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In this thesis we provide a deeper insight into the fundamental phase transition between isotropic and nematic states of liquid crystals. In particular, we focus on the problem of classifying all minimising states of the Onsager free-energy functional in three dimensions, a problem which has drawn a lot of attention since 1949. We study the bifurcation diagram of the Onsager free-energy functional for liquid crystals with orientation parameter on the sphere. In particular, we concentrate on the bifurcations from the isotropic solution for a general class of two-body interaction potentials including the Onsager kernel. Reformulating the problem as a non-linear eigenvalue problem for the kernel operator, we prove that spherical harmonics are the corresponding eigenfunctions and we present a direct relationship between the coefficients of the Taylor expansion of this class of interaction potentials and their eigenvalues. We find explicit expressions for all bifurcation points corresponding to bifurcations from the isotropic state of the Onsager free-energy functional equipped with the Onsager interaction potential. A substantial amount of our analysis is based on the use of spherical harmonics and a special algorithm for computing expansions of products of spherical harmonics in terms of spherical harmonics is presented. Using a Lyapunov-Schmidt reduction, we derive a bifurcation equation depending on five state variables. The dimension of this state space is further reduced to two dimensions by using the rotational symmetry of the problem and the invariant theory of groups. On the basis of these results, we show that the first bifurcation from the isotropic state of the Onsager interaction potential is a transcritical bifurcation and that the corresponding solution is uniaxial. In addition, we prove some global properties of the bifurcation diagram such as the fact that the trivial solution is the unique local minimiser if the bifurcation parameter is high, that it is not a local minimiser if the bifurcation parameter is small, the boundedness of all equilibria of the functional and that the bifurcation branches are either unbounded or that they meet another bifurcation branch.

The work in this thesis is inspired by the fabrication of Zn45Au30Cu25. This is the first alloy undergoing ultra-reversible martensitic transformations and closely satisfying the cofactor conditions, particular conditions of geometric compatibility between phases, which were conjectured to influence reversibility. With the aim of better understanding reversibility, in this thesis we study the martensitic microstructures arising during thermal cycling in Zn45Au30Cu25, which are complex and different in every phase transformation cycle. Our study is developed in the context of continuum mechanics and nonlinear elasticity, and we use tools from nonlinear analysis. The first aim of this thesis is to advance our understanding of conditions of geometric compatibility between phases. To this end, first, we further investigate cofactor conditions and introduce a physically-based metric to measure how closely these are satisfied in real materials. Secondly, we introduce further conditions of compatibility and show that these are nearly satisfied by some twins in Zn45Au30Cu25. These might influence reversibility as they improve compatibility between high and low temperature phases. Martensitic phase transitions in Zn45Au30Cu25 are a complex phenomenon, especially because the crystalline structure of the material changes from a cubic to a monoclinic symmetry, and hence the energy of the system has twelve wells. There exist infinitely many energy-minimising microstructures, limiting our understanding of the phenomenon as well as our ability to predict it. Therefore, the second aim of this thesis is to find criteria to select physically-relevant energy minimisers. We introduce two criteria or selection mechanisms. The first involves a moving mask approximation, which allows one to describe some experimental observations on the dynamics, while the second is based on using vanishing interface energy. The moving mask approximation reflects the idea of a moving curtain covering and uncovering microstructures during the phase transition, as appears to be the case for Zn45Au30Cu25, and many other materials during thermally induced transformations. We show that the moving mask approximation can be framed in the context of a model for the dynamics of nonlinear elastic bodies. We prove that every macroscopic deformation gradient satisfying the moving mask approximation must be of the form 1 + a(x) ⊗ n(x), for a.e. x. With regards to vanishing interface energy, we consider a one-dimensional energy functional with three wells, which simplifies the physically relevant model for martensitic transformations, but at the same time highlights some key issues. Our energy functional admits infinitely many minimising gradient Young measures, representing energy-minimising microstructures. In order to select the physically relevant ones, we show that minimisers of a regularised energy, where the second derivatives are penalised, generate a unique minimising gradient Young measure as the perturbation vanishes. The results developed in this thesis are motivated by the study of Zn45Au30Cu25, but their relevance is not limited to this material. The results on the cofactor conditions developed here can help for the understanding of new alloys undergoing ultra-reversible transformations, and as a guideline for the fabrication of future materials. Furthermore, the selection mechanisms studied in this work can be useful in selecting physically relevant microstructures not only in Zn45Au30Cu25, but also in other materials undergoing martensitic transformations, and other phenomena where pattern formation is observed.

