Cette thèse présente une étude de films monomoléculaires d'acides gras à l'interface eau-air, principalement par microscopie à l'angle de Brewster. Cette technique permet l'observation directe des coexistences de phases lors de transitions de phases du premier ordre; de plus elle est sensible à l'anisotropie optique des films. La première partie contient une étude du diagramme de phase des acides gras, qui comporte des mésophases analogues aux phases de cristaux liquides smectiques. Nous avons montré que lors des transitions de phases entre phases denses, la texture (l'arrangement moléculaire)des phases est réversible et nous avons obtenu des informations sur l'ordre des transitions. Nous avons aussi mis en évidence, dans deux phases constituées de molécules verticales, une très faible anisotropie optique due à la forme rectangulaire du réseau moléculaire. Nous avons ensuite étudié une film d'acide myristique adsorbé à la surface d'une solution aqueuse d'acide myristique. Nous avons observé des domaines d'une mésophase "inclinée" contenant des lignes de défauts d'orientationmoléculaire présentant des fluctuations thermiques. La mesure de leur amplitude a permis de déterminer la tension de ces lignes. Nous nous sommes aussi intéressés aux constantes élastiques qui gouvernent la forme et la texture à l'équilibre des domaines de mésophase. Nous avons d'une part déterminé la tension de ligne de l'interface entre une phase liquide et une mésophase "inclinée" en étudiant la forme des domaines. Celle-ci résulte de l'équilibre entre la tension de ligne, qui favorise des domaines circulaires, et les interactions répulsives à longue portée entre dipoles moléculaires, qui forment les domaines. Des mesures de potentiel de surface ont permis de calculer l'intensité des forces dipolaires. d'autre part, la texture d'autre domaines de cette mésophaseinclinée nous a renseigné sur la valeur du rapport entre l'élasticité de courbure de la direction moléculaire et l'anisotropie de la tension de ligne.

La microscopie à l'angle de Brewster est une nouvelle et tres puissante technique d'étude des films monomoléculaires à la surface de l'eau. Son principe est basé sur les propriétés de réflectivité des interfaces. Elle est sensible à l'épaisseur, la densité et l'anisotropie optique des films. Cette technique a été appliquée à l'étude de couches adsorbées à la surface de solutions aqueuses d'acides gras (acides palmitiques et myristiques). Ces couches traversent pendant leur formation des transitions de phases. Le nombre, la nature et la morphologie de ces phases dépendent de nombreux paramètres dont le pH. Nous avons entre autres observé des phases optiquement anisotropes, contituées de molécules inclinées par rapport à la normale à la solution.Ces phases sont sans doute des mésophases "verrouillées", c'est à dire que la direction des molécules est fixée par rapport aux directions intermoléculaires. Elles présentent différents types de défauts d'orientation, dont des structures en étoile. L'existence de telles structures est expliquée par application d'un modèle d'élasticité continue développé pour l'étude des films minces de cristaux liquides smectiques. Des structures en zig-zag, en spirales et en bandes de largeur déterminée ont également été observées. Nous avons également étudié les couches d'un polymère (le PDMS) à la surface de l'eau. Nous y avons observé la séparation latérale en domaines de densités de surface différentes, à la fois dans le régime monocouche et le régime multicouches.

Bewegungen von prozessiven molekularen Motoren des Zytoskeletts sind durch ein Wechselspiel von gerichteter Bewegung entlang von Filamenten und Diffusion in der umgebenden Lösung gekennzeichnet. Diese eigentümlichen Bewegungen werden in der vorliegenden Arbeit untersucht, indem sie als Random Walks auf einem Gitter modelliert werden. Ein weiterer Gegenstand der Untersuchung sind Effekte von Wechselwirkungen zwischen den Motoren auf diese Bewegungen. <br /> <br /> Im einzelnen werden vier Transportphänomene untersucht: <br /> (i) Random Walks von einzelnen Motoren in Kompartimenten verschiedener Geometrien, <br /> (ii) stationäre Konzentrationsprofile, die sich in geschlossenen Kompartimenten infolge dieser Bewegungen einstellen,<br /> (iii) randinduzierte Phasenübergänge in offenen röhrenartigen Kompartimenten, die an Motorenreservoirs gekoppelt sind, und <br /> (iv) der Einfluß von kooperativen Effekten bei der Motor-Filament-Bindung auf die Bewegung. Alle diese Phänomene sind experimentell zugänglich, und mögliche experimentelle Realisierungen werden diskutiert. / Movements of processive cytoskeletal motors are characterized by an interplay between directed motion along filament and diffusion in the surrounding solution. In the present work, these peculiar movements are studied by modeling them as random walks on a lattice. An additional subject of our studies is the effect of motor-motor interactions on these movements. <br /> <br /> In detail, four transport phenomena are studied: <br /> (i) Random walks of single motors in compartments of various geometries, <br /> (ii) stationary concentration profiles which build up as a result of these movements in closed compartments, <br /> (iii) boundary-induced phase transitions in open tube-like compartments coupled to reservoirs of motors, and <br /> (iv) the influence of cooperative effects in motor-filament binding on the movements. All these phenomena are experimentally accessible and possible experimental realizations are discussed.

