The Singular Set of Higher Dimensional Unstable Obstacle Type Problems

In this paper we will investigate the singular points of the following unstable free boundary problem: {equation}\label{Eq} \Delta u= -\chi_{\{u>0\}} \quad\quad\textrm{in} B_1(0) {equation} where $\chi_{\{u>0\}}$ is the characteristic function of the set $\{u>0\}$

Interpolymer complexation of a polymer brush

Controllable macromolecular gating between nanoscopic compartments is of high interest for single molecule studies of biological macromolecules. By definition, a good macromolecular gate should completely stop biomolecules, such as proteins, from crossing between compartments in its closed state while letting them pass in its open state. Polymer brushes of poly(ethylene glycol) have been proven excellent barriers for proteins in previous work, but are limited in terms of stimuli-responsive behaviour needed for macromolecular gating. In this thesis work, the pH reversible interpolymer complexation between a poly(ethylene glycol) brush and poly(methacrylic acid) was investigated as a potential macromolecular gating mechanism. Conclusions were based on the evaluation of the resulting surface complex using three surface sensitive characterisation techniques. Upon complexing at low pH, the polymer layer was found to adopt a shrunken state with significant behavioural changes, while completely reversing back to its initial state after a neutral pH had been introduced. This pH reversible interaction show great promise as a pH controlled macromolecular gating mechanism and calls for further studies with suitable nanostructures. To this end, the fabrication and properties of a new solid-state nanopore device is also presented, together with the direction needed for future work towards a new macromolecular gating system

Functional polymer brush coatings for nanoscale devices

Nanobiotechnology is an interdisciplinary field that has garnered considerable attention for offering exciting new opportunities of studying and manipulating biomolecules at the nanoscale. This prospect bears large potential benefits in the field of medicine and the whole life science sector in general. Fabrication of different nanostructure devices that can handle liquids at the scale of biomolecules, such as nanochannels or nanopores, provide a good basis within nanobiotechnology. However, the materials of nanostructures tend to not interact with complex biomolecules in ways that are sufficiently specific or controlled. This issue can be avoided by functionalising the surface of nanostructures with different organic coatings, and polymer brushes have shown a diverse range of functionality in this regard. This thesis summarises efforts towards designing functional polymer brush coatings for nanoscale devices. Surface sensitive techniques are used to characterise the grafting of dense poly(ethylene glycol) brushes to various noble metals and silicon dioxide. The new functionalisation protocol for polymer brushes on silicon dioxide provides excellent biofunctionality and is demonstrated to be compatible with two different nanostructures. The specific hydrogen-bond mediated interaction between a poly(ethylene glycol) brush and poly(methacrylic acid) in solution at low pH is shown to make the polymer brush reversibly stimuli-responsive. Preliminary results further demonstrate how this interaction can be controlled electrochemically and indicates its suitability as a macromolecular gating mechanism for nanosized openings. Finally, characterisation and fabrication of plasmonic nanopore arrays with separately functionalisable compartments using electron beam lithography techniques is presented

A parabolic free boundary problem with Bernoulli type condition on the free boundary

Consider the parabolic free boundary problem $$\Delta u - \partial_t u = 0 \textrm{in} \{u>0\}, |\nabla u|=1 \textrm{on} \partial\{u>0\} .$$ For a realistic class of solutions, containing for example {\em all} limits of the singular perturbation problem $$\Delta u_\epsilon - \partial_t u_\epsilon = \beta_\epsilon(u_\epsilon) \textrm{as} \epsilon\to 0,$$ we prove that one-sided flatness of the free boundary implies regularity. In particular, we show that the topological free boundary $\partial\{u>0\}$ can be decomposed into an {\em open} regular set (relative to $\partial\{u>0\}$) which is locally a surface with H\"older-continuous space normal, and a closed singular set. Our result extends the main theorem in the paper by H.W. Alt-L.A. Caffarelli (1981) to more general solutions as well as the time-dependent case. Our proof uses methods developed in H.W. Alt-L.A. Caffarelli (1981), however we replace the core of that paper, which relies on non-positive mean curvature at singular points, by an argument based on scaling discrepancies, which promises to be applicable to more general free boundary or free discontinuity problems