Classical Rate Theory Research Articles

Predicting Selectivity with a Bifurcating Surface: Inaccurate Model or Inaccurate Statistics of Dynamics?

Reactions on post-transition-state bifurcation (PTSB) energy surfaces are an important class of reaction in which classical rate theories, such as the transition state theory, fail to account for the selectivity. Quasiclassical trajectory molecular dynamic (QCT-MD) simulation is an important computational approach to understanding reactions mechanisms, especially for reactions that cannot be predicted from conventional rate theories. However, the applicability of direct dynamic simulations is hampered by huge computational costs for collecting a statistically meaningful set of trajectories, making it difficult to compare simulation results with theoretical or physical insights-based predictions (non-MD predictions). In this work, we examine the PTSB of Schmidt-Aubé reactions studied by Tantillo and co-workers. With machine-learning using kernel-ridge regression (KRR) to predict atomic forces, statistical reliability was enhanced by significantly increasing the number of trajectories. With KRR, the bottleneck of simulating dynamics (atomic forces in QCT-MD with density functional theory) was accelerated more than 100-fold. We found that this KRR-aided QCT-MD approach is successful in predicting branching ratios with a much larger number of trajectories. With our approach, statistical errors are greatly reduced, and hypothetical non-MD models for predicting selectivity are tested with much higher confidence. By comparison with non-MD models, dynamical properties that affect branching ratios become more clearly described.

Mechanism of hydrogen isotope exchange for tritium removal in plasma-facing materials: a multi-scale investigation

The safety of future fusion reactors is critically dependent on the tritium (T) retention in plasma-facing materials. Hydrogen isotope (HI) exchange offers a method to redistribute HIs within solid materials, presenting a feasible approach for removing T from bulk materials and trapped by strong trapping sites. Nonetheless, unraveling the intricate mechanism behind HI exchange remains an urgent yet formidable challenge. This study undertakes a comprehensive investigation into the mechanism of HI exchange in tungsten materials across multiple scales. First, we developed a multi-component hydrogen isotope transport and exchange model (HIDTX) based on classical rate theory. The model validation was further carried out, demonstrating good consistency with the well-controlled laboratory experiments. From the results of different comparative models in HIDTX, it is found that the reduction in deuterium retention due to HI exchange was primarily driven by three synergistic effects: competitive re-trapping, collision, and swapping effects. Through molecular dynamics (MD) and first-principles calculations, the microscopic mechanism of HI exchange was revealed to be that the presence of hydrogen atoms in the interstitial sites surrounding a vacancy in tungsten decreased the binding energy between the vacancy and hydrogen. Meanwhile, we discovered that the combination of thermal desorption and HI exchange can significantly lower the temperature required for the hydrogen removal and enhance the removal rate. Particularly, the hydrogen removal time can be shortened by approximately 95% with simultaneous HI exchange compared to that with only thermal desorption. This work provides a practical guideline for comprehending and subsequently designing for efficient T removal in future nuclear fusion materials.

Cavity-induced bifurcation in classical rate theory

We show how coupling an ensemble of bistable systems to a common cavity field affects the collective stochastic behavior of this ensemble. In particular, the cavity provides an effective interaction between the systems, and parametrically modifies the transition rates between the metastable states. We predict that the cavity induces a phase transition at a critical temperature which depends linearly on the number of systems. It shows up as a spontaneous symmetry breaking where the stationary states of the bistable system bifurcate. We observe that the transition rates slow down independently of the phase transition, but the rate modification vanishes for alternating signs of the system-cavity couplings, corresponding to a disordered ensemble of dipoles. Our results are of particular relevance in polaritonic chemistry where the presence of a cavity has been suggested to affect chemical reactions.

Numerical study of critical straight, frown, and chevron incisions in small incision cataract surgery

Introduction: When an intraocular lens (IOL) injector is inserted through a pre-cut corneal incision (e.g., an empirical size of 2.2 mm) during small incision cataract surgery, uncontrollable tearing to the corneal tissue may occur, which is highly associated with the incision shape, size, and location. The goal of this numerical study was to investigate the optimal incision scheme amongst three typical shapes, i.e., straight, frown, and chevron incisions using mechanical modeling and finite element analysis.Methods: Assuming that the damage is caused by the tissue fracture at the incision tips and is governed by the classical energy release rate (ERR) theory which compares the current ERR value subject to IOL injection and the material’s intrinsic parameter, critical ERR Gc.Results: It was found that for chevron incisions, the incision shape with an angle of 170° was superior which induced minimal ERR value, while for frown incisions, the shape with a central angle of 6° was optimal. Both chevron and frown incisions could allow a larger size of injector to inject through than a straight pre-cut. In particular, the frown incision performed the best due to its lowest corresponding ERR and easy operation.Discussion: It was also observed that regions where the embedded fibrils are more dispersed and exhibit high isotropy were more favorable. If necessary, the chevron incision was recommended to be more aligned with the direction exhibiting a larger modulus, for example, along the circumferential direction near the limbus. This study provides useful knowledge in operation design and a deep insight into mechanical damage to corneal wounds in small incision cataract surgery.