The present thesis reports computer simulation studies of several phase transition related phenomena in a range of soft-condensed matter systems. A coherent unifying theme of the thesis is the understanding of dynamics of phase transitions through free energy calculations using recently developed efficient non-Boltzmann sampling methods. Based on the system/phenomena of interest, the thesis has been classified into four major parts: I. Isotropic-nematic (IN) phase transition in liquid crystals. II. Nucleation phenomena in gas-liquid transition with particular emphasis on the systems close to the spinodal curve. III. Collapse transition in linear hydrocarbon (n-alkane) chains for a varying range of length, solvent and temperature. IV. Crystallization of unbranched polymer chains in dilute solution, with particular emphasis on the temperature dependent crossover between the rod-like crystalline state and spherical molten globule state. The thesis has been further divided into ten chapters running through the four parts mentioned before. In the following we provide a brief chapter-wise outline of the thesis. Part I deals with the power law relaxation and glassy dynamics in thermotropic liquid crystals close to the IN transition and consists of two chapters. To start with, Chapter I.1 provides an introduction to thermotropic liquid crystals. Here we briefly introduce various liquid crystalline phases, the order parameter used to characterize the IN transition, a few well established theoretical models, and we conclude with describing the recent experimental and computer simulation studies that have motivated the work described in the next chapter. In Chapter I.2, we present our molecular dynamics simulation studies on single particle and collective orientational dynamics across the IN transition for Lebwohl Lasher model, which is a well-known lattice model for thermotropic liquid crystals. Even this simplified model without any translational degrees of freedom successfully captures the short-tointermediate time power law decay recently observed in optical heterodyne detected optical Kerr effect (OHDOKE) measurements near the IN transition. The angular velocity time correlation function also exhibits a rather pronounced power law decay near the IN boundary. In the mean squared angular displacement at comparable time scales, we observe the emergence of a sub-diffusive regime which is followed by a super-diffusive regime before the onset of the longtime diffusive behavior. We observe signature of dynamical heterogeneity through pronounced non-Gaussian behavior in the orientational motion particularly at lower temperatures. Interestingly, this behavior closely resembles what is usually observed in supercooled liquids. We obtain the free energy as a function of orientational order parameter by the use of recently developed transition matrix Monte Carlo (TMMC) method. The free energy surface is flat for the system considered here and the barrier between isotropic and nematic phases is vanishingly small for this weakly first-order transition, hence allowing for large scale, collective, and correlated orientational density fluctuations. We attribute this large scale fluctuations as the reason for the observed power law decay of the orientational time correlation functions. Part II consists of three chapters, where we focus on the age old problem of nucleation and growth, both from the perspective of thermodynamics and kinetics. We account for the rich history of the problem in the introductory Chapter II.1. In this chapter we describe various types and examples of the nucleation phenomena, and a brief account of the major theoretical approaches used so far. We begin with the most successful Classical Nucleation Theory (CNT), and then move on to more recent applications of Density Functional Theory (DFT) and other mean-field types of models. We conclude with a comparison between the experiments, theories and computational studies. In the next chapter (Chapter II.2) we attempt to elucidate the mechanism of nucleation near the gas-liquid spinodal from a microscopic point of view. Here we construct a multidimensional free energy surface of nucleation of the liquid phase from the parent supercooled and supersaturated vapor phase near the gas-liquid spinodal. In particular, we remove the Becker-Doring constraint of having only one growing cluster in the system. The free energy, as a function of the size of the largest cluster, develops a pronounced minimum at a subcritical cluster size close to the spinodal. This signifies a two step nature of the process of nucleation, where the rapid formation of subcritical nuclei is followed by further growth by slower density fluctuations on an uphill free energy surface. An alternative free energy pathway involving the participation of many subcritical clusters is envisaged near the spinodal where the growth of the nucleus is found to be promoted by a coalescence mechanism in contrast to the single particle addition assumption within CNT. The growth of the stable phase becomes progressively collective and spatially diffuse, and the significance of a “critical nucleus” is lost for deeper quenches. In this chapter we present our studies both in 3dimensional Lennard-Jones (LJ) system and Ising model (both 2and 3dimensions). Our general findings seem to be independent of the model chosen. While the previous chapter focuses on relatively well-studied 3-dimensional (3D) LJ system, in Chapter II.3 we present our studies on the characteristics of the nucleation phenomena in 2dimensional (2D) Lennard-Jones fluid. To the best of our knowledge this is the first extensive computer simulation study to check the accuracy of CNT in 2D. Using various Monte Carlo methods, we calculate the free energy barrier for nucleation, line tension, and bulk densities of equilibrium