A molecular dynamics study of the microscopic properties of simple dense fluids

The microscopic properties of a two-dimensional model dense fluid of Lennard-Jones disks have been studied using the so-called "molecular dynamics" method. Analyses of the computer-generated simulation data in terms of "conventional" thermodynamic and distribution functions verify the physical validity of the model and the simulation technique.

The radial distribution functions g(r) computed from the simulation data exhibit several subsidiary features rather similar to those appearing in some of the g(r) functions obtained by X-ray and thermal neutron diffraction measurements on real simple liquids. In the case of the model fluid, these "anomalous" features are thought to reflect the existence of two or more alternative configurations for local ordering.

Graphical display techniques have been used extensively to provide some intuitive insight into the various microscopic phenomena occurring in the model. For example, "snapshots" of the instantaneous system configurations for different times show that the "excess" area allotted to the fluid is collected into relatively large, irregular, and surprisingly persistent "holes". Plots of the particle trajectories over intervals of 2.0 to 6.0 x 10^-12 sec indicate that the mechanism for diffusion in the dense model fluid is "cooperative" in nature, and that extensive diffusive migration is generally restricted to groups of particles in the vicinity of a hole.

A quantitative analysis of diffusion in the model fluid shows that the cooperative mechanism is not inconsistent with the statistical predictions of existing theories of singlet, or self-diffusion in liquids. The relative diffusion of proximate particles is, however, found to be retarded by short-range dynamic correlations associated with the cooperative mechanism--a result of some importance from the standpoint of bimolecular reaction kinetics in solution.

A new, semi-empirical treatment for relative diffusion in liquids is developed, and is shown to reproduce the relative diffusion phenomena observed in the model fluid quite accurately. When incorporated into the standard Smoluchowski theory of diffusion-controlled reaction kinetics, the more exact treatment of relative diffusion is found to lower the predicted rate of reaction appreciably.

Finally, an entirely new approach to an understanding of the liquid state is suggested. Our experience in dealing with the simulation data--and especially, graphical displays of the simulation data--has led us to conclude that many of the more frustrating scientific problems involving the liquid state would be simplified considerably, were it possible to describe the microscopic structures characteristic of liquids in a concise and precise manner. To this end, we propose that the development of a formal language of partially-ordered structures be investigated.

The tensile fracture of mild steel

The fibrous and cleavage tensile fracture of an annealed mild steel was investigated. Round tensile specimens of two geometries, one straight and one with a circumferential notch, were pulled at temperatures between room temperature and liquid nitrogen temperature. Tensile fractures occurred at average strains from 0.02 to 0.87. The mechanism of fibrous fracture at room temperature was investigated metallographically. The stress-strain values at which fibrous and cleavage fractures are initiated were determined.

Many fine microcracks, which are associated with pearlite colonies and inclusion stringers, develop prior to fibrous fracture. The macrofracture, which leads to final separation of the tensile specimen, is initiated by the propagation of a microcrack beyond the microstructural feature with which it is associated. Thus, the fibrous fracture of mild steel does not develop by the gradual growth and coalescence of voids that are large enough to be visible in the optical microscope. When the microcracks begin to open and propagate, final fracture quickly follows. Axial cracks are a prominent feature of the macrofracture that forms in the interior of the specimen immediately before final fracture.

The Bridgman distribution of stresses is not valid in a notched tensile specimen. Fibrous and cleavage fractures occur at approximately the same value of maximum tensile stress. When the maximum tensile stress that is necessary for cleavage fracture is plotted against the corresponding maximum tensile strain, the result is an unique locus.

Signatures of Topological Superconductors

Topological superconductors are particularly interesting in light of the active ongoing experimental efforts for realizing exotic physics such as Majorana zero modes. These systems have excitations with non-Abelian exchange statistics, which provides a path towards topological quantum information processing. Intrinsic topological superconductors are quite rare in nature. However, one can engineer topological superconductivity by inducing effective p-wave pairing in materials which can be grown in the laboratory. One possibility is to induce the proximity effect in topological insulators; another is to use hybrid structures of superconductors and semiconductors.

