The semi-classical black hole tunnelling method is a useful technique to calculate black hole temperature and understand black hole thermodynamics. I will investigate the black hole tunnelling method in detail. I will compare two different approaches used to calculate black hole tunnelling. The tunnelling method can be applied to a broad range of spacetimes and I will show this explicitly in order to demonstrate the robustness of the tunnelling technique. In particular, I will apply the tunnelling method to spacetimes including: Rindler (the method can recover the Unruh temperature), and more general spacetimes (such as Kerr-Newman and Taub-NUT). I will also discuss the 5d Kerr-Gödel spacetimes in detail (while showing a previous unobserved property of these spaces). Once the parameter space of Kerr-Gödel is understood in detail, I will show how the tunnelling method can also be successfully applied to the Kerr-Gödel black hole. Finally, the key result of my thesis involves extending the tunnelling method to model fermion emission. The previous tunnelling calculations all involved the emission of scalar particles. I will model the emission of spin-1/2 fermions from various spacetimes including the Rindler spacetime and general non-rotating black holes. I will also model the emission of charged spin-1/2 fermions from the Kerr-Newman spacetime to show that the method is also applicable to rotating spacetimes. In all these cases I show that the correct Hawking temperature (Unruh temperature in the case of Rindler) is recovered for spin-1/2 fermion emission. Although this final result is not surprising, it is an important result because it confirms that Dirac particles will radiate from the black hole at the same temperature as scalar particles. It has always been assumed that this is the case but there is very little literature involving fermion radiation of black holes. So the results of my calculations are twofold, I demonstrate that Dirac particles are emitted at the same temperature as scalar particles from a black hole and it shows how robust the semi-classical tunnelling technique is.

The semi-classical black hole tunnelling method is a useful technique to calculate black hole temperature and understand black hole thermodynamics. I will investigate the black hole tunnelling method in detail. I will compare two different approaches used to calculate black hole tunnelling. The tunnelling method can be applied to a broad range of spacetimes and I will show this explicitly in order to demonstrate the robustness of the tunnelling technique. In particular, I will apply the tunnelling method to spacetimes including: Rindler (the method can recover the Unruh temperature), and more general spacetimes (such as Kerr-Newman and Taub-NUT). I will also discuss the 5d Kerr-Gödel spacetimes in detail (while showing a previous unobserved property of these spaces). Once the parameter space of Kerr-Gödel is understood in detail, I will show how the tunnelling method can also be successfully applied to the Kerr-Gödel black hole. Finally, the key result of my thesis involves extending the tunnelling method to model fermion emission. The previous tunnelling calculations all involved the emission of scalar particles. I will model the emission of spin-1/2 fermions from various spacetimes including the Rindler spacetime and general non-rotating black holes. I will also model the emission of charged spin-1/2 fermions from the Kerr-Newman spacetime to show that the method is also applicable to rotating spacetimes. In all these cases I show that the correct Hawking temperature (Unruh temperature in the case of Rindler) is recovered for spin-1/2 fermion emission. Although this final result is not surprising, it is an important result because it confirms that Dirac particles will radiate from the black hole at the same temperature as scalar particles. It has always been assumed that this is the case but there is very little literature involving fermion radiation of black holes. So the results of my calculations are twofold, I demonstrate that Dirac particles are emitted at the same temperature as scalar particles from a black hole and it shows how robust the semi-classical tunnelling technique is.

