A statistical-mechanical theory of self-diffusion in glass-forming liquids

A statistical-mechanical theory of self-diffusion in glass-forming liquids is presented. A non-Markov linear Langevin equation is derived from a Newton equation by employing the Tokuyama-Mori projection operator method. The memory function is explicitly written in terms of the force-force correlation functions. The equations for the mean-square displacement, the mean-fourth displacement, and the non-Gaussian parameter are then formally derived. The present theory is applied to the glass transitions in the glass-forming liquids to discuss the crossover phenomena in the dynamics of a single particle from a short-time ballistic motion to a long-time self-diffusion process via a β (caging) stage. The effects of the renormalized friction coefficient on self-diffusion are thus explored with the aid of analyses of the simulation results by the mean-field theory proposed recently by the present author. It is thus shown that the relaxation time of the renormalized memory function is given by the β-relaxation time. It is also shown that for times longer than the β-relaxation time the dynamics of a single particle is identical to that discussed in the suspensions.     

A statistical-mechanical theory of slow dynamics near the glass transition

A statistical-mechanical theory of slow dynamics near the glass transition in two kinds of glass-forming systems, (M) molecular systems and (S) suspensions of colloids, is presented from a unified point of view based on the Tokuyama-Mori projection operator method. The exact diffusion equations for the coherent- and the incoherent-intermediate scattering functions are first derived, whose memory functions are convolutionless in time and contain the correlation effects due to the hydrodynamic interactions in (S). The analytic expressions of the memory functions are then calculated within the mode-coupling theory (MCT) approximation and are shown to coincide with the conventional ones obtained by MCT. Alternative mode-coupling equations are thus obtained in (M) and (S) separately. Self-diffusion is also discussed. Alternative equations for the mean-square displacement and the non-Gaussian parameter are also derived within MCT approximation. All results in both the systems are compared with those obtained by MCT.     

Mapping from a fragile glass-forming system to a simpler one near their glass transitions

Extensive molecular-dynamics simulations on two different glass-forming systems are performed to test the prediction proposed recently by the mean-field theory (MFT) of glass transition that a fragile system can be mimicked by a simpler one near their glass transitions. One is a simulation for a hard-sphere fluid with size polydispersity where the volume fraction φ is a control parameter and the other is for a Lennard-Jones binary mixture where the inverse temperature 1/T is a control parameter. Then, their mean-square displacements are fully analyzed by MFT. Thus, we show that both results are collapsed on a master curve given by MFT when , where is a long-time self-diffusion coefficient. We also investigate the non-singular behavior of consistently from a unified standpoint based on MFT. Thus, we show that macroscopic physical quantities in a fragile system can be transformed into those in a simpler one through a universal parameter .     

Theoretical approaches to the glass transition in simple liquids

Theoretical approaches to the development of an understanding of the behaviour of simple supercooled liquids near the structural glass transition are reviewed and our work on this problem, based on the density functional theory of freezing and replicated liquid state theory, are summarized in this context. A few directions for further work on this problem are suggested.     

A mode-mode coupling theory of chemical reaction in a dense fluid

A study is made of the coupling between chemical reaction and diffusion in a dense fluid. Our analysis utilizes the projection operator formalism and a generalized Langevin equation that is based on irreversible, phenomenological equations of motion instead of conventional Hamiltonian mechanics. It also is shown that this same non-Hamiltonian theory provides a simple way of deriving Kawasaki's mode-mode coupling theory of diffusion.This research was supported by a grant from the National Science Foundation.     

A DFT‐based exploration augmented by X‐ray and NMR of the stereoselectivity in the 1,3‐dipolar cycloaddition of 1‐pyrroline‐1‐oxide to methyl cinnamate and benzylidene acetophenone

A B3LYP/6–31G* study was carried out for the reactions of 1‐pyrroline‐1‐oxide (N1) with methyl cinnamate (E1) and benzylidene acetophenone (E2) for getting a quantitative rationalization of the experimental findings. The product ratios were determined by NMR studies of the crude reaction mixtures. The conformation and stereochemistry of the isolated cycloadducts were finally confirmed by 2D NMR and X‐ray diffraction. The endo/exo‐selectivities were predicted through the computation of activation parameters on the basis of assumed concerted mechanism. The regioselectivity and reactivity were amply predicted by local and global electrophilicity indices and were found to be in good agreement with the experimental findings which were supportive of polar character and of the direction of charge transfer (CT) accompanying the cycloaddition. It was found that the cycloaddition involving methyl cinnamate was endo‐selective, while that with benzylidene acetophenone produced the exo‐isomer as the major adduct. Copyright © 2010 John Wiley & Sons, Ltd.     

Analytical approach to quantum phase transitions of ultracold Bose gases in bipartite optical lattices using the generalized Green’s function method

In order to investigate the quantum phase transitions and the time-of-flight absorption pictures analytically in a systematic way for ultracold Bose gases in bipartite optical lattices, we present a generalized Green’s function method. Utilizing this method, we study the quantum phase transitions of ultracold Bose gases in two types of bipartite optical lattices, i.e., a hexagonal lattice with normal Bose–Hubbard interaction and a d-dimensional hypercubic optical lattice with extended Bose–Hubbard interaction. Furthermore, the time-of-flight absorption pictures of ultracold Bose gases in these two types of lattices are also calculated analytically. In hexagonal lattice, the time-of-flight interference patterns of ultracold Bose gases obtained by our analytical method are in good qualitative agreement with the experimental results of Soltan-Panahi, et al. [Nat. Phys. 7, 434 (2011)]. In square optical lattice, the emergence of peaks at \(\left( { \pm \frac{\pi }{a}, \pm \frac{\pi }{a}} \right)\) in the time-of-flight absorption pictures, which is believed to be a sort of evidence of the existence of a supersolid phase, is clearly seen when the system enters the compressible phase from charge-density-wave phase.     

A low-temperature tilter system and its application to the measurement of the anisotropy of optical rotation in K2Zncl4 in the vicinity of the phase transition at 145 k

The ferroelectric/ferroelastic phase transition of K₂ZnCl₄ at 145 K has been investigated employing the tilter method (Kaminsky and Glazer (1996) Ferroelectrics 183, 133) which was adapted to low-temperature experiments. We were able to observe the anisotropy of optical rotation in K₂ZnCl₄ which shows a distinct discontinuity at the transition temperature. A precursor-optical rotation in the ferroelectric phase is discussed in connection with a cluster ordering scheme suggested in an earlier X-ray study. The rotation of the optical indicatrix which accompanies the transition from the orthorhombic to the monoclinic system shows a Landau-type temperature dependence. Model calculations based on the dipole-dipole interaction between the atoms of the structures at different temperatures are in qualitative agreement with the experimental results.