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|Title:||The role of topological defects and textures in the kinetics of phase ordering|
|Doctoral Committee Chair(s):||Goldbart, Paul M.; Oono, Yoshitsugu|
|Department / Program:||Physics|
|Degree Granting Institution:||University of Illinois at Urbana-Champaign|
Physics, Condensed Matter
|Abstract:||In this thesis, I present the results of a theoretical investigation of ordering processes induced by symmetry-breaking quenches in two physical systems. Both systems investigated possess a rich homotopy structure of the order-parameter space, which results in numerous topologically stable objects being generated during the quench, and influencing the properties of the system during the subsequent approach to equilibrium. The results reported are mostly computational in nature. The two systems investigated are (i) nematic liquid crystals, which support topologically stable abelian (in the uniaxial nematic case) and non-abelian (in the biaxial nematic case) singular defects, and (ii) the O(3)-symmetric vector (i.e., Heisenberg-type) system in 2 spatial dimensions, which supports topologically stable, but non-singular objects--topological textures.
In the case of nematic systems, the numerical investigation concentrates on the phase-ordering process and point defect dynamics following a quench into both the uniaxial and biaxial nematic phases of a quasi-2-dimensional liquid crystalline system. By comparing the growth laws for the characteristic length scales extracted from the order-parameter correlations and from the total number of topological defects in the system, it is determined that weak violations of dynamical scaling occur in the system, even at the latest times studied. The observed scaling violations are attributed to the presence of a logarithmic correction to the asymptotic power-law growth of the average inter-defect separation. Following the quench to the biaxial nematic phase, there are four topologically distinct defect species present in the system, the populations of which are studied in detail.
In addition to the computational investigation of the phase ordering process in 2-dimensional nematic systems, analytical derivations of the singular (power-law) short-distance behavior of the contribution to the structure factor (i.e., the light scattering intensity) for all types of topologically stable defects encountered in 2- and 3-dimensional uniaxial and biaxial nematics are presented.
The second system studied--the Heisenberg-type model in 2 spatial dimensions--is first implemented numerically as the discretized O(3) nonlinear $\sigma$-model with the standard form of free energy and with purely dissipative dynamics. Two distinct mechanisms for the decay of the order-parameter variations--single texture unwinding, and topological charge annihilation--are identified and characterized in this system. By examining the correlations in the order parameter and in the topological charge density, it is shown that dynamical scaling is strongly violated during the phase-ordering process, and multiple characteristic length-scales growing as distinct power-laws in time are identified.
In order to study in detail the origins of the observed multi-scaling behavior, the phase-ordering process is then studied within a modified O(3) nonlinear $\sigma$-model with an additional free energy term (analogous to the so-called Skyrme term, familiar in high-energy physics) that stabilizes the textures against shrinking and unwinding. It is found that this modification influences the multi-scaling properties of the system in a dramatic way, and that with single-texture unwindings suppressed, the form of the spectrum of exponents characterizing the decay of the moments of the topological charge density distribution can be predicted successfully by a simple two-length-scale argument.
|Rights Information:||Copyright 1996 Zapotocky, Martin|
|Date Available in IDEALS:||2011-05-07|
|Identifier in Online Catalog:||AAI9712498|
This item appears in the following Collection(s)
Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois