Band structures of ultrathin films of heavy elements, £\-Sn and Sb, were investigated using first-principle calculations with the inclusion of spin-orbit coupling. The band structures were gradually varied as the physical parameters were adjusted. The band inversion was obtained at the high symmetry point in Brillouin zone, making a topological phase transition. In this study, the band inversion at £F point of the Brillouin zone was predicted in single bi-layer of Sb(111) and single bi-layer and two bi-layers of £\-Sn(111). The topological phase transition is from trivial insulator to topological insulator for single bi-layer of Sb(111). Finally, the topological phase transition is from trivial semi-metal to topological semi-metal for single bi-layer of £\-Sn(111), whereas as it is from topological semi-metal to trivial metal for two bi-layers of £\-Sn(111).

Non-stoichiometric barium titanate (BaTiO3) powder of TiO2-excess compositions has been investigated using both reducing sintering and acceptor-doping. Crystalline phases were analysed by XRD. Attention has been paid to the analysis of the corresponding sintered microstructure by adopting scanning and transmission electron microcopy. Reducing sintering was in the low oxygen partial pressure, so as to dominate the oxygen-deficient. According to the defect chemistry, the purpose of acceptor-doping was the same as reducing sintering. We look out for phenomena which may be indicative that oxygen vacancies generated by acceptor-doping and reducing sintering have resulted in the metastable retention of high temperature hexagonal-BaTiO3 to an ambient temperature. In the Mg-doped study investigated the possibility that Mg2+ substitutes on Ti4+ site rather than the Ba2+ site, as expected from the radii. According to the unknown phase was indexed a supercell of MgTiO3, that showed evidence of Mg2+ dissolves in BaTiO3 and occupies the Ba2+ site. To reduce in a hydrogen atmosphere was a high dark conductivity. The Ti3+ content was determined via colorimetry. Because of the defect chemistry led to oxygen-deficient h-BaTiO3, i.e.BaTi1-xTixO3-x/2. The observed volume expansion behavior under Ar-H2 atmosphere demonstrates the possibility of having various microstructures via control of oxygen partial pressure. The transformation matrix described the relation between the two reciprocal lattices of the twinning. Investigation of reciprocal lattices was shown that ordering oxygen deficient on the BaO3 layer in the twin boundary. There was evidence of XRD patterns and surface energy that explained more and more twins in the microstructure via control of the low oxygen partial pressure. According to this theory, lamellae twins were generated by oxygen-deficient. The hexagonal phase might be also expressed as the cubic BaTiO3 containing twin boundary at BaO3 planes every three layers. That demonstrates the possibility of hexagonal phase retention in BaTiO3 was oxygen vacancies.

Abstract In this study, we employed the Monte Carlo simulation method to investigate the q-state coupled XY model based on the Landau free energy of couple hexatic order and herringbone order proposed by Bruinsma and Aeppli. On two-dimensional triangular lattices simulation results reveal that the q-state coupled XY model will generate a q-state clock phase transition and a XY transition. The unique generated q-state clock phase transition and XY transition will couple in some coupling parameter domain. The novel coupled transitions behavior agree with the phase transition of some kinds of liquid crystal. For example, the three-state Potts phase transition generated by a 3-state coupled XY model and the Sm-A ¡÷Hex-B transition of free ¡V standing two layers liquid crystal are matched. Their heat capacity anomaly is similar and the heat capacity exponent is both closed to £\¡Ü0.3. We also investigated the system of coupled ferromagnetic order and antiferromagnetic order. Adapted the positive coupling parameter on the Hamiltonian of 3-state coupled XY model, the simulation results reveal that the system generate an antiferromagnetic three-state Potts transition. In some parameter domain the antiferromagnetic three-state Potts transition and XY transition are coupled, and become a novel transition. The novel transition may explain the Sm-A ¡÷Hex-B transition of some kinds of liquid crystal which lack herringbone order.