The photoactive component of a polymeric organic solar cell can be produced by drying a mixture consisting of a volatile solvent and non-volatile polymers. As the solvent evaporates, the polymers demix and self-assemble into microscale structures, the morphology of which plays a pivotal role in determining the efficiency of the resulting device. Thus, a detailed understanding of the physical mechanisms that drive and influence structure formation in evaporating solvent-polymer mixtures is of high scientific and industrial value. This thesis explores several problems that aim to produce novel insights into the dynamics of evaporating solvent-polymer mixtures. First, the role of compositional Marangoni instabilities in slowly evaporating binary mixtures is studied using the framework of linear stability theory. The analysis is non-trivial because evaporative mass loss naturally leads to a time-dependent base state. In the limit of slow evaporation compared to diffusion, a separation of time scales emerges in the linear stability problem, allowing asymptotic methods to be applied. In particular, an asymptotic solution to linear stability problems that have slowly evolving base states is derived. Using this solution, regions of parameter space where an oscillatory instability occurs are identified and used to formulate appropriate conditions for observing this phenomenon in future experiments. The second topic of this thesis is the use of multiphase fluid models to study the dynamics of evaporating solvent-polymer mixtures. A two-phase model is used to assess the role of compositional buoyancy and to examine the formation of a polymer-rich skin at the free surface. Then, a three-phase model is used to conduct a preliminary investigation of the link between evaporation and phase separation. Finally, this thesis explores the dynamics of a binary mixture that is confined between two horizontal walls using a diffusive phase-field model and its sharp-interface and thin-film approximations. We first determine the conditions under which a homogeneous mixture undergoes phase separation to form a metastable bilayer. We then present a novel mechanism for generating a repeating lateral sequence of alternating A-rich and B-rich domains from this bilayer.

This dissertation is a theoretical treatment of the electric field dependence of optical properties such as Quantum Confined Stark (QCS) shifts, Photoluminescence Quenching (PLQ), and Excitonic Mixing in quantum well heterostructures. The reduced spatial dimensionality in heterostructures greatly enhances these optical properties, more than in three dimensional semiconductors. Charge presence in the quantum well from doping causes the potential to bend and deviate from the ideal square well potential. A potential bending that varies as the square of distance measured from the heterostructure interfaces is derived self-consistently. This potential is used to solve the time-independent Schrodinger equation for bound state energies and wave functions within the framework of the Wentzel, Kramers, and Brillouin (WKB) approximation. The theoretical results obtained from the WKB approximation are limited to wide gap semiconductors with large split off bands such as gallium arsenide-gallium aluminum arsenide and indium gallium arsenide—indium phosphide. Quantum wells with finite confinement heights give rise to an energy dependent WKB phase. External electric and magnetic fields are incorporated into the theory for two different geometries. For electric fields applied perpendicular to the heterostructure multilayers, QCS shifts and PLQ are found to be in excellent agreement with the WKB calculations. Orthogonality between electrons and holes gives rise to interband mixing in the presence of an external electric field. On the contrary, intraband mixing between light and heavy holes is not sufficiently accounted for in the WKB approximation.

A rich variety of discs are found orbiting massive bodies in the universe. These could be accretion discs composed of gas around stellar mass compact objects fueling micro-quasar activity; protoplanetary discs, mainly composed of dust and gas, are the progenitors for planet formation; accretion discs composed of stars and gas around super-massive black holes at the centers of galaxies fueling the active galactic nuclei activity; discs in spiral galaxies; and many more. Structural and kinematic properties of these discs in several astrophysical systems are correlated to the global properties; for example, over a sample of thousands of galaxies, a correlation has been found between lopsidedness, black hole growth, and the presence of young stellar populations in the centers of galaxies. Galaxy formation and evolution of the central BH are some of the contexts in which such correlations become important. Studying the dynamics of these discs helps to explain their structural properties and is thus of paramount importance. In most astrophysical discs(a notable exception being the stellar discs in spiral galaxies),the dynamics are usually dominated by the gravity of the central object, and is thus nearly Keplerian. However, there is a small contribution to the total force experienced by the disc due to the disc material. Discs mentioned above differ from each other due to different underlying force that dominates the non-Keplerian dynamics of these discs. Two important numbers which are useful in describing physical properties of any disc structure in astrophysics are: (1) Mach number M , and(2) Toomre Q parameter. If thermal pressure gradient and/or random motion dominate the non-Keplerian forces, then M « Q, and in the case when the self-gravity of the disc is more important then Particles constituting the disc orbit under Keplerian potential due to central object, plus the small contribution from the non-Keplerian potential due to disc self-gravity, or the thermal pressure gradient. For a Keplerian potential, the radial and azimuthal frequencies are in 1 : 1 ratio w.r.t. each other and hence there is no precession in the orbits. In case of nearly Keplerian potential(when non-Keplerian contributions are small), the orbits precess at a rate proportional to the non-Keplerian forces. It is this non-zero but small precession that allows the existence of modes whose frequencies are proportional to the precession rate. These modes are referred to as slow modes (Tremaine 2001). Such modes are likely to be the only large-scale or long-wavelength modes. The damping they suffer due to viscosity, collisions, Landau damping, or other dissipative processes is also relatively less. Hence, these modes can dominate the overall appearance of discs. In this thesis we intend to study slow modes for nearly Keplerian discs. Slow modes innear-Kepleriandiscscantobethereasonforvariousnon-axisymmetricfeatures observed in many systems: 1 Galactic discs: Of the few galaxies for which the observations of galactic nuclei exist, two galaxies: NGC4486B(an elliptical galaxy) andM31(spiral galaxy), show an unusual double-peak distribution of stars at their centers. In order to explain such distributions, Tremaine in 1995 proposed an eccentric disc model for M31; this model was then further explored by many authors. In addition, lopsidedness is observed in many galaxies on larger scales, and such asymmetries need to be explained via robust modeling of galactic discs. 2 Debris disc: Many of the observed discs show non-axisymmetric structures, such as lopsided distribution in brightness of scattered light, warp, and clumps in the disc around β Pictoris; spiral structure inHD141569A,etc. Most of these features have been attributed to the presence of planets, and in some cases planets have also been detected. However, Jalali & Tremaine(2012)proposed that most of these structures can be formed also due to slow (m =1 or 2) modes. 3 Accretion Discs around stellar mass binaries have also been found to be asymmetric. One plausible reason for this asymmetry can be m =1slowmodes in these systems. Slow modes are studied in detail in this thesis. The main approaches that we have used, and the major conclusions from this work are as follows: Slow pressure modes in thin accretion disc Earlier work on slow modes assumed that the self-gravity of the disc dominates the pressure gradient in the discs. However, this assumption is not valid for thin and hot accretion discs around stellar mass compact objects. We begin our study of slow modes with the analysis of modes in thin accretion discs around stellar mass compact objects. First, the WKB analysis is used to prove the existence of these modes. Next, we formulate the eigenvalue equation for the slow modes, which turns out to be in the Sturm-Liouville form; thus all the eigenvalues are real. Real eigenvalues imply that the disc is stable to these perturbations. We also discuss the possible excitation mechanisms for these modes; for instance, excitation due to the stream of matter from the secondary star that feeds the accretion disc, or through the action of viscous forces. Slow modes in self-gravitating, zero-pressure fluid disc We next generalize the study of slow m = 1 modes for a single self-gravitating disc of Tremaine(2001) to a system of two self-gravitating counter–rotating, zero-pressure fluid discs, where the disc particles interact via softened-gravity. Counter– rotating streams of matter are susceptible to various instabilities. In particular, Touma(2002)found unstable modes in counter–rotating ,nearly Keplerian systems. These modes were calculated analytically for a two-ring system, and numerically for discs modeled assuming a multiple–ring system. Motivated by this, the corresponding problem for continuous discs was studied by Sridhar & Saini(2010),who proposed a simple model, with dynamics that could be studied largely analytically in the local WKB approximation. Their work, however, had certain limitations; they could construct eigenmodes only for η =0&12, where η is the mass fraction in the retrograde disc. They could only calculate eigenvalues but not the eigen functions. To overcome the above mentioned limitations, we formulate and analyze the full eigenvalue problem to understand the systematic behaviour of such systems. Our general conclusions are as follows 1 The system is stable for m = 1 perturbations in the case of no-counter rotation. 2. For other values of mass fraction , the eigenvalues are generally complex, and the discs are unstable. For η =12,theeigenvalues are imaginary, giving purely growing modes. 