Dynamic Bond Percolation Model Research Articles

Chemically Specific Dynamic Bond Percolation Model for Ion Transport in Polymer Electrolytes

We introduce a coarse-grained approach for characterizing the long-timescale dynamics of ion diffusion in general polymer electrolytes using input from short molecular dynamics trajectories. The approach includes aspects of the dynamic bond percolation model [J. Chem. Phys. 1983, 79, 3133−3142] by treating ion diffusion in terms of hopping transitions on a fluctuating lattice. We extend this well-known approach by using short (i.e., 10 ns) molecular dynamics (MD) trajectories to predict the distribution of ion solvation sites that comprise the lattice and to predict the rate of hopping among the lattice sites. This yields a chemically specific dynamic bond percolation (CS-DBP) model that enables the description of long-timescale ion diffusion in polymer electrolytes at a computational cost that makes feasible the screening of candidate materials. We employ the new model to characterize lithium-ion diffusion properties in six polyethers that differ by oxygen content and backbone stiffness: poly(trimethylen...

Multilevel unipolar resistance switching in TiO2 thin films

We report on multilevel switching behavior in the unipolar resistance switching of TiO2 thin films. Multiple metastable states were observed during the reset process by measuring I-V curves. As observed using a conducting atomic force microscope, the multilevel resistance switching was accompanied by decreases in area and in the conductance of the top surface conducting regions. These experimental observations at both the macroscopic and microscopic levels could be explained by using the “random circuit breaker network” model, which is a dynamic bond percolation model.

Segmental Dynamics and Ionic Conduction in Poly(vinyl methyl ether)−Lithium Perchlorate Complexes

Broadband dielectric spectroscopy was used to investigate the segmental dynamics and ionic conduction in LiClO4/PVME complexes with Li/O from 0.1/100 to 10/100, at temperatures from Tg to approximately Tg + 80 degrees C. Although no microphase separation is observed via DSC, dielectric experiments reveal two segmental relaxations and one localized ion motion process. The fastest process is attributed to relaxations of segments in ion-depleted domains and it slows down with increasing salt content, as does the ion motion. The segmental relaxation of PVME chains in ion-rich domains is even slower than the ion motion process, and about 104 times slower than the fast segmental process in the 0.5/100 complex. This process becomes faster with increasing LiClO4 content, despite the concurrent increase in the bulk Tg. Maximum molar conductivity is obtained in the 2/100 complex and the ionic conduction is about 10-9 S/cm at 30 degrees C. By using the dynamic bond percolation model, it was found that the ions move about 0.8 nm for the 0.5/100 complex at 25 degrees C at the time scale of the slow segmental relaxation, assuming that structural renewal is realized by the latter. This size, together with the strong correlation between the ionic conduction and the slow segmental relaxation, supports the idea that hopping from one segment to another one is probably the effective fundamental step giving rise to macroscopic conduction.

Optimizing the Design of Polyelectrolytes Using Monte Carlo Simulations

Monte Carlo calculations were performed to simulate ion diffusion through polymer matrices. The parameters can be chosen so that a broad range of polymer electrolytes can be modeled. The focus of the present study is optimizing conductivity in polyelectrolytes. A dynamic bond percolation model was extended to include local harmonic motion of covalently bound anions in polyelectrolyte systems. Local motion of these anions facilitates cation escape from potential wells, and thus, improves overall conductivity, while maintaining a cation transference number of one. Simulations show that ion correlation has a significant effect on diffusion. Increasing the temperature or the dielectric constant of the medium reduces the dependence on ion interaction, while increasing the density of ions in the polymer matrix increases the dependence. At high densities, physical blocking significantly interferes with diffusion at all temperatures. The cation conductivity was found to maximize at a lower density in polyelectrolytes than in polymer-salt complexes. © 2001 The Electrochemical Society. All rights reserved.

Diffusion on a rearranging lattice

In this paper we present a computer simulation of a random walk (RW) for diffusion on a rearranging lattice. The lattice consists of two types of sites - one highly conducting (type 1) and the other poorly conducting (type 2), distributed at random. The two types of site are assigned different waiting times ( for type 1 and for type 2). We assume that at intervals of time the site distribution changes. The effect of this rearrangement on the diffusion coefficient is studied with varying . We study this effect for different ratios of dwell time of the two types of site (R) and also for different fractions (X) of the less conducting sites. An empirical relation for is suggested. We have employed the well model and considered diffusion controlled by sties, rather than bonds. So our approach is different from the dynamic bond percolation model, which studies these aspects. Our results show that the diffusion coefficient D may change by a factor of up to 3 (approximately) for rapid rearrangement, and there is a considerable effect of varying X and R on the range of variation of D, where X is the fraction of poorly conducting sites, and R is the ratio of the dwell times for types of site. Further for ( is the time unit for the random walk) the effect of rearrangement becomes negligible. The results may be useful for studying diffusion and conduction of ion conducting polymers.