liquid and vapor phases, and also investigate the size and shape of the critical nucleus. The study is carried out at an intermediate level of supersaturation (away from the spinoidal limit). In 2D, a surprisingly large cutoff (rc ≥ 7.0σ where σ is the diameter of LJ particles) in the truncation of the LJ potential is required to obtain converged results. A lower cutoff leads to a substantial error in the values of the line tension, nucleation barrier, and characteristics of the critical cluster. Note that typically 2.5σ is sufficient for 3D LJ fluids. We observe that in 2D system CNT fails to provide a reliable estimate of the free energy barrier. While it is known to slightly overestimate the nucleation barrier in 3D, it underestimates the barrier by as much as 50% at the saturation ratio S = 1.1(defined as S = P/Pc, where Pc is the coexistence pressure) and at the reduced temperature T* = 0.427(defined as T* = KBT/ ε, where ε is the depth of the potential well). The reason for the marked inadequacy of the CNT in 2D can be attributed to the non-circular nature of the critical clusters. Although the shape becomes increasingly circular and the clusters become more compact with increase in cutoff radius, an appreciable non-circular nature remains even without any cutoff to make the simple CNT inaccurate. Part III again consists of three chapters and focuses on the conformational equilibria. Collapse transition and self-organized structures of n-alkanes in solution. In Chapter III.1 we carry out a brief survey of the existing theories of polymer in solution, with particular emphasis on the collapse process in poor solvents. We also introduce the concept of “hydrophobicity” and “hydrophobic collapse”, which is now a subject enormous interest, partly because it my help in understanding the initial processes involved in protein folding. We briefly discuss the subject of formation of beautiful self-organized structures by block copolymers, and also simple homopolymers which is essentially the focus of the work embodied in the next two chapters. In Chapter III.2 we demonstrated a chain length dependent crossover in the structural properties of linear hydrocarbon (n-alkane) chains using detailed atomistic simulations in explicit water. We identify a number of exotic structures o the polymer chain through energy minimization of representative snapshots collected from molecular dynamics trajectory. While the collapsed state is ring-like(circular) for small chains(CnH2n+2; n ≤ 20) and spherical for very long ones( n = 100), we find the emergence of ordered helical structures at intermediate lengths (n ~ 40). We find different types of disordered helices and toroid-like structures at n = 60. We also report a sharp transition in the stability of the collapsed state as a function of the chain length through relevant free energy calculations. While the collapsed state is only marginally metastable for C20H42, a clear bistable free energy surface emerges only when the chain is about 30 monomers long. For n = 30, the polymer exhibits an intermittent oscillation(characterized by well-developed 1/f noise, where f is the frequency ) between the collapsed and the coil structures, characteristic of two stable states separated by a small barrier. This appears to support a weakly first order phase transition between the extended and the collapsed states. Chapter III.3 extends the study of previous chapter to much longer chains (n ≥ 100), which irreversibly collapse in water into globular forms. Even though the collapsed form has a nearly spherical shape, close inspection shows a propensity towards local ordering in the alignment of the polymer segments. This tendency to maintain alignment in order to maximize the number of contacts leads to a core-shell like structure, where the shell is often characterized by a bent rod-like shape consisting of two adjacent segments running in parallel. A key event associated with the initial stage of collapse seems to be the formation of a skewed ring (or loop) that serves as a “nucleation center” for rest of the chain to collapse into. Time evolution of the radial distribution function of water surrounding the polymer, shows that the density of neighboring water decreases by only about 15-20% from that of bulk water. Even though interior of the ting-like structures is fully devoid of water, solvent accessible surface representation shows that these regions are geometrically/spatially inaccessible to water molecules. We suggest that the role of water is to stabilize such ring-like structures once formed by natural conformational fluctuations of the polymer chain. This view is confirmed by observation of spontaneous formation and melting away of such ring-like entities in a polar aprotic solvent(DMSO). We also comment on the role of the flexibility of polymer chains in determining the collapse kinetics. The last part(Part IV) of the thesis consists of two chapters that deal with the crystallization of linear polymer chains from dilute solution. The way long chain polymers crystallize is drastically different from their small molecule counterparts due to their topological connectivity. Linear polymers often crystallize from dilute solution in the form of thin lamellae with well-defined crystallographic features. In Chapter IV.1 we briefly survey the current theoretical understanding and confusions associated with the highly debated field of polymer crystallization. While the last few decades have seen the development of many successful phenomenological theories, the molecular mechanism of formation of such self-organized lamellae is extremely complex and very poorly understood. There are clearly two distinct steps in polymer crystallization. Firstly, the individual linear polymers must self-organize into bundles of somewhat regular structures. These structures then