Diffusion Percolation Research Articles

Phase transitions in the node, edge, bootstrap, and diffusion percolation models on the Sierpiński carpet

We investigate four types of percolation models — node, edge, bootstrap, and diffusion percolation — in three fractal graphs constructed on the Sierpiński carpet, employing the Monte Carlo method based on the Newman–Ziff algorithm. For each case, we calculate the percolation threshold and critical exponents (ν, γ, and β) through the crossing of percolation probabilities and the finite-size scaling analysis, incorporating correction-to-scaling effects. Our results reveal that critical exponents of the percolation phase transition in the three fractal graphs exhibit universality across all four percolation models. Furthermore, we demonstrate that the hyperscaling relation dν=γ+2β is also valid in the percolation phase transition on the Sierpiński carpet if the spatial dimension d is replaced by the Hausdorff dimension.

Induced diffusion percolation model: Examining the role of superactive nodes in the diffusion of innovations

Building on a recent study on scientific collaboration networks, we propose an induced diffusion percolation model that brings superactive nodes into focus. Defined as active nodes surrounded by at least k active or superactive neighbors, superactive nodes play a key role in innovation diffusion by inducing their neighbors to adopt an innovation. We investigate the induced diffusion percolation model using the modified Newman–Ziff algorithm on two-dimensional lattices (square and triangular lattices) and regular random networks with and without clustering. The induction by superactive nodes leads to a first-order percolation phase transition in two-dimensional lattices and a double transition – a continuous percolation transition followed by a discontinuous jump of the order parameter of the largest cluster’s strength – in regular random networks. Whereas clustering increases the percolation threshold in the classical percolation model on regular random networks, it decreases the critical initial activation probability that triggers a discontinuous jump of the induced activation.

Phase transition in the diffusion and bootstrap percolation models on regular random and Erdős-Rényi networks

The diffusion and bootstrap percolation models were studied in regular random and Erdős-Rényi networks using the modified Newman-Ziff algorithms. We calculated the percolation threshold and the order parameter of the percolation transition (strength of the giant cluster) and its derivatives. The percolation transitions are classified by the results. The diffusion percolation with a small k has a double transition, and the bootstrap percolation with m≥3 has the first-order percolation transition. The diffusion percolation with a large k and the bootstrap percolation with a small m show the second-order percolation transition. Particularly, third-order percolation transitions were discovered in the bootstrap percolation of m=2 in regular random networks.

Bootstrap and diffusion percolation transitions in three-dimensional lattices

We study the bootstrap and diffusion percolation models in the simple-cubic (sc), body-centered cubic (bcc), and face-centered cubic (fcc) lattices using the Newman–Ziff algorithm. The percolation threshold and critical exponents were calculated through finite-size scaling with high precision in the three lattices. In addition to the continuous and first-order percolation transitions, we found a double transition, which is a continuous transition followed by a discontinuity of the order parameter. We show that the continuous transitions of the bootstrap and diffusion percolation models have the same critical exponents as the classical percolation within error bars and they all belong to the same universality class.

Tunable chemical complexity to control atomic diffusion in alloys

In this paper we report a new fundamental understanding of chemically-biased diffusion in Ni–Fe random alloys that is tuned/controlled by the intrinsic quantifiable chemical complexity. Development of radiation-tolerant alloys has been a long-standing challenge. Here we show how intrinsic chemical complexity can be utilized to guide the atomic diffusion and suppress radiation damage. The influence of chemical complexity is shown by the example of interstitial atom (IA) diffusion that is the most important defect in radiation effects. We use μs-scale molecular dynamics to reveal sluggish diffusion and percolation of IAs in concentrated Ni–Fe alloys. We develop a mean field diffusion model to take into account the effect of migrating defect energy properties on diffusion percolation, which is verified by a new kinetic Monte Carlo approach addressing detailed processes. We demonstrate that the local variations in the ground state energy of IA configurations in alloys, reflecting the chemical difference between alloying components, drives the percolation effects for atomic diffusion. Percolation, chemically-biased and sluggish diffusion are phenomena that are directly related to the chemical complexity intrinsically to multicomponent alloys.

Newman-Ziff algorithm for the bootstrap percolation: Application to the Archimedean lattices

We propose very efficient algorithms for the bootstrap percolation and the diffusion percolation models by extending the Newman-Ziff algorithm of the classical percolation (M.E.J. Newman and R.M. Ziff (2000) [27]). Using these algorithms and the finite-size-scaling, we calculated with high precision the percolation threshold and critical exponents in the eleven two-dimensional Archimedean lattices. We present the condition for the continuous percolation transition in the bootstrap percolation and the diffusion percolation, and show that they have the same critical exponents as the classical percolation within error bars in two dimensions. We conclude that the bootstrap percolation and the diffusion percolation almost certainly belong to the same universality class as the classical percolation.

