Landau-Teller Model Research Articles

Experimental and numerical studies on the thermal nonequilibrium behaviors of CO with Ar, He, and H2

The time-dependent rotational and vibrational temperatures were measured to study the shock-heated thermal nonequilibrium behaviors of CO with Ar, He, and H2 as collision partners. Three interference-free transition lines in the fundamental vibrational band of CO were applied to the fast, in situ, and state-specific measurements. Vibrational relaxation times of CO were summarized over a temperature range of 1110–2820 K behind reflected shocks. The measured rotational temperature instantaneously reached an equilibrium state behind shock waves. The measured vibrational temperature experienced a relaxation process before reaching the equilibrium state. The measured vibrational temperature time histories were compared with predictions based on the Landau–Teller model and the state-to-state approach. The state-to-state approach treats the vibrational energy levels of CO as pseudo-species and accurately describes the detailed thermal nonequilibrium processes behind shock waves. The datasets of state-specific inelastic rate coefficients of CO–Ar, CO–He, CO–CO, and CO–H2 collisions were calculated in this study using the mixed quantum-classical method and the semiclassical forced harmonic oscillator model. The predictions based on the state-to-state approach agreed well with the measured data and nonequilibrium (non-Boltzmann) vibrational distributions were found in the post-shock regions, while the Landau–Teller model predicted slower vibrational temperature time histories than the measured data. Modifications were applied to the Millikan–White vibrational relaxation data of the CO–Ar and CO–H2 systems to improve the performance of the Landau–Teller model. In addition, the thermal nonequilibrium processes behind incident shocks, the acceleration effects of H2O on the relaxation process of CO, and the characterization of vibrational temperature were highlighted.

Simulations of CO2 multi-temperature vibrational kinetics on the basis of new relaxation time models

Vibrational relaxation of a single-component carbon dioxide in adiabatic thermal bath is studied in multi-temperature approaches using several models for the vibrational relaxation time. The energy production terms in two- and three-temperature vibrational energy relaxation equations are written in the frame of the common Landau-Teller model as well using corrections based on introducing multi-temperature relaxation times for various vibrational energy exchanges within and between CO2 modes. For the relaxation time evaluation, two theoretical models for the transition probabilities are implemented: the first-order perturbation theory and the forced harmonic oscillator model; several experimental data sets are also assessed. It is shown that the forced harmonic oscillator model provides satisfactory agreement with experimental relaxation times in a wide temperature range. Solutions obtained using the two theoretical models for the transition probabilities show essentially different trends, incubation time and relaxation rate.

Linear stability analysis of one-dimensional detonation coupled with vibrational relaxation

The linear stability of one-dimensional detonations with one-reaction chemistry coupled with molecular vibration nonequilibrium is investigated using the normal mode approach. The chemical kinetics in the Arrhenius form depend on an averaged temperature model that consists of translational–rotational mode and vibrational mode. The Landau–Teller model is applied to specify the vibrational relaxation. A time ratio is introduced to denote the ratio between the chemical time scale and the vibrational time scale in this study, which governs the vibrational relaxation rate in this coupling kinetics. The stability spectrum of disturbance eigenmodes is obtained by varying the bifurcation parameters independently at a different time ratio. These parameters include the activation energy, the degree of overdrive, the characteristic vibrational temperature, and the heat release. The results indicate that the neutral stability limit shifts to higher activation energy on the vibrational nonequilibrium side with a smaller time ratio, implying that the detonation is stabilized. A similar observation is seen at a lower degree of overdrive. Compared with the above two parameters, the characteristic vibrational temperature plays a minor role in the stabilization of detonation, and no change in the number of eigenmodes is identified throughout the selected range. By plotting the neutral stability curves relating the heat release to the above parameters, the decreases in instability ranges are obviously seen under vibrational nonequilibrium. The thermal nonequilibrium effect on detonation stability is clearly demonstrated. The analysis presented in this paper is ultimately justified by comparing the results with numerical simulation.