further aggregate to lamellar form and crystallize into a lattice. In this respect , it has marked similarity to the problem of protein crystallization. In chapter IV.2 we present Brownian dynamics simulation studies of a single polythelene chain of length 500. Such systems can reasonably mimic the process of crystallization from dilute solutions. Our simulations could successfully reproduce some of the interesting phenomena observed in experiments and very recent computer simulation studies, including multi-center nucleation of rod-like structures within a single polymer chain, an inverse relation between lamellar thickness and temperature etc. But our primary focus has been to understand the nature of the phase transition as one traverses along the melting temperature and the underlying free energy surface. Near the melting temperature we observe a very intriguing fluctuation between the disordered molten globule state and the ordered rod-like crystalline, where these two forms have highly different shape and structure. These fluctuations have strong signature of 1/f noise or intermittency. This clearly indicates the existence of a weakly first order transition, where two widely different states with large difference in values of order parameter are separated by a rather small free energy barrier. This can be related to the experimentally observed density fluctuations that resemble spinodal decomposition. It is important to note that very similar fluctuations have been observed in our previous studies on liquid crystals (Chapter 1.2) and intermediate sized alkalines in water(Chapter III.2) that signifies a universal underlying energy landscape for these systems. We have discussed the scope of future work at the end of each chapter whenever appropriate.

The understanding of complex condensed matter systems is an area of intense study. In this thesis, some properties of simple liquids with strong preference for tetrahedral local ordering are explored. These liquids are amenable to supercooling, and give complex crystalline structures on eventual crystallisation. All liquids studied are simple, monatomic and are similar to real metallic liquids. The vibrational density of states of a glass created in simulation is calculated. We show a correspondence between the vibrational properties of the crystal and the glass, indicating that the vibrational spectra of crystals can be used to understand the more complex vibrational spectra of the glass of the same substance. The dynamics of supercooled liquids is investigated using a previously not implemented comprehensive measure of structural relaxation. This new measure decays more slowly in the deeply supercooled domain than the commonly used measure. A new atomic model for octagonal quasicrystals is presented. The model is based on findings from a molecular dynamics simulation that resulted in 45˚ twinned β-Mn. A decoration is derived from the β-Mn unit cell and the unit cell of the intermediate structure found at the twinning interface. Extensive simulations are used to explore the phase diagram of a liquid at low densities. The resulting phase diagram shows a spinodal line and a phase coexistence region between a liquid and a crystalline phase ending in a critical point. This contradicts the old conclusion of the Landau theory -- that continuous transitions between liquids and crystals cannot exist The same liquid is explored at higher densities. Upon cooling the liquid performs a first order liquid-liquid phase transition. The low temperature liquid is shown to be strong and to have very good glass forming abilities. This result offers new insights into fragile to strong transitions and suggests the possibility of a good metallic glass former. / At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 2: Submitted. Paper 3: In progress.

El trabajo resumido en esta tesis se ha llevado a cabo dentro del grupo de Altas Presiones y Espectroscopia de la Universidad de Cantabria y tiene dos objetivos principales. Por un lado, se han estudiado las propiedades ópticas de distintos iones de metales de transición y tierras raras en diversas redes aislantes. Se han realizado medidas experimentales de luminiscencia, absorción, tiempo de vida y espectroscopia resuelta en tiempo. Concretamente, uno de los objetivos principales ha sido establecer los mecanismos responsables de la luminiscencia de upconversion de los materiales estudiados, y los requerimientos estructurales para aumentar su eficiencia. Para ello se han comparado diversos métodos de síntesis así como diferentes concentraciones de impurezas. Asimismo, se han investigado transiciones de fase a alta presión en semiconductores con tamaño de partícula en el rango de los nanómetros mediante absorción óptica, espectroscopia Raman y difracción de rayos X. La síntesis de las muestras representa también una parte muy importante de este trabajo. / This work has been performed within the High Pressure and Spectroscopy group at the University of Cantabria. The aim of this thesis is two-fold; firstly, this work has been devoted to the study of the microscopic origin of the optical properties of diverse transition metal and rare-earth ions in different insulating host lattices. Concretely, one of the main goals has been to establish the mechanisms responsible for the upconversion luminescence properties of the studied materials, as well as the structural requirements to increase their quantum efficiency. Different synthesis methods and impurity concentrations have been compared. Secondly, phase transitions under high pressure in semiconductor nanocrystals or quantum dots have been studied by means of optical absorption, Raman spectroscopy and X-ray diffraction. Besides, the synthesis of the samples is a remarkable point since we have prepared ourselves all the samples studied in this thesis work.