Microstructure modeling of nuclear structural materials: Recent progress and future directions

Modeling and simulation of microstructures are essential to understand the complex responses and behaviors of nuclear materials in extreme environments. The needs to assess the extended life operation as well as the growing interest in accelerating nuclear materials development and qualification have stimulated the use of high-fidelity multiscale models aided by empirical and ab initio data. This paper reviews the role of various models across different length and time scales in investigating irradiation effects on microstructure evolution and degradation, in particular the embrittlement caused by radiation induced or enhanced formation of nanoscale chemical heterogeneities. The strength and limitations of these models, including classical rate theories, cluster dynamics, phase-field methods, and atomistic models informed by ab initio energies, are discussed with seminal examples. Challenges regarding the lack of thermo-kinetic data and theoretical treatments considering chemical complexities and magnetic excitations, as well as the stabilizing effect by excess point defects in nuclear structural materials are presented, along with potential solutions based on ab initio informed surrogate energy models and statistical sampling by Monte Carlo simulations. The review then highlights the opportunities to leverage the advantages of different methods by establishing hybrid models by shared variables or coupled codes and applications. Finally, the review concludes with forward-looking remarks on how the use of physics-based models can aid the improvement of machine-learning models of property degradation and vice versa.

Mixed quantum classical reaction rates based on the phase space formulation of the hierarchical equations of motion

In a previous work [J. Chem. Phys. 140, 174105 (2014)], we have shown that a mixed quantum classical (MQC) rate theory can be derived to investigate the quantum tunneling effects in the proton transfer reactions. However, the method is based on the high temperature approximation of the hierarchical equation of motion (HEOM) with the Debye-Drude spectral density, and results in a multistate Zusman type of equation. We now extend this theory to include quantum effects of the bath degrees of freedom. By writing the full HEOM into a multidimensional partial differential equation in phase space, we can define a new reaction coordinate, and the previous method can be generalized to the full quantum regime. The validity of the new method is demonstrated by using numerical examples, including the spin-Boson model, and the double well model for proton transfer reaction. The new method is found to resolve some key problems of the previous theory based on high temperature approximation, including possible numerical instability in long time simulation and wrong rate constant at low temperatures.

Interest Rate Theory in Islamic Economic Perspective

Inflation is a condition in which there is a sharp (absolute) increase in prices that lasts continuously for a long period of time followed by a decline in the real (intrinsic) value of a country's currency. circulating through an increase in bank interest rates. For this reason, the government carries out monetary policy by suppressing the money supply through increasing bank interest rates. However, the interest rate has various kinds, namely classical interest rate theory, neo classical, modern keyness and hicks. Therefore, this study focuses more on interest rates and the theory of interest rates by experts. So that it produces descriptive research.

Measurements of Low Temperature Rate Coefficients for the Reaction of CH with CH2O and Application to Dark Cloud and AGB Stellar Wind Models

Abstract Rate coefficients have been measured for the reaction of CH radicals with formaldehyde, CH2O, over the temperature range of 31–133 K using a pulsed Laval nozzle apparatus combined with pulsed laser photolysis and laser-induced fluorescence spectroscopy. The rate coefficients are very large and display a distinct decrease with decreasing temperature below 70 K, although classical collision rate theory fails to reproduce this temperature dependence. The measured rate coefficients have been parameterized and used as input for astrochemical models for both dark cloud and Asymptotic Giant Branch stellar outflow scenarios. The models predict a distinct change (up to a factor of two) in the abundance of ketene, H2CCO, which is the major expected molecular product of the CH + CH2O reaction.