Monte Carlo Simulation of Correlation Effects in a Random SC Alloy via Interstitialcy Mechanisms

In this paper some recent progress in the area of Monte Carlo simulation of diffusion via the interstitialcy mechanism in a randomly ordered binary alloy is reviewed. Topics discussed include the calculation of tracer correlation factorsfA and fBas a function of composition and jump frequency ratiowA/wBand interstitialcy correlation factors fI; which play a crucial role in the interpretation of ion-conductivity data. The percolation behavior of fI when wA ≪ wB is analysed in detail and limits of the tracer diffusivity ratios bD A/bD B for alloy compositions below thepercolation threshold are presented. Allowance for non-collinear jumps (partly) replacing concurrent collinear site exchanges leads to a reduction of diffusion correlation effects.This goes along with a shift of the diffusion percolation threshold to lower concentrations of the (more) mobile component B. Even stronger changes of mass and charge transport compared to an exclusively collinear interstitialcy scheme are observed for additional contributions of direct interstitial jumps. It is remarkable that for both extensions of interstitialcy-mediated diffusion the Haven ratio appears to be greater than unity in certain compositionranges poor in B.

Soil water retention, air flow and pore structure characteristics after corn cob biochar application to a tropical sandy loam

Soil structure is a key soil physical property that affects soil water balance, gas transport, plant growth and development, and ultimately plant yield. Biochar has received global recognition as a soil amendment with the potential to ameliorate the structure of degraded soils. We investigated how corn cob biochar contributed to changes in soil water retention, air flow by convection and diffusion, and derived soil structure indices in a tropical sandy loam. Intact soil cores were taken from a field experiment that had plots without biochar (CT), and plots each with 10tha−1 (BC-10), 20tha−1 without or with phosphate fertilizer (BC-20 and BC-20+P respectively). Soil water retention was measured within a pF range of 1 to 6.8. Gas transport parameters (air permeability, ka, and relative gas diffusivity, Dp/D0) were measured between pF 1.5 and 3.0. Application of 20tha−1 led to significant increase in soil water retention compared to the CT and BC-10 as a result of increased microporosity (pores <3μm) whereas for soil specific surface area, biochar had minimal impact. No significant influence of biochar was observed for ka and Dp/D0 for the BC treatments compared to the CT despite the larger values for the two properties in the 20tha−1 treatments. Although not significant, the diffusion percolation threshold reduced by 34% and 18% in the BC-20 and BC-20+P treatments, respectively, compared to the CT. Similarly, biochar application reduced the convection percolation threshold by 15 to 85% in the BC-amended soils. The moderate impact of corn cob biochar on soil water retention, and minimal improvements in convective and diffusive gas transport provides an avenue for an environmentally friendly disposal of crop residues, particularly for corn cobs, and structural improvement in tropical sandy loams.

A Monte Carlo study of ionic transport in a simple cubic random alloy via the interstitialcy mechanism: effects of non-collinear and direct interstitial jumps

Self-diffusion and ionic conduction via the interstitialcy mechanism in a simple cubic, binary random alloy AB were investigated as a function of composition using Monte Carlo simulation. It was found that allowance for non-collinear jumps (partly) replacing concurrent collinear site exchanges leads to a reduction in diffusion correlation effects. This goes along with a shift of the diffusion percolation threshold to lower concentrations of the (more) mobile component B. Even stronger changes of mass and charge transport compared to an exclusively collinear interstitialcy scheme are observed for additional contributions of direct interstitial jumps. It is remarkable that for both extensions of interstitialcy mediated diffusion, the Haven ratio appears to be greater than unity in certain composition ranges poor in B. All results rely on the calculation of tracer and interstitialcy correlation factors in the simplest possible three-dimensional lattice structure. Yet they may have more general relevance to the interpretation of tracer self-diffusion data and ionic conductivity measurements on crystalline materials.