Uno de los aspectos más fascinantes del mundo que nos rodea es la gran variedad de escalas a las que tienen lugar los diversos fenómenos. En muchos casos esta diversidad pone de manifiesto la estructura fractal de la Naturaleza y podemos hablar entonces de fenómenos complejos, en los que eventos de diferentes magnitudes no pueden analizarse de manera independiente. Dicha complejidad emerge como un fenómeno cooperativo a escalas microscópicas, que produce un complejo comportamiento macroscópico caracterizado por correlaciones de largo alcance e invarianza de escala. Aparecen así conceptos como leyes de escalado, universalidad y renormalización, pilares fundamentales dentro de la Física Estadística.El abanico de fenómenos complejos es muy amplio, y abarca sistemas de muy diversas disciplinas que van desde la Físicamás ortodoxa hasta la Biología, Sociología, Geología e, incluso, Economía. Esta Tesis se centra en fenómenos complejos extendidos en el espacio. En concreto hemos focalizado nuestra labor en tres grandes temas que constituyen importantes focos de interés dentro de la Mecánica Estadística: Crecimiento de Interfases, Sociofísica y Redes Complejas. / The ubiquity of complexity in Nature provides examples of a huge variety of systems to be analyzed by means of Statistical Mechanics and leads to the interconnection among various scientific disciplines. This Thesis focuses on three highlight topics of spatially extended complex systems: Interface Growth,Sociophysics, and Complex Networks. The document has been partitioned in three separated parts according to those topics.The first part deals with far-from-equilibrium growing interfaces. This subject represents one of the main fields in which fractal geometry has been widely applied, and is nowadays of great interest in Condensed Matter Physics. The Chapter 2 provides a brief and basic introduction to interface growth. We introduce some fractal and scaling concepts, as well as the main universality classes in presence of annealed disorder (EW and KPZ) in terms of both growth equations and discrete models. In Chapter 3 we focus on the elastic interface dynamics in disordered media, i.e., in presence of quenched randomness. This Chapter contains original research based on cellular automata simulations. We carry out a novel study of the dynamics by focusing on the discrete activity patterns that the interface sites describe during therelaxation toward the steady state. We analyze the spatio-temporal correlations of such patterns as the temperature is varied. We observe that, for some range of low temperatures, the out-of-equilibrium relaxation can be understood in the context of creep dynamics.The second part of the Thesis focus on Sociophysics. This discipline attends to the social interactions among individuals -most often mapped onto networks to provide them a topological structure- and has recently attracted much interest in the physics community. Social interactions give rise to adaptive systems that exhibit complex features as self-organization and cooperation. Therefore, Statistical Mechanics provides the necessary tools to analyze the behavior of such groups of agentsin a first level of simplification. The topics that Sociophysics deals with are quite a number, and we particularly focus on processes of opinion formation. The Chapter 4 presents a basic classification of the different opinion formation models present in the literature. In Chapter 5 we provide some analytical and numerical own results to describe the effect that the social temperature- understood as a simplified description of the interplay between an agent, its surroundings, and a collective climate parameter- may exert on such opinion formation processes. The thermal effect can be implemented in different ways. In the first part of the Chapter we work on a simple opinion formation model that, according to some procedural rules, reproduces the Sznajd dynamics. We include the thermal effect by means of some probability that the agents adopt the opposite opinion that the one indicated by such rules. In the second part of the Chapterwe consider a system with three different interacting groups of individuals, where the thermal effect is implemented as certain probability of spontaneous changes of the agents opinion. We exploit the van Kampen's expansion approach to analyze the macroscopic behavior of the different supporter group densities as well as the fluctuations around such macroscopic behavior.The third and last part of the document concerns Complex Networks, which have recently prompted the scientific community to investigate the mechanisms that determine their topology and dynamical properties.The rapid development of networks like the Internet and the World-Wide-Web, which represent today the basic substrate for all sort of communications at planetary level, has given rise to a number of interdisciplinary studies with highly technological applications. We first provide an introduction to complex networks in Chapter 6, where we introduce some basic concepts as scale-free graphs, mixing patterns, clustering coefficient, and small-world effect. In Chapter 7 we deal with traffic processes on networks, and specifically we focus on optimization of the routing protocols that define the connecting paths among all the pair of nodes. Such optimization pursues to avoid the traffic jams that emerge for huge quantities of matter or information flowing inthe graph. We propose an optimization algorithm that, in order to avert jamming, minimizes the number of paths that go through the most visited node (maximal betweenness) while keeping the path length as short as possible, i.e., in the proximities of the length distribution of the initial shortest-path protocol.

