In several applications of casting, dendritic microstructure is not desirable as it results in poor mechanical properties. Enhancing the fluid flow in the mushy zone by stirring is one of the means to suppress this dendritic growth. The strong fluid flow detaches the dendrites from the solid-liquid interface and carries them into the mold to form slurry. The detached dendrites coarsen in the slurry and form into rosette or globular particles based on processing conditions. This slurry offers less resistance to flow even at a high solid fraction and easily flow into the die-cavity. The above principle is the basis of a new manufacturing technology called “semi-sold forming” (SSF), in which metal alloys are cast in the semi-solid state. This technique has several advantages over other existing commercial casting processes, such as reduction of macrosegregation, reduction of porosity and low forming efforts. A major challenge existing in semisolid manufacturing is the production of metallic slurry in a consistent manner. The main difficulty arises because of the presence of a wide range of process parameters affecting the quality of the final product. An established method of producing slurry is by stirring the alloy using an electromagnetic stirrer. From an elaborate review of literature, it is apparent that solidification in presence of electromagnetic stirring involves a wide range of shear and cooling rates variation. However, the CFD models found in the literature are generally not based on accurate rheological properties, which are known to be functions of the relevant process parameters. Hence, there is a clear need for a comprehensive numerical model for such a solidification process, involving accurate rheological data for the semisolid slurry subjected to a range of processing conditions. The objective of the present work is to develop a numerical model for studying the transport phenomena during solidification with linear electromagnetic stirring. The study is presented in the context of a billet making process in a cylindrical mould using linear electromagnetic stirring. The mould consists of two parts: the upper part of the mould is surrounded by a linear electromagnetic stirrer forming the zone of active stirring, and the lower part of the mould is used to cool the liquid metal. The material chosen for the study is Al-7.32%Si (A356) alloy, commonly used for die casting applications. A complete numerical model will therefore have two major components: one dealing with rheological behavior of the semisolid slurry, and the other involving macroscopic modeling of the process using computational fluid dynamics (CFD) techniques. For the latter part of the model, determination of rheological behavior of the slurry is a pre-requisite. The rheological characteristics of the stirred slurry, as a function of shear rate and cooling rate, is determined experimentally using a concentric cylinder viscometer. Two different series of experiments are performed. In the first series, the liquid metal is cooled at a constant cooling rate and sheared with different shear rates to get the effect of shear rate on viscosity. In the second series of experiments, the liquid metal is cooled at different cooling rates and sheared at a constant shear rate to obtain the effect of cooling rate on viscosity. During all these experiments, the shear rate is calculated from the measured angular velocity of spindle using inductive position sensor; viscosity of the slurry is calculated based on the torque applied to the slurry and angular velocity of the spindle; and the solid fraction is calculated from measured temperature of the slurry based on Schiel equation. From these data, a constitutive relation for variable viscosity is established, which is subsequently used in a numerical model for simulating the transport phenomena associated with the solidification process. The numerical model uses a set of single-phase governing equations of mass, momentum, energy and species conservation. The set of governing equations is solved using a pressure based finite volume technique, along with an enthalpy based phase change algorithm. The numerical simulation of this process also involves modeling of Lorentz force field. The numerical study involves prediction of temperature, velocity, species and solid fraction distribution. First, studies are performed for a base case with a moderate stirring intensity of 250A primary current and 50 Hz frequency. It is found that the electromagnetic forces have maximum values near the mould periphery, which results in an ascending movement of the slurry near the mould periphery. Because of continuity, this slurry comes down along the axis of the mould. Stirring produces a strong fluid flow which results good mixing in the melt. Correspondingly, a homogenized temperature distribution is found in the domain. Because of strong stirring, the solid fraction in the slurry is found to be distributed almost uniformly. It is also found that fragmentation of dendrites increases solid fraction in the slurry with processing time. During processing, the continuous rejection of solute makes the liquid progressively solute enriched. It is predicted from the present study that the remaining liquid surrounding the primary solid phase finally solidifies with a near-eutectic composition, which is desirable from the point of view of semisolid casting. Correspondingly, a set of experiments are performed to validate the numerically predicted results. The numerical predictions of temperature variations are in good agreement with experiments, and the predicted flow field evolution correlate well with the microstructures obtained through experiments at various locations, as observed in the numerical results. Subsequently the study is extended to predict the effect of process parameters such as stirring intensity and cooling rate on the distributions of solid fraction and solute in the domain. It is found, from the simulation, that the solidification process is significantly affected by stirring intensity. At increasing primary excitation current, the magnitude of Lorentz force increases and results in increase of slurry velocity. Correspondingly, the fragmentation of dendrites from the solid/liquid is more during solidification at higher stirring intensity, which increases the fraction of solid in the slurry to a high value. It is also found that the solute and fraction of solid in the liquid mixes well under stirring action. Thus, a near uniform distribution of solute and solid fraction is found in the domain. It is found that stirring at high currents produces high solid fraction in the liquid. Also, at very low cooling rate, the solid fraction in the liquid increases. The present study focuses on the model development and experimental validation for solidification with linear electromagnetic stirring for producing a rheocast billet. Further studies highlighting the effects of various process parameters on the thermal history and microstructure formation are also presented.

