W. Dieterich

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”.

Percolation in Composites

Many properties of composite materials such as diffusion, electrical conduction, dielectric response as well as elasticity, are intimately related to the geometrical arrangement of the constitutive phases, including the geometry of the respective interfaces. Percolation theory, whose objective is to characterize the connectivity properties in random geometries and to explore them with respect to physical processes, thus provides a natural frame for the theoretical description of random composites. This article explains basic concepts of static percolation theory and percolative transport, which subsequently are applied to specific experiments on heterogeneous ionic conductors.

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.

Ion dynamics in structurally disordered materials: effects of random Coulombic traps

Abstract A stochastic lattice gas model is presented pertaining to classical charge transport in doped solid materials. Specifically, we investigate the diffusive dynamics of particles of charge q in an energy-landscape determined by immobile centers of charge — q which are distributed randomly in space. Results from extensive Monte Carlo simulations for tracer diffusion, conductivity and nuclear-spin relaxation are discussed and compared in detail with experiments on ion conduction in alkali-doped network glasses. Emphasis is put on the dependence of transport properties on carrier concentration and on non-exponential, slow relaxational effects which manifest themselves in different experimental spectra.

Percolation model for mixed alkali effects in solid ionic conductors

We investigate a random walk model for describing mixed alkali effects in ionic conductors of the β‐alumina type. In our model, the observed drastic variation in the transport properties of the mixed crystals is related to critical properties near a percolation threshold, which originates from the blocking of conduction paths by complexes containing the substituted ions. The motion of a tracer ion and the conductivity are obtained by using the Monte Carlo technique and are discussed in relation to experiments. It is pointed out that anomalous diffusion, which occurs near percolation, leads to a characteristic frequency dependence of the dynamic conductivity, which should be experimentally accessible.

TIME-DEPENDENT DENSITY FUNCTIONAL THEORY AND THE KINETICS OF LATTICE GAS SYSTEMS IN CONTACT WITH A WALL

We develop an improved mean-field theory which allows us to describe the diffusive dynamics near phase transformations in condensed systems. Starting from a master equation for a stochastic lattice gas we obtain evolution equations on the single-particle level, whose stationary solutions in principle are consistent with the exact equilibrium statistics. Our method, which generalizes an approach proposed earlier, is based on a combination of a local equilibrium assumption and the lattice version of classical density functional theory. In the continuum limit, which is worked out for attractive interactions, generalized Cahn–Hilliard-type equations are recovered. Microscopic kinetic coefficients can be identified, which in general depend on the instantaneous local correlations in the nonequilibrium state. Moreover we study semi-infinite systems interacting with a planar wall and derive the appropriate boundary conditions to be imposed on the continuum equations. Applications to problems of the kinetics of ph...

Hopping transport in the presence of site-energy disorder: Temperature and concentration scaling of conductivity spectra

Recent measurements on ion conducting glasses have revealed that conductivity spectra for various temperatures and ionic concentrations can be superimposed onto a common master curve by an appropriate rescaling of the conductivity and frequency. In order to understand the origin of the observed scaling behavior, we investigate by Monte Carlo simulations the diffusion of particles in a lattice with site energy disorder for a wide range of both temperatures and concentrations. While the model can account for the changes in ionic activation energies upon changing the concentration, it in general yields conductivity spectra that exhibit no scaling behavior. However, for typical concentrations and sufficiently low temperatures, a fairly good data collapse is obtained analogous to that found in experiment.

Soft particle model for block copolymers

A soft particle model for diblock (AB) copolymer melts is proposed. Each molecule is mapped onto two soft spheres built by Gaussian A- and B-monomer distributions. An approximate analytical expression for the joint distribution function for the distance between both spheres and their radii of gyration is derived, which determines the entropic contribution to the intramolecular free energy. Adding a mean-field expression for the intermolecular interactions, we obtain the total free energy of the system. Based on this free energy, Monte Carlo simulations are carried out to study the kinetics of microphase ordering in the bulk and its effect on molecular diffusion. This is followed by an analysis of thin films, with emphasis on pattern transfer from walls with a periodic structure. It is shown that the level of coarse graining in the soft particle model is suitable to describe structural and kinetic properties of copolymers on mesoscopic scales.