Power-law Kinetics Research Articles

Random walks with fractally correlated traps: Stretched exponential and power-law survival kinetics.

We consider the survival probability f(t) of a random walk with a constant hopping rate w on a host lattice of fractal dimension d and spectral dimension d_{s}≤2, with spatially correlated traps. The traps form a sublattice with fractal dimension d_{a}<d and are characterized by the absorption rate w_{a} which may be finite (imperfect traps) or infinite (perfect traps). Initial coordinates are chosen randomly at or within a fixed distance of a trap. For weakly absorbing traps (w_{a}≪w), we find that f(t) can be closely approximated by a stretched exponential function over the initial stage of relaxation, with stretching exponent α=1-(d-d_{a})/d_{w}, where d_{w} is the random walk dimension of the host lattice. At the end of this initial stage there occurs a crossover to power-law kinetics f(t)∼t^{-α} with the same exponent α as for the stretched exponential regime. For strong absorption w_{a}≳w, including the limit of perfect traps w_{a}→∞, the stretched exponential regime is absent and the decay of f(t) follows, after a short transient, the aforementioned power law for all times.

Kinetics and Thermodynamics of Synthesis of Hybrid Oligomer‐Acrylated Cycloaliphatic Epoxy in the Presence of Triphenylphosphine and Triethylamine Catalysts

ABSTRACTThe kinetics for the esterification of cycloaliphatic epoxy resin with acrylic acid was investigated in the presence of two different catalysts, namely triethylamine (TEA) and triphenylphosphine (TPP). The reactions were carried out at four different temperatures in the range of 80–110°C and 60–90°C for TPP and TEA catalysts, respectively. The power law kinetics model was adopted for analyzing experimental data to determine reaction equations. Both differential and integral methods were applied to determine the reaction orders and reaction constants. The overall reaction order for the reaction systems was found to be 2 using both differential and integral methods. According to the differential method, the reaction orders with respect to epoxide and carboxylic acid groups were calculated to be 1.2 and 0.8, respectively. The reactions could be considered as first‐order reactions with respect to each group based on integral method analysis. The activation energies and frequency factors of these esterification reactions were found to be 77.2 kJ·mol−1 and 1.72E+10 L·mol−1·min−1 in the presence of TPP and 65.3 kJ·mol−1 and 1.44E+7 L·mol−1·min−1 in the presence of TEA. The thermodynamic parameters of reaction in the presence of TPP and TEA were calculated. The results confirmed that both reactions were spontaneous and irreversible. It was found that the reaction in the presence of TEA produced a highly ordered activated complex.

A Comparative Study of Defect Energy Distribution and Its Impact on Degradation Kinetics in GeO2/Ge and SiON/Si pMOSFETs

The high mobility germanium (Ge) channel is considered as a strong candidate for replacing Si in pMOSFETs in the near future. It has been reported that the conventional power-law degradation kinetics of Si devices is inapplicable to Ge. In this paper, further investigation is carried out on defect energy distribution, which clearly shows that this is because the defects in GeO2/Ge and SiON/Si devices have different physical properties. The three main differences are: 1) energy alternating defects (EAD) exist in Ge devices but are insignificant in Si; 2) the distribution of as-grown hole traps has a tail in the Ge bandgap but not in Si, which plays an important role in the degradation kinetics and device lifetime prediction; and 3) EAD generation in Ge devices requires the injected charge carriers to overcome a second energy barrier, but not in Si. Taking the above differences into account, the power-law kinetics of EAD generation can be successfully restored by following a new procedure, which can assist in the Ge process/device optimization.

Insight Into Electron Traps and Their Energy Distribution Under Positive Bias Temperature Stress and Hot Carrier Aging

The access transistor of SRAM can suffer both positive bias temperature instability (PBTI) and hot carrier aging (HCA) during operation. The understanding of electron traps (ETs) is still incomplete and there is little information on their similarity and differences under these two stress modes. The key objective of this paper is to investigate ETs in terms of energy distribution, charging and discharging properties, and generation. We found that both PBTI and HCA can charge ETs which center at 1.4 eV below conduction band ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$E_{c}$ </tex-math></inline-formula> ) of high-k dielectric, agreeing with theoretical calculation. For the first time, clear evidences are presented that HCA generates new ETs, which do not exist when stressed by PBTI. When charged, the generated ETs’ peak is 0.2 eV deeper than that of preexisting ETs. In contrast with the power law kinetics for charging the preexisting ETs, filling the generated ETs saturates in seconds, even under an operation bias of 0.9 V. ET generation shortens device lifetime and must be included in modeling HCA. A cyclic and antineutralization ETs model is proposed to explain PBTI and HCA degradation, which consists of preexisting cyclic ETs, generated cyclic ETs, and antineutralization ETs.