The subject is to study two-dimension phase transition. The nano-magnetic particles dispersed in magnetic fluid can aggregate to form magnetic columns under external perpendicular magnetic field. At some appropriate condition, these magnetic columns are able to form two-dimension lattices. It is a novel mesoscopic system for studying two-dimensional melting transition. By controlling external magnetic field, we explore phenomena of phase transition and defect dynamics. This article can be divided to three parts. In the first part, we study the phase transition with two-dimension lattice forming with magnetic fluid, then we classify the crystal, hexatic and liquid phases in the melting process with translational correlation function and bond-orientational correlation function. In the second part, we analyze the defect dynamics within the lattice with Burgers Vector. In the third part of this article, the external DC magnetic field is replaced with the AC magnetic field, we find that the two-dimension lattices are also formed in low frequency region. The relationship between the two-dimension lattice forming and the AC frequency are studied.

Many phenomena associated with nonlinear optics are produced by the light-matter interaction in liquid crystals. Nematic liquid crystals possess the properties of the birefringence and that refractive indexes of nematic liquid crystal vary with temperature. As a light beam propagates in liquid crystals, the light beam experiences changes of refractive indexes because the optical field reorientates molecules or the optical intensity changes the temperature of liquid crystal. Then, some interesting phenomena of optical nonlinearity produce in liquid crystals. This study investigates mainly the nonlinear behaviors in nematic liquid crystals. By etching ITO glasses to control distribution of electric field, we discuss applications in photo-electric field. These works are described as follows: First, a low voltage was applied to a planar nematic liquid crystal cell; the director field can be reoriented using a low intensity. Then, the self-focusing effect produces due to a variation of refractive indexes. The light beam in nematic liquid crystal forms a spatial soliton by producing the effect of the self-focusing to balance the diffraction. Additionally, we study the interaction between solitons. One soliton creates a potential well of refractive index, anther one will be attracted in the potential well. As the separated distance between two solitons and the pumping angle are appropriate, two solitons propagate in the form of spiral. Second, we study the behavior of light in a periodic refractive index medium. The director field of the nematic liquid crystal (NLC) is reorientated in a grating¡Vlike indium-tin-oxide electrode cell by applying a controllable-voltage. The variation of refractive index with voltage varied 0v to 10v was observed by a conoscopic method. Numerical simulations have reproduced the main features of the gradient distribution of refractive index in the waveguide. Several phenomena of a polarized laser beam that propagated in the waveguide with different incident angles and positions have observed by a CCD camera, including solitons, undulate beam, the total internal reflection and beam coupling. Third, at the temperature close to nematic-isotropic phase transition temperature, the variation of refractive index in the liquid crystal becomes obvious to the change of temperature. And, a laser beam can easily reorientate molecules. We changed beam intensity in liquid crystal cell, different nonlinear phenomena were observed. Besides, A combined microscopic and conoscopic technique was used in experiments as a convenient way to analyze the optical nonlinearity that is associated with the molecular configuration of nematic liquid crystal.

Markov chains are algorithms that can provide critical information from exponentially large sets efficiently through random sampling. These algorithms are ubiquitous across numerous scientific and engineering disciplines, including statistical physics, biology and operations research. In this thesis we solve sampling problems at the interface of theoretical computer science with applied computer science, discrete mathematics, statistical physics, chemistry and economics. A common theme throughout each of these problems is the use of bias. The first problem we study is biased permutations which arise in the context of self-organizing lists. Here we are interested in the mixing time of a Markov chain that performs nearest neighbor transpositions in the non-uniform setting. We are given "positively biased'' probabilities $\{p_{i,j} \geq 1/2 \}$ for all $i < j$ and let $p_{j,i} = 1-p_{i,j}$. In each step, the chain chooses two adjacent elements~$k,$ and~$\ell$ and exchanges their positions with probability $p_{ \ell, k}$. We define two general classes of bias and give the first proofs that the chain is rapidly mixing for both. We also demonstrate that the chain is not always rapidly mixing by constructing an example requiring exponential time to converge to equilibrium. Next we study rectangular dissections of an $n \times n$ lattice region into rectangles of area $n$, where $n=2^k$ for an even integer $k.$ We consider a weighted version of a natural edge flipping Markov chain where, given a parameter $\lambda > 0,$ we would like to generate each rectangular dissection (or dyadic tiling)~$\sigma$ with probability proportional to $\lambda^{|\sigma|},$ where $|\sigma|$ is the total edge length. First we look at the restricted case of dyadic tilings, where each rectangle is required to have the form $R = [s2^{u},(s+1)2^{u}]\times [t2^{v},(t+1)2^{v}],$ where $s, t, u$ and~$v$ are nonnegative integers. Here we show there is a phase transition: when $\lambda < 1,$ the edge-flipping chain mixes in time $O(n^2 \log n)$, and when $\lambda > 1,$ the mixing time is $\exp(\Omega({n^2}))$. The behavior for general rectangular dissections is more subtle, and we show the chain requires exponential time when $\lambda >1$ and when $\lambda <1.$ The last two problems we study arise directly from applications in chemistry and economics. Colloids are binary mixtures of molecules with one type of molecule suspended in another. It is believed that at low density typical configurations will be well-mixed throughout, while at high density they will separate into clusters. We characterize the high and low density phases for a general family of discrete interfering colloid models by showing that they exhibit a "clustering property" at high density and not at low density. The clustering property states that there will be a region that has very high area to perimeter ratio and very high density of one type of molecule. A special case is mixtures of squares and diamonds on $\Z^2$ which correspond to the Ising model at fixed magnetization. Subsequently, we expanded techniques developed in the context of colloids to give a new rigorous underpinning to the Schelling model, which was proposed in 1971 by economist Thomas Schelling to understand the causes of racial segregation. Schelling considered residents of two types, where everyone prefers that the majority of his or her neighbors are of the same type. He showed through simulations that even mild preferences of this type can lead to segregation if residents move whenever they are not happy with their local environments. We generalize the Schelling model to include a broad class of bias functions determining individuals happiness or desire to move. We show that for any influence function in this class, the dynamics will be rapidly mixing and cities will be integrated if the racial bias is sufficiently low. However when the bias is sufficiently high, we show the dynamics take exponential time to mix and a large cluster of one type will form.

