teller vibronic coupling is 100 times more efficient at inducing radiationless transitions.

The results of the radiationless transition rate calculation are used to calculate line broadening in several of the excited electronic states of benzene. The calculated line broadening in all cases is in qualitative agreement with experimental line widths.

III. ³B_1u←¹A_1g Absorption Spectra

The ³B_1u←¹A_1g absorption spectra of C₆H₆ and C₆D₆ at 4.2˚K have been obtained at high resolution using the phosphorescence photoexcitation method. The spectrum exhibits very clear evidence of a pseudo-Jahn-Teller distortion of the normally hexagonal benzene molecule upon excitation to the triplet state. Factor group splitting of the 0 – 0 and 0 – 0 + v exciton bands have also been observed. The position of the mean of the 0 – 0 exciton band of C₆H₆ when compared to the phosphorescence origin of a C₆H₆ guest in a C₆D₆ host crystal indicates that the “static” intermolecular interactions between guest and hose are different for C₆H₆ and C₆D₆. Further investigation of this difference using the currently accepted theory of isotopic mixed crystals indicates that there is a 2cm^-1 shift of the ideal mixed crystal level per hot deuterium atom. This shift is observed for both the singlet and triplet states of benzene.

IV. ³E_1u←¹A_1g, Absorption Spectra

The ³E_1u←¹A_1g absorption spectra of C₆H₆ and C₆D₆ at 4.2˚K have been obtained using the phosphorescence photoexcitation technique. In both cases the spectrum is broad and structureless as would be expected from the line broadening calculations.

Dielectric recovery of short A-C arcs between low-boiling-point electrodes

The re-ignition characteristics (variation of re-ignition voltage with time after current zero) of short alternating current arcs between plane brass electrodes in air were studied by observing the average re-ignition voltages on the screen of a cathode-ray oscilloscope and controlling the rates of rise of voltage by varying the shunting capacitance and hence the natural period of oscillation of the reactors used to limit the current. The shape of these characteristics and the effects on them of varying the electrode separation, air pressure, and current strength were determined.

The results show that short arc spaces recover dielectric strength in two distinct stages. The first stage agrees in shape and magnitude with a previously developed theory that all voltage is concentrated across a partially deionized space charge layer which increases its breakdown voltage with diminishing density of ionization in the field-tree space. The second stage appears to follow complete deionization by the electric field due to displacement of the field-free region by the space charge layer, its magnitude and shape appearing to be due simply to increase in gas density due to cooling. Temperatures calculated from this second stage and ion densities determined from the first stage by means of the space charge equation and an extrapolation of the temperature curve are consistent with recent measurements of arc value by other methods. Analysis or the decrease with time of the apparent ion density shows that diffusion alone is adequate to explain the results and that volume recombination is not. The effects on the characteristics of variations in the parameters investigated are found to be in accord with previous results and with the theory if deionization mainly by diffusion be assumed.

1. Ultrasonic studies of binary liquid structure in the critical region. Theory and experiment for the 2,6-lutidine/water system. 2. Hartree-Fock calculations of electric polarizabilities of some simple atoms and molecules, and their practicality. 3. Calculation of vibrational transition probabilities in collinear atom-diatom and diatom-diatom collisions with Lennard-Jones interaction

Part 1. Many interesting visual and mechanical phenomena occur in the critical region of fluids, both for the gas-liquid and liquid-liquid transitions. The precise thermodynamic and transport behavior here has some broad consequences for the molecular theory of liquids. Previous studies in this laboratory on a liquid-liquid critical mixture via ultrasonics supported a basically classical analysis of fluid behavior by M. Fixman (e. g., the free energy is assumed analytic in intensive variables in the thermodynamics)--at least when the fluid is not too close to critical. A breakdown in classical concepts is evidenced close to critical, in some well-defined ways. We have studied herein a liquid-liquid critical system of complementary nature (possessing a lower critical mixing or consolute temperature) to all previous mixtures, to look for new qualitative critical behavior. We did not find such new behavior in the ultrasonic absorption ascribable to the critical fluctuations, but we did find extra absorption due to chemical processes (yet these are related to the mixing behavior generating the lower consolute point). We rederived, corrected, and extended Fixman's analysis to interpret our experimental results in these more complex circumstances. The entire account of theory and experiment is prefaced by an extensive introduction recounting the general status of liquid state theory. The introduction provides a context for our present work, and also points out problems deserving attention. Interest in these problems was stimulated by this work but also by work in Part 3.