Dans cette étude, nous proposons une étude théorique et numérique de l'influence des vibrations de hautes fréquences et de faibles amplitudes sur la séparation thermo-gravitationnelle des constituants d'un mélange binaire saturant un milieu poreux. La cellule considérée est horizontale, de grand rapport d'aspect et placée dans le champ de pesanteur. La formulation mathématique est obtenue en utilisant le formalisme des équations moyennées. Les conditions aux limites imposées au niveau des parois horizontales diffèrent de celles du problème de Rayleigh Bénard. On considère respectivement le cas où un flux constant est imposé sur ces parois puis le cas où un flux thermique constant est imposé sur la paroi inférieure alors qu'une température constante est imposée sur la paroi supérieure. Dans les deux configurations, onmet en évidence de solutions stables monocellulairesconduisant à la séparation des constituants du mélange et ce pour une large gamme des paramètres adimensionnels régissant le problème. Les résultats analytiques et de simulations numériques directes sont en très bon accord. Dans les deux cas, une étude de stabilité linéaire de la solution d'équilibre et de la solution monocellulaireest réalisée par méthode spectrale. / In this study, the influence of vertical vibrations on species thermo-gravitational separation of a binary fluid, saturating a porous medium, is presented. The cell is horizontal of large aspect ratio and situated in the gravity field. A formulation using time average equations is used. Two configurations have been considered and compared. In the first one, a constant heat flux is imposed on the horizontal walls and in the second case, a constant heat flux is imposed on the bottom wall while a constant temperature is imposed on the top wall. For each configuration, stable unicellular solutions leading to species separation are obtained, depending on the dimensionless parameters of the problem. The analytical results are in good agreement with those obtained by direct numerical simulations. In both cases, a linear stability analysis of the equilibrium solution and the unicellular one is presented using a spectral method.

La théorie de couplage de mode (MCT) est l'une des méthodes les plus utilisées pour étudier les transitions vitreuses dans les fluides classiques. Ses prédictions sont en général en accord semi quantitatif avec les simulations. Sa mise en oeuvre nécessite la détermination de la structure statique, généralement par résolution des équations d'Orsntein-Zernike avec une fermeture adéquate. Partant de fermetures utilisant des fonctions « bridges » déduites de la fonctionnelle de référence du mélange de sphères dures, notre travail a consisté d'abord à étudier l'influence de la qualité de cette structure statique sur les prédictions relatives aux états non ergodiques dans des mélanges binaires dissymétriques. Nous avons ensuite considéré les résultats de la théorie du couplage de modes dans sa version naïf (NMCT ) et complète, afin d'analyser les mécanismes d'arrêt, les comparer au fluide effectif et aux approches stochastiques (équations de Langevin généralisées). Enfin, nous proposons une version pragmatique de cette méthode qui fournit des prédictions en meilleur accord quantitatif avec les résultats des simulations pour une variété de potentiels d'interaction / The mode coupling theory (MCT) is one of the most widely used methods for studying the glass transition in classical fluids. Its predictions are usually in semi-quantitative agreement with simulation. Its implementation requires the determination of the static structure usually from the Ornstein-Zernike equations with a suitable closure. Starting from closures that use bridge functions deduced from the hard-sphere reference functional, our work consisted first in studying the influence of the quality of this static structure on the predictions concerning the non-ergodic states in asymmetric binary mixtures. We next considered the results of the mode coupling theory in its naive and full versions, in order to analyze the arrest mechanisms and compare them to the effective fluid and the stochastic approaches (generalized Langevin equations). Finally, we propose a pragmatic version of this method that provides predictions in better quantitative agreement with simulations for a variety of interaction potentials

Model-H is used to describe structures found in the phase separation in films of binary liquid mixture that have a surface that is free to deform and also may energetically prefer one of the components. The film rests on a solid smooth substrate that has no preference for any component. On the one hand the study focuses on static aspects by investigating steady states that are characterised by their concentration and film height profiles. A large variety of such states are systematically analysed by numerically constructing bifurcation diagrams in dependence of a number of control parameters. The numerical method used is based on minimising the free energy functional at given constraints within a finite element method for a variable domain shape. The structure of the bifurcation diagrams is related to the symmetry properties of the individual solutions on the various branches. On the other hand the full time dependent model-H is linearised about selected steady states, in particular, the laterally invariant, i.e.