Chemical reactions can be understood as transitions from basin to basin on a high dimensional potential energy landscape. Varying temperature only changes the average kinetic energy of the system. While applying voltages or external pressures directly tilts the landscape and drives the reactions in desired directions. In solids at relatively low temperature, where the entropy term is approximately invariant, the reaction spontaneity is determined by the energy difference between the reactant and product basins and the reaction rate can be calculated from the barriers in between. To achieve sufficient accuracy to explain experimental observations we are interested in, density functional theory (DFT) is usually employed to calculate energies. There are two types of reactions I have studied: the first type of reaction only involves a few number of individual atoms, corresponding to traveling in a small volume in the high dimensional configuration space; the other type involves a large amount of atoms moving in a concerted pattern, and the distance traveled in the configuration space is significantly longer. The scopes of these two in the energy landscapes are in different scales and thus proper metrics for distance measurements are required. In the first case, I have mainly studied Li/Na behaviors in the cathode materials of secondary batteries. Here resolving the energy landscape step by step with detailed information is possible and useful. By analyzing the energy landscapes with DFT plus the Hubbard U correction, I have explained several phenomena related to the degradation of lithium-rich layered oxides, rate performance of surface modified LiFePO₄, and capacity of vanadium-based fluorophosphates. Predictions on both thermodynamic and kinetic properties of materials are also made based on the calculation results and some are confirmed by experiments. In the second case, my focus is on solid-solid phase transitions. With a tremendous long reaction pathway, examining every possible atomic step is too expensive. By adopting periodic boundary conditions, a small supercell can represent the main feature of the energy landscape in a coarse grained way, where the connection between phases is easier to explore. After the big picture of a phase transition mechanism learned from this simplified model, details along the reaction pathway, like new phase nucleation and growth, could be resolved by using a larger supercell. In the above treatment, two types of variables, the cell vectors and atomic positions, span a generalized configuration space. Special consideration is required to balance these two to keep consistency under different supercells and avoid biases. A solid-state NEB (SSNEB) and a solid-state dimer (SSD) method are then developed to locate saddle points in the generalized configuration space. With the methodology well justified, we are able to efficiently find possible nucleation mechanisms, for examples the CdSe rock salt to wurtzite and Mo A15 to BCC phase transitions. SSNEB is also applied in studying phases transitions under pressures, including the graphite to diamond, and CaIrO₃ perovskite to post-perovskite transitions. Combined with the adaptive kinetic Monte Carlo (AKMC) algorithm, SSD shows the ability to find new polymorphs of CdSe and the connecting barriers between them. / text