Vibrational–translational relaxation in nitrogen discharge plasmas: Master equation modeling and Landau–Teller model revisited

Vibrational–translational (V–T) relaxation is quite common in molecular nitrogen discharge non-equilibrium plasmas. In this paper, the energy relaxation in V–T transition is investigated by master equation modeling on all vibrational levels below the dissociation limit. The state-to-state transition rates are calculated by a forced harmonic oscillator (HO)-free rotating model. Meanwhile, the classic Landau–Teller model based on the HO of vibrational levels is revisited. First, the V–T relaxation in a heat bath is compared between the HO model, Morse’s anharmonic oscillator (AHO) model, and realistic vibrational levels by a direct-potential-fit analysis of spectroscopic data. The relaxation of average vibrational energy using the AHO model is faster than that using the HO model. Then, the influence of more frequent vibrational–vibrational (V–V) collision on the V–T transition in the heat bath is investigated by using different numbers of vibrational levels. The anharmonic effect is significant with more vibrational levels. Finally, the V–T energy transfer is modeled by a coupled solution to master equations and gas heating. The stronger the non-equilibrium between vibrational and translational temperature in the beginning, the larger the difference that can be obtained between the HO model (Landau–Teller theory) and realistic vibrational levels.

Improvement of the Landau-Teller model for CO2 on the basis of the Chapman–Enskog method

Vibrational relaxation of carbon dioxide in the three-temperature kinetic-theory approach is studied. The focus is on the evaluation of the vibrational energy production terms in the coupled symmetric-bending and asymmetric modes. Using the generalized Chapman – Enskog method, the energy production terms are expressed as functions of the gas temperature, vibrational temperatures of CO2 modes and probabilities of corresponding transitions. It is shown that a modification of the Landau–Teller formula can be obtained from the rigorous theoretical expressions using several limiting assumptions. Whereas VT relaxation in the bending mode can be described by the Landau–Teller approximation with a satisfactory accuracy, its application to the inter-mode exchange requires introducing relaxation times depending on the vibrational temperatures of the modes involved to the transitions.

Numerical simulation of a CO2, CO, O2, O, C mixture: Validation through comparisons with results obtained in a ground-based facility and thermochemical effects

Various issues of numerical simulation of high-enthalpy non-equilibrium flows of a gas mixture of CO2, CO, O2, O, and C are discussed. Navier–Stokes equations are considered within the framework of one-, two-, and three-temperature approximations with allowance for chemical reactions. In the one-temperature approximation, chemical reactions are modelled by the Arrhenius law. In the multi-temperature approximations, simulations are performed with the use of the Treanor–Marrone model for carbon dioxide dissociation and the Park model for dissociation of diatomic gas molecules, which take into account vibrational non-equilibrium of molecules. The Arrhenius law is also used for calculating recombination reactions and also forward and backward exchange reactions. Translational-vibrational and vibrational-vibrational energy exchange is simulated by a simplified Landau–Teller model. Some examples of flows with the use of the above-mentioned approximations are calculated. It is demonstrated that the one-temperature approach ensures an adequate reproduction of experimental data on the shock standoff distance, heat flux, and pressure coefficient on the cone, which are obtained in a high-enthalpy wind tunnel. The calculation of the flow around a cylinder under Martian re-entry conditions reveals a significant effect of thermochemical non-equilibrium.

Dual-pulse laser ignition model

A two-dimensional mathematical model of dual-pulse laser ignition that self-consistently integrates Navier-Stokes, translational and vibrational energy, and neutral and charged species equations has been presented. The results showed that the ignition kernel dynamics depends on the shape and initial energy distribution in the energy spot created by the first ultraviolet laser pulse. The results also suggest that the ignition delay time and the flame kernel development depend on the laser intensity, vibrational-nonequilibrium, and initial electron number density. For the high initial degree of ionization, we have obtained ignition of the lean methane–air mixture with the equivalence ratio of 0.6. Vibrational-nonequilibrium taken into account by the Landau-Teller model leads to the slower ignition kernel growth and the significant increase in the ignition delay time. For the case modeled, a change in the overlap of the second laser with the focal point of the first laser pulse leads to the early split up of the kernel and the flame extinguishment.

Chemical reaction mechanism related vibrational nonequilibrium effect on the Zel'dovich−von Neumann−Döring (ZND) detonation model

