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Some problems in the theory of crystalline surfaces
No full textItem marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding ([email protected]) on 2011-05-07T15:02:10Z
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Reason: ETDs are only available to UIUC Users without author permissionETDs are only available to UIUC Users without author permissionU of I OnlyThis thesis is concerned with the statics and dynamics of free crystalline surfaces. Both unstable and relaxational dynamics are treated with particular attention paid to elastic interactions. An equation of motion for the thermal faceting of an unstable planar crystalline surface is proposed, from which the behavior in the linear region of the surface structure function, initial domain size, and velocity of domain spreading are extracted. Elastic interactions are shown to modify the location of the spinodal line significantly.Solutions for the dynamics of the evaporative decay of one-dimensional sine waves, isolated step packets, and isolated facets on surfaces below their roughening transition are presented, which have only the initial time as a free parameter. The solutions are tested against mesoscopic models with excellent agreement found. Intrinsic and reconstruction-driven isolated facet growth are identified as distinct processes, with different equations of motion. The decay of a sine wave is shown to generate a far from equilibrium layer at the top of the sine wave, which has to be separated as a boundary layer.The late stages of thermal faceting of a vicinal surface are shown to lead to a spontaneous breakdown of the coarse-grained free energy. Finite size effects and interface energies resulting from elastic interactions are calculated. Elastic interactions are shown to freeze one-dimensional domain ripening in thermal faceting at late times. The elastic displacement field and the corresponding stress and strain tensors of an isolated step are computed and used to measure the surface dipole moments of Si as: d\sb{x} = 1.46eV/A and d\sb{z} = 0.58eV/A. The displacement field of a stepped surface is computed. Surface steps on Pb are shown to melt due to the elastic field of a step and as a local wetting transition. The melting of steps, vacancies, and diffusers is shown to account for the anomalous behavior of the Pb equilibrium crystal shape.Made available in DSpace on 2011-05-07T14:02:09Z (GMT). No. of bitstreams: 2
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Approach to equilibrium in systems with continuous symmetries
No full textIn this thesis, we consider the approach to equilibrium of quenched systems with continuous symmetry, whose relaxational dynamics is dominated by topological defects. The general aspects of the problem are discussed in chapter 1. In chapters 2 and 3, we report the results of two and three dimensional simulations of a simple model with non-conserved order parameter and the symmetry of a planar ferromagnet. A transient behavior is observed at early times in two dimensions. Finite-size scaling of the scattering function is demonstrated and it is shown that dynamical scaling is satisfied not only by the correlation functions of the order parameter but also by the correlation functions of the defects (point-vortices in two dimensions and vortex-strings in three dimensions). In the three dimensional case, the effect of a bias in the initial conditions is considered. The introduction of a bias (or external field) leads to exponential relaxation and the break-down of dynamical scaling. In chapter 4, we consider a system with conserved order parameter, which is proposed as a model of crystal surface relaxation. Multiscaling behavior for the scattering function is investigated, with negative results. A comparison of the correlation functions in the conserved and non-conserved case indicates that the conservation constraint significantly affects the vortex dynamics. In chapter 5, we discuss a model of the superconducting transition. A linear stability analysis of the normal-superconductor interface for type I superconductors indicates the presence of an instability, analogous to that responsible for dendritic patterns in solidification. A simple mean-field picture of the transition kinetics of type II superconductors suggests the existence of two dynamical regimes, characterized by a power-law and a logarithmic growth of ordered (superconducting) domains in the system. Numerical simulations of type II superconductors in the spinodal regime bear out this prediction. Selected computer programs are given in the appendices.Made available in DSpace on 2011-05-07T11:52:32Z (GMT). No. of bitstreams: 2
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Previous issue date: 1991Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding ([email protected]) on 2011-05-07T14:33:40Z
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Pattern formation in systems far from equilibrium