Scale-up from batch to flow-through wet milling process for injectable depot formulation

Injectable depot formulations are aimed at providing long-term sustained release of a drug into systemic circulation, thus reducing plasma level fluctuations and improving patient compliance. The particle size distribution of the formulation in the form of suspension is a key parameter that controls the release rate. In this work, the process of wet stirred media milling (ball milling) of a poorly water-soluble substance has been investigated with two main aims: (i) to determine the parametric sensitivity of milling kinetics; and (ii) to develop scale-up methodology for process transfer from batch to flow-through arrangement. Ball milling experiments were performed in two types of ball mills, a batch mill with a 30ml maximum working volume, and a flow-through mill with a 250ml maximum working volume. Milling parameters were investigated in detail by methodologies of QbD to map the parametric space. Specifically, the effects of ball size, ball fill level, and rpm on the particle breakage kinetics were systematically investigated at both mills, with an additional parameter (flow-rate) in the case of the flow-through mill. The breakage rate was found to follow power-law kinetics with respect to dimensionless time, with an asymptotic d50 particle size in the range of 200–300nm. In the case of the flow-through mill, the number of theoretical passes through the mill was found to be an important scale-up parameter.

Lattice Boltzmann models for the grain growth in polycrystalline systems

In the present work, lattice Boltzmann models are proposed for the computer simulation of normal grain growth in two-dimensional systems with/without immobile dispersed second-phase particles and involving the temperature gradient effect. These models are demonstrated theoretically to be equivalent to the phase field models based on the multiscale expansion. Simulation results of several representative examples show that the proposed models can effectively and accurately simulate the grain growth in various single- and two-phase systems. It is found that the grain growth in single-phase polycrystalline materials follows the power-law kinetics and the immobile second-phase particles can inhibit the grain growth in two-phase systems. It is further demonstrated that the grain growth can be tuned by the second-phase particles and the introduction of temperature gradient is also an effective way for the fabrication of polycrystalline materials with grained gradient microstructures. The proposed models are useful for the numerical design of the microstructure of materials and provide effective tools to guide the experiments. Moreover, these models can be easily extended to simulate two- and three-dimensional grain growth with considering the mobile second-phase particles, transient heat transfer, melt convection, etc.

Pd-Ag/Al2O3 catalyst: Stages of deactivation in tail-end acetylene selective hydrogenation

The deactivation of a Pd-Ag/α-Al2O3 catalyst (2.5mm in diameter, containing 0.03wt.% Pd and 0.18wt.% Ag) with egg-shell distribution was studied for acetylene selective hydrogenation. The reaction was performed in a fixed bed reactor using C2H2/C2H4/H2 feed mixture (1:67.5:1.5 molar ratio). The long-term deactivation behaviour of the catalyst was studied by differential method of analysis using appropriate power law kinetics for the main reaction and catalyst decay. Based on the analysis of model results, it was proposed that the catalyst deactivation proceeded in two stages. The initial stage (Stage I) was relatively short (<20h) and characterized by rapid catalyst deactivation, followed by second stage (Stage II) of much slower deactivation rate exhibiting a deactivation rate constant one order of magnitude smaller than Stage I. It was attributed to the dual role of carbonaceous deposits on catalyst that exhibited sequential activation and deactivation effects. The TG and FT-IR analysis of the spent catalyst revealed formation of aliphatic (soft) coke after about 100h on stream.

