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Beginning in the early 1990's, the study of binary mixtures containing simple molecules under high pressure led to the discovery of a solid van der Waals compound: a weakly bound molecular solid whose cohesion is primarily due to van der Waals forces. The formation of this type of compound, such as He(N2)11 discovered in 1992 in the helium-nitrogen system by Vos et al. [Vos, 1992], is predicted in systems such as those containing nitrogen and methane. The nitrogen-methane binary system is studied at room temperature under high pressure in order to construct the pressure-concentration phase diagram up to 16 GPa. Over 20 mixtures of varying concentration have been studied within a diamond anvil cell through Raman spectroscopy and powder X-ray diffraction using synchrotron radiation. Within the phase diagram, phases exist that resemble the pure species at similar thermodynamic conditions, and additionally, two novel van der Waals compounds are observed.

It is impossible to enumerate all the areas where diffusion plays a crucial role. Numerous studies of diffusion have been made since the observation of brownian motion by Robert Brown in 1827. In the last decades, the development of computers made it possible to carry numerical studies of diffusion. Monte Carlo algorithms are often used to model the random walk of a particle in a given system in order to measure its diffusion coefficient. In the last years, Dr Slater's research group developed an exact calculation method that allows one to compute diffusion coefficients with high precision. This calculation method, even if major modifications were necessary, is the base of the two projects presented in this thesis. The first project is a sequel of an article published in 2006 by Hickey et al. These authors derived an expression that predicts the diffusion coefficient of a point-like particle in a two-phase system, such as a hydrogel made of gelatin with maltodextrin viscous inclusions. This expression works well for a simple two-phase system but neglects numerous characteristics such hydrogels can present. In this thesis' first project, we modify this expression in order to include the possible interactions between the particle and the gel structure, the interfacial steric effects between phases and the possible incomplete phase separation. We validate these modifications by comparing them with exact numerical calculations. In preceding studies made by the research group of Dr Slater, it was assumed that the system was quenched, i.e. the obstacles didn't move. However, it is logical to believe that gel fibers inside a real hydrogel are subject to thermal motion and that they vibrate around a mean position. The second project presents a new numerical method allowing one to investigate the effect of obstacles motion on the diffusion of a particle. We vary different parameters such as the vibration frequency compared to the diffusion time scale of the particle, the amplitude of vibration, the obstacle concentration as well as their configuration (periodic or random). This new method is innovative because it makes it possible to study in details the transition of a system from a quenched state (fixed obstacles) to an annealed state (obstacles vibrating much faster than the diffusion time scale of the particle).

Majorana fermions promise potential applications in quantum computing, superconductivity, and related fields. In this thesis, an analysis of A. Y. Kitaev's “Kitaev Chain”, a quasi-one-dimensional quantum wire in contact with a p-wave superconductor, designed as a model exhibiting unpaired Majoranas, is performed. Described by tunneling of spinless fermions between quantum dots, and formation of Cooper pairs on neighboring dots, Kitaev's chain Hamiltonian serves as a basis for emergent Majorana Zero Modes (zero energy Majorana fermions localized at either end of the chain) and artificial gauges (phases) to appear. By exact diagonalization, energy spectra and wavefunctions of a chain of spinless fermions on discrete quantum dots described by Kitaev's Hamiltonian are generated. By transforming the system into a basis of Majorana fermions and "bond fermions", where Majoranas on neighboring dots are paired, emergent Majorana Zero Modes (MZMs) are found at the ends of the chain. These emergent MZMs are paired in a non-local, zero energy bond fermion, which is found to allow degenerate energy states of the system to occur. Joining the ends of the chain by allowing tunneling and pairing of fermions on end sites, a ring topology is considered, where an "artificial gauge" emerges. This artificial gauge, or phase, causes a phase change on tunneling and Cooper pairing Hamiltonian matrix elements as a result of operator ordering within the Hamiltonian's ring terms. These required operator orderings are derived by comparison of energy spectra of the Kitaev ring in the fermion and bond fermion bases. Matching of calculated energy spectra in the Majorana and fermionic bases is used to confirm the presence of the artificial gauge, where this phase is found to be necessary in order to maintain a consistent energy spectra across the transformation between bases. This analysis is performed in order to understand the concept of Majorana Zero Modes and the emergence of Majorana fermions in 1D chains. By doing so, it is determined what Majorana fermions are, where they come from, and why Majorana Zero Modes are considered to be zero energy. These results contribute to the understanding of Kitaev chains and rings, as well as serve as a starting point for discussions regarding physical implications of the artificial gauge's effect, fermion statistics, and the emergence of Majorana Zero Modes in quasi-one-dimensional systems.

Bimolecular recombination and complete photocurrent decay in metallophthalocyanine thin films

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