No full textWe study two representative problems related to the dynamics of pattern formation in non-linear, dissipative systems far from equilibrium. The first problem addresses the dynamical mechanism of velocity and shape selection in dendritic crystal growth. Using the boundary-layer model of solidification, we demonstrate the new solvability theory of velocity selection. We perform the linear stability analysis for the needle crystal steady states and establish the linear stability of the fastest needle crystal, which accounts for the unique dendrite selection in experiments. We then show, for the first time, that the competition between surface tension and kinetic anisotropy leads to the tip-splitting/sidebranching instability of the needle crystal. This explains the presence of a morphological transition and the dense-branching morphology. Finally, an efficient lattice model is developed to study the late stages of diffusion-controlled growth. We establish the existence of an asymptotic dense-branching morphology and relate it to the diffusion-limited aggregation. A clear morphological transition from kinetic effect dominated growth to surface tension dominated growth is observed, marked by a difference in the way growth velocity scales with undercooling. Scaling behaviour in the evolution of interfacial instability is found, in a planar geometry, indicating a non-linear selection of a unique length-scale, insensitive to short length-scale fluctuations.The second problem we study regards the dynamics of phase separation in block copolymer melts. By numerical minimization of the free energy for a block copolymer melt, we calculate the scaling exponents for the way in which the equilibrium lamellar thickness of the microdomains varies with the degree of polymerization. We propose a scaling theory of the approach to equilibrium, from which we relate the exponents in block copolymer systems to the dynamical exponents in spinodal decomposition. We also study the lamellar pattern formed by a propagating front. The selection of the unique front velocity and wavelength agrees well with the predictions of the marginal-stability hypothesis.Made available in DSpace on 2011-05-07T13:41:26Z (GMT). No. of bitstreams: 2
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Previous issue date: 1990Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding ([email protected]) on 2011-05-07T14:57:57Z
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Renormalization group theory for systems far from equilibrium
No full textThis thesis is largely concerned with the long-time or large-scale asymptotic behavior of a variety of non-equilibrium (physical) systems in the absence of noise, to which the renormalization group theory (RG) and the structural stability hypothesis are systematically applied.The first type of problems I study are examples of systems possessing physical scale invariance, for which I focus on similarity solutions, where the exponents are anomalous, and traveling-wave problems, where there is an apparent non-uniqueness of solutions of front propagation. For anomalous nonlinear diffusion and propagation of fronts in porous media, I show that the presence of anomalous dimensions far from equilibrium can be explained systematically. The exponents and related scaling laws are calculated analytically by perturbative renormalization group (RG) theory. I study the propagation of superfluid turbulence, and find that preliminary experimental data are in qualitative agreement with the theoretical RG predictions: in particular, a sharp propagating front is observed.For the velocity selection in pattern formation systems, it is proposed that only structurally stable fronts are reproducibly observable in practice. Combining the structural stability hypothesis with RG techniques enables one to predict the uniquely selected velocity. The results apply to both the linear-marginal-stability and nonlinear-marginal-stability cases under very general conditions. A variational principle is also implemented to identify the transition between different regimes. I also present a numerical implementation of the RG for the problems mentioned above, constructing similarity solutions and traveling waves. I show that the numerical RG method is computationally more efficient than direct numerical integration of equations, and yields more accurate results than previous analytical perturbative RG calculations, especially when (perturbation) parameters are not small.The second type of problems I investigate are singular perturbation problems, classical or quantum, without apparent physical scale invariance. Conventionally, numerous (ad hoc) singular perturbation methods are made use of to solve them. A systematic renormalization group (RG) theory is developed as a unified framework for global singular perturbation methods. This is the first time that the direct correspondence between singular perturbation and RG is established. This study reveals several very general conclusions: (1) Singular perturbation methods are naturally understood as renormalized perturbation methods in physics, and (2) Amplitude equations are nothing but RG equations, and (3) in several aspects, the RG method is technically superior.Made available in DSpace on 2011-05-07T13:44:47Z (GMT). No. of bitstreams: 2
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Previous issue date: 1994Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding ([email protected]) on 2011-05-07T14:58:39Z
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Reason: ETDs are only available to UIUC Users without author permissionETDs are only available to UIUC Users without author permissionU of I Onl
Renormalization group theory for systems far from equilibrium
