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Title: Continuum modelling and numerical approaches for energy minimisation in diblock copolymers
Author: Parsons, Quentin
ISNI:       0000 0004 7652 9939
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2018
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We use a continuum phase field model to study pattern formation in Diblock Copolymers (DBCs), with an emphasis on terracing in thin films. Phase separation in DBCs is an important process in nanotechnology that yields an array of highly regular structures on very fine length scales that are of enormous utility. Terraces are ubiquitous in thin film DBC melts. It was long believed that modelling these structures was an intractable problem, but recent attempts to do so using the Self Consistent (Mean) Field Theory (SCFT) were successful. In this thesis, we use the simpler and more flexible Ohta-Kawasaki (non-local Cahn-Hilliard) Density Functional Theory (DFT) model, derived from the SCFT, with obstacle homogeneous, and long-range energy components. Our main result is that such models can be used to capture terraces in the simplified 2D scenarios we examine, and that these structures form as an inevitable consequence of a simple energy minimisation process. We consider a 2-phase model for DBC micro-structure evolution, which we analyse rigorously and comprehensively. We study stationary and dynamic (gradient flow) variants of the problem, using the latter as the basis for an efficient but provably convergent numerical method for simulating DBC evolution. Our theoretical analysis and numerical method are based on a common Moreau-Yosida regularisation of the homogeneous energy component. We use the Semi-Smooth Newton (SSN) method to resolve the non-linearities in our numerical method. Comprehensive numerical experiments confirm our theoretical predictions, including a Method of Manufactured Solutions (MMS) test for the order of convergence. We propose a complete specification of the equally spaced global energy minimisers of the symmetric, 2-phase problem in 1D, and offer a practical algorithm for finding them. We explicitly characterise the Order-Disorder Transition (ODT) in parameter space. We extend our 2-phase model and methods to a 3-phase situation involving a copolymer/homopolymer blend, in which the free surface of the copolymer can adjust to form terraces. We develop energy estimates for energy-minimising 1D morphologies in the sharp interface limit, which we then combine via a double-tangent construction to analyse low energy, 2D structures. We argue that thin films with terraces have a lower free energy than their flat counterparts. Time-dependent numerical simulations in 2D confirm these predictions and provide insight into the evolution of these structures. We replicate and extend recent SCFT results, by performing a numerical experiment in which a terrace forms from a flat initial state. The model, tools and methods developed in this thesis should serve as a solid foundation for researching more complex, multi-step 3D terraces.
Supervisor: Kay, David ; Muench, Andreas Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Materials science ; Applied mathematics ; Numerical analysis