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.     

Dynamic light scattering by optically anisotropic colloidal particles in polyacrylamide gels

The translational and rotational motions of optically anisotropic spherical particles embedded in cross-linked polyacrylamide gels is studied by dynamic light scattering. The particles are liquid crystal droplets solidified in the nematic phase. The amount of cross linkers is varied to cross the sol-gel transition where the system becomes nonergodic for both translational and rotational diffusion modes of the probes. The translational and rotational dynamic correlation functions are obtained by measuring the intensity correlation function between crossed polarizers in the parallel and perpendicular geometries. Data from nonergodic systems is analyzed using an extension, to include rotations, of the method of Pusey and van Megen [Physica A 157, 705 (1989)]. Both diffusion modes are observed to be arrested as the rigidity of the gel increases.     

Electronic structure of CeF from frozen-core four-component relativistic multiconfigurational quasidegenerate perturbation theory

We have investigated the ground state and the two lowest excited states of the CeF molecule using four-component relativistic multiconfigurational quasidegenerate perturbation theory calculations, assuming the reduced frozen-core approximation. The ground state is found to be (4f(1))(5d(1))(6s(1)), with Omega = 3.5, where Omega is the total electronic angular momentum around the molecular axis. The lowest excited state with Omega = 4.5 is calculated to be 0.104 eV above the ground state and corresponds to the state experimentally found at 0.087 eV. The second lowest excited state is experimentally found at 0.186 eV above the ground state, with Omega = 3.5 based on ligand field theory calculations. The corresponding state having Omega = 3.5 is calculated to be 0.314 eV above the ground state. Around this state, we also have the state with Omega = 4.5. The spectroscopic constants R(e), omega(e), and nu(1-0) calculated for the ground and first excited states are in almost perfect agreement with the experimental values. The characteristics of the CeF ground state are discussed, making comparison with the LaF(+) and LaF molecules. We denote the d- and f-like polarization functions as d(*) and f(*). The chemical bond of CeF is constructed via {Ce(3.6+)(5p(6)d(*0.3)f(*0.1))F(0.6-)(2p(5.6))}(3+) formation, which causes the three valence electrons to be localized at Ce(3.6+).     

Calculation of the fine structure and intensity of the singlet-triplet transitions in the imidogen radical

The singlet-triplet transition moments are calculated for the NH radical by multiconfiguration self-consistent field (MCSCF) method with a quadratic response (QR) technique. The band systems in the visible region (b(1)Sigma(+)-->X(3)Sigma(-) and a(1)Delta-->X(3)Sigma(-)) of the NH radical are analyzed in comparison with previous ab initio treatments and with the recent experimental data in attempt to solve some discrepancies. The b(1)Sigma(+)-->X(3)Sigma(Omega)(-) transition moments ratio for the two spin sublevels Omega = 1 and Omega=0 of the ground state is well reproduced and the radiative lifetime of the b(1)Sigma(+) state (tau(b)=58 ms) is obtained in a good agreement with the experimental value tau(b)=53((-13)(+17)) ms. The A(3)Pi<--a(1)Delta transition probability is calculated for the first time and found to be in an excellent agreement with the recent optical pumping measurements of the NH radical in a molecular beam, where population transfer from the metastable a(1)Delta state to the ground X(3)Sigma(-) state is achieved. For the a(1)Delta-->X(3)Sigma(-) transition some improvement is achieved in comparison with the previous ab initio results, but the calculated radiative lifetime (tau(a)=3.9 s) is still much lower than the recent measurement provides (tau(a)=12.5 s). The zero field splitting and spin-rotation coupling constants are calculated for the ground state by different methods and advantage of the density functional theory is stressed.     

Micellar spheres in a high frequency oscillatory field

The viscoelasticity of aqueous micellar solutions of two oxyethylene/oxybutylene block copolymers (E(92)B(18) and B(20)E(510)) has been investigated using a torsional resonator operated at 26 kHz. For both systems considered, values of the dynamic viscosity (eta'(infinity)) point to partial draining of the micellar corona induced by the high-frequency oscillatory field. At low effective volume fractions, values of the elastic modulus (G'(infinity)) indicate that the repulsive interactions between micelles can be modeled by a power law function u(r) proportional to 1/r(nu) with exponents close to 13 and 6 for copolymers E(92)B(18) and B(20)E(510) respectively. At a critical copolymer concentration (c*) plots of log(G'(infinity)) against log(c) deviate from the straight lines established at low concentrations, implying that the systems undergo ergodic/nonergodic transitions.     