Condensed matter physics provides us with an opportunity to explore a large variety of systems with diverse properties. Central to the understanding of these systems is a characterization of the nature of their ground states and low energy excitation. Often, such systems show various forms of emergent properties that are absent in the microscopic level. Identification of such emergent phases of condensed matter form an important avenue of research in the field. In this thesis example of such phases and their associated phase transitions have been studied. The work presented here may be broadly divided into two themes: construction of the theoretical framework for understanding materials already studied experimentally, and, trying to provide new theoretical avenues which may be relevant for understanding future experiments. In these studies we shall explore some unconventional phases and phase transitions that may occur in condensed matter systems. A comprehensive understanding of the properties of such unconventional phases and phase transitions is important in the context of the large array of experimentally studied materials that regularly defy conventional wisdom in more than one way. The thesis consists of two distinct parts. In the first part we study three problems in frustrated magnets. The second part consists of studies of the tunnelling spectroscopy of metal-insulator-superconductor junctions in graphene. Studies in frustrated magnets have opened up the possibility of existence of a whole range of phases beyond the already known magnetically ordered ones. Some of these new phases, like the spin nematic or the valence bond solid, display some other conventional order themselves. Others, like the much sort after spin liquid phases displays a whole new kind of order that cannot be captured through the celebrated Landau’s classification of phases on the basis of symmetry breaking and associated order parameters. The phase transitions in these systems are also equally interesting and lead to intriguing possibilities that demand new modes of analysis. In this part of the thesis we shall study the different properties of three magnets with spin-1/2, 1 and 3/2 respectively. We start by providing an introduction to frustrated spin systems in Chapter [1]. The origin of antiferromagnetic interactions in Mott insulators is discussed and the concept of frustration of magnetic interaction is explained. We also point out the causes that may destroy magnetic order in spin systems, particularly the role of quantum fluctuations in presence or absence of magnetic frustration. This is followed with a brief outline of various magnetically ordered and disordered ground states with particular emphasis on the description of the later. We also give a brief outline of various properties of such phases and associated quantum phase transitions particularly noting the influences of quantum interferences encoded in the Berry phase terms. A brief description of the finite temperature properties is also provided. We end an outline of various experimentally relevant compounds that requires comprehensive understanding, some of which have been addressed in this thesis. In Chapter [2] we study the properties of a spin-nematic state in context of the recently discovered spin-1 Mott insulator Nickel Gallium Sulphide (NiGa2S4). This isotropic triangular lattice compound shows no spin ordering till low temperatures. We propose that it may have a particular type of spin-nematic ground state and explain the experimentally observed properties of the compound on the basis of our proposal. Starting from a two band Hubbard model description, relevant for the compound, we derive the Bilinear Biquadratic spin Hamiltonian. We then show, within mean field theory, that this Hamiltonian describes a transition from the spiral state to a ferro-nematic state as a function of the ratio of bilinear and biquadratic couplings. We also study the possible effects of small pinning disorder andmagnetic field and suggest experiments that can possibly distinguish the proposed nematic state from others. In Chapter [3] we explore the effects of the magneto-elastic coupling in the spin-3/2 B-site chromite spinel Cadmium Chromite (CdCr2O4). In this compound the spins form a pyrochlore lattice. Nearest neighbour spins interact antiferromagnetically. Due to frustration the system does not order at low temperatures and instead goes into a classical spin liquid state. Such a cooperative paramagnet is very susceptible to external perturbations which may relieve their frustration. In CdCr2O4, at lower temperatures the magnetic frustration is relieved by distorting the lattice through a first order