Understanding Photoinduced Charge Transfer Dynamics of Single Perylenediimide Dyes in a Polymer Matrix by Bin-Time Dependence of their Fluorescence Blinking Statistics

Fluorescence on–off blinking of single perylenediimide (DMP–PDI) dyes embedded in a disordered matrix of poly(methyl methacrylate) (PMMA) was investigated by single-molecule fluorescence spectroscopy. In particular, we examined the bin-time dependencies of the complementary cumulative distribution functions (cCDF) of the on-time and off-time durations. It appears that intersystem crossing (ISC) within DMP–PDI competes with charge transfer between DMP–PDI and PMMA. The single-molecule and ensemble cCDFs of the on-time and off-time durations (induced by charge transfer) were best described by Weibull functions and log-normal functions, respectively. The dispersive kinetics of the on-time duration were attributed to radical ion pair ISC from triplet charge separation (3CS) state to the corresponding singlet (1CS) state via charge hopping between charge trap sites in PMMA. This process should compete with triplet charge recombination (3CR), which restores the triplet state. On-to-off switching occurs only when 3CS → 1CS ISC overcomes the 3CR process. The resultant Weibull distribution (with A < 1) reflects that the probability of charge hopping rapidly decreases with time. On the other hand, the off-time log-normal kinetics of single-molecules are explained by a Gaussian distribution of activation energies for charge recombination (the so-called Albery model). A simulation study revealed that the ensemble off-time cCDF included power-law kinetics of charge hopping in the polymer matrix.

Unified kinetics of n-heptane hydroisomerisation over various Pt/zeolite catalysts

Hydroisomerisation of n-heptane has been carried out over six Pt-loaded zeolite-based catalysts. The activity and selectivity of each catalyst under various operating conditions of temperature, pressure and time were studied. Generally, a low pressure and high temperature favoured the overall n-heptane conversion while the selectivity towards isomers decreased at high temperature as the overall conversion increased to a high value. Based on the initial rates, kinetic modelling of the experimental data was carried out using power law kinetics and a simplified dual-site Langmuir–Hinshelwood–Hougen–Watson (LHHW) model. The LHHW approach was found more appropriate in representing the experimental data obtained over each catalyst and a kinetic model equation developed on the basis of surface reaction rate controlling best fitted to the experimental data. The activation energies were found to be in the range 94–143 kJ mol–1.

A comparative study of LHHW and ER kinetic models for NO oxidation over Co3O4 catalyst

Experimental data for the NO oxidation to NO2 at atmospheric pressure over Co3O4 catalyst were subjected to various kinetic rate models. The power law kinetics, Langmuir-Hinshelwood-Hougen-Watson (LHHW) kinetics, and Eley-Rideal (ER) kinetics were employed to suggest a possible reaction mechanism and a suitable kinetic rate equation. The kinetic rate equation based on the LHHW kinetics with adsorption of NO as the rate-controlling step and the kinetic rate equation based on the ER kinetics with surface reaction as the rate-limiting step were first equivalently found as the best-fit kinetic rate equations. In each case similar activation energies were resulted and that the oxygen adsorption was found inhibiting the rate of the oxidation reaction. However, the final discrimination based on the literature findings suggested the ER model as the best-fit kinetic model with activation energy of 58.3kJ/mol and enthalpy of dissociative adsorption of oxygen as −2.10kJ/mol.

Power-law kinetics of methanol synthesis from carbon dioxide and hydrogen on copper–zinc oxide catalysts with alumina or zirconia supports

Kinetics of methanol synthesis from carbon dioxide and hydrogen were studied on two catalysts, a copper–zinc oxide–alumina catalyst (CuZA) and a copper–zinc oxide–zirconia (CuZZ) catalyst. Although both catalysts show similar turnover frequencies for the methanol synthesis reaction, CuZZ is more selective for methanol synthesis because the reverse water gas shift reaction occurs more slowly on this catalyst. The results of the catalytic tests were modeled with power-law equations which highlight the strong positive impact of hydrogen partial pressure on methanol synthesis activity and selectivity. The comparison of the experimental results with thermodynamic equilibrium allows separating thermodynamic and kinetic driving forces.