The proposal of interfacing s-wave superconductors with quantum spin Hall systems provides a promising route to engineered topological superconductivity. Given the exciting recent progress on the fabrication side, identifying experiments that definitively expose the topological superconducting phase (and clearly distinguish it from a trivial state) raises an increasingly important problem. With this goal in mind, we proposed a detection scheme to get an unambiguous signature of topological superconductivity, even in the presence of ordinarily detrimental effects such as thermal fluctuations and quasiparticle poisoning. We considered a Josephson junction built on top of a quantum spin Hall material. This system allows the proximity effect to turn edge states in effective topological superconductors. Such a setup is promising because experimentalists have demonstrated that supercurrents indeed flow through quantum spin Hall edges. To demonstrate the topological nature of the superconducting quantum spin Hall edges, theorists have proposed examining the periodicity of Josephson currents respect to the phase across a Josephson junction. The periodicity of tunneling currents of ground states in a topological superconductor Josephson junction is double that of a conventional Josephson junction. In practice, this modification of periodicity is extremely difficult to observe because noise sources, such as quasiparticle poisoning, wash out the signature of topological superconductors. For this reason, We propose a new, relatively simple DC measurement that can compellingly reveal topological superconductivity in such quantum spin Hall/superconductor heterostructures. More specifically, We develop a general framework for capturing the junction's current-voltage characteristics as a function of applied magnetic flux. Our analysis reveals sharp signatures of topological superconductivity in the field-dependent critical current. These signatures include the presence of multiple critical currents and a non-vanishing critical current for all magnetic field strengths as a reliable identification scheme for topological superconductivity.

This system becomes more interesting as interactions between electrons are involved. By modeling edge states as a Luttinger liquid, we find conductance provides universal signatures to distinguish between normal and topological superconductors. More specifically, we use renormalization group methods to extract universal transport characteristics of superconductor/quantum spin Hall heterostructures where the native edge states serve as a lead. Interestingly, arbitrarily weak interactions induce qualitative changes in the behavior relative to the free-fermion limit, leading to a sharp dichotomy in conductance for the trivial (narrow superconductor) and topological (wide superconductor) cases. Furthermore, we find that strong interactions can in principle induce parafermion excitations at a superconductor/quantum spin Hall junction.

As we identify the existence of topological superconductor, we can take a step further. One can use topological superconductor for realizing Majorana modes by breaking time reversal symmetry. An advantage of 2D topological insulator is that networks required for braiding Majoranas along the edge channels can be obtained by adjoining 2D topological insulator to form corner junctions. Physically cutting quantum wells for this purpose, however, presents technical challenges. For this reason, I propose a more accessible means of forming networks that rely on dynamically manipulating the location of edge states inside of a single 2D topological insulator sheet. In particular, I show that edge states can effectively be dragged into the system's interior by gating a region near the edge into a metallic regime and then removing the resulting gapless carriers via proximity-induced superconductivity. This method allows one to construct rather general quasi-1D networks along which Majorana modes can be exchanged by electrostatic means.

Apart from 2D topological insulators, Majorana fermions can also be generated in other more accessible materials such as semiconductors. Following up on a suggestion by experimentalist Charlie Marcus, I proposed a novel geometry to create Majorana fermions by placing a 2D electron gas in proximity to an interdigitated superconductor-ferromagnet structure. This architecture evades several manufacturing challenges by allowing single-side fabrication and widening the class of 2D electron gas that may be used, such as the surface states of bulk semiconductors. Furthermore, it naturally allows one to trap and manipulate Majorana fermions through the application of currents. Thus, this structure may lead to the development of a circuit that enables fully electrical manipulation of topologically-protected quantum memory. To reveal these exotic Majorana zero modes, I also proposed an interference scheme to detect Majorana fermions that is broadly applicable to any 2D topological superconductor platform.