Interdiffusion of solvent into glassy polymer films: a molecular dynamics study

Large scale molecular dynamics and grand canonical Monte Carlo simulation techniques are used to study the behavior of the interdiffusion of a solvent into an entangled polymer matrix as the state of the polymer changes from a melt to a glass. The weight gain by the polymer increases with time t as t(1/2) in agreement with Fickian diffusion for all cases studied, although the diffusivity is found to be strongly concentration dependent especially as one approaches the glass transition temperature of the polymer. The diffusivity as a function of solvent concentration determined using the one-dimensional Fick's model of the diffusion equation is compared to the diffusivity calculated using the Darken equation from simulations of equilibrated solvent-polymer solutions. The diffusivity calculated using these two different approaches are in good agreement. The behavior of the diffusivity strongly depends on the state of the polymer and is related to the shape of the solvent concentration profile.     

A study of pore blockage in silicalite zeolite using free energy perturbation calculations

Binary systems consisting of large coadsorbed molecules (n-hexane, cyclohexane, and benzene) with smaller penetrant molecules (methane) were simulated to investigate the mechanisms of pore blockage in the zeolite silicalite. Benzene and cyclohexane trap the methane molecules in the zeolite channels on the time scales of molecular dynamics simulations. Minimum energy paths for methane diffusion past the blocking molecules were determined, and free energy perturbation calculations were carried out along the paths to get the rate constants of methane hopping past coadsorbed benzene and cyclohexane molecules, which adsorb in the channel intersections. Three principal diffusion pathways were found in both the methane/benzene and methane/cyclohexane systems. Minima which were connected by low-energy pathways were grouped together into macrostates. Using the calculated hopping rates between macrostates, kinetic Monte Carlo was then used to obtain the diffusivity of methane with a coadsorbate benzene loading such that all channel intersections are filled by benzene - conditions where molecular dynamics simulations fail. Passage of methane across cyclohexane molecules involved pushing the cyclohexane molecules into the channels from their preferred channel intersection positions.     

Hybrid Monte Carlo: An efficient algorithm for condensed matter simulation

A detailed comparison has been made of the performance of molecular dynamics and hybrid Monte Carlo simulation algorithms for calculating thermodynamic properties of 2D Lennard-Jonesium. The hybrid Monte Carlo simulation required an order of magnitude fewer steps than the molecular dynamics simulation to calculate reproducible values of the specific heat. The ergodicity of the two algorithms was compared via the use of intermediate scattering functions. For classical systems the intermediate scattering functions should be real; however, a simple analysis demonstrates that this function will have a significant imaginary component when ergodicity breaks down. For q vectors near the zone boundary, the scattering functions are real for both algorithms. However, for q vectors near the zone center (i.e., harmonic, weakly coupled modes), the scattering function calculated via molecular dynamics had a significantly larger imaginary component than that calculated using hybrid Monte Carlo. Therefore, the hybrid Monte Carlo algorithm is more ergodic and samples phase space more efficiently than molecular dynamics for simulations of 2D Lennard-Jonesium. © 1994 by John Wiley & Sons, Inc.     

Photon counting statistics for blinking CdSe-ZnS quantum dots: a Lévy walk process

We analyze photon statistics of blinking CdSe-ZnS nanocrystals interacting with a continuous wave laser field, showing that the process is described by a ballistic Lévy walk. In particular, we show that Mandel's Q parameter, describing the fluctuations of the photon counts, is increasing with time even in the limit of long time. This behavior is in agreement with the theory of Silbey and co-workers (Jung et al. Chem. Phys. 2002, 284, 181), and in contrast to all existing examples where Q approaches a constant, independent of time in the long time limit. We then analyze the distribution of the time averaged intensities, showing that they exhibit a nonergodic behavior, namely, the time averages remain random even in the limit of a long measurement time. In particular, the distribution of occupation times in the on-state compares favorably to a theory of weak ergodicity breaking of blinking nanocrystals. We show how our data analysis yields information on the amplitudes of power-law decaying on and off time distributions, information not available using standard data analysis of on and off time histograms. Photon statistics reveals fluctuations in the intensity of the bright state indicating that it is composed of several states. Photon statistics exhibits a Lévy walk behavior also when an ensemble of 100 dots is investigated, indicating that the strange kinetics can be observed already at the level of small ensembles.     

Some properties of aqueous xanthan gels formed under the action of aluminium(III) ions

Aqueous xanthan solutions form gels when aluminium salts are added and the solutions are heated above 45 °C. The gelation process was followed by dynamic light scattering. Characterization was based on the heterodyne and nonergodic approaches. Both techniques gave the same fast relaxation times, but for the slow motion much larger values were found in the heterodyne than in the nonergodic approach. The heterodyne fraction 1-X was found to correlate closely with the plateau height of the time correlation function (TCF) at large delay times in the nonergodic experiments. Three methods of gel point determination are demonstrated: (i) onset of heterodyne/nonergodic behavior, (ii) observation of a sharp maximum for the fast relaxation time at the gel point, (iii) observation of power-law behavior of the TCF. The statistics of nonergodic fluctuations were examined and evaluated. The potential of this procedure for detailed structure evaluation of inhomogeneities in the gel is emphasized.     

Statistical mechanics of time independent nondissipative nonequilibrium states

We examine the question of whether the formal expressions of equilibrium statistical mechanics can be applied to time independent nondissipative systems that are not in true thermodynamic equilibrium and are nonergodic. By assuming that the phase space may be divided into time independent, locally ergodic domains, we argue that within such domains the relative probabilities of microstates are given by the standard Boltzmann weights. In contrast to previous energy landscape treatments that have been developed specifically for the glass transition, we do not impose an a priori knowledge of the interdomain population distribution. Assuming that these domains are robust with respect to small changes in thermodynamic state variables we derive a variety of fluctuation formulas for these systems. We verify our theoretical results using molecular dynamics simulations on a model glass forming system. Nonequilibrium transient fluctuation relations are derived for the fluctuations resulting from a sudden finite change to the system's temperature or pressure and these are shown to be consistent with the simulation results. The necessary and sufficient conditions for these relations to be valid are that the domains are internally populated by Boltzmann statistics and that the domains are robust. The transient fluctuation relations thus provide an independent quantitative justification for the assumptions used in our statistical mechanical treatment of these systems.     

Electroosmotic flow in microchannels with arbitrary geometry and arbitrary distribution of wall charge

General solutions are developed for direct current (DC) and alternating current (AC) electroosmotic flows in microfluidic channels with arbitrary cross-sectional geometry and arbitrary distribution of wall charge (zeta potential). The applied AC electric field can also be of arbitrary waveform. By proposing a nondimensional time scale varpi defined as the ratio of the diffusion time of momentum across the electric double-layer thickness to the period of the applied electric field, we demonstrate analytically that the Helmholtz-Smoluchowski electroosmotic velocity is an appropriate slip condition for AC electroosmotic flows in typical microfluidic applications. With this slip condition approach, electroosmotic flows in rectangular and asymmetric trapezoidal microchannels with nonuniform wall charge, as examples, are investigated. The unknown constants in the proposed general solutions are numerically determined with a least-squares method through matching the boundary conditions. We find that the wall charge affects significantly the electroosmotic flow while the channel geometry does not. Moreover, the flow feature is characterized by another nondimensional time scale Omega defined as the ratio of the diffusion time of momentum across the channel hydraulic radius to the period of the applied electric field. The onset of phase shift between AC electroosmotic velocity and applied electric field is also examined analytically.     

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.     

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.     

