Philipp Maass

Fundamental questions relating to ion conduction in disordered solids

A number of basic scientific questions relating to ion conduction in homogeneously disordered solids are discussed. The questions deal with how to define the mobile ion density, what can be learnt from electrode effects, what the ion transport mechanism is, the role of dimensionality and what the origins of the mixed-alkali effect, the time-temperature superposition, and the nearly constant loss are. Answers are suggested to some of these questions, but the main purpose of the paper is to draw attention to the fact that this field of research still presents several fundamental challenges.

The dynamic structure model for ion transport in glasses

Abstract A model is developed for ion transport in glass which involves the creation of fluctuating pathways within a dynamically determined structure. Key features include a site memory effect which introduces vacancies appropriate to each kind of mobile ion, and a mismatch energy which emerges whenever an ion attempts to enter a different kind of site. The exploration of this model by numerical methods leads (i) to a power law relationship between ionic conductivity and cation content (now confirmed in the literature) and (ii) to the elucidation of many facets of the mixed alkali effect. It is suggested that this ‘dynamic structure’ model could form the basis for a comprehensive theory of vitreous electrolytes.

Anomalous fluctuations in the dynamics of complex systems: from DNA and physiology to econophysics

We discuss examples of complex systems composed of many interacting subsystems. We focus on those systems displaying nontrivial long-range correlations. These include the one-dimensional sequence of base pairs in DNA, the sequence of flight times of the large seabird Wandering Albatross, and the annual fluctuations in the growth rate of business firms. We review formal analogies in the models that describe the observed long-range correlations, and conclude by discussing the possibility that behavior of large numbers of humans (as measured, e.g., by economic indices) might conform to analogs of the scaling laws that have proved useful in describing systems composed of large numbers of inanimate objects.

Non-Debye relaxations in disordered ionic solids

Abstract Motions of charged defects in ionic solids, including glassy ionic conductors, defective crystals and composite materials, imply slow relaxation processes, which are observable within a wide range of timescales larger than microscopic (vibrational) times. These processes manifest themselves in numerous dynamical probes, like ac-conductivity, nuclear spin-relaxation, quasi-elastic neutron scattering and mechanical relaxation. The present theoretical understanding of the corresponding response functions is reviewed. Stochastic models based on ion hopping are the most natural approach for systems with structural disorder on microscopic length scales, but more coarse-grained, phenomenological schemes are addressed as well. Macroscopically inhomogeneous systems and interfacial problems are modeled by random impedance networks. Generally, non-exponential relaxation gets enhanced when Coulomb interactions between ions are taken into account. This is demonstrated by large-scale Monte Carlo simulations of disordered lattice gases for ion diffusion and is supported further by new results on random dipolar systems in the context of the “nearly constant dielectric loss response”.

SCALING BEHAVIOR IN ECONOMICS : THE PROBLEM OF QUANTIFYING COMPANY GROWTH

Inspired by work of both Widom and Mandelbrot, we analyze the Computstat database comprising all publicly traded United States manufacturing companies in the years 1974–1993. We find that the distribution of company size remains stable for the 20 years we study, i.e., the mean value and standard deviation remain approximately constant. We study the distribution of sizes of the “new” companies in each year and find it to be well approximated by a log- normal. We find (i) the distribution of the logarithm of the growth rates, for a fixed growth period of T years, and for companies with approximately the same size S displays an exponential “tent-shaped” form rather than the bell-shaped Gaussian, one would expect for a log-normal distribution, and (ii) the fluctuations in the growth rates — measured by the width of this distribution σT — decrease with company size and increase with time T. We find that for annual growth rates (T = 1), σT ∼ S−β, and that the exponent β takes the same value, within the error bars, for several measures of the size of a company. In particular, we obtain β = 0.20 ± 0.03 for sales, β = 0.18 ± 0.03 for number of employees, β = 0.18±0.03 for assets, β = 0.18 ± 0.03 for cost of goods sold, and β = 0.20 ± 0.03 for propert, plant, and equipment. We propose models that may lead to some insight into these phenomena. First, we study a model in which the growth rate of a company is affected by a tendency to retain an “optimal” size. That model leads to an exponential distribution of the logarithm of growth rate in agreement with the empirical results. Then, we study a hierarchical tree-like model of a company that enables us to relate β to parameters of a company structure. We find that β = −1n Π/1nz, where z defines the mean branching ratio of the hierarchical tree and Π is the probability that the lower levels follow the policy of higher levels in the hierarchy. We also study the output distribution of growth rates of this hierarchical model. We find that the distribution is consistent with the exponential form found empirically. We also discuss the time dependence of the shape of the distribution of the growth rates.