No full textThis thesis is largely concerned with the long-time or large-scale asymptotic behavior of a variety of non-equilibrium (physical) systems in the absence of noise, to which the renormalization group theory (RG) and the structural stability hypothesis are systematically applied.The first type of problems I study are examples of systems possessing physical scale invariance, for which I focus on similarity solutions, where the exponents are anomalous, and traveling-wave problems, where there is an apparent non-uniqueness of solutions of front propagation. For anomalous nonlinear diffusion and propagation of fronts in porous media, I show that the presence of anomalous dimensions far from equilibrium can be explained systematically. The exponents and related scaling laws are calculated analytically by perturbative renormalization group (RG) theory. I study the propagation of superfluid turbulence, and find that preliminary experimental data are in qualitative agreement with the theoretical RG predictions: in particular, a sharp propagating front is observed.For the velocity selection in pattern formation systems, it is proposed that only structurally stable fronts are reproducibly observable in practice. Combining the structural stability hypothesis with RG techniques enables one to predict the uniquely selected velocity. The results apply to both the linear-marginal-stability and nonlinear-marginal-stability cases under very general conditions. A variational principle is also implemented to identify the transition between different regimes. I also present a numerical implementation of the RG for the problems mentioned above, constructing similarity solutions and traveling waves. I show that the numerical RG method is computationally more efficient than direct numerical integration of equations, and yields more accurate results than previous analytical perturbative RG calculations, especially when (perturbation) parameters are not small.The second type of problems I investigate are singular perturbation problems, classical or quantum, without apparent physical scale invariance. Conventionally, numerous (ad hoc) singular perturbation methods are made use of to solve them. A systematic renormalization group (RG) theory is developed as a unified framework for global singular perturbation methods. This is the first time that the direct correspondence between singular perturbation and RG is established. This study reveals several very general conclusions: (1) Singular perturbation methods are naturally understood as renormalized perturbation methods in physics, and (2) Amplitude equations are nothing but RG equations, and (3) in several aspects, the RG method is technically superior.U of I OnlyETDs are only available to UIUC Users without author permissio
Computational Studies Of Dendritic Crystal Growth
No full textSubmitted by Meng Tao ([email protected]) on 2012-10-23T19:13:37Z
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Previous issue date: 2003-12Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Meng Tao ([email protected]) on 2012-10-23T19:13:37Z
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Approach to equilibrium in systems with continuous symmetries
No full textIn this thesis, I consider the approach to equilibrium of quenched systems
with continuous symmetry, whose relaxational dynamics is dominated by topological
defects. The general aspects of the problem and relevant theoretical,
numerical and experimental results from the literature are discussed in chapter
1. In chapters 2 and 3, I report the results of two and three dimensional simulations
of a simple model with non-conserved order parameter and the symmetry
of a planar ferromagnet. A transient behavior is observed at early times in two
dimensions, indicating that the vortex annihilation dynamics significantly affects
the initial ordering process in the system. Finite-size scaling of the scattering
function is demonstrated and it is shown that dynamical scaling is satisfied not
only by the correlation functions of the order parameter but also by the correlation
functions of the defects (point-vortices in two dimensions and vortex-strings
in three dimensions). In the three dimensional case, the effect of a bias in the
initial conditions is considered. The introduction of a bias (or external field)
leads to exponential relaxation and the break-down of dynamical scaling. An
experiment is suggested, which could reproduce the conditions of the simulation
in bulk samples of quenched nematic liquid crystals. Possible relevance to superfluids
systems is also discussed. In chapter 4, I consider a system with conserved
order parameter, which is proposed as a model of crystal surface relaxation. The
observed value for the growth of order in the system is in agreement with arecent
theoretical prediction. Multiscaling behavior for the scattering function is
investigated, with negative results. A comparison of the correlation functions
in the conserved and non-conserved case indicates that, while the conservation
constraint does not influence the structure of the vortex defects, it significantly
affects their dynamics. In chapter 5, I discuss a model of the superconducting
transition. A linear stability analysis of the normal-superconductor interface for
type I superconductors is presented. The presence of an instability analogous
to that responsible for dendritic patterns in solidification is pointed out. Numerical
simulations of the phase propagation in type I superconductors confirm
the indications of the linear stability analysis. A simple mean-field picture of
the transition kinetics of type II superconductors suggests the existence of two
dynamical regimes, characterized by a power-law and a logarithmic growth of
ordered (superconducting) domains in the. system. These two regimes can be
understood in terms of the spatial dependence of the vortex-string interaction.