Xe gas bubbles evolution in UO<sub>2</sub> fuels–A phase field simulation

Owing to the large formation energy of vacancies and inert gas atoms (Xe and Kr) in nuclear fuel (UO2), the thermodynamic equilibrium concentrations of these species are extremely low in the UO2 matrix, which makes it extremely difficult to conduct quantitative study of gas bubbles evolution by phase-field method (PFM). In this study, a more physics based quantitative phase-field model has been proposed. The free energy density of the system was derived according to the principles of thermodynamics and KKS model, the UO2 tri-vacancy, Xe gas atoms and gas bubbles were considered in the system. This model enables one to study the gas bubble growth with extremely low concentrations of vacancy and Xe gas atom in the UO2 matrix. The influence of temperature, vacancy and Xe gas atom generation rates on single and multi-gas bubbles evolution were studied. At high temperature and with high generation rates of vacancies and Xe gas atoms, the gas bubbles had higher growth rate. In addition, the effect of temperature gradient on gas bubble migration was also studied by adding a temperature gradient term in the Cahn-Hillard equations. The gas bubble preferred to migrate to high temperature area. Their shape changed from initial circular shape to a prolate shape along the direction of temperature gradient, which is consistent with the experimental results. The simulation results confirmed the formation of center cavity in the nuclear fuel pellet. The simulation results are consistent with the classical rate theory and experimental observations.

Multi-Phase Field Model of Localized Corrosion Kinetics with Corrosion Products Formation

Localized corrosion is one of the complex forms of corrosion which makes it difficult to detect and design-against. During metal corrosion in the corrosive environment, corrosion products are also formed as a result of electrochemical reactions inside the electrolyte. These products can precipitate on the corroding surface and stop/ slowdown the overall corrosion process. In order to understand this complex phenomenon, a multi-phase field model is proposed to simulate metal corrosion with corrosion products formation. Both anodic (metal oxidation) and cathodic (oxygen reduction) reactions along with other electrochemical reactions in the electrolyte are considered. The free energy of the system is described in terms of its metal ion concentration and the order parameters. Rather than considering linear kinetics (Allen-Cahn equation), inspired from classical rate theory, non-linear (Butler-Volmer) kinetics is considered to describe the temporal evolution of order parameters. The time dependent evolution of ionic species is governed by Nernst-Plank equations while electrostatic potential is governed by Poisson equation. The model results are compared with experimental findings and several examples are presented to show the practical applications of this model. One interesting example, which can be studied in detail with this methodology, is the formation of metal oxide nanowires (zinc/ aluminum oxide).

Correlation between the Arrhenius crossover and the glass forming ability in metallic glasses

The distinctive characteristic of the metallic glass-forming system is that the variation in viscosity with temperature obeys Vogel-Fulcher-Tammann (VFT) relationship in the undercooled state and Arrhenius relationship in the high temperature region. A dimensionless index has thus been proposed based on the Arrhenius-VFT crossover and the classical nucleation rate and growth rate theory to evaluate the glass-forming ability (GFA). The indicator G(a) is expressed with the combination of Tg, the glass transition temperature, Tx, the onset crystallization temperature, Tl, the liquidus temperature, T0, the VFT temperature, and a a constant that could be determined according to the best correlation between G(a) and the critical cooling rate (Rc). Compared with other GFA indexes, G(a) shows the best fit with Rc, with the square of the correlation coefficient (R2) being 0.9238 when a = 0.15 for the 23 various alloy systems concerned about. Our results indicate the crossover in the viscosity variation has key effect on GFA and one can use the index G(a) to predict Rc and GFA for different alloys effectively.

A qualitative quantum rate model for hydrogen transfer in soybean lipoxygenase.

The hydrogen transfer reaction catalysed by soybean lipoxygenase (SLO) has been the focus of intense study following observations of a high kinetic isotope effect (KIE). Today high KIEs are generally thought to indicate departure from classical rate theory and are seen as a strong signature of tunnelling of the transferring particle, hydrogen or one of its isotopes, through the reaction energy barrier. In this paper, we build a qualitative quantum rate model with few free parameters that describes the dynamics of the transferring particle when it is exposed to energetic potentials exerted by the donor and the acceptor. The enzyme's impact on the dynamics is modelled by an additional energetic term, an oscillatory contribution known as "gating." By varying two key parameters, the gating frequency and the mean donor-acceptor separation, the model is able to reproduce well the KIE data for SLO wild-type and a variety of SLO mutants over the experimentally accessible temperature range. While SLO-specific constants have been considered here, it is possible to adapt these for other enzymes.