Ionization equilibrium at the transition from valence-band to acceptor-band migration of holes in boron-doped diamond

A quasi-classical model of ionization equilibrium in the p-type diamond between hydrogen-like acceptors (boron atoms which substitute carbon atoms in the crystal lattice) and holes in the valence band (v-band) is proposed. The model is applicable on the insulator side of the insulator–metal concentration phase transition (Mott transition) in p-Dia:B crystals. The densities of the spatial distributions of impurity atoms (acceptors and donors) and of holes in the crystal are considered to be Poissonian, and the fluctuations of their electrostatic potential energy are considered to be Gaussian. The model accounts for the decrease in thermal ionization energy of boron atoms with increasing concentration, as well as for electrostatic fluctuations due to the Coulomb interaction limited to two nearest point charges (impurity ions and holes). The mobility edge of holes in the v-band is assumed to be equal to the sum of the threshold energy for diffusion percolation and the exchange energy of the holes. On the basis of the virial theorem, the temperature Tj is determined, in the vicinity of which the dc band-like conductivity of holes in the v-band is approximately equal to the hopping conductivity of holes via the boron atoms. For compensation ratio (hydrogen-like donor to acceptor concentration ratio) K ≈ 0.15 and temperature Tj, the concentration of “free” holes in the v-band and their jumping (turbulent) drift mobility are calculated. Dependence of the differential energy of thermal ionization of boron atoms (at the temperature 3Tj/2) as a function of their concentration N is calculated. The estimates of the extrapolated into the temperature region close to Tj hopping drift mobility of holes hopping from the boron atoms in the charge states (0) to the boron atoms in the charge states (−1) are given. Calculations based on the model show good agreement with electrical conductivity and Hall effect measurements for p-type diamond with boron atom concentrations in the range from 3 × 1017 to 3 × 1020 cm−3, i.e., up to the Mott transition. The model uses no fitting parameters.

The Water‐Induced Linear Reduction Gas Diffusivity Model Extended to Three Pore Regions

Characterization of soil functional pore structure is an essential prerequisite to understand key gas transport processes in variably saturated soils in relation to soil ecosystems, climate, and environmental services. In this study, the water-induced linear reduction (WLR) soil gas diffusivity model originally developed for sieved, repacked soil was extended to two simple, linear regions to characterize gas diffusion and functional pore-network structure also in intact, structured soil systems. Based on the measurements in soils with markedly different pore regions, we showed that the two linear regions can denote a percolation threshold where soil gas diffusion ceases due to interconnected water films, preferential gas diffusion in fracture networks (e.g., fractured limestone), and intra-aggregate or intramatrix gas diffusion. From measured or three-region WLR (3WLR) modeled gas diffusivity, we derived a simple pore connectivity index, Cip (ranging from 0 to 1), that showed a linear behavior with air-filled porosity (ε) for sieved, repacked soils ranging from 6 to 54% clay. We suggest that deviation from this Cip‒ε line is a direct measure of soil structure. The new 3WLR model could accurately describe gas diffusivity from moist to dry conditions across differently structured porous media, including narrow soil size fractions, perforated plastic blocks, fractured limestone, peaty soils, aggregated volcanic ash soils, and particulate substrates for Earth- or space-based applications. The new Cip function provided distinct soil structural fingerprints from moist to dry conditions for all porous media. We further used the 3WLR and Cip analyses to discuss the decreasing trend in gas diffusion percolation threshold with soil compaction.

Garnet related lithium ion conductor processed by spark plasma sintering for all solid state batteries

Oxide-based all solid state battery (ASSB) systems require highly enhanced cell assemblies technology in the light of the sintering issue, although various inorganic-based lithium ion conductors have been proposed and attempted to be adopted in battery systems. Here, the compatibility of Li–La–(Zr–Ta)–O garnet related lithium ion conductors with ASSB systems processed by spark plasma sintering (SPS) is reported. Further, the challenges with respect to the material and processing are also discussed. Li7−xLa3Zr1.5Ta0.5O12−d processed by SPS shows a high ionic conductivity of 1.35 × 10−3 S cm−1 and 1.23 × 10−2 S cm−1 at 25 °C and 80 °C, respectively. In addition, the material also shows good structural stability, which is a critical challenge encountered in garnet structured materials treated under severe SPS process conditions. An ASSB assembled by SPS using the Li4Ti5O12–Li7−xLa3Zr1.5Ta0.5O12−d composite electrode shows a surface capacity of 0.2 mAh cm−2 at 80 °C. The battery shows a well-organized structure with a good lithium diffusion percolation path and the low solid contact resistance without any defects and undesirable reactions arising from the sintering method.