Simulation of ionic diffusion in solid polymer electrolytes with correlated chain motion

Based on the dynamic bond percolation model, we try to regenerate the non-Arrhenius temperature dependence for the ionic conductivity of the polyethylene oxide type polymer electrolytes by considering continuous, correlated motions of the host polymer matrix. To simulate the chain movements, periodic rotations of bond groups of several sizes were introduced into a two-dimensional square lattice. Although the Arrhenius temperature dependence of the rotation frequencies were assumed such that the rotation periods ∝ exp(E/kT), where E increases with the bond group size, the resulting logarithmic diffusivity plotted against reciprocal temperature produces a convex curvature—the typical feature observed by the experiments.

The effects of chain segment motion on ionic diffusion in solid polymer electrolytes

The “vehicular effects” of chain segment motion on ionic diffusion in solid polymer electrolytes have been investigated via numerical simulation on a two-dimensional square lattice where the dynamical variation of chain configuration is presented by translational or rotational bond movement. It is found that (a) both types of bond motion promote continuous diffusion when the fraction ( p) of available bonds is below the static percolation threshold of p=0.5 in two dimensions; (b) translational motion of bonds parallel to the direction of diffusion produces larger diffusion coefficients ( D) than that by random renewal of the dynamic bond percolation model (DBPM), while the perpendicular motion or rotational motion gives smaller values of D; (c) Smooth lines instead of “stair-case like” curves generated by DBPM are obtained in the mean-squared displacement versus time plot, when bonds are shifting along the diffusion route. The dependence of diffusion coefficients on the variation of motion patterns of bonds is expected to be related to the temperature change under which these patterns are excited accordingly, such that VTF behavior of certain polymer electrolytes may be deduced.

Ionic conductivity in poly(propylene glycol) complexed with lithium and sodium triflate

Conductivity and viscosity measurements have been made for poly(propylene glycol)-MCF3SO3 (M=Li, Na) complexes in order to examine more closely the Vogel–Tammann–Fulcher (VTF) empirical relationship which has been found in previous reports to provide a good fit to the experimental data. Further, a dynamic bond percolation model of ion conduction in polymer electrolytes has predicted VTF behavior and an inverse relationship between molar conductivity and viscosity or Walden ‘‘rule’’ behavior. We find that deviations occur from both the VTF and Walden empirical relationships and propose a modest alteration in the form of the dynamic percolation model for ions moving in polyether systems.

Polymer Electrolytes: Hopping, Domain Structures and Frequency-Dependent Conductivity

AbstractThe dynamic bond percolation model was developed to deal with dynamic disorder, treating ion mobility by a percolation model in which the assignment of any site-to-site jump as allowed or forbidden changes on a timescale related to the local reorganizational dynamics of the polymer segments (the renewal time). Here we discuss the special cases of highfrequency spectra and partially crystalline electrolytes. At high frequencies, the present hopping model yields unphysical behavior (frequencyindependent response); we trace this back to the incorrect treatment of short-time dynamics, and show how it can be corrected. For partially crystalline materials, we show that a rollover feature in the spectrum, in the microwave range, can be expected when ions are trapped in isolated regions of high conductivity, such as amorphous pockets in largely crystalline PEO.

Collision-induced light scattering from growing clusters. Depolarization by fractals

It is argued that collision-induced light scattering (CILS) will be a uniquely good probe of the structure of growing clusters, and especially of any fractal character of that growth. Computer calculations of CILS from a dynamic bond percolation model are presented which support this assertion in detail. The first application of the real-space renormalization group (RSRG) to CILS is given, resulting in a formula for the exponent of a power fit of the CILS intensity of a fractal vs. the number of monomers which is in excellent agreement with the computer calculations.

Particle motion through a dynamically disordered medium: The effects of bond correlation and application to polymer solid electrolytes