Modeling thermal spike driven reactions at low temperature and application to zirconium carbide radiation damage

Abstract The development of TEM-visible damage in materials under irradiation at cryogenic temperatures cannot be explained using classical rate theory modeling with thermally activated reactions since at low temperatures thermal reaction rates are too low. Although point defect mobility approaches zero at low temperature, the thermal spikes induced by displacement cascades enable some atom mobility as it cools. In this work a model is developed to calculate “athermal” reaction rates from the atomic mobility within the irradiation-induced thermal spikes, including both displacement cascades and electronic stopping. The athermal reaction rates are added to a simple rate theory cluster dynamics model to allow for the simulation of microstructure evolution during irradiation at cryogenic temperatures. The rate theory model is applied to in-situ irradiation of ZrC and compares well at cryogenic temperatures. The results show that the addition of the thermal spike model makes it possible to rationalize microstructure evolution in the low temperature regime.

Li+ solvation and kinetics of Li+-BF4-/PF6- ion pairs in ethylene carbonate. A molecular dynamics study with classical rate theories.

Using our polarizable force-field models and employing classical rate theories of chemical reactions, we examine the ethylene carbonate (EC) exchange process between the first and second solvation shells around Li+ and the dissociation kinetics of ion pairs Li+-[BF4] and Li+-[PF6] in this solvent. We calculate the exchange rates using transition state theory and correct them with transmission coefficients computed by the reactive flux, Impey, Madden, and McDonald approaches, and Grote-Hynes theory. We found that the residence times of EC around Li+ ions varied from 60 to 450 ps, depending on the correction method used. We found that the relaxation times changed significantly from Li+-[BF4] to Li+-[PF6] ion pairs in EC. Our results also show that, in addition to affecting the free energy of dissociation in EC, the anion type also significantly influences the dissociation kinetics of ion pairing.

Electrochemical Kinetics and Dimensional Considerations at the Nanoscale

A critical understanding of thermodynamics inherent to charge transfer is necessary for scientific insights, as well as for applications ranging from biochemical reactions, electrocatalysis, and charge storage in capacitors and batteries. In this letter, we first examine issues related to conventional classical reaction rate theories, such as the Marcus-Hush models for homogeneous electron transfer, and the extensions by Chidsey to heterogeneous electron kinetics. While the foundational attributes have almost always been reckoned in terms of one-electron based charge transfer, much of the theoretical and experimental analysis has only obliquely referred to dimensionality considerations. We have found a major role of the dimensionality dependent density of states (DOS) in mesoscale and nanoscale materials in modulating electron kinetics. We have then discovered several novel characteristics that have never been reported before, such as, (i) the observation of DOS dependent reaction rate constant and electrical current oscillations, coupled with (ii) the prediction of tests of dimensionality/quasi-dimensionality. We posit a chirality dependent electron transfer rate constant in nanotubes and graphene-like structures, considering the coupling of the energy level spacing in the nanostructured electrode with that in the ambient, e.g., electrolyte. The manifestation of novel features in electron transfer kinetics as a function of dimensionality probes new frontiers in the tunability of electrochemical characteristics. We expect our work to generate very wide interest in the scientific community due to the importance of heterogeneous charge transfer across a variety of disciplines.

Rate theory on water exchange in aqueous uranyl ion

We report a classical rate theory approach to predict the exchange mechanism that occurs between water and aqueous uranyl ion. Using our water and ion-water polarizable force field and molecular dynamics techniques, we computed the potentials of mean force for the uranyl ion-water pair as a function of different pressures at ambient temperature. These potentials of mean force were used to calculate rate constants using transition rate theory; the transmission coefficients also were examined using the reactive flux method and Grote-Hynes approach. The computed activation volumes are positive; thus, the mechanism of this particular water-exchange is a dissociative process.

Rate theory of solvent exchange and kinetics of Li(+) - BF4 (-)/PF6 (-) ion pairs in acetonitrile.

In this paper, we describe our efforts to apply rate theories in studies of solvent exchange around Li(+) and the kinetics of ion pairings in lithium-ion batteries (LIBs). We report one of the first computer simulations of the exchange dynamics around solvated Li(+) in acetonitrile (ACN), which is a common solvent used in LIBs. We also provide details of the ion-pairing kinetics of Li(+)-[BF4] and Li(+)-[PF6] in ACN. Using our polarizable force-field models and employing classical rate theories of chemical reactions, we examine the ACN exchange process between the first and second solvation shells around Li(+). We calculate exchange rates using transition state theory and weighted them with the transmission coefficients determined by the reactive flux, Impey, Madden, and McDonald approaches, and Grote-Hynes theory. We found the relaxation times changed from 180 ps to 4600 ps and from 30 ps to 280 ps for Li(+)-[BF4] and Li(+)-[PF6] ion pairs, respectively. These results confirm that the solvent response to the kinetics of ion pairing is significant. Our results also show that, in addition to affecting the free energy of solvation into ACN, the anion type also should significantly influence the kinetics of ion pairing. These results will increase our understanding of the thermodynamic and kinetic properties of LIB systems.