The Experiment on the Features of Ultrasonic Vibration Stimulating the Absorption and Desorption of Methane in Coal

Through adopting the self-explored experimental system of ultrasonic vibration, this experiment studied the adsorption and desorption characteristics of methane in coal under the sound field varying in frequency and sound intensity. Results of the research showed that the methane absorption in coal reduces when being placed in the sound field, and the amount of absorption reduces as the sound intensity increases. The amount of methane desorbed in coal increases under the working of sound field, which indicates that the effect of sound field can promote methane desorption and diffusion. the adsorption 、desorption law of methane in coal are the same under the condition of sound field and no sound field, The absorption model can be described by the Eequation D-R、Langmuir Equation、Freundlich Equation and Equation BET. Plus, the Equation D-R、Langmuir Equation can better describe the law of methane absorption in coal. Desorption model can be expressed by empirical formula、diffusion model and percolation model, and the diffusion model can better describe the desorbing characteristics of methane. These results can provide theoretical principles to improve the rate of coal seam extraction under ultrasonic vibration.

Structural investigations of migration pathways in lithium ion-conducting La 2/3− xLi 3 xTiO 3 perovskites

The average structure and microstructure of a lithium-ion conducting perovskite La 2/3− x Li 3 x TiO 3 ( x = 0.12) were investigated using neutron diffraction and transmission electron microscopy (TEM). The obtained results were compared with those of previous studies on La 2/3− x Li 3 x TiO 3 series compounds, and the relationship between their structures and ionic conduction was discussed. The Rietveld refinement using neutron diffraction data reveals that the average structure (space group: Cmmm) of furnace-cooled Li-rich La 2/3− x Li 3 x TiO 3 ( x = 0.12) involves alternative ordered arrangement of La along the c-axis and anti-phase-tilt of TiO 6 octahedra along the b-axis as with Li-poor La 2/3− x Li 3 x TiO 3( x = 0.05). It was found that the strong correlation between structure and percolative diffusion pathways in the perovskites is primarily referred to the ordered arrangement of La ions, and the “bottleneck” square—surrounded by four oxygen ions determined by the distortion and tilt of TiO 6 octahedra. In addition, the 90°-oriented micro-domain structure, which also influences the percolative diffusion pathways in La 2/3− x Li 3 x TiO 3, was observed by TEM.

From Atomic Scale Ordering to Mesoscale Spatial Patterns in Surface Reactions: A Heterogeneous Coupled Lattice-Gas Simulation Approach

A current challenge in the modeling of surface reaction-diffusion systems is to connect the length scales from a realistic atomistic treatment of local ordering of adsorbed reactants and of reaction kinetics to an "exact" description of mesoscale spatial pattern formation. We discuss a heterogeneous coupled lattice-gas approach which utilizes parallel kinetic Monte Carlo simulations of lattice-gas models to simultaneously determine the local reaction kinetics and diffusive transport properties at various macroscopic "points" distributed across the surface. These simulations are periodically coupled to reflect macroscopic mass transport via surface diffusion. We place particular emphasis on the key issue of correctly describing the associated chemical or collective diffusion flux. This method is general, but it is discussed here only in the context of simple models for the bistable CO oxidation reaction on surfaces, where macroscopic mass transport involves only diffusion of adsorbed CO. Detailed analysis is presented for a canonical model of CO oxidation which incorporates nearest-neighbor exclusion between adsorbed oxygen, leading to superlattice ordering of this reactant. In this model, interactions of adsorbed CO with other adspecies are ignored. This reduces CO surface transport from a many-particle to a single-particle problem, although one with the complexity of "ant-in-the-labyrinth"-type percolative diffusion (the labyrinth being formed by coadsorbed oxygen).

Multiscale modeling of oxygen diffusion through the oxide during silicon oxidation

We investigate at the atomic scale the oxygen diffusion process occurring during silicon oxidation. First, we address the energetics of several oxygen species in the oxide using density-functional calculations. Our results support the interstitial ${\mathrm{O}}_{2}$ molecule as the most stable oxygen species. We then adopt a classical scheme for describing the energetical and topological properties of the percolative diffusion of the ${\mathrm{O}}_{2}$ molecule through the interstitial network of the oxide. By studying a large set of disordered oxide structures, we derive distributions of energy minima, transition barriers, and the number of connections between nearest-neighbor minima. These distributions are then mapped onto a lattice model to study the long-range ${\mathrm{O}}_{2}$ diffusion process by Monte-Carlo simulations. The resulting activation energy for diffusion is found to be in agreement with experimental values. We also extend our atomic-scale approach to an oxide of higher density, finding a significant decrease of the diffusivity. To address the ${\mathrm{O}}_{2}$ diffusion directly at the $\mathrm{Si}\ensuremath{-}\mathrm{Si}{\mathrm{O}}_{2}$ interface, we construct a lattice model of the interface which incorporates the appropriate energetic and connectivity properties in a statistical way. In particular, this lattice model shows a thin oxide layer of higher density at the interface, in accord with x-ray reflectivity data. We carry out Monte-Carlo simulations of the ${\mathrm{O}}_{2}$ diffusion for this model and obtain the dependence of the diffusion rate on oxide thickness. For oxide thicknesses down to about $2\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$, we find that the presence of an oxide layer of higher density at the $\mathrm{Si}\ensuremath{-}\mathrm{Si}{\mathrm{O}}_{2}$ interface causes a drop of the ${\mathrm{O}}_{2}$ diffusion rate with respect to its value in bulk $\mathrm{Si}{\mathrm{O}}_{2}$, in qualitative agreement with the observed oxidation kinetics.