magnetoelastic transition. Thus the compound presents a case where the relevant perturbation to the frustrated spin interactions is provided by spin-phonon coupling. An effect of such perturbations on a cooperative paramagnet is of general interest and all aspects of this are not understood presently. We take the initial step of characterizing the spin-phonon interaction in detail. Based on recent sound velocity experiments, we construct a microscopic theory for the sound velocity renormalization due to the spin-phonon coupling and explain the recent experimental data obtained by S. Zherlitsyn et al. using our theory we can explain the dependence of the sound velocity on temperature as well as magnetic field. We also construct a Landau theory to explain (qualitatively) the behaviour of sound velocity across the magneto-structural transition. Further, we discuss the effects due to the small Dzyaloshinskii-Moriya interaction that may be present in these compounds. In Chapter [4] we study the possibility of a direct second order quantum phase transition from spiral to dimer phase in two dimensional antiferromagnets. Such transitions between phases with incompatible symmetries are forbidden within conventional Landau Ginzburg-Wilson paradigm of critical phenomena. Early works showed that when the spiral is destroyed by long wavelength fluctuations a fractionalized Z2 spin liquid is obtained. In this work we show an alternative way–the quantum destruction of the spiral magnet. We argue that, when the defects of the spiral phase proliferate and condense, their associated Berry phase automatically leads to dimerization. We apply our theory to study concrete lattice models where such transitions may be observed. This transition is an example of a Landau forbidden deconfined quantum phase transition. The proposed critical theory is naturally written in terms of fractional degrees of freedom which emerge right at the critical point. These fractional particles interact with each other through emergent gauge fields and are deconfined right at the critical point (but are confined in either of the two adjoining phases). We argue, based on existing results, that the monopoles of the gauge field are dangerously irrelevant right at the critical point rendering the later noncompact. The critical point is characterized by an emergent global U (1) conservation law that is absent in the microscopic model, a typical feature of a deconfined quantum critical point. The resultant field theory belongs to the class of anisotropic NCCP3 class which may be studied numerically in future to understand its critical properties. In modern condensed matter physics the emergence of new and novel phases of matter have often been associated with the presence of strong correlations. Indeed, strongly correlated systems seem to harbour in them the potential to realize some of the most unconventional and exotic emergent phases of matter. However in graphene, which is a single layer of graphite, the emergence of novel properties, as present experiments suggest, is due to its unique band structure and not a fallout of intricate correlation effects. Band structure studies of graphene suggest that the material is a zero gap semiconductor with the low energy excitations resembling massless Dirac quasi-particles. The consequence of this is immediate and interesting. It has lead to the possibility of exploring the physics of relativistic fermions in two spatial dimensions and much of this has been studied with great vigour in the last five years. In our studies, presented in Chapter [5], we explore one of the many consequence of this emergent Dirac structure of the low energy quasi-particles, namely the properties of metal-insulator-superconductor junctions of graphene. The twin effect of Klein tunneling of Dirac fermions (and associated transmission resonances) and Andreev reflection (both specular and retro) sets them aside from their conventional counterparts. The graphene normal metal-insulator-superconductor (NIS) junctions show strikingly different properties like oscillations in the sub-gap tunneling conductance as a function of both barrier strength and width. We make a detailed study of this for arbitrary barrier strengths and widths with and without Fermi-surface mismatch between the normal and the superconducting sides. The amplitude of these oscillations are maximum for aligned Fermi surface and vanishes for large Fermi surface mismatch. We provide an understanding for this unconventional behaviour of graphene NIS junctions. We also suggest experimental tests for our theory. Such experimental verification will reveal one more remarkable emergent property in a condensed matter system.