Discrepancies in detonation cell size are always found between experiments and numerical simulations. The recent numerical simulation by Shi et al. (2016) included the vibrational relaxation effect in molecules and revealed that the vibrational nonequilibrium effect may be one of the reasons for the above discrepancy. In this study, a steady one-dimensional Zel'dovich–von Neumann–Döring (ZND) model is modified to account for the vibrational nonequilibrium effect in gaseous detonation. By introducing a new parameter α, which is defined as the ratio of translational-rotational mode of the specific heat at constant volume to the overall specific heat at constant volume, the translational-rotational energy and the vibrational energy are treated separately in the energy conservation equation. Both the single step and the two step Arrhenius models are used to describe the chemical reaction progress, and the Landau–Teller model and Park's two-temperature model are applied to describe the vibrational energy relaxation and the coupling between the molecular vibration and the chemical reaction, respectively. The parametric study on the modified ZND model is conducted and compared with the conventional ZND one, including the activation energy in chemical model, the ratio of the chemical time scale to the vibrational relaxation time scale and the characteristic vibrational temperature. The results show that the half reaction length is relatively not sensitive to the variation of characteristic vibrational temperature applied in the model (compared with the activation energy and the time ratio), but is increased to a significant extent with the increase of the activation energy and the decrease of the time ratio, when considered the vibrational nonequilibrium. The simulated half reaction lengths in the modified model are in good agreement with those in the conventional model, if the chemical time scale is larger than the vibrational relaxation time scale above a critical ratio when the vibrational nonequilibrium effect is insignificant.

Estimating heat transfer coefficients for solid–gas interfaces using the Landau–Teller model

In this work, a new approach to estimate heat transfer coefficients for solid–gas interfaces is proposed. The new formulation takes into account that the photon transfer process between solid walls and gases is not instantaneous, but involves electronic transitions, even when low frequency radiation is exchanged. The delay required to electron cloud transitions, called relaxation time, which plays a crucial role in the heat transfer process, is evaluated using the Landau–Teller model.

Back-Influence of Molecular Motion on Energy Transfer in the Landau-Teller Model of Atom Molecule Scattering.

This year we celebrate the 80th anniversary of the Landau-Teller model for energy exchange in a collinear collision of an atom with a harmonic diatomic molecule. Even after 80 years though, the analytic theory to date has not included in it the back-influence of the oscillator's motion on the energy transfer between the approaching particle and the molecule. This is the topic of the present paper. The back-influence can be obtained by employing classical second-order perturbation theory. The second-order theory is used in both a classical and semiclassical context. Classically, analytic expressions are derived for the final phase and action of the diatom, after the collision. The energy loss of the atom is shown to decrease linearly with the increasing energy of the oscillator. The magnitude of this decrease is a direct consequence of the back-reaction of the oscillator on the translational motion. The qualitative result is universal, in the sense that it is not dependent on the details of the interaction of the atom with the oscillator. A numerical application to a model collision of an Ar atom with a Br2 diatom demonstrates the importance and accuracy of the second-order perturbation theory. The same results are then used to derive a second-order perturbation theory semiclassical expression for the quantum transition probability from initial vibrational state ni to final vibrational state nf of the oscillator. A comparison of the theory with exact quantum data is presented for a model collision of Br2 with a hydrogen molecule, where the hydrogen molecule is considered as a single approaching particle.

Electron-vibration relaxation in oxygen plasmas

An ideal chemical reactor model is used to study the vibrational relaxation of oxygen molecules in their ground electronic state, X3Σg-, in presence of free electrons. The model accounts for vibrational non-equilibrium between the translational energy mode of the gas and the vibrational energy mode of individual molecules. The vibrational levels of the molecules are treated as separate species, allowing for non-Boltzmann distributions of their population. The electron and vibrational temperatures are varied in the range [0–20,000]K. Numerical results show a fast energy transfer between oxygen molecules and free electron, which causes strong deviation of the vibrational distribution function from Boltzmann distribution, both in heating and cooling conditions. Comparison with Landau–Teller model is considered showing a good agreement for electron temperature range [2000–12,000]K. Finally analytical fit of the vibrational relaxation time is given.

Advanced models for vibrational–chemical coupling in multi-temperature flows

In this paper, self-consistent models for coupled vibrational relaxation and dissociation in multi-temperature gas mixture flows are proposed on the basis of the kinetic theory methods. The Treanor–Marrone dissociation model is generalized taking into account the dependence of model parameter on the vibrational state. Multi-temperature dissociation rate coefficients are calculated on the basis of the improved Treanor–Marrone model using the Boltzmann and Treanor vibrational distributions; comparison with traditional models shows a significant difference in the dissociation rate coefficients for high temperatures. Generalization of the well known Landau–Teller model overcoming limitations of the original model is proposed. The developed models are validated against experimental data and assessed in non-equilibrium shock heated flows of O2/O and N2/N mixtures. It is shown that the proposed models provide a good accuracy in the wide range of flow conditions while being simple and computationally efficient.