Kinetics of spreading of synergetic surfactant mixtures in the case of partial wetting

Spreading kinetics of mixtures of hydrocarbon surfactant, octaethylene glycol monododecyl ether, and fluoro-surfactant, Zonyl FSN-100, on highly hydrophobic substrate was experimentally studied. The mixtures reveal a synergism in their wetting properties with equilibrium contact angle of mixtures being 15–20° lower than that of individual surfactant solutions. The synergism is due to different affinity of the surfactants to the liquid/air and liquid/solid interface. Both individual surfactants and their mixtures demonstrate power law kinetics of spreading over the time span of tens of seconds. The spreading exponent is lower than that for pure liquids, but spreading exponent of mixtures is higher than that of individual solutions. The maximum in the spreading exponent is observed for the mixtures demonstrating the lowest equilibrium contact angles. For these mixtures the spreading exponent is close to that of pure liquids.

High-temperature spin dynamics studied by solid-state nuclear resonance and electron paramagnetic resonance in 29Si:B crystals

The relaxation of nuclear magnetization in the 29Si:B crystals obeys power-law kinetics at 300 K due to direct electron–nuclear relaxation. Admixture of the exponential relaxation associated with spin diffusion was revealed at higher temperatures. The inhomogeneous distribution of linear deformation defects was revealed by electron paramagnetic resonance. This factor mainly contributes to power-law kinetics of nuclear spin relaxation.

Chemical reaction network approaches to Biochemical Systems Theory

This paper provides a framework to represent a Biochemical Systems Theory (BST) model (in either GMA or S-system form) as a chemical reaction network with power law kinetics. Using this representation, some basic properties and the application of recent results of Chemical Reaction Network Theory regarding steady states of such systems are shown. In particular, Injectivity Theory, including network concordance [36] and the Jacobian Determinant Criterion [43], a “Lifting Theorem” for steady states [26] and the comprehensive results of Müller and Regensburger [31] on complex balanced equilibria are discussed. A partial extension of a recent Emulation Theorem of Cardelli for mass action systems [3] is derived for a subclass of power law kinetic systems. However, it is also shown that the GMA and S-system models of human purine metabolism [10] do not display the reactant-determined kinetics assumed by Müller and Regensburger and hence only a subset of BST models can be handled with their approach. Moreover, since the reaction networks underlying many BST models are not weakly reversible, results for non-complex balanced equilibria are also needed.

Influence of temperature on the mechanical alloying of Cu–Nb powder mixtures

This study focuses on the mechanical alloying behaviour of equiatomic Cu–Nb powder mixtures subjected to ball milling at temperatures between 300 and 900K. A homogeneous amorphous phase forms at room temperature. The increase of processing temperature promotes the formation of mixtures of amorphous and nanostructured Cu and Nb phases below 650K, and of nanostructured Cu–Nb composites above 650K. The grain size of Cu and Nb phases increases with temperature according to a power-law kinetics.

1-Butanol dehydration in microchannel reactor: Kinetics and reactor modeling

A structured microchannel reactor (MCR) coated with pure γ-Al2O3 was applied to investigate 1-butanol dehydration under atmospheric and isothermal conditions. Kinetic experiments were carried out and the advantages related to mass transfer properties with the application of MCR were explored under different reaction parameters and conditions. It was revealed that with the average coating thickness of 25μm, the operation was free of diffusional limitations and hence intrinsic kinetics could be determined by describing the reactor as pseudohomogeneous PFR with reaction kinetic expressions incorporated. Experimental data was regressed using power-law kinetics with a simplified reaction scheme for 1-butanol dehydration to butenes and dibutylether as well as with mechanistic model involving surface butoxies as relevant species. Both models were able to describe the experimental observations and the estimated values for kinetic parameters were in physically meaningful order of magnitude. A dynamic mathematical model was developed including diffusional mass transport of components and reaction inside coating, whereas, in free channel, plug flow mass transport was considered. The proposed model besides reproducing the experimental results was also able to predict presence of diffusional limitations when coating thickness is increased beyond 40μm. Furthermore, the simulation results show that for specific operation regimes Microchannel Reactor (MCR) outperforms packed-bed reactor, as mass transfer limitations can be appreciably reduced for 1-butanol dehydration.