The fields of statistical physics, discrete probability, combinatorics, and theoretical computer science have converged around efforts to understand random structures and algorithms. Recent activity in the interface of these fields has enabled tremendous breakthroughs in each domain and has supplied a new set of techniques for researchers approaching related problems. This thesis makes progress on several problems in this interface whose solutions all build on insights from multiple disciplinary perspectives. First, we consider a dynamic growth process arising in the context of DNA-based self-assembly. The assembly process can be modeled as a simple Markov chain. We prove that the chain is rapidly mixing for large enough bias in regions of Z^d. The proof uses a geometric distance function and a variant of path coupling in order to handle distances that can be exponentially large. We also provide the first results in the case of fluctuating bias, where the bias can vary depending on the location of the tile, which arises in the nanotechnology application. Moreover, we use intuition from statistical physics to construct a choice of the biases for which the Markov chain M_mon requires exponential time to converge. Second, we consider a related problem regarding the convergence rate of biased permutations that arises in the context of self-organizing lists. The Markov chain M_nn in this case is a nearest-neighbor chain that allows adjacent transpositions, and the rate of these exchanges is governed by various input parameters. It was conjectured that the chain is always rapidly mixing when the inversion probabilities are positively biased, i.e., we put nearest neighbor pair x<y in order with bias 1/2 <= p_{xy} <= 1 and out of order with bias 1-p_{xy}. The Markov chain M_mon was known to have connections to a simplified version of this biased card-shuffling. We provide new connections between M_nn and M_mon by using simple combinatorial bijections, and we prove that M_nn is always rapidly mixing for two general classes of positively biased {p_{xy}}. More significantly, we also prove that the general conjecture is false by exhibiting values for the p_{xy}, with 1/2 <= p_{xy} <= 1 for all x< y, but for which the transposition chain will require exponential time to converge. Finally, we consider a model of colloids, which 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. This clustering has proved elusive to verify, since all local sampling algorithms are known to be inefficient at high density, and in fact a new nonlocal algorithm was recently shown to require exponential time in some cases. We characterize the high and low density phases for a general family of discrete {it interfering binary mixtures} 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, very small perimeter, and high density of one type of molecule. Special cases of interfering binary mixtures include the Ising model at fixed magnetization and independent sets.

This thesis is mainly concerned with some important properties of complex fluids, and how these properties are influenced by structures in the nano/mesoscopic scale. Short-range assembly of the constituent molecules results in an amazing variety of phase behavior in these systems. Liquid-liquid phase transitions, or transitions from a homogeneous(mixed) phase to an immiscible phase (two-phase coexistence), are the outcome of a competition between entropy and short-ranged attractive forces, and form an important part of this thesis. A rich phase behavior is uncovered by a detailed study of liquid-liquid phase transitions in a mixture of ethanol(E) and water(W), induced by the addition of ammonium sulfate(AS) ions (E and W are otherwise completely soluble in each other). This is the main motivation for choosing this system. Furthermore, experimental evidence of the presence of supramolecular association in alcohol-water mixtures [J.-H. Guo et al., Phys. Rev Lett, 91, 15401(2003)] enhances our interest to study the phase behavior in more detail. The presence of a critical point, at which there is a second order phase transition, is quite common in complex fluids. An issue which has been the subject of extensive scientific research in recent years is the influence of nano/mesoscopic structure on the critical behavior of these fluids corresponds to the Ising universality class. However, the approach to the asymptotic regime is governed by a competition between the correlation length of critical concentration fluctuations and the additional length scale arising due to structuring., which results in a crossover from the universal Ising behavior to the mean-field behavior, sometimes within the critical domain. This phenomenon of crossover criticality is presently explored in the E + W + AS system. A significant portion of the thesis presents explorations on the critical behavior in the vicinity of special critical points (SCP), which are formed by the coalescence of two or more critical points. Recentrant liquid-liquid phase transitions observed in the E + W + AS system, furnishes an unique opportunity for the realization of three SCPs – the double critical point(DCP) and the critical double point(CDP) formed by the merger of two critical points , and a critical inflection point(CIP), formed by the merger of three critical points. A CIP had not been experimentally realized prior to the studies presented in this thesis. Apart from the above studies investigations are also carried out on the conformational changes of a technologically important conducting polymer, polyethylene dioxythiophene doped with polystyrene suflonate (PEDOT-PSS), in various solvents. The electrical and optical properties of the polymer films get enhanced when solution processed with specific solvents. The experiments presented in this thesis are directed at unraveling the role of conformational modifications in the electrical and optical properties of these systems. The experimental techniques that were employed in the present studies are: Laser light scattering, small-angle X-ray scattering(SAXS) measurements and visual observations. The eoexistence surface of the system E + W + AS was determined by visual observations. Laser light scattering measurements were conducted to study the critical behavior of osmotic susceptibility (xr) of E + W + As, whereas SAXS studies were conducted to ascertain the existence, and quantify the spatial extent of the additional length scale in the two systems investigated. The main objectives of this research were: (i) to study the phase behavior of the ternary mixture E + W + AS at atmospheric pressure; (ii) to check the existence of crossover from 3-D Ising to mean-field critical behavior while moving away from Tc in this system; (iii) to determine the nature (monotonic or nonmonotonic) of crossover; (iv) to provide some insight into the origin of this crossover behavior in terms of an additional length scale characteristic of the system; (v) to understand the evolution of the critical behavior in the proximity of CDP, and DCP; (vi) to experimentally realize the CIP; and (vii) to investigate the presence of solvent-induced conformational changes in conducting polymer.