Phase transitions abound in the physical world, from the subatomic length scales of quark condensation to the decoupling forces in the early universe. In the Bose-Einstein condensation phase transition, a gas of trapped bosonic atoms is cooled to a critical temperature. Below this temperature, a macroscopic number of atoms suddenly starts to occupy a single quantum state; these atoms comprise the Bose-Einstein condensate (BEC). The dynamics of the BEC phase transition are the focus of this dissertation and the experiments described here have provided new information on the details of BEC formation. New theoretical developments are proving to be valuable tools for describing BEC phase transition dynamics and interpreting new experimental results. With their amenability to optical manipulation and probing along with the advent of new microscopic theories, BECs provide an important new avenue for gaining insight into the universal dynamics of phase transitions in general.Spontaneous symmetry breaking in the system's order parameter may be one result of cooling through a phase transition. A potential consequence of this is the spontaneous formation of topological defects, which in a BEC appear as vortices. We experimentally observed and characterized the spontaneous formation of vortices during BEC growth. We attribute vortex creation to coherence length limitations during the initial stages of the phase transition. Parallel to these experimental observations, theory collaborators have used the Stochastic Gross-Pitaevski Equation formalism to simulate the growth of a condensate from a thermal cloud. The experimental and theoretical statistical results of the spontaneous formation of vortex cores during the growth of the condensate are in good quantitative agreement with one another, supporting our understanding of the dynamics of the phase transition. We believe that our results are also qualitatively consistent with the Kibble-Zurek mechanism, a universal model for topological defect formation.Ultimately, our understanding of the dynamics of the BEC phase transition may lead to a broader understanding of phase transitions in general, and provide new insight into the development of coherence in numerous systems.

Understanding the behaviour of nuclear materials in extreme conditions is of prime importance for the analysis of the operation limits of nuclear fuels, and prediction of possible nuclear reactor accidents, relevant to the general objectives of nuclear safety research. The main purpose of this thesis is the study of high temperature phase transitions in nuclear materials, with special attention to the candidate fuel materials for the reactors of the 4th Generation. In this framework, material properties need to be investigated at temperatures higher than 2500K, where equilibrium conditions are difficult to obtain. Laser heating combined with fast pyrometer is the method used at the European Institute for Transuranium Elements (JRC - ITU). It is associated to a novel process used to determine phase transitions, based on the detection, via a suited low-power (mW) probe laser, of changes in surface reflectivity that may accompany solid/liquid phase transitions. Fast thermal cycles, from a few ms up to the second, under almost container-free conditions and control atmosphere narrow the problem of vaporisation and sample interactions usually meet with traditional method. This new experimental approach has led to very interesting results. It confirmed earlier research for material systems known to be stable at high temperature (such as U-C) and allowed a refinement of the corresponding phase diagrams. But it was also feasible to apply this method to materials highly reactive, thus original results are presented on PuO2, NpO2, UO2-PuO2 and Pu-C systems.

In this thesis several classes of directed paths are considered as models of linear polymers in a dilute solution. We obtain the generating functions for each model by considering factorization arguments. Information about the polymer behaviour can be extracted from the singularity structure of the associated generating functions. By using modified versions of these models we study the adsorption and localization of polymer molecules, the behaviour of polymers subject to a tensile force, the effects of stiffness, as well as the behaviour of polymers in confined geometries. In each of these situations the resulting generating functions contain at least two physical singularities. We identify the phase transitions in these systems by a changeover in the dominant singularity of the generating function. In the study of localization and polymers subject to a force, we utilize both homopolymer and random copolymer models. For copolymers, the physically relevant properties are obtained by considering a quenched average of the free energy over all possible monomer sequences. This procedure is intractable even for the simplest models. By considering the Morita approximation for several walk models we obtain results which give a bound on the corresponding features of the quenched system. We use a mapping between a simple model of duplex DNA and an adsorbing Motzkin path in order to study the mechanical unzipping of duplex DNA. From this model, we obtain force-temperature diagrams which show re-entrant behaviour of the force. We also develop a simple low temperature theory to describe the behaviour of the force close to T=0 and find that the shape of the force-temperature curve is associated with entropy in the ground state of the system. We consider the effect of stiffness on polymer adsorption and find that the phase transition is second order for all finite stiffness parameters. For systems of polymers in confined geometries, we find that the behaviour of the polymer depends on the distance between the confining surfaces and the associated interactions with each surface. In this problem, there exist regimes where the polymer exerts a force on the surfaces which can be attractive, repulsive or zero.