Power-law versus exponential relaxation of 29Si nucleus spins in Si:B crystals

The Si:B micro-crystals enriched with 29Si isotope have been studied by high resolution nuclear magnetic resonance (NMR) in the 300–800K temperature range. The recovery of nuclear magnetization saturated by radiofrequency impulses follows pure power-law kinetics at 300K, while admixture of exponential relaxation takes place at 500K. The power-law relaxation corresponds to direct electron–nuclear relaxation due to the inhomogeneous distribution of paramagnetic centers, while exponential kinetics corresponds to the nuclear spin diffusion mechanism. The inhomogeneous distribution of deformation defects is a most probable reason of the power-law kinetics of nuclear spin relaxation.

Effect of mineralogy and temperature on atmospheric acid leaching and rheological behaviour of model oxide and clay mineral dispersions

The atmospheric acid leaching mechanism and kinetics and rheological behaviour of model iron oxide (hematite and goethite) and clay (kaolinite and smectite) mineral dispersions were investigated. Isothermal, batch leaching tests were conducted at pH1 for 4h at 25 and 70°C. Both clays displayed incongruent leaching where lattice substituted species (e.g., Na, Ca, Mg, K) leached faster than AlSiO framework species (Al and Si) initially. For the iron oxides, Fe released from goethite was considerably greater than hematite. Temperature elevation to 70°C led to dramatically enhanced leaching rates for the iron oxides but had subtle effect on that of clay minerals. Furthermore, the clays consumed more acid than the iron oxides, reflecting former's greater acid neutralization capacity. During leaching, all dispersions displayed non-Newtonian, shear thinning, rheological behaviour. Generally, lower pulp shear yield stresses (<5Pa) and viscosities (<10mPas) were displayed by the iron oxides compared with clays (5-20Pa and 15–65mPas). Whilst the yield stresses and viscosities of the hematite and kaolinite dispersions were time-independent, those of goethite and smectite increased and attenuated with increasing leaching time, respectively. The leaching mechanism of kaolinite and smectite followed a porous layer, volume diffusion-controlled shrinking core model with activation energies (Ea) of 13.8 and 21.6kJ/mol, respectively. Hematite and goethite leaching mechanism, on the other hand, was both chemical and diffusion-reaction controlled, following an empirical, shrinking particle power law kinetics of order 1.5 with Ea of 28.7 and 24.6kJ/mol, respectively.

Rehydration and Rehydroxylation in Ancient Ceramics: New Constraints from Mass Gain Analyses Versus Annealing Temperatures

We present a series of new mass‐gain data acquired on ancient pottery following heating at increasing temperatures from 60°C to 590°C to further constrain our understanding of the rehydration and rehydroxylation (RHX) processes acting within archeological fired‐clay artifacts. The ancient pottery, deliberately chosen for its low‐porosity, is second century ad Samian ware fragments originating from Lezoux in central France. The mass‐gain data, after each temperature step, adequately define a (time)1/N power law. We demonstrate that values of the 1/N exponent vary systematically with heating temperature. These results suggest that the sorption of chemisorbed and rehydroxylated water into the clay matrix occurs simultaneously. The 1/N exponent values obtained are consistent with power law kinetics resulting from the combination of both one‐dimensional and Brownian diffusion processes. The balance between these two end‐member processes, in particular after heating at 500°C, may depend on the nature of the clay and on the manufacturing conditions of the studied ceramics. The results presented herein confirm previous claims that major complexities vitiate the RHX dating method proposed by Wilson et al.

Cycle life modeling and the capacity fading mechanisms in a graphite/LiNi0.6Co0.2Mn0.2O2 cell

A 3.0 Ah pouch-type lithium-ion cell with a high energy density (200 Wh kg−1) was studied to establish a cycle life model of the battery. The cells consisted of graphite and LiNi0.6Co0.2Mn0.2O2 electrodes, and they were cycled at 1 C over a 100 % depth of discharge at different temperatures (25, 35, and 45 °C). A semi-empirical cycle life model was developed using experimental data by adopting a simple time power law and Arrhenius kinetics. The cycle life model agreed well with the experimental data but featured a much higher power law factor (1.3) compared to the value of 0.5 associated with the general t 1/2 rule based on the rate of solid electrolyte interphase (SEI) growth. The capacity fading mechanisms that operated within the cells were investigated using electrochemical, physical, and chemical diagnostic methods. These results revealed that the loss of active lithium ions via SEI formation as well as electrode degradation, especially cathode degradation, was responsible for the capacity fading in the cells.