Analytical models of calcium binding in a calcium channel.

The anomalous mole fraction effect of L-type calcium channels is analyzed using a Fermi like distribution with the experimental data of Almers and McCleskey [J. Physiol. 353, 585 (1984)] and the atomic resolution model of Lipkind and Fozzard [Biochemistry 40, 6786 (2001)] of the selectivity filter of the channel. Much of the analysis is algebraic, independent of differential equations. The Fermi distribution is derived from the configuration entropy of ions and water molecules with different sizes, different valences, and interstitial voids between particles. It allows us to calculate potentials and distances (between the binding ion and the oxygen ions of the glutamate side chains) directly from the experimental data using algebraic formulas. The spatial resolution of these results is comparable with those of molecular models, but of course the accuracy is no better than that implied by the experimental data. The glutamate side chains in our model are flexible enough to accommodate different types of binding ions in different bath conditions. The binding curves of Na(+) and Ca(2+) for [CaCl2] ranging from 10(-8) to 10(-2) M with a fixed 32 mM background [NaCl] are shown to agree with published Monte Carlo simulations. The Poisson-Fermi differential equation-that includes both steric and correlation effects-is then used to obtain the spatial profiles of energy, concentration, and dielectric coefficient from the solvent region to the filter. The energy profiles of ions are shown to depend sensitively on the steric energy that is not taken into account in the classical rate theory. We improve the rate theory by introducing a steric energy that lumps the effects of excluded volumes of all ions and water molecules and empty spaces between particles created by Lennard-Jones type and electrostatic forces. We show that the energy landscape varies significantly with bath concentrations. The energy landscape is not constant.

Spectral Rate Theory for Two-State Kinetics.

Classical rate theories often fail in cases where the observable(s) or order parameter(s) used is a poor reaction coordinate or the observed signal is deteriorated by noise, such that no clear separation between reactants and products is possible. Here, we present a general spectral two-state rate theory for ergodic dynamical systems in thermal equilibrium that explicitly takes into account how the system is observed. The theory allows the systematic estimation errors made by standard rate theories to be understood and quantified. We also elucidate the connection of spectral rate theory with the popular Markov state modeling approach for molecular simulation studies. An optimal rate estimator is formulated that gives robust and unbiased results even for poor reaction coordinates and can be applied to both computer simulations and single-molecule experiments. No definition of a dividing surface is required. Another result of the theory is a model-free definition of the reaction coordinate quality. The reaction coordinate quality can be bounded from below by the directly computable observation quality, thus providing a measure allowing the reaction coordinate quality to be optimized by tuning the experimental setup. Additionally, the respective partial probability distributions can be obtained for the reactant and product states along the observed order parameter, even when these strongly overlap. The effects of both filtering (averaging) and uncorrelated noise are also examined. The approach is demonstrated on numerical examples and experimental single-molecule force-probe data of the p5ab RNA hairpin and the apo-myoglobin protein at low pH, focusing here on the case of two-state kinetics.

About a deficit in low-order convergence rates on the example of autoconvolution

We revisit in -spaces the autoconvolution equation with solutions which are real-valued or complex-valued functions defined on a finite real interval, say . Such operator equations of quadratic type occur in physics of spectra, in optics and in stochastics, often as part of a more complex task. Because of their weak nonlinearity, deautoconvolution problems are not seen as difficult and hence little attention is paid to them wrongly. In this paper, we will indicate on the example of autoconvolution a deficit in low-order convergence rates for regularized solutions of nonlinear ill-posed operator equations with solutions in a Hilbert space setting. So, for the real-valued version of the deautoconvolution problem, which is locally ill-posed everywhere, the classical convergence rate theory developed for the Tikhonov regularization of nonlinear ill-posed problems reaches its limits if standard source conditions using the range of fail. On the other hand, convergence rate results based on Hölder source conditions with small Hölder exponent and logarithmic source conditions or on the method of approximate source conditions are not applicable since qualified nonlinearity conditions are required which cannot be shown for the autoconvolution case according to current knowledge. We also discuss the complex-valued version of autoconvolution with full data on and see that ill-posedness must be expected if unbounded amplitude functions are admissible. As a new detail, we present situations of local well-posedness if the domain of the autoconvolution operator is restricted to complex -functions with a fixed and uniformly bounded modulus function.