To study the diffusion of small particles through a dynamically disordered medium, a dynamic bond percolation model has been developed. This differs from the standard percolation theory, in that the lattice is no longer static but undergoes rearrangements which reassign the open and closed bonds. Physically, these rearrangements correspond to orientational motions of the (polymer) host lattice. An interesting feature of this model is that, even below the percolation threshold, diffusive behavior can occur, as long as the renewal time τ ren, the time characteristic of the rearrangement of bonds, is short compared to the observation time. Ionic conductivity in polymeric electrolytes is one of the systems for which this model is useful. In these materials, carrier ions diffuse through a medium (the polymer) which is undergoing dynamic motion caused by configurational motions of the polymer. Since polymer chain motions will affect several ion binding sites simultaneously or serially, the model is further elaborated to account for correlations in the segmental motions of the polymer host. The renewal of the bonds is changed from occurring randomly to include simple correlation effects. Of principal concern in this study is the effect that correlated renewals have on the transport behavior of the model. Simulations were done on a 1-D lattice and a diffusion coefficient calculated. The values of the diffusion coefficients from the two systems (with and without correlated renewal) are studied and their behavior as a function of the fraction of available bonds, f, and the renewal time is compared. For both correlated and uncorrelated renewals, the systems were diffusive. The diffusion coefficients, in both cases, increased with increasing f and decreasing τ ren, corresponding to an increase in the free volume, the configurational entropy, and the temperature of the polymer systems. The diffusion coefficient from the correlated systems were always smaller than those from the uncorrelated systems, except for the limit f = 100% and τ ren ⪡ τ hop. The ratio of the diffusion coefficients for the correlated and uncorrelated systems was studied as a function of τ ren and f. This ratio falls off to a constant value as τ ren is increased and reaches minimum value at f = 50%. This behavior of the ratio as a function of f can be explained by considering the diffusion of the bonds in the lattice for the correlated case.

Applications of dynamic bond percolation theory to the dielectric response of polymer electrolytes

Our recently-developed dynamic bond percolation model is extended and applied to polymer electrolytes by assuming an approximate form for the relaxation of the carrier mean-square displacement to its asymptotic value below the percolation threshold, with similar assumptions for the short-range motion of the ionic charges bound to the polymer host. The parameters characterizing the long-range-motion part of the carrier response are shown to be fully determined in terms of the bond percolation parameters, thereby allowing comparison with the exact analytic solution presently available only in one dimension. The behavior of the dielectric response based on the assumed functional form of 〈r 2> 0(t) (the mean-square carrier displacement from its initial position in a frozen lattice) is shown to be given by a sum of Debye dielectric loss peaks, and is compared with frequency-dependent data for PEO·NaSCN.

Generalized hopping model for frequency-dependent transport in a dynamically disordered medium, with applications to polymer solid electrolytes.

Protonic diffusion in hydrogen-bonded networks, ionic conduction in polymeric solid electrolytes, and other processes in which the carrier transport mechanism involves motion of the host medium on a time scale comparable to that of the carrier motion itself require generalization of the usual models based on carrier hopping in a static medium. Under the assumption that this concurrent motion of the host can be modeled by a random reassignment (or ``renewal'') of hopping probabilities, with a constant probability \ensuremath{\lambda} per unit time for renewal to occur, the effects of host motion on the frequency-dependent diffusion coefficient D(\ensuremath{\omega}) are now considered. We consider both the dynamic bond-percolation model (in which the site-to-site hopping probability is randomly assigned either the value w or the value 0) and the more general model based on a possibly continuous distribution of hopping rates randomly assigned between different pairs of sites. Under these assumptions, the diffusion coefficient D(\ensuremath{\omega}) with renewal is shown to be obtainable from D(\ensuremath{\omega}) without renewal through the formal substitution i\ensuremath{\omega}\ensuremath{\rightarrow}\ensuremath{\lambda}+i\ensuremath{\omega}. For the \ensuremath{\omega}=0 limit, an expression is obtained for the time-dependent mean-square displacement with renewal in terms of the mean-square displacement without renewal. These general formal results are applied to the one-dimensional dynamic percolation model, for which specific exact analytic results are thereby obtained, and D(\ensuremath{\omega}) is calculated and studied for this case.

Dynamic bond percolation theory: A microscopic model for diffusion in dynamically disordered systems. I. Definition and one-dimensional case

A dynamic bond percolation model is defined and studied. The model is intended to describe diffusion of small particles (ions, electrons) in a medium which is statistically disordered (as in ordinary bond percolation), but which is also undergoing dynamic rearrangement processes on a timescale short compared to the observation time. The model should be applicable to polymeric solid electrolytes, where the orientational motions of the polymer (which are responsible for configurational entropy) cause the dynamic motion of the medium (polymer) in which the small particles (alkali ions) diffuse. The model is characterized by three parameters: an average hopping rate w which appears in the master equation for hopping, a percentage of available bonds f, and a mean renewal time τ̄ren for dynamic motion of the medium to rearrange the assignments of closed and open bonds. We show that the behavior is always diffusive for observation times long compared to τ̃ren, in agreement with experiment on polymeric solid electrolytes. We also derive a closed-form expression for the diffusion coefficient. For observation times smaller than the renewal time there is no diffusion, again in accord with the behavior of polymeric solid electrolytes below the glass transition temperature. The diffusion coefficient is a monotonically increasing function of the inverse renewal time and hence of the free volume, the configurational entropy, and the temperature.