Dynamics of Quantum Supercooled Liquids: A Mode Coupling Approach

Abstract

In this thesis, I use the quantum mode coupling theory (QMCT) formulation to study the dynamics in a supercooled liquid to understand how quantum fluctuations affect the liquid-glass transition. I calculate the Kubo-transformed density correlation function in order to avoid the difficulties associated with the direct calculation of the quantum correlation functions. The Kubo and the quantum correlation functions are connected through a mapping of quantum particles into classical ring-polymers. The radius of gyration of the polymer is directly related to the uncertainty in the position of the quantum particle. The position uncertainty increases with the quantumness of the system, which is quantified in terms of the thermal de-Broglie wavelength. I propose a perturbative method to simplify the form of the self-consistent equations of QMCT, which makes it feasible to study the relaxation dynamics directly in the time-domain, significantly reducing the computational cost. Implementing the perturbative calculation in the hard-sphere supercooled liquid, I find that moderate quantum fluctuations can cause enhanced caging, leading to liquid-glass transition at densities smaller than the classical transition density. The relaxation time associated with the density fluctuations shows power-law divergence with increasing density, similar to the classical HS system. However, the power-law exponents show a linear rise with increasing quantum fluctuations, which suggests a dynamic nature of the quantum effect. Further, at a fixed density, far from transition point, relaxation time shows an exponential (Vogel-Fulcher-Tamman) increase with the quantum fluctuation, which crosses over to a power-law like divergence as the transition point is approached. Extending the study to higher quantum regime, I observe that an enhanced tunneling effect leads to a re-entrant transition from glass to liquid phase. The intervening glass phase allows us to divide the liquid phase into two: low and xii high quantum liquids. The relaxation time in the high quantum regime shows a power-law like divergence as the liquid-glass transition point is approached. The study shows faster relaxation in the higher quantum regime due to enhanced tunneling. I further analyze the frequency-dependent specific heat in the supercooled quantum liquid. Liquid-glass transition is generally thought of as a second order phase transition; thus specific heat measurement is one of the important tools to detect it. I use QMCT and Zwanzig’s formalism to express specific heat in terms of the longitudinal viscosity of the liquid. I find a substantial variation in frequency-dependence of the specific heat as the quantumness of the liquid is changed, and this variation becomes more significant as the density of the system is increased. Near the glass transition point, slower dynamical modes contribute to the specific heat in quantum liquids as compared to the classical liquids. Another fundamental observable to analyze relaxation processes in liquids is the tagged-particle dynamics. The tagged-particle density correlation acts as the generating function of the moments of tagged-particle displacement. I derive a coupled set of equations for the second and the fourth moments (Kubotransformed) of tagged-particle displacement using QMCT. The most interesting results for these moments are obtained in the short times which are related to the uncertainty due to quantum fluctuation. The non-zero values of the moments at zero-time due to quantum uncertainty stands out from the classical case. The non-Gaussian nature of the particle distribution function at short (ballistic) times leading to strong dynamical heterogeneity in the tagged-particle motion further reflects the enhanced quantum effect. I derive an analytic expression for diffusion coefficient which shows non-monotonous behavior with increasing quantumness and qualitatively reflects the re-entrant diffusive behavior observed in Lennard-Jones simulations.