Towards a theory for the mixed alkali effect in glasses

A general theoretical approach for the mixed alkali effect in glasses is discussed based on the idea that the covalent host network creates different structural energy landscapes for different types of mobile ions. By deriving a simplified model from this approach, it is shown how the experimentally observed changes in ion mobilities both upon mixing of two types of ions and upon changing the total concentration of mobile ions can be understood. Computer simulations of the model suggest that Coulomb forces have to be taken into account for explaining the differing behavior of activation energies in single modified and mixed ion glasses. It is further shown how the mixed alkali internal friction peak results from diffusional exchange processes of unlike ions in the glassy network.

Colloquium: Cluster growth on surfaces : Densities, size distributions, and morphologies

Understanding and control of cluster and thin film growth on solid surfaces is a subject of intensive research to develop nanomaterials with new physical properties. In this Colloquium we review basic theoretical concepts to describe submonolayer growth kinetics under non-equilibrium conditions. It is shown how these concepts can be extended and further developed to treat self-organized cluster formation in material systems of current interest, such as nanoalloys and molecular clusters in organic thin film growth. The presentation is focused on ideal flat surfaces to limit the scope and to discuss key ideas in a transparent way. Open experimental and theoretical challenges are pointed out.

Phase separation in confined geometries: Solving the Cahn-Hilliard equation with generic boundary conditions

Abstract We apply implicit numerical methods to solve the Cahn–Hilliard equation for confined systems. Generic boundary conditions for hard walls are considered, as they are derived from physical principles. Based on a detailed stability analysis an automatic time step control could be implemented, which makes it possible to explore the demixing kinetics of two thermodynamically stable phases over many orders in time with good space resolution. The power of the method is demonstrated by investigating spinodal decomposition in two-dimensional systems. At early times of the decomposition process the numerical results are in excellent agreement with analytical predictions based on the linearized equations. Due to the efficiency of the variable time step procedure it is possible to monitor the process until a stable equilibrium is reached.

Description of far-from-equilibrium processes by mean-field lattice gas models

Mean-field kinetic equations are a valuable tool to study the atomic dynamics and spin dynamics of simple lattice gas and Ising models. They can be derived from the microscopic master equation of the system and contain analytical expressions for kinetic coefficients and thermodynamic quantities which are usually introduced phenomenologically. We review several methods to obtain such equations, and discuss applications to the dynamics of order–disorder transitions, spinodal decomposition, and dendritic growth in the isothermal or chemical model. In the case of dendritic growth we show that the mean-field kinetic equations are equivalent to standard continuum equations for this problem and derive expressions for macroscopic quantities, e.g. the surface tension and kinetic coefficients, as functions of the microscopic order parameters. In spinodal decomposition, we focus our attention on the vacancy mechanism, which is a more faithful picture of diffusion in solids than the more widely examined exchange mechanism. We study the interfaces between an unstable mixture and a stable ‘vapour’ phase, and analyse surface modes that lead to specific surface patterns. For order–disorder transitions, studied in the framework of a repulsive two-sublattice model, we derive sets of coupled equations for the mean concentration (a conserved quantity) and for the occupational difference between the two sublattices emerging from the symmetry breaking due to ordering (non-conserved order parameter). These equations are applied to transport in the presence of ordered domains. Finally, we discuss the possibilities of improving the simple mean-field approximation by density functional theories and various forms of the dynamic pair approximation, including the path-probability method.

Mixed alkali effects in ionic conductors : a new model and computer simulations

Abstract A new model theory of the mixed alkali effect that is consistent with the most recent EXAFS data of cation environments in mixed alkali silicate glass is presented. The dynamics of ion transport and the memory effects of site occupancy are brought out by computer simulations based on the model in an infinite percolation cluster. The salient features of the mixed alkali effect in the diffusion coefficients of the two cations are reproduced in the results of computer simulations.