Numerical simulations of type II superconductors in the spinodal regime bear
out this prediction, confirming that the quenched dynamics of this system is well
described by the effective interaction among the defects.Submitted by Carolyn Mead ([email protected]) on 2011-05-03T15:08:05Z
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Renormalization group methods for dynamics of spatially extended systems
No full textThe renormalization group (RG) provides a powerful tool and concept in the study of dynamics of spatially extended systems: different microscopic dynamics may be related, via coarse graining, to the same macroscopic dynamics. This thesis makes two contributions in this direction: first, we
obtain the correct description of the dynamics of a continuous system on a discrete lattice, and second, we study the macroscopic sharp interface limit of various phase field models of dendritic growth. We demonstrate the virtue of using coarse grained variables in simulations, as opposed
to uniformly sampling the underlying continuous configuration. By coarse graining a system repeatedly, the 'perfect linear' operator is obtained as the fixed point of a RG flow. The basin of attraction defines the dynamic universality class with respect to detailed microscopic interactions. In linear problems, the perfect linear operator gives the exact result of physical quantities down to the chosen lattice grid size, subject to the inherent numerical error of a simulation. It is therefore more efficient than the conventional discretization, which requires smaller grid size. The direct application of the perfect linear operator to nonlinear Model A dynamics also leads to improvement. A modified perfect operator that has a short range of interaction is empirically determined and gives results comparable to that from using the perfect linear operator. In the dendritic growth
problem, a variety of microscopic phase field models converge to the same macroscopic dynamics in the sharp interface limit. The boundary layer calculation, where the interface width is a small parameter, shows the importance of the phase field's exponential decaying behavior outside the
interface region. Subject to this constraint, there is a universality class of phase field models regarding the form of potential functions in the model. The potential functions are defined in terms of
a new set of functions, which ensures the existence of first order solutions and thus the universality of the model. The outer temperature field is shown to be discontinuous at the second order.Submitted by Elizabeth Kent ([email protected]) on 2012-05-21T18:07:37Z
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Previous issue date: 1999Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Elizabeth Kent ([email protected]) on 2012-05-21T18:07:37Z
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Dynamic Critical Phenomena of the Superconducting Transition
No full text87 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2002.One of the reasons why the Monte Carlo results do not accurately reproduce the dynamic exponent of the system is that a key assumption is found not to hold true. Equating Monte Carlo time to real time is believed to reproduce relaxational dynamics, but this is found not to be the case for the vortex loop model. A generalization of the standard Monte Carlo algorithm is proposed to deal with situations where the assumption breaks down, the study of which is ongoing.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD
Dynamic Critical Phenomena of the Superconducting Transition
No full textOne of the reasons why the Monte Carlo results do not accurately reproduce the dynamic exponent of the system is that a key assumption is found not to hold true. Equating Monte Carlo time to real time is believed to reproduce relaxational dynamics, but this is found not to be the case for the vortex loop model. A generalization of the standard Monte Carlo algorithm is proposed to deal with situations where the assumption breaks down, the study of which is ongoing.Made available in DSpace on 2015-09-25T20:02:42Z (GMT). No. of bitstreams: 2
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Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDsRestricted to the U of I community idenfinitely during batch ingest of legacy ETDsU of I Only87 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2002
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