Reaction and internal energy relaxation rates in viscous thermochemically non-equilibrium gas flows

In the present paper, reaction and energy relaxation rates as well as the normal stress are studied for viscous gas flows with vibrational and chemical non-equilibrium. Using the modified Chapman-Enskog method, multi-temperature models based on the Treanor and Boltzmann vibrational distributions are developed for the general case taking into account all kinds of vibrational energy transitions, exchange reactions, dissociation, and recombination. Integral equations specifying the first-order corrections to the normal mean stress and reaction rates are derived, as well as approximate systems of linear equations for their numerical computation. Generalized thermodynamic driving forces associated with all non-equilibrium processes are introduced. It is shown that normal stresses and rates of non-equilibrium processes can be expressed in terms of the same driving forces; the symmetry of kinetic coefficients in these expressions is proven. The developed general model is applied to a particular case of a pure N2 viscous flow with slow VT relaxation. Normal stress and rates of vibrational relaxation are studied for various ratios of vibrational and translational temperatures. The cross effects between different vibrational transitions in viscous flows are evaluated, along with the influence of anharmonicity and flow compressibility on the first-order corrections to the relaxation rate. Limits of validity for the widely used Landau–Teller model of vibrational relaxation are indicated.

Energy transfer models in nitrogen plasmas: analysis of N₂(X¹Σg⁺)-N(⁴S(u))-e⁻ interaction.

The relaxation of N₂(X¹Σg⁺) molecules in a background gas composed of N((4)S(u)) atoms and free electrons is studied by using an ideal isochoric and isothermic chemical reactor. A rovibrational state-to-state model is developed to study energy transfer process induced by free electron and atomic collisions. The required cross sections and the corresponding rate coefficients are taken from two well-known kinetic databases: NASA Ames kinetic mechanism for the description of the N₂(X¹Σg⁺)-N((4)S(u)) processes and the Phys4Entry database for the electron driven processes, N₂(X¹Σg⁺)-e(-). The evolution of the population densities of each individual rovibrational level is explicitly determined via the numerical solution of the master equation for temperatures ranging from 10000 to 30,000 K. It was found that the distribution of the rovibrational energy levels of N₂(X¹Σg⁺) is strongly influenced by the electron driven collisional processes, which promote the excitation of the low lying vibrational levels. The macroscopic vibrational energy relaxation is governed by the molecule-atom collisions, when free electrons, initially cold are relaxing to the final heat-bath temperature. Thus, the main role of the free electrons is to ensure the equilibration of vibrational and free electron excitation, thus validating the existence of the local equilibrium T(V)-T(e). However, if electrons and heavy particles are assumed to be in equilibrium at the heat bath temperature, electron driven processes dominate the vibrational relaxation. Finally, we have assessed the validity of the Landau-Teller model for the description of the inelastic energy transfer between molecules and free electrons. In the case of free-electron temperatures lower than 10,000 K, Landau-Teller relaxation model gives an accurate description of the vibrational relaxation, while at higher temperatures the error in the predictions can be significant and the model should not be used.

Computational Investigation of Thermal Nonequilibrium Effects in Scramjet Geometries

To characterize the impact of thermal nonequilibrium in shock-induced combustion scramjets, several thermal equilibrium and nonequilibrium computational fluid dynamics simulations have been performed. Specifically, the effects that would be encountered in shock-tunnel testing are of interest. Therefore, thermal equilibrium and nonequilibrium simulations are run for a Mach 6 shock-tunnel nozzle and a radical farming scramjet model with total enthalpies between 3.3 and . Simulating both the nozzle and scramjet with thermal nonequilibrium represents the shock-tunnel test, whereas equilibrium inflow into a scramjet simulated in thermal nonequilibrium provides the scenario encountered in actual flight. For all nonequilibrium simulations, the Landau–Teller model in combination with modified Millikan and White thermal relaxation coefficients is used. It is shown that modeling the entire flow as being in thermal equilibrium results in higher effective temperatures within the local high-temperature regions where ignition occurs, which causes the ignition length to decrease and thus the combustion process to be more efficient compared with the nonequilibrium cases. Additionally, the conditions for ignition in simulations representing shock-tunnel experiments are found to be more favorable than in the simulations representing a flight test, assuming the same enthalpy and boundary conditions. Furthermore, these differences are found to be insensitive to the total enthalpy.

