My name is Marsel Gokovi and I am currently a student at Griffith University. I am studying a Bachelor of Science (Accelerated) with Honours with a major in Mathematics. I am also knowledgeable in various fields in Physics (up to a third year university level) . I enjoy mathematics and its applications. I especially enjoy it in the context of explaining the fundamental phenomena that occur in/out of this world. Mathematics can essentially be used to explain anything, and so during my research career I would like to use it to explain anything from the behaviour of particles to statistical modelling. I am interested in research and improvement!
The Travelling time across Microporous Potentials and the Diffusion Coefficient
Microporous materials – materials with pore sizes on the nanoscale – have long been used for their storage and filtration properties. Such membranes have traditionally been challenging to study, because of the typically large variability in pore structures (pore widths and orientations). However, with the advent of fabrication technologies that can produce membranes with highly regular pores of quite precise structure, models that describe the transport of fluids through such pores have an important role to play. Not only can they be used to predict how fluids pass through these pores, but they can help in improving the design of membranes for specific purposes. Some applications, such as gas filtration or separation, commonly involve transport at very low densities, known as the Knudsen regime. Under these conditions, molecules behave like oscillators that repeatedly cross the pore as they move along it: the interaction between pore wall and molecule is much more significant than the relatively rare intermolecular interactions. Knudsen developed a model over a century ago to describe low-density transport of rare gases along pipes. A similar model can be used to describe low-density transport in micropores, but it much be altered to accommodate the long-rang influence of pore-molecule interactions on the molecules’ dynamics (described using an interaction potential). Once this influence is included in the model, the permeability of molecules through a given membrane, described by the collective diffusion coefficient, is strongly dependent on the nature of the corresponding interaction potential, due to its effect on the mean crossing time of molecules oscillating across the pore. The aim of this project is to understand the dependence of these crossing times on the interaction potential. This in turn allows us to predict the collective diffusion coefficient for a given interaction potential, which is a key part in understanding the design of a membrane, affects its transport properties. Expressions already exist to define the collective diffusion coefficient in terms of the crossing times, and the crossing times in terms of the interaction potential: currently the relations between these quantities have only been determined numerically. By studying these quantities analytically, we can understand their interdependence more easily, and quite possibly calculate them more quickly as well. There are two broad categories of interaction potential that require different approaches. Narrower pores have U-shaped potentials, with a central minimum, while broader pores have a W-shaped potential with two local minima on either side of a central maximum. These two pores require different approaches, in order to achieve our aims, because the variation of the crossing times has a fundamentally different nature. For the U-shaped pores, the crossing times can be expressed as an infinite sum of terms. Our aim is to calculate these terms, and use them to provide estimates (and errors) for the collective transport coefficient. For the W-shaped pores, the crossing-time blow-up to infinity at finite energies (although the mean crossing time converges), in much the same way that a pendulum will take an infinite time to reach its unsteady equilibrium (the ‘upside-down pendulum’) if it is pushed with exactly the right force (given exactly the right energy), but will return to its lowest point in finite time when pushed with any other force (given any other energy). This finite-time blow-up means that no infinite sum of terms exists for the crossing times of the W-shaped potential. However, there is a well-established body of theory that describes the oscillation of the pendulum, including for the unstable equilibrium. We aim to use this literature to develop expressions for the crossing time and thus the collective diffusion coefficient. We will compare the estimates that we obtain for the collective diffusion coefficient (and crossing times, where possible), using the techniques described above, with results already appearing in the scientific literature, pertaining to numerical simulation or experimental data of gas permeation through membranes comprising slit or cylindrical pores.