\ layered states. The resulting dispersion relations are discussed and related to the corresponding bifurcation points of the steady states. In general, the results do well agree and confirm each other. The described analysis is performed for a number of important cases whose comparison allows us to gain an advanced understanding of the system behaviour: We distinguish the critical and off-critical case that correspond to zero and non-zero mean concentration, respectively. In the critical case the investigation focuses on (i) flat films without surface bias, (ii) flat films with surface bias, (iii) height-modulated films without surface bias, and (iv) height-modulated films with surface bias. Each case is analysed for several mean film heights and (if applicable) energetic bias at the free surface using the lateral domain size as main control parameter. Linear stability analyses of layered films and symmetry considerations are used to understand the structures of the determined bifurcation diagrams. For off-critical mixtures our study is more restricted. There we consider height-modulated films without and with surface bias for several mean film heights and (if applicable) energetic bias employing the mean concentration as main control parameter.

This thesis, which contains fourteen chapters in two parts, presents theoretical and computer simulation studies of dynamics in supercooled liquids and thermotropic liquid crystals. These two apparently diverse physical systems are unified by a startling similarity in their complex slow dynamics. Part I consists of six chapters on supercooled liquids while Part II comprises seven chapters on thermotropic liquid crystals. The fourteenth chapter provides a concluding note. Part I starts with an introduction to supercooled liquids given in chapter 1. This chapter discusses basic features of supercooled liquids and the glass transition and portrays some of the theoretical frameworks and formalisms that are widely recognized to have contributed to our present understanding. Chapter 2 introduces a new model of binary mixture in order to study dynamics across the supercooled regime. The system consists of an equimolar mixture of the Lennard-Jones spheres and the Gay-Berne ellipsoids of revolution, and thus one of its components has orientational degrees of freedom (ODOF). A decoupling between trans-lational diffusion and rotational diffusion is found to occur below a temperature where the second rank orientational correlation time starts showing a steady deviation from the Arrhenius temperature behavior. At low temperatures, the optical Kerr effect (OKE) signal derived from the system shows a short-to-intermediate time power law decay with a very weak dependence on temperature, if at all, of the power law exponent as has been observed experimentally. At the lowest temperature investigated, jump motion is found to occur in both the translational and orientational degrees of freedom. Chapter 3 studies how the binary mixture, introduced in the previous chapter, explores its underlying potential energy landscape. The study reveals correlations between the decoupling phenomena, observed almost universally in supercooled molecular liquids, and the manner of exploration of the energy landscape of the system. A significant deviation from the Debye model of rotational diffusion in the dynamics of ODOF is found to begin at a temperature at which the average inherent structure energy of the system starts falling as the temperature decreases. Further, the coupling between rotational diffusion and translational diffusion breaks down at a still lower temperature, where a change occurs in the temperature dependence of the average inherent structure energy. Chapters 4-6 describe analytical and numerical approaches to solve kinetic models of glassy dynamics for various observables. The β process is modeled as a thermally activated event in a two-level system and the a process is described as a β relaxation mediated cooperative transition in a double-well. The model resembles a landscape picture, conceived by Stillinger [Science 267, 1935 (1995)], where the a process is assumed to involve a concerted series of the β processes, the latter being identified as elementary relaxations involving transitions between contiguous basins. For suitable choice of parameter values, the model could reproduce many of the experimentally observed features of anomalous heat capacity behavior during a temperature cycle through the glass transition as described in chapter 4. The overshoot of the heat capacity during the heating scan that marks the glass transition is found to be caused by a delayed energy relaxation. Chapter 5 shows that the model can also predict a frequency dependent heat capacity that reflects the two-step relaxation behavior. The high-frequency peak in the heat capacity spectra appears with considerably larger amplitude than the low-frequency peak, the latter being due to the a relaxation. The model, when simplified with a modified description of the a process that involves an irreversible escape from a metabasin, can be solved analytically for the relaxation time. This version of the model captures salient features of the structural relaxation in glassy systems as described in chapter 6. In Part II, thermotropic liquid crystals are studied in molecular dynamics simulations using