Revised model coefficients for vibrational relaxation in a nitrogen–oxygen gas mixture

A numerical investigation of thermal non-equilibrium flows requires species specific relaxation rates, which are often calculated using the Landau–Teller model. This model requires the determination of collision specific relaxation times, which can be computed using Millikan and White’s empirical formula. The coefficients used in this formula for each specific collision pair form a set of coefficients, which are assessed here. The focus of the investigation lies on their performance in hypersonic low-temperature (300–2,500 K) flows that occur at shock-tunnel nozzle exits or in supersonic combustion ramjets (scramjets) before combustion. Two experimental validation cases are chosen; a shock-tunnel nozzle and a sharp cone in hypersonic cross-flow experiment. A comparison of the experimentally measured vibrational temperatures at the nozzle exit against numerical data shows large discrepancies for two commonly used coefficient sets. A revised set of coefficients is proposed that greatly improves the agreement between the numerical and experimental results. Furthermore, the numerically generated shock shape over the sharp cone using the revised set of coefficients correlates well with the experimental measurements.

70 years of Landau–Teller theory for collisional energy transfer. Semiclassical three-dimensional generalizations of the classical collinear model

This article, in historical retrospective, describes the development of the celebrated Landau-Teller (LT) model of 1936 for vibrational-translational energy exchange in collisions of an atom with a diatomic molecule. We discuss semiclassical generalizations of the classical LT model and generalizations of the collinear LT model to account for the effects of rotation of the diatom on the vibrational relaxation rate. The former is based on the recovery of the Landau semiclassical exponent from the classical LT encounter time, and the latter on the definition of a 1-D driving mode within the manifold of the translational and rotational degrees of freedom of the colliding partners. The utility of generalized LT models is illustrated by three case studies that exemplify weak and strong effects of the rotation as well as the efficiencies of different driving modes in the vibrational relaxation of highly asymmetric diatoms.

Lévy flights in the Landau-Teller model of molecular collisions.

We consider the Landau-Teller model, which is a prototype for the exchanges of energy, in molecular collisions, between internal degrees of freedom and those of the center of mass. We show that the statistics of the energy exchanges computed through the dynamics over a finite time is of the Lévy type for high enough frequencies of the internal motions, while it reduces to the familiar Gaussian one in the limit of low frequencies. The relevance for the definition of the times of relaxation to equilibrium is also pointed out.

A Study of Vibrational Relaxation of B-State Carbon Monoxide in the Heme Pocket of Photolyzed Carboxymyoglobin

The vibrational energy relaxation of dissociated carbon monoxide in the heme pocket of sperm whale myoglobin has been studied using equilibrium molecular dynamics simulation and normal mode analysis methods. Molecular dynamics trajectories of solvated myoglobin were run at 300K for both the δ- and ∈-tautomers of the distal histidine, His64. Vibrational population relaxation times were estimated using the Landau–Teller model. For carbon monoxide (CO) in the myoglobin ∈-tautomer, for a frequency of ω0=2131cm−1 corresponding to the B1 state, T1∈(B1)=640±185 ps, and for a frequency of ω0=2119cm−1 corresponding to the B2 state, T1∈(B2)=590±175 ps. Although the CO relaxation rates in both the ∈- and δ-tautomers are similar in magnitude, the simulations predict that the vibrational relaxation of the CO is faster in the δ-tautomer. For CO in the myoglobin δ-tautomer, it was found that the relaxation times were identical within error for the two CO substate frequencies, T1δ(B1)=335±115 ps and T1δ(B2)=330±145 ps. These simulation results are in reasonable agreement with experimental results of Anfinrud and coworkers (unpublished results). Normal mode calculations were used to identify the dominant coupling between the protein and CO molecules. The calculations suggest that the residues of the myoglobin pocket, acting as a first solvation shell to the CO molecule, contribute the primary “doorway” modes in the vibrational relaxation of the oscillator.

Level-dependent damping in intermolecular vibrations: Linear spectroscopy

A treatment of stimulated Raman intermolecular spectroscopy is presented that employs a Landau–Teller model of damping. This model incorporates a quantum-number dependence to population relaxation and pure dephasing, thereby introducing a specific temperature and frequency dependence into the damping in the intermolecular spectrum. Optical-heterodyne detected Raman-induced Kerr effect data obtained in CS2 and acetonitrile over a broad temperature range are shown to agree with the basic predictions of the model.