Uniformly convergent finite element and finite difference methods for singularly perturbed ordinary differential equations

This thesis is concerned with uniformly convergent finite element and finite difference methods for numerically solving singularly perturbed two-point boundary value problems. We examine the following four problems: (i) high order problem of reaction-diffusion type; (ii) high order problem of convection-diffusion type; (iii) second order interior turning point problem; (iv) semilinear reaction-diffusion problem. Firstly, we consider high order problems of reaction-diffusion type and convection-diffusion type. Under suitable hypotheses, the coercivity of the associated bilinear forms is proved and representation results for the solutions of such problems are given. It is shown that, on an equidistant mesh, polynomial schemes cannot achieve a high order of convergence which is uniform in the perturbation parameter. Piecewise polynomial Galerkin finite element methods are then constructed on a Shishkin mesh. High order convergence results, which are uniform in the perturbation parameter, are obtained in various norms. Secondly, we investigate linear second order problems with interior turning points. Piecewise linear Galerkin finite element methods are generated on various piecewise equidistant meshes designed for such problems. These methods are shown to be convergent, uniformly in the singular perturbation parameter, in a weighted energy norm and the usual L2 norm. Finally, we deal with a semilinear reaction-diffusion problem. Asymptotic properties of solutions to this problem are discussed and analysed. Two simple finite difference schemes on Shishkin meshes are applied to the problem. They are proved to be uniformly convergent of second order and fourth order respectively. Existence and uniqueness of a solution to both schemes are investigated. Numerical results for the above methods are presented.

Design, fabrication and characterisation of components for microfluidic enzymatic biofuel cells

The concept of a biofuel cell takes inspiration from the natural capability of biological systems to catalyse the conversion of organic matter with a subsequent release of electrical energy. Enzymatic biofuel cells are intended to mimic the processes occurring in nature in a more controlled and efficient manner. Traditional fuel cells rely on the use of toxic catalysts and are often not easily miniaturizable making them unsuitable as implantable power sources. Biofuel cells however use highly selective protein catalysts and renewable fuels. As energy consumption becomes a global issue, they emerge as important tools for energy generation. The microfluidic platforms developed are intended to maximize the amount of electrical energy extracted from renewable fuels which are naturally abundant in the environment and in biological fluids. Combining microfabrication processes, chemical modification and biological surface patterning these devices are promising candidates for micro-power sources for future life science and electronic applications. This thesis considered four main aspects of a biofuel cell research. Firstly, concept of a miniature compartmentalized enzymatic biofuel cell utilizing simple fuels and operating in static conditions is verified and proves the feasibility of enzyme catalysis in energy conversion processes. Secondly, electrode and microfluidic channel study was performed through theoretical investigations of the flow and catalytic reactions which also improved understanding of the enzyme kinetics in the cell. Next, microfluidic devices were fabricated from cost-effective and disposable polymer materials, using the state-of-the-art micro-processing technologies. Integration of the individual components is difficult and multiple techniques to overcome these problems have been investigated. Electrochemical characterization of gold electrodes modified with Nanoporous Gold Structures is also performed. Finally, two strategies for enzyme patterning and encapsulation are discussed. Several protein catalysts have been effectively immobilized on the surface of commercial and microfabricated electrodes by electrochemically assisted deposition in sol-gel and poly-(o-phenylenediamine) polymer matrices and characterised with confirmed catalytic activity.