primarily the family of the Gay-Berne model systems. To start with, chapter 7 provides a brief introduction to thermotropic liquid crystals, especially from the perspective of the issues discussed in the following chapters. This chapter ends up with a detail description of the family of the Gay-Berne models. Chapter 8 demonstrates that a model system for calamitic liquid crystal (comprising rod-like molecules) could capture the short-to-intermediate time power law decay in the OKE signal near the isotropic-nematic (I-N) phase transition as observed experimentally. The single-particle second rank orientational time correlation function (OTCF) for the model liquid crystalline system is also found to sustain a power law decay regime in the isotropic phase near the I-N transition. On transit across the I-N phase boundary, two power law decay regimes, separated by a plateau, emerge giving rise to a step-like feature in the single-particle second rank OTCF. When the time evolution of the rotational non-Gaussian parameter is monitored as a diagnostic of spatially heterogeneous dynamics, a dominant peak is found to appear following a shoulder at short times, signaling the growth of pseudonematic domains. These observations are compared with those relevant ones obtained for the supercooled binary mixture, as discussed in chapter 2, in the spirit of the analogy suggested recently by Fayer and coworkers [J. Chem. Phys. 118, 9303 (2003)]. In chapter 9, orientational dynamics across the I-N transition are investigated in a variety of model systems of thermotropic liquid crystals. A model discotic system that consists of disc-like molecules as well as a lattice system have been considered in the quest of a universal short-to-intermediate time power law decay in orientational relaxation, if any. A surprisingly general power law decay at short to intermediate times in orientational relaxation is observed in all these systems. While the power law decay of the OKE signal has been recently observed experimentally in calamitic systems near the I-N phase boundary and in the nematic phase by Fayer and coworkers [J. Chem. Phys. 116, 6339 (2002), J. Phys. Chem. B 109, 6514 (2005)], the prediction for the discotic system can be tested in experiments. Chapter 10 presents the energy landscape view of phase transitions and slow dynamics in thermotropic liquid crystals by determining the inherent structures of a family of one-component Gay-Berne model systems. This study throws light on the interplay between the orientational order and the translational order in the mesophases the systems exhibit. The onset of the growth of the orientational order in the parent phase is found to induce a translational order, resulting in a smectic-like layer in the underlying inherent structures. The inherent structures, surprisingly, never seem to sustain orientational order alone if the parent nematic phase is sandwiched between the high-temperature isotropic phase and the low-temperature smectic phase. The Arrhenius temperature dependence of the orientational relaxation time breaks down near the I-N transition and this breakdown is found to occur at a temperature below which the system explores increasingly deeper potential energy minima. There exists a remarkable similarity in the manner of exploration of the potential energy landscape between the Gay-Berne systems studied here and the well known Kob-Andersen binary mixture reported previously [Nature, 393, 554 (1998)]. In search of a dynamical signature of the coupling between orientational order and translational order, anisotropic translational diffusion in the nematic phase has been investigated in the Gay-Berne model systems as described in chapter 11. The translational diffusion coefficient parallel to the director D// is found to first increase and then decrease as the temperature drops through the nematic phase. This reversal occurs where the smectic order parameter of the underlying inherent structures becomes significant for the first time. The non-monotonic temperature behavior of D// can thus be viewed from an energy landscape analysis as a dynamical signature of the coupling between orientational and translational order at the microscopic level. Such a view is likely to form the foundation of a theoretical framework to explain the anisotropic translation diffusion. Chapter 12 investigates the validity of the Debye model of rotational diffusion near the I-N phase boundary with a molecular dynamics simulation study of a Gay-Berne model system for calamitic liquid crystals. The Debye model is found to break down near the I-N phase transition. The breakdown, unlike the one observed in supercooled molecular liquids where a jump diffusion model is often invoked, is attributed to the growth of orientational pair correlation. A mode-coupling theory analysis is provided in support of the explanation. Chapter 13 presents a molecular dynamics study of a binary mixture of prolate ellipsoids of revolution with different aspect ratios interacting with each other through a generalized Gay-Berne potential. Such a study allows to investigate directly the aspect ratio dependence of the dynamical behavior. In the concluding note, chapter 14 starts with a brief summary of the outcome of the thesis and ends up with suggestion of a few relevant problems that may prove worthwhile to be addressed in future.