I. Quantum mechanics of one-dimensional two-particle models. II. A quantum mechanical treatment of inelastic collisions

Part I

Solutions of Schrödinger’s equation for system of two particles bound in various stationary one-dimensional potential wells and repelling each other with a Coulomb force are obtained by the method of finite differences. The general properties of such systems are worked out in detail for the case of two electrons in an infinite square well. For small well widths (1-10 a.u.) the energy levels lie above those of the noninteresting particle model by as much as a factor of 4, although excitation energies are only half again as great. The analytical form of the solutions is obtained and it is shown that every eigenstate is doubly degenerate due to the “pathological” nature of the one-dimensional Coulomb potential. This degeneracy is verified numerically by the finite-difference method. The properties of the square-well system are compared with those of the free-electron and hard-sphere models; perturbation and variational treatments are also carried out using the hard-sphere Hamiltonian as a zeroth-order approximation. The lowest several finite-difference eigenvalues converge from below with decreasing mesh size to energies below those of the “best” linear variational function consisting of hard-sphere eigenfunctions. The finite-difference solutions in general yield expectation values and matrix elements as accurate as those obtained using the “best” variational function.

The system of two electrons in a parabolic well is also treated by finite differences. In this system it is possible to separate the center-of-mass motion and hence to effect a considerable numerical simplification. It is shown that the pathological one-dimensional Coulomb potential gives rise to doubly degenerate eigenstates for the parabolic well in exactly the same manner as for the infinite square well.

Part II

A general method of treating inelastic collisions quantum mechanically is developed and applied to several one-dimensional models. The formalism is first developed for nonreactive “vibrational” excitations of a bound system by an incident free particle. It is then extended to treat simple exchange reactions of the form A + BC →AB + C. The method consists essentially of finding a set of linearly independent solutions of the Schrödinger equation such that each solution of the set satisfies a distinct, yet arbitrary boundary condition specified in the asymptotic region. These linearly independent solutions are then combined to form a total scattering wavefunction having the correct asymptotic form. The method of finite differences is used to determine the linearly independent functions.

The theory is applied to the impulsive collision of a free particle with a particle bound in (1) an infinite square well and (2) a parabolic well. Calculated transition probabilities agree well with previously obtained values.

Several models for the exchange reaction involving three identical particles are also treated: (1) infinite-square-well potential surface, in which all three particles interact as hard spheres and each two-particle subsystem (i.e. BC and AB) is bound by an attractive infinite-square-well potential; (2) truncated parabolic potential surface, in which the two-particle subsystems are bound by a harmonic oscillator potential which becomes infinite for interparticle separations greater than a certain value; (3) parabolic (untruncated) surface. Although there are no published values with which to compare our reaction probabilities, several independent checks on internal consistency indicate that the results are reliable.

CP and the three pion decay of the K°

The time distribution of the decays of an initially pure K° beam into π+π-π° has been analyzed to determine the complex parameter W (also known as Ƞ^+-° and (x + iy)). The K° beam was produced in a brass target by the interactions of a 2.85 GeV/c π- beam which was generated on an internal target in the Lawrence Radiation Laboratory (LRL) Bevatron. The counters and hodoscopes in the apparatus selected for events with a neutral (K°) produced in the brass target, two charged secondaries passing through a magnet spectrometer and a ɣ-ray shower in a shower hodoscope.

From the 275K apparatus triggers, 148 K → π+π-π° events were isolated. The presence of a ɣ-ray shower in the optical shower chambers and a two-prong vee in the optical spark chambers were devices used to isolate the events. The backgrounds were further reduced by reconstructing the momenta of the two charged secondaries and applying kinematic constraints.

The best fit to the final sample of 148 events distributed between .3 and 7.0 K_S lifetimes gives:

ReW = -.05 ±.17

ImW = +.39 +.35/-.37

This result is consistent with both CPT invariance (ReW = 0) and CP invariance (W = 0). Backgrounds are estimated to be less than 10% and systematic effects have also been estimated to be negligible.

An analysis of the present data on CP violation in this decay mode and other K° decay modes has estimated the phase of ɛ to be 45.3 ± 2.3 degrees. This result is consistent with the super weak theories of CP violation which predicts the phase of ɛ to be 43°. This estimate is in turn used to predict the phase of Ƞ°° to be 48.0 ± 7.9 degrees. This is a substantial improvement on presently available measurements. The largest error in this analysis comes from the present limits on W from the world average of recent experiments. The K → πuʋ mode produces the next largest error. Therefore further experimentation in these modes would be useful.

Surface markers of regionalization in the vertebrate nervous system

In order to identify new molecules that might play a role in regional specification of the nervous system, we generated and characterized monoclonal antibodies (mAbs) that have positionally-restricted labeling patterns.

The FORSE-1 mAb was generated using a strategy designed to produce mAbs against neuronal cell surface antigens that might be regulated by regionally-restricted transcription factors in the developing central nervous system (CNS). FORSE-1 staining is enriched in the forebrain as compared to the rest of the CNS until E18. Between E11.5-E13.5, only certain areas of the forebrain are labeled. There is also a dorsoventrally-restricted region of labeling in the hindbrain and spinal cord. The mAb labels a large proteoglycan-like cell-surface antigen (>200 kD). The labeling pattern of FORSE-1 is conserved in various mammals and in chick.

To determine whether the FORSE-1 labeling pattern is similar to that of known transcription factors, the expression of BF-1 and Dlx-2 was compared with FORSE-1. There is a striking overlap between BF-1 and FORSE-1 in the telencephalon. In contrast, FORSE-1 and Dlx-2 have very different patterns of expression in the forebrain, suggesting that regulation by Dlx-2 alone cannot explain the distribution of FORSE-1. They do, however, share some sharp boundaries in the diencephalon. In addition, FORSE-1 identifies some previously unknown boundaries in the developing forebrain. Thus, FORSE-1 is a new cell surface marker that can be used to subdivide the embryonic forebrain into regions smaller than previously described, providing further complexity necessary for developmental patterning.

I also studied the expression of the cell surface protein CD9 in the developing and adult rat nervous system. CD9 is implicated in intercellular signaling and cell adhesion in the hematopoetic system. In the nervous system, CD9 may perform similar functions in early sympathetic ganglia, chromaffin cells, and motor neurons, all of which express the protein. The presence of CD9 on the surfaces of Schwann cells and axons at the appropriate time may allow the protein to participate in the cellular interactions involved in myelination.

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.

Optimization and Control of Power Flow in Distribution Networks

Climate change is arguably the most critical issue facing our generation and the next. As we move towards a sustainable future, the grid is rapidly evolving with the integration of more and more renewable energy resources and the emergence of electric vehicles. In particular, large scale adoption of residential and commercial solar photovoltaics (PV) plants is completely changing the traditional slowly-varying unidirectional power flow nature of distribution systems. High share of intermittent renewables pose several technical challenges, including voltage and frequency control. But along with these challenges, renewable generators also bring with them millions of new DC-AC inverter controllers each year. These fast power electronic devices can provide an unprecedented opportunity to increase energy efficiency and improve power quality, if combined with well-designed inverter control algorithms. The main goal of this dissertation is to develop scalable power flow optimization and control methods that achieve system-wide efficiency, reliability, and robustness for power distribution networks of future with high penetration of distributed inverter-based renewable generators.

Proposed solutions to power flow control problems in the literature range from fully centralized to fully local ones. In this thesis, we will focus on the two ends of this spectrum. In the first half of this thesis (chapters 2 and 3), we seek optimal solutions to voltage control problems provided a centralized architecture with complete information. These solutions are particularly important for better understanding the overall system behavior and can serve as a benchmark to compare the performance of other control methods against. To this end, we first propose a branch flow model (BFM) for the analysis and optimization of radial and meshed networks. This model leads to a new approach to solve optimal power flow (OPF) problems using a two step relaxation procedure, which has proven to be both reliable and computationally efficient in dealing with the non-convexity of power flow equations in radial and weakly-meshed distribution networks. We will then apply the results to fast time- scale inverter var control problem and evaluate the performance on real-world circuits in Southern California Edison’s service territory.

The second half (chapters 4 and 5), however, is dedicated to study local control approaches, as they are the only options available for immediate implementation on today’s distribution networks that lack sufficient monitoring and communication infrastructure. In particular, we will follow a reverse and forward engineering approach to study the recently proposed piecewise linear volt/var control curves. It is the aim of this dissertation to tackle some key problems in these two areas and contribute by providing rigorous theoretical basis for future work.