Topics in the statistical mechanics of extended objects

Abstract

In this thesis, we study some aspects of the equilibrium and nonequilibrium statistical mechanics of physical systems with extended spatial degrees of freedom. The specific systems we model are: The Abrikosov fiux lattice in high-Tc superconductors, melts of living, semiflexible polymers and the sponge phases of bilayer membranes. The first three chapters of this thesis deal with some properties of the mixed pheise of high-Tc superconductors. In this phase, an externally applied magnetic field penetrates the superconductor in the form of flux lines. Interest in this phase hcis been stimulated by the discovery that flux lines in high-Tc superconductors undergo a phase transition (generally believed to be a transition between solid and liquid phases) as the externally applied fleld or the temperature is varied. In Chapter 1 of this thesis we present a density-functional analysis of this novel melting transition. Our theory is based on the mapping of the flux-line system in very anisotropic, layered superconductors to a system of planar vortices confined to move on superconducting layers and interacting via a three-dimensional pair potential. This system of vortices is treated as a classical liquid whose static correlations and freezing transition can be studied by generalizing methods of liquid-state and density-functional theories. An earlier attempt (by Sengupta et al.), at modelling the fiux-lattice-melting transition, calculated liquid-state correlations in the Hyper-Netted-Chain (HNC) approximation. In this Chapter we demonstrate the need for going beyond the HNC approximation to describe correlations in the flux liquid accurately. The results we present here are based on an improved liquid-state theory for the flux liquid, which extends the Rogers-Young approximation (proposed initially for three-dimensional liquids) to the two-dimensional, one-component plasma. The freezing curve we calculate improves substantially on the earlier results in terms of agreement with experiments. Calculations of field-field correlation functions, obtainable via neutron-scattering experiments, are also presented. Our theory predicts a weakly first-order, fiux-lattice melting transition, in agreement with recent experiments on fiuxn lattice melting in YBCO. In Chapter 2, we propose a theory for the equal-time, two-point correlation functions of a weakly disordered liquid, using a replica approach. The freezing transition of such a liquid is studied using a replica-symmetric density-functional theory. Our analysis represents, to the best of our knowledge, the first extension of liquid-state and densityfunctional methods to disordered liquids. We apply our formalism to the study of the mixed phase in the presence of weak, randomly positioned, point pinning sites. Our theory predicts a suppression of the first-order melting phase boundary in the B — T plane to lower temperatures (relative to the phase boundary in the pure system), as the field is increased. For small disorder strengths and at low fields, this suppression is very small. This result explains recent experiments which see the remnants of this first-order phase boundary in the pure system even in the presence of quenched disorder. We also present, for the first time, calculations of the off-diagonal (in replica space) correlation functions of a disordered liquid. In Chapter 3, we draw on results obtained in Chapters 1 and 2 to analyse the effects of thermal fiuctuations and quenched disorder on muon-spin-rotation spectra in the mixed phase of anisotropic superconductors. We describe time-averaged densities in the flux solid using density-functional theory. Our results account for over 50 % of the linewidth narrowing, over and above that predicted assuming an Abrikosov lattice at zero temperature, in sufficiently anisotropic, layered superconductors. Our results thus provide a partial resolution of the controversy siu-rounding the interpretation of muon-spin-rotation spectra in BSCCO. Given certain assumptions about time scales, we calculate the moments of the magnetic-field distribution function in the liquid phase of the flux system. We present an approximate analysis of the effects of short-range correlations in determining muon-spin-rotation spectra in the flux-liquid phase. We also present an analysis of muon-spin-rotation spectra in the weakly pinned flux liquid. In Chapter 4, we study a lattice model for the crystallization of living, semiflexible polymers via extensive Monte Carlo simulations and the analysis of limits in which it can be solved exactly. Living polymers have attracted considerable experimental and theoretical interest in recent years - in these systems polymer lengths are not fixed but can fluctuate, attaining an equilibrium distribution. Our model generalizes Flory’s lattice model for the crystallization of conventional polymers to include polydispersity and the possibility of polymerization. Our model predicts a continuous freezing transition to an orientationally ordered, crystalline state in two dimensions. (This transition is of 2-d Ising type on the square lattice.) We analyse polymer-length distributions and show that they can be an important diagnostic for the nature of the phase. Our analysis of limits in which this model can be solved exactly (the F-model limit and the free-fermion Umit in two dimensions) indicates that correlation lengths can become very leirge over much of the phase diagram due to the proximity to the power-law, high-temperature phase of the F-model. We believe this result accounts for the considerable discrepancy in results obtained in simulations on related lattice models for polymer crystallization in two dimensions. Our model exhibits a first-order transition in three dimensions, in qualitative agreement with the mean-field prediction of Flory. We show how various thermodynamic functions, correlation functions, and polymer-length distributions behave in the vicinity of this transition. In Chapter 5, we study glass formation in our three-dimensional model for living, semiflexible polymers following quenches to temperatures below the equilibrium crystallization temperature. Our model exhibits logarithmic relaxation out of quenched configurations, an apparently continuous glass-crystal transition as well as an exotic lamellar glass. We propose a novel Monte Carlo analog of a scanning calorimetry experiment and use it to study the thermal properties of the glasses we obtain. We can tune glassy behavior in om’ model by varying a single parameter, which we interpret as the analogue of a “frustration” parameter. Our study suggests that this system is an interesting model glass, on which theories of the glass transition based on the hypothesis of an underlying equilibrium phase transition can be tested. In Chapter 6, we propose and study a lattice model for random surfaces and use this model to study transitions between symmetric sponge, asymmetric sponge and spongewith- free-edges phases in bilayer membrane systems. This model is studied through extensive Monte Carlo simulations in the grand canonical ensemble. Our model is more general than the model proposed by Huse and Leibler for the same system and is more amenable to simulations. We demonstrate, through the first numerical study of correlation functions in such models, that, at least in the Huse-Leibler limit, there is no trace of the synmaetric sponge to sponge-with-free-edges transition in the correlation functions wliich are measured in experiments