2 The pattern speed appears to be non-negative for all values of , with the growth(or damping) rate being larger for larger values of pattern speed. 3 Perturbed surface density profile is generally lopsided, with an overall rotation of the patterns as they evolve in time, with the pattern speed given by the real part of the eigenvalue. Local WKB analysis for Keplerian stellar disc We next urn to stellar discs, whose dynamics is richer than softened gravity discs. Jalali & Tremaine(2012)derived the dispersion relation for short wavelength slow modes for a single disc with Schwarzschild distribution function. In contrast to the softened gravity discs(which have slow modes only for m = 1), stellar discs permit slow modes for m 1. The dispersion relation derived by Jalali & Tremaine makes it evident that all m 1 slow modes are neutrally stable. We study slow modes for the case of two counter–rotating discs, each described by Schwarzschild distribution function, and derive the dispersion relation for slow m 1 modes in the local WKB limit and study the nature of the instabilities. One of the important applications of the dispersion relation derived in this chapter is the stability analysis of the modes. For fluid discs, it is well known that the stability of m = 0 modes guarantees the stability of higher m modes; and the stability criterion for such discs is the well known Toomre stability criterion. However, this is not the case for collisionless discs. Even if the discs are stable for axisymmetric modes, they can still be unstable for non-axisymmetric modes. The stability of axisymmetric modes is governed by the Toomre stability criterion The non-axisymmetric perturbations were found to be unstable if the mass in the retrograde component of the disc is non-zero. We next solve the dispersion relation using the Bohr-Sommerfeld quantization condition to obtain the eigen-spectrum for a given unperturbed surface density profile and velocity dispersion. We could obtain only the eigenvalues for no counter– rotation η = 0, where η is the mass fraction in the retrograde disc and equal counter–rotation(η =12). All the eigenvalues obtained were real for no counter– rotation, and purely growing/damping for equal counter–rotation. The eigenvalue trends that we get favour detection of high ω and low m modes observationally. We also make a detailed comparison between the eigenvalues for m = 1 modes that we obtain with those obtained after solving the integral eigenvalue problem for the softened gravity discs for no counter–rotation and equal counter–rotation. The match between the eigenvalues are quite good, confirming the assertion that softened gravity discs can be reasonable surrogates for collisionless disc for m =1 modes. Non-local WKB theory for eigenmodes One major limitation of the above method is that eigenfunctions cannot be obtained as directly as in quantum mechanics because the dispersion relation is transcendental in radial wave number . We overcome this difficulty by dropping the assumption of locality of the relationship between perturbed self-gravity and surface density. Using the standard WKB analysis and epicyclic theory, together with the logarithmic-spiral decomposition of surface density and gravitational potential, we formulate an integral equation for determining both WKB eigenvalues and eigenfunctions. The application of integral equation derived is not only restricted to Keplerian disc; it could be used to study eigenmodes in galactic discs where the motion of stars is not dominated by the potential due to a central black hole (however we have not pursued the potential application in this thesis). We first verify that the integral equation derived reduces to the well known WKB dispersion relation under the local approximation. We next specialize to slow modes in Keplerian discs. The following are some of the general conclusions of this work 1 We find that the integral equation for slow modes reduces to a symmetric eigenvalue problem, implying that the eigenvalues are all real, and hence the disc is stable. 2 All the non-singular eigenmodes we obtain are prograde, which implies that the density waves generated will have the same sense of rotation as the disc, albeit with a speed which is compared to the the rotation speed of the disc. 3 Eigenvalue ω decreases as we go from m =1 to 2. In addition, for a given , the number of nodes for m =1 are larger than those for m =2. 4 The fastest pattern speed is a decreasing function of the heat in the disc. Asymmetric features in various types of discs could be due to the presence of slow m =1 or 2 modes. In the case of debris discs, these asymmetric features could also be due to the presence of planets. Features due to the presence of slow modes or due to planets can be distinguished from each other if the observations are made for a long enough time. The double peak nucleus observed in galaxies like M31 and NGC4486B differ from each other: stellar distribution in NGC4486B is symmetric w.r.t. its photocenter in contrast to a lopsided distribution seen in M31. It is more likely that the double peak nucleus in NGC4486B is due to m = 2 mode, rather than m = 1 mode as is the case for M31. NGC4486B being an elliptical galaxy, it is possible that the excitation probability for m =2 mode is higher.