Heat capacity, enthalpy fluctuations, and configurational entropy in broken ergodic systems.

A common assumption in the glass science community is that the entropy of a glass can be calculated by integration of measured heat capacity curves through the glass transition. Such integration assumes that glass is an equilibrium material and that the glass transition is a reversible process. However, as a nonequilibrium and nonergodic material, the equations from equilibrium thermodynamics are not directly applicable to the glassy state. Here we investigate the connection between heat capacity and configurational entropy in broken ergodic systems such as glass. We show that it is not possible, in general, to calculate the entropy of a glass from heat capacity curves alone, since additional information must be known related to the details of microscopic fluctuations. Our analysis demonstrates that a time-average formalism is essential to account correctly for the experimentally observed dependence of thermodynamic properties on observation time, e.g., in specific heat spectroscopy. This result serves as experimental and theoretical proof for the nonexistence of residual glass entropy at absolute zero temperature. Example measurements are shown for Corning code 7059 glass.

Composition dependence of glass transition temperature and fragility. II. A topological model of alkali borate liquids.

Glass transition temperature and fragility are two important properties derived from the temperature dependence of the shear viscosity of glass-forming melts. While direct calculation of these properties from atomistic simulations is currently infeasible, we have developed a new topological modeling approach that enables accurate prediction of the scaling of both glass transition temperature and fragility with composition. A key feature of our approach is the incorporation of temperature-dependent constraints that become rigid as a liquid is cooled. Using this approach, we derive analytical expressions for the composition (x) dependence of glass transition temperature, T(g)(x), and fragility, m(x), in binary alkali borate systems. Results for sodium borate and lithium borate systems are in agreement with published values of T(g)(x) and m(x). Our modeling approach reveals a natural explanation for the presence of the constant T(g) regime observed in alkali borate systems.

Impact of fragility on enthalpy relaxation in glass.

The macroscopic properties of a glass are continually relaxing toward their equilibrium supercooled liquid values. Experimentally, the shape of the relaxation function in a glass is known to depend on the fragility of the supercooled liquid. In this paper, we investigate the impact of fragility on the relaxation behavior of glasses in the enthalpy landscape framework. We show that the fragility of a supercooled liquid is a direct result of the interplay of enthalpic and entropic effects in the enthalpy landscape. Through proper adjustment of the transition barriers in an enthalpy landscape, the fragility of a system can be adjusted while maintaining the same glass transition temperature. By modeling a set of systems with identical glass transition temperatures but varying values of fragility, we show that supercooled liquid fragility has a significant impact on the enthalpy relaxation behavior of a glass. In particular, the magnitude of enthalpy relaxation decreases dramatically with increasing fragility. Finally, we discuss how in the limit of infinite fragility the glass transition becomes an ideal second-order phase transition where no relaxation is possible in the glassy state.

Metabasin approach for computing the master equation dynamics of systems with broken ergodicity.

We propose a technique for computing the master equation dynamics of systems with broken ergodicity. The technique involves a partitioning of the system into components, or metabasins, where the relaxation times within a metabasin are short compared to an observation time scale. In this manner, equilibrium statistical mechanics is assumed within each metabasin, and the intermetabasin dynamics are computed using a reduced set of master equations. The number of metabasins depends upon both the temperature of the system and its derivative with respect to time. With this technique, the integration time step of the master equations is governed by the observation time scale rather than the fastest transition time between basins. We illustrate the technique using a simple model landscape with seven basins and show validation against direct Euler integration. Finally, we demonstrate the use of the technique for a realistic glass-forming system (viz., selenium) where direct Euler integration is not computationally feasible.

Continuously broken ergodicity.

A system that is initially ergodic can become nonergodic, i.e., display "broken ergodicity," if the relaxation time scale of the system becomes longer than the observation time over which properties are measured. The phenomenon of broken ergodicity is of vital importance to the study of many condensed matter systems. While previous modeling efforts have focused on systems with a sudden, discontinuous loss of ergodicity, they cannot be applied to study a gradual transition between ergodic and nonergodic behavior. This transition range, where the observation time scale is comparable to that of the structural relaxation process, is especially pertinent for the study of glass transition range behavior, as ergodicity breaking is an inherently continuous process for normal laboratory glass formation. In this paper, we present a general statistical mechanical framework for modeling systems with continuously broken ergodicity. Our approach enables the direct computation of entropy loss upon ergodicity breaking, accounting for actual transition rates between microstates and observation over a specified time interval. In contrast to previous modeling efforts for discontinuously broken ergodicity, we make no assumptions about phase space partitioning or confinement. We present a hierarchical master equation technique for implementing our approach and apply it to two simple one-dimensional landscapes. Finally, we demonstrate the compliance of our approach with the second and third laws of thermodynamics.

Monte Carlo method for computing density of states and quench probability of potential energy and enthalpy landscapes.

The thermodynamics and kinetics of a many-body system can be described in terms of a potential energy landscape in multidimensional configuration space. The partition function of such a landscape can be written in terms of a density of states, which can be computed using a variety of Monte Carlo techniques. In this paper, a new self-consistent Monte Carlo method for computing density of states is described that uses importance sampling and a multiplicative update factor to achieve rapid convergence. The technique is then applied to compute the equilibrium quench probability of the various inherent structures (minima) in the landscape. The quench probability depends on both the potential energy of the inherent structure and the volume of its corresponding basin in configuration space. Finally, the methodology is extended to the isothermal-isobaric ensemble in order to compute inherent structure quench probabilities in an enthalpy landscape.

Split-step eigenvector-following technique for exploring enthalpy landscapes at absolute zero.

The mapping of enthalpy landscapes is complicated by the coupling of particle position and volume coordinates. To address this issue, we have developed a new split-step eigenvector-following technique for locating minima and transition points in an enthalpy landscape at absolute zero. Each iteration is split into two steps in order to independently vary system volume and relative atomic coordinates. A separate Lagrange multiplier is used for each eigendirection in order to provide maximum flexibility in determining step sizes. This technique will be useful for mapping the enthalpy landscapes of bulk systems such as supercooled liquids and glasses.

A simplified eigenvector-following technique for locating transition points in an energy landscape.

We derive an eigenvector-following technique for locating transition points in an N-dimensional energy landscape. A separate Lagrange multiplier is used for each eigendirection to provide maximum flexibility in determining step sizes. In contrast to previous techniques based on a similar approach, we provide a simple algorithm for choosing specific values of these Lagrange multipliers. We demonstrate the robustness of the algorithm using two-dimensional Cerjan-Miller and Adams landscapes. The technique has also been applied to the S(12) molecular cluster.