I started writing this thesis not only to obtain a doctoral degree, but also to compile in a particular way all the work that I have done during this time. The articles published during these years can only give a short overview of my research task. I decided to give my own perspective of the things I have learned and the results I have obtained. Some sections are directly the published articles, but some other are not and contain a significant amount of unpublished data. Even in some cases the published plots have been modified / altered to provide more insight or to maintain consistency. Historical perspectives often provide a deep understanding of the problems and have been briefly discussed in some chapters. This thesis contains theoretical and computer simulation studies to under-stand effects of spatial correlation on dynamics in several complex systems. Based on the different phenomena studied, the thesis has been divided into three major parts: I. Pair hydrophobicity, composition-dependent anomalies and structural trans-formations in aqueous binary mixtures II. Microscopic analysis of hydrophobic force law in a two dimensional (2D) water-like model system III. Diffusion of a tagged particle on a rugged energy landscape with spatial correlations The three parts have been further divided into ten chapters. In the following we provide part-wise and chapter-wise outline of the thesis. Part I consists of six chapters, where we focus on several important aqueous binary mixtures of amphiphilic molecules. To start with, Chapter 1 provides an introduction to non-ideality often encountered in aqueous binary mixtures. Here we briefly discuss the existing ideas of structural transformations associated with solvation of a foreign molecule in water, with particular emphasis on the classic “iceberg” model. Over the last decade, several investigations, especially neutron scattering and diffraction experiments, have questioned the validity of existing theories and have given rise to an alternate molecular picture involving micro aggregation of amphiphilic co-solvents in their aqueous binary mixtures. Such microheterogeneity was also supported by other experiments and simulations. In Chapter 2, we present our calculation of the separation dependence of potential of mean force (PMF) between two methane molecules in water-dimethyl sulfoxide (DMSO) mixture, using constrained molecular dynamics simulation. It helps us to understand the composition-dependence of pair hydrophobicity in this binary solvent. We find that pair hydrophobicity in the medium is surprisingly enhanced at DMSO mole fraction xDMSO ≈ 0.15, which explains several anomalous properties of this binary mixture – including the age-old mystery of DMSO being a protein stabilizer at lower concentration and protein destabilizer at higher concentration. Chapter 3 starts with discussion of non-monotonic composition dependence of several other properties in water-DMSO binary mixture, like diffusion coefficient, local composition fluctuation and fluctuations in total dipole moment of the system. All these properties exhibit weak to strong anomalies at low solute concentration. We attempt to provide a physical interpretation of such anomalies. Previous analyses often suggested occurrence of a “structural transformation” (or, microheterogeneity) in aqueous binary mixtures of amphiphilic molecules. We show that this structural transformation can be characterized and better understood under the purview of percolation theory. We define the self-aggregates of DMSO as clusters. Analysis of fractal dimension and cluster size distribution with reference to corresponding “universal” scaling exponents, combined with calculation of weight-averaged fraction of largest cluster and cluster size weight average, reveal a percolation transition of the clusters of DMSO in the anomalous concentration range. The percolation threshold appears at xDMSO ≈ 0.15. The molecular picture suggests that DMSO molecules form segregated islands or micro-aggregates at concentrations below the percolation threshold. Close to the critical concentration, DMSO molecules start forming a spanning cluster which gives rise to a bi-continuous phase (of water-rich region and DMSO-rich region) beyond the threshold of xDMSO ≈ 0.15. This percolation transition might be responsible for composition-dependent anomalies of the binary mixture in this low concentration regime. Similar phenomenon is observed for another amphiphilic molecule – ethanol, as discussed in Chapter 4. We again find composition dependent anomalies in several thermophysical properties, such as local composition fluctuation, radial distribution function of ethyl groups and self-diffusion co-efficient of ethanol. Earlier experiments often suggested distinct structural regimes in water-ethanol mixture at different concentrations. Using the statistical mechanical techniques introduced in the previous chapter, we show that ethanol clusters undergo a percolation transition in the anomalous concentration range. Despite the lack of a precise determination of the percolation threshold, estimate lies in the ethanol mole fraction range xEtOH ≈ 0.075 - 0.10. This difficulty is probably due to transient nature of the clusters (as will be discussed in Chapter 6) and finite size of the system. The scaling of ethanol cluster size distribution and the fractal behavior of ethanol clusters, however, conclusively demonstrate their “spanning” nature. To develop a unified understanding, we further study the composition-dependent anomalies and structural