Dynamic light scattering by permanent gels: Partial heterodyne and nonergodic medium methods for data evaluation

Four relationships between the intensity and field correlation functions have been used in the analysis of the measured intensity correlation function in dynamic light scattering for four different polymer gel samples. The evaluation of diffusion coefficients was performed using inverse Laplace transformation (ILT) and the cumulante method, respectively. The diffusion coefficients derived from the nonergodic medium method are almost the same as those from the partial heterodyne method when ILT is used. ILT can give similar results by all four procedures even when large baselines for the nonergodic medium method exist. The cumulants method leads to incorrect diffusion coefficients from the correlation functions derived using the nonergodic medium method.     

Quasi-equilibria in reduced Liouville spaces

The quasi-equilibrium behaviour of isolated nuclear spin systems in full and reduced Liouville spaces is discussed. We focus in particular on the reduced Liouville spaces used in the low-order correlations in Liouville space (LCL) simulation method, a restricted-spin-space approach to efficiently modelling the dynamics of large networks of strongly coupled spins. General numerical methods for the calculation of quasi-equilibrium expectation values of observables in Liouville space are presented. In particular, we treat the cases of a time-independent Hamiltonian, a time-periodic Hamiltonian (with and without stroboscopic sampling) and powder averaging. These quasi-equilibrium calculation methods are applied to the example case of spin diffusion in solid-state nuclear magnetic resonance. We show that there are marked differences between the quasi-equilibrium behaviour of spin systems in the full and reduced spaces. These differences are particularly interesting in the time-periodic-Hamiltonian case, where simulations carried out in the reduced space demonstrate ergodic behaviour even for small spins systems (as few as five homonuclei). The implications of this ergodic property on the success of the LCL method in modelling the dynamics of spin diffusion in magic-angle spinning experiments of powders is discussed.     

Effects of translational and rotational diffusion on the association in kinetic model of nucleation

Kinetics of an association and dissociation of single elements with the effects of translational and rotational diffusion and angular limitations is discussed. Separated clusters embedded in a solution of orientable single elements are considered.Steady-state positional and angular distribution of single elements is calculated from the equation of translational-rotational diffusion and the boundary conditions proposed for orientation-limited association. Although spherical orientable elements are assumed, the model can be used for non-spherical particles with aspect ratios close to unity.Diffusion-limited rate constants of association and dissociation are proposed which depend on translational and rotational diffusion constants of single elements, the tolerance angle of the association, and the cluster size.Effective concentration of single elements and effective rate constants are expressed by the equilibrium and diffusion-limited rate constants. Effects of finite diffusion rates and finite tolerance angle are discussed.The equations of the kinetic model of nucleation are modified due to the diffusion-limited rate of the association.     

Molecular dynamics investigation of oxygen vacancy diffusion in rutile

Oxygen vacancy diffusion in rutile was studied by Born-Oppenheimer molecular dynamics techniques in the framework of the semiempirical molecular orbital method MSINDO. Migration of an oxygen vacancy from the rutile (110) surface towards the bulk was simulated. The metadynamics technique was employed to accelerate the diffusion processes. In this way, transition state structures and activation energies for the diffusion processes were obtained. Rate constants and the time scale of diffusion processes were estimated for different temperatures using the calculated activation energy. It was found that the vacancies in the bulk are less stable than on the surface. The feasibility of oxygen vacancy diffusion under experimental conditions is discussed.     

Dynamical arrest of electron transfer in liquid crystalline solvents

We argue that electron transfer reactions in slowly relaxing solvents proceed in the nonergodic regime, making the reaction activation barrier strongly dependent on the solvent dynamics. For typical dielectric relaxation times of polar nematics, electron transfer reactions in the subnanosecond time scale fall into nonergodic regime in which nuclear solvation energies entering the activation barrier are significantly lower than their thermodynamic values. The transition from isotropic to nematic phase results in weak discontinuities of the solvation energies at the transition point and the appearance of solvation anisotropy weakening with increasing solute size. The theory is applied to analyze experimental kinetic data for the electron transfer kinetics in the isotropic phase of 5CB liquid crystalline solvent. We predict that the energy gap law of electron transfer reactions in slowly relaxing solvents is characterized by regions of fast change of the rate at points where the reaction switches between the ergodic and nonergodic regimes. The dependence of the rate on the donor-acceptor separation may also be affected in a way of producing low values for the exponential falloff parameter.