Dependence of the O2diffusion rate on oxide thickness during silicon oxidation

We address the rate of O2 diffusion through the oxide layer at Si–SiO2 interfacesusing an atomic-scale approach. In particular, we investigate the combined effectof a percolative diffusion mechanism and of a dense oxide layer located close tothe silicon substrate. We first extend our atomic-scale description of O2 diffusionin amorphous SiO2 to the case of a densified oxide. This yields an activationenergy which compares well with the experimental result. Next, we investigatethe dependence of the O2 diffusion rate on oxide thickness at Si–SiO2interfaces using Monte Carlo simulations. We consider both homogeneous andnonhomogeneous oxide layers. The nonhomogeneous oxide is composed oftwo layers, a normal and a densified one. The thickness and the massdensity of the densified layer are taken from experiment. In the case of anormal oxide, we find that the O2 diffusion rate increases with decreasingthickness, as a result of the percolative nature of the diffusion mechanism.When a densified layer is inserted, the diffusion coefficient drops below itsvalue for bulk amorphous SiO2, for oxide thicknesses larger than 2 nm.This result is consistent with the experimental behaviour of the oxidationkinetics.

Surface diffusion in mixed overlayers with superlattice ordering: Percolative transport around obstacles and along domain boundaries

To elucidate surface diffusion in the presence of a coadsorbate with superlattice ordering, we consider particle hopping on a square lattice with some fraction, θB, of quenched blocking sites arranged with checkerboard or c(2×2) ordering. Behavior for low θB corresponds to diffusion around isolated obstacles, and can be described by exact density expansions. Behavior for high θB corresponds to percolative diffusion along (or sometimes away from) domain boundaries. The connectivity of these domain boundaries is closely related to the existence of symmetry breaking [i.e., long-range c(2×2) order] in the distribution of blocking sites. In some cases, symmetry breaking induces critical behavior for diffusive transport which is fundamentally different from that for the conventional “ant in the labyrinth” problem. Our results apply to recently developed models for CO oxidation, where CO(ads) diffuses rapidly through coadsorbed relatively immobile c(2×2)-O(ads). The characterization of CO diffusion in these systems is key to describing spatial pattern formation.

The Kubo number as a parameter governing the level of chaos in magnetic turbulence

It is well known that the Kubo number R allows to classify the transport regimes in turbulent systems. A small Kubo number leads to the so-called quasilinear diffusion coefficient, while a large Kubo number corresponds to the percolative diffusion coefficient. Here we show, by means of a numerical simulation of magnetic field line transport in a three-dimensional anisotropic magnetic turbulence, in which the magnetic fluctuating level and the correlation lengths can be varied independently of each other, that the Kubo number also determines the level of chaos of the magnetic-field lines. We find weak chaos, closed magnetic surfaces, and anomalous transport regimes for R⪡1; widespread chaos, destroyed magnetic surfaces, and quasilinear scaling of the diffusion coefficient for R≳0.3; and global stochasticity as well as the percolation scaling of the diffusion coefficient for R⪢1.

Properties of Hot-Melt Extruded Theophylline Tablets Containing Poly(Vinyl Acetate)

The objectives of this study were to investigate the properties of poly(vinyl acetate) (PVAc) as a retardant polymer and to study the drug release mechanism of theophylline from matrix tablets prepared by hot-melt extrusion. A physical mixture of drug, polymer, and drug release modifiers was fed into the equipment and heated inside the barrel of the extruder. The cylindrical extrudates were either cut into tablets or ground into granules and compressed with other excipients into tablets. Due to the low glass transition temperature of the PVAc, the melt extrusion process was conducted at approximately 70°C. Theophylline was used as the model drug in this study. Theophylline was present in the extrudate in its crystalline form and was released from the tablets by diffusion. The Higuchi diffusion model and percolation theories were applied to the dissolution data to explain the drug release properties of the matrix systems. The release rate was shown to be dependent on the granule size, drug particle size, and drug loading in the tablets. Water-soluble polymers were demonstrated to be efficient release rate modifiers for this system.