transformations in another amphiphilic molecule, tertiary butyl alcohol (TBA) in Chapter 5. Similar to the above-mentioned aqueous binary mixtures of DMSO and ethanol, we demonstrate here that the anomalies occur due to local structural changes involving self-aggregation of TBA molecules and percolation transition of TBA clusters at xTBA ≈ 0.05. At this percolation threshold, we observe a lambda-type divergence in the fluctuation of the size of the largest TBA cluster, reminiscent of a critical point. Interestingly, water molecules themselves exhibit a reverse percolation transition at higher TBA concentration ≈ 0.45, where large spanning water clusters now break-up into small clusters. This is accompanied by significant divergence of the fluctuations in the size of the largest water cluster. This second transition gives rise to another set of anomalies around. We conclude this part of the thesis with Chapter 6, where we introduce a novel method for understanding the stability of fluctuating clusters of DMSO, ethanol and TBA in their respective aqueous binary mixtures. We find that TBA clusters are the most stable, whereas ethanol clusters are the most transient among the three representative amphiphilic co-solvents. This correlates well with the amplitude of anomalies observed in these three binary mixtures. Part II deals with the topic of hydrophobic force law in water. In the introductory Chapter 7 of this part, we briefly discuss the concept of hydrophobicity which is believed to be of importance in understanding / explaining the initial processes involved in protein folding. We also discuss the experimental observations of Israelachvili (on the force between hydrophobic plates) and the empirical hydrophobic force law. We briefly touch upon the theoretical back-ground, including Lum-Chandler-Weeks theory. We conclude this chapter with a brief account of relevant and important in silico studies so far. In Chapter 8, we present our studies on Mercedes-Benz (MB) model – a two dimensional model system where circular disks interact with an anisotropic potential. This model was introduced by Ben-Naim and was later parametrized by Dill and co-workers to reproduce many of the anomalous properties of water. Using molecular dynamics simulation, we show that hydrophobic force law is indeed observed in MB model, with a correlation length of ξ=3.79. The simplicity of the model enables us to unravel the underlying physics that leads to this long range force between hydrophobic plates. In accordance with Lum-Chandler-Weeks theory, density fluctuation of MB particles (leading to cavitation) between the hydrophobic rods is clearly distinguishable – but it is not sufficiently long ranged, with density correlation extending only up to ζ=2.45. We find that relative orientation of MB molecules plays an important role in the origin of the hydrophobic force in long range. We define appropriate order parameters to capture the role of orientation, and briefly discuss a plausible approach of an orientation-dependent theory to explain this phenomenon. Part III consists of two chapters and focuses on the diffusion of a Brownian particle on a Gaussian random energy landscape. We articulate the rich history of the problem in the introductory Chapter 9. Despite broad applicability and historical importance of the problem, we have little knowledge about the effect of ruggedness on diffusion at a quantitative level. Every study seems to use the expression of Zwanzig [Proc. Natl. Acad. U.S.A, 85, 2029 (1988)] who derived the effective diffusion coefficient, Deff =D0 exp (-β2ε2 )for a Gaussian random surface with variance ε, but validity of the same has never been tested rigorously. In Chapter 10, we introduce two models of Gaussian random energy surface – a discrete lattice and a continuous field. Using computer simulation and theoretical analyses, we explore many different aspects of the diffusion process. We show that the elegant expression of Zwanzig can be reproduced ex-actly by Rosenfeld diffusion-entropy scaling relationship. Our simulations show that Zwanzig’s expression overestimates diffusion in the uncorrelated Gaussian random lattice – differing even by more than an order of magnitude at moderately high ruggedness (ε>3.0). The disparity originates from the presence of “three-site traps” (TST) on the landscape – which are formed by deep minima flanked by high barriers on either side. Using mean first passage time (MFPT) formalism, we derive an expression for the effective diffusion coefficient, Deff =D0 exp ( -β2ε2)[1 +erf (βε/2)]−1 in the presence of TSTs. This modified expression reproduces the simulation results accurately. Further, in presence of spatial correlation we derive a general expression, which reduces to Zwanzig’s form in the limit of infinite spatial correlation and to the above-mentioned equation in absence of correlation. The Gaussian random field has an inherent spatial correlation. Diffusion coefficient obtained from the Gaussian field – both by simulations and analytical methods – establish the effect of spatial correlation on random walk. We make special note of the fact that presence of TSTs at large ruggedness gives rise to an apparent breakdown of ergodicity of the type often encountered in glassy liquids. We characterize the same using non-Gaussian order parameter, and show that this “breakdown” scales with ruggedness following an asymptotic power law. We have discussed the scope of future work at the end of each chapter when-ever appropriate.

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