The thesis concerns the mathematical study of the calculus of variations and its application to liquid crystals. In the first chapter we examine vectorial problems in the calculus of variations with an additional pointwise constraint so that any admissible function n ε W1,1(ΩM), and M is a manifold of suitable regularity. We formulate necessary and sufficient conditions for any given state n to be a strong or weak local minimiser of I. This is achieved using a nearest point projection mapping in order to use the more classical results which apply in the absence of a constraint. In the subsequent chapters we study various static continuum theories of liquid crystals. More specifically we look to explain a particular cholesteric fingerprint pattern observed by HP Labs. We begin in Chapter 2 by focusing on a specific cholesteric liquid crystal problem using the theory originally derived by Oseen and Frank. We find the global minimisers for general elastic constants amongst admissible functions which only depend on a single variable. Using the one-constant approximation for the Oseen-Frank free energy, we then show that these states are global minimisers of the three-dimensional problem if the pitch of the cholesteric liquid crystal is sufficiently long. Chapter 3 concerns the application of the results from the first chapter to the situations investigated in the second. The local stability of the one-dimensional states are quantified, analytically and numerically, and in doing so we unearth potential shortcomings of the classical Oseen-Frank theory. In Chapter 4, we ascertain some equivalence results between the continuum theories of Oseen and Frank, Ericksen, and Landau and de Gennes. We do so by proving lifting results, building on the work of Ball and Zarnescu, which relate the regularity of line and vector fields. The results prove to be interesting as they show that for a director theory to respect the head to tail symmetry of the liquid crystal molecules, the appropriate function space for the director field is S BV2 (Ω,S^{2,/sup>). We take this idea and in the final chapter we propose a mathematical model of liquid crystals based upon the Oseen-Frank free energy but using special functions of bounded variation. We establish the existence of a minimiser, forms of the Euler-Lagrange equation, and find solutions of the Euler-Lagrange equation in some simple cases. Finally we use our proposed model to re-examine the same problems from Chapter 2. By doing so we extend the analysis we were able to achieve using Sobolev spaces and predict the existence of multi-dimensional minimisers consistent with the known experimental properties of high-chirality cholesteric liquid crystals.}

The application of techniques from nonlinear analysis to materials science has seen great developments in the recent years and it has really been a driving force for substantial mathematical research in the area of partial differential equations and the multi-dimensional calculus of variations. This thesis has been motivated by two recent and remarkable experimental observations of H. Seiner in shape-memory alloys which we attempt to interpret mathematically. Much of the work is original and has given rise to deep problems in the calculus of variations. Firstly, we study the formation of non-classical austenite-martensite interfaces. Ball & Carstensen (1997, 1999) theoretically investigated the possibility of the occurrence of such interfaces and studied the cubic-to-tetragonal case extensively. In this thesis, we present an analysis of non-classical austenite-martensite interfaces recently observed by Seiner et al.~in a single crystal of a CuAlNi shape-memory alloy, undergoing a cubic-to-orthorhombic transition. We show that these can be described by the general nonlinear elasticity model and we make some predictions regarding the admissible volume fractions of the martensitic variants involved, as well as the habit plane normals. Interestingly, in the above experimental observations, the interface between the austenite and the martensitic configuration is never exactly planar, but rather slightly curved, resulting from the pattern of martensite not being exactly homogeneous. However, it is not clear how one can reconstruct the inhomogeneous configuration as a stress-free microstructure and, instead, a theoretical approach is followed. In this approach, a general method is provided for the construction of a compatible curved austenite-martensite interface and, by exploiting the structure of quasiconvex hulls, the existence of curved interfaces is shown in two and three dimensions. As far as the author is aware of, this is the first construction of such a curved austenite-martensite interface. Secondly, we study the nucleation of austenite in a single crystal of a CuAlNi shape-memory alloy consisting of a single variant of stabilized 2H martensite. The nucleation process is induced by localized heating and it is observed that, regardless of where the localized heating is applied, the nucleation points are always located at one of the corners of the sample - a rectangular parallelepiped in the austenite. Using a simplified nonlinear elasticity model, we propose an explanation for the location of the nucleation points by showing that the martensite is a local minimizer of the energy with respect to localized variations in the interior, on faces and edges of the sample, but not at some corners, where a localized microstructure can lower the energy. The result for the interior, faces and edges is established by showing that the free-energy function satisfies a set of quasiconvexity conditions at the stabilized variant throughout the specimen, provided this is suitably cut. The proofs of quasiconvexity are based on a rigidity argument and are specific to the change of symmetry in the phase transformation. To the best of the author's knowledge, quasiconvexity conditions at edges and corners have not been considered before.