Part 2. Among variational theories of electronic structure, the Hartree-Fock theory has proved particularly valuable for a practical understanding of such properties as chemical binding, electric multipole moments, and X-ray scattering intensity. It also provides the most tractable method of calculating first-order properties under external or internal one-electron perturbations, either developed explicitly in orders of perturbation theory or in the fully self-consistent method. The accuracy and consistency of first-order properties are poorer than those of zero-order properties, but this is most often due to the use of explicit approximations in solving the perturbed equations, or to inadequacy of the variational basis in size or composition. We have calculated the electric polarizabilities of H₂, He, Li, Be, LiH, and N₂ by Hartree-Fock theory, using exact perturbation theory or the fully self-consistent method, as dictated by convenience. By careful studies on total basis set composition, we obtained good approximations to limiting Hartree-Fock values of polarizabilities with bases of reasonable size. The values for all species, and for each direction in the molecular cases, are within 8% of experiment, or of best theoretical values in the absence of the former. Our results support the use of unadorned Hartree-Pock theory for static polarizabilities needed in interpreting electron-molecule scattering data, collision-induced light scattering experiments, and other phenomena involving experimentally inaccessible polarizabilities.

Part 3. Numerical integration of the close-coupled scattering equations has been carried out to obtain vibrational transition probabilities for some models of the electronically adiabatic H₂-H₂ collision. All the models use a Lennard-Jones interaction potential between nearest atoms in the collision partners. We have analyzed the results for some insight into the vibrational excitation process in its dependence on the energy of collision, the nature of the vibrational binding potential, and other factors. We conclude also that replacement of earlier, simpler models of the interaction potential by the Lennard-Jones form adds very little realism for all the complication it introduces. A brief introduction precedes the presentation of our work and places it in the context of attempts to understand the collisional activation process in chemical reactions as well as some other chemical dynamics.

I. Fundamental limitations in microelectronics. II. Power Schottky diode design and comparison with the junction diode. III. Permittivity of strontium titanate

Part I

The physical phenomena which will ultimately limit the packing density of planar bipolar and MOS integrated circuits are examined. The maximum packing density is obtained by minimizing the supply voltage and the size of the devices. The minimum size of a bipolar transistor is determined by junction breakdown, punch-through and doping fluctuations. The minimum size of a MOS transistor is determined by gate oxide breakdown and drain-source punch-through. The packing density of fully active bipolar or static non-complementary MOS circuits becomes limited by power dissipation. The packing density of circuits which are not fully active such as read-only memories, becomes limited by the area occupied by the devices, and the frequency is limited by the circuit time constants and by metal migration. The packing density of fully active dynamic or complementary MOS circuits is limited by the area occupied by the devices, and the frequency is limited by power dissipation and metal migration. It is concluded that read-only memories will reach approximately the same performance and packing density with MOS and bipolar technologies, while fully active circuits will reach the highest levels of integration with dynamic MOS or complementary MOS technologies.

Part II

Because the Schottky diode is a one-carrier device, it has both advantages and disadvantages with respect to the junction diode which is a two-carrier device. The advantage is that there are practically no excess minority carriers which must be swept out before the diode blocks current in the reverse direction, i.e. a much faster recovery time. The disadvantage of the Schottky diode is that for a high voltage device it is not possible to use conductivity modulation as in the p i n diode; since charge carriers are of one sign, no charge cancellation can occur and current becomes space charge limited. The Schottky diode design is developed in Section 2 and the characteristics of an optimally designed silicon Schottky diode are summarized in Fig. 9. Design criteria and quantitative comparison of junction and Schottky diodes is given in Table 1 and Fig. 10. Although somewhat approximate, the treatment allows a systematic quantitative comparison of the devices for any given application.

Part III

We interpret measurements of permittivity of perovskite strontium titanate as a function of orientation, temperature, electric field and frequency performed by Dr. Richard Neville. The free energy of the crystal is calculated as a function of polarization. The Curie-Weiss law and the LST relation are verified. A generalized LST relation is used to calculate the permittivity of strontium titanate from zero to optic frequencies. Two active optic modes are important. The lower frequency mode is attributed mainly to motion of the strontium ions with respect to the rest of the lattice, while the higher frequency active mode is attributed to motion of the titanium ions with respect to the oxygen lattice. An anomalous resonance which multi-domain strontium titanate crystals exhibit below 65°K is described and a plausible mechanism which explains the phenomenon is presented.