Investigation of micro-devices for neurobiological applications

The aim of this project is to integrate neuronal cell culture with commercial or in-house built micro-electrode arrays and MEMS devices. The resulting device is intended to support neuronal cell culture on its surface, expose specific portions of a neuronal population to different environments using microfluidic gradients and stimulate/record neuronal electrical activity using micro-electrode arrays. Additionally, through integration of chemical surface patterning, such device can be used to build neuronal cell networks of specific size, conformation and composition. The design of this device takes inspiration from the nervous system because its development and regeneration are heavily influenced by surface chemistry and fluidic gradients. Hence, this device is intended to be a step forward in neuroscience research because it utilizes similar concepts to those found in nature. The large part of this research revolved around solving technical issues associated with integration of biology, surface chemistry, electrophysiology and microfluidics. Commercially available microelectrode arrays (MEAs) are mechanically and chemically brittle making them unsuitable for certain surface modification and micro-fluidic integration techniques described in the literature. In order to successfully integrate all the aspects into one device, some techniques were heavily modified to ensure that their effects on MEA were minimal. In terms of experimental work, this thesis consists of 3 parts. The first part dealt with characterization and optimization of surface patterning and micro-fluidic perfusion. Through extensive image analysis, the optimal conditions required for micro-contact printing and micro-fluidic perfusion were determined. The second part used a number of optimized techniques and successfully applied these to culturing patterned neural cells on a range of substrates including: Pyrex, cyclo-olefin and SiN coated Pyrex. The second part also described culturing neurons on MEAs and recording electrophysiological activity. The third part of the thesis described integration of MEAs with patterned neuronal culture and microfluidic devices. Although integration of all methodologies proved difficult, a large amount of data relating to biocompatibility, neuronal patterning, electrophysiology and integration was collected. Original solutions were successfully applied to solve a number of issues relating to consistency of micro printing and microfluidic integration leading to successful integration of techniques and device components.

Beyond simple imaging with energetic beams – nanopatterning, correlative microscopy and statistical analysis of inclusions

Electron microscopy (EM) has advanced in an exponential way since the first transmission electron microscope (TEM) was built in the 1930’s. The urge to ‘see’ things is an essential part of human nature (talk of ‘seeing is believing’) and apart from scanning tunnel microscopes which give information about the surface, EM is the only imaging technology capable of really visualising atomic structures in depth down to single atoms. With the development of nanotechnology the demand to image and analyse small things has become even greater and electron microscopes have found their way from highly delicate and sophisticated research grade instruments to key-turn and even bench-top instruments for everyday use in every materials research lab on the planet. The semiconductor industry is as dependent on the use of EM as life sciences and pharmaceutical industry. With this generalisation of use for imaging, the need to deploy advanced uses of EM has become more and more apparent. The combination of several coinciding beams (electron, ion and even light) to create DualBeam or TripleBeam instruments for instance enhances the usefulness from pure imaging to manipulating on the nanoscale. And when it comes to the analytic power of EM with the many ways the highly energetic electrons and ions interact with the matter in the specimen there is a plethora of niches which evolved during the last two decades, specialising in every kind of analysis that can be thought of and combined with EM. In the course of this study the emphasis was placed on the application of these advanced analytical EM techniques in the context of multiscale and multimodal microscopy – multiscale meaning across length scales from micrometres or larger to nanometres, multimodal meaning numerous techniques applied to the same sample volume in a correlative manner. In order to demonstrate the breadth and potential of the multiscale and multimodal concept an integration of it was attempted in two areas: I) Biocompatible materials using polycrystalline stainless steel and II) Semiconductors using thin multiferroic films. I) The motivation to use stainless steel (316L medical grade) comes from the potential modulation of endothelial cell growth which can have a big impact on the improvement of cardio-vascular stents – which are mainly made of 316L – through nano-texturing of the stent surface by focused ion beam (FIB) lithography. Patterning with FIB has never been reported before in connection with stents and cell growth and in order to gain a better understanding of the beam-substrate interaction during patterning a correlative microscopy approach was used to illuminate the patterning process from many possible angles. Electron backscattering diffraction (EBSD) was used to analyse the crystallographic structure, FIB was used for the patterning and simultaneously visualising the crystal structure as part of the monitoring process, scanning electron microscopy (SEM) and atomic force microscopy (AFM) were employed to analyse the topography and the final step being 3D visualisation through serial FIB/SEM sectioning. II) The motivation for the use of thin multiferroic films stems from the ever-growing demand for increased data storage at lesser and lesser energy consumption. The Aurivillius phase material used in this study has a high potential in this area. Yet it is necessary to show clearly that the film is really multiferroic and no second phase inclusions are present even at very low concentrations – ~0.1vol% could already be problematic. Thus, in this study a technique was developed to analyse ultra-low density inclusions in thin multiferroic films down to concentrations of 0.01%. The goal achieved was a complete structural and compositional analysis of the films which required identification of second phase inclusions (through elemental analysis EDX(Energy Dispersive X-ray)), localise them (employing 72 hour EDX mapping in the SEM), isolate them for the TEM (using FIB) and give an upper confidence limit of 99.5% to the influence of the inclusions on the magnetic behaviour of the main phase (statistical analysis).