Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Marcus equation”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Numerical methods for fractional Fokker–Planck equation with multiplicative Marcus Lévy noises

The Fokker–Planck equation (FPE) is an important deterministic tool for investigating stochastic dynamical systems. In this paper, we consider the space-time fractional FPE driven by multiplicative Marcus Lévy noises. Efficient numerical schemes are presented to solve the equations. Stability and convergence of the methods are also discussed. We give some numerical experiments to validate our schemes, and examine the effects of parameters on solutions. Additionally, we analyze the maximal likely trajectories and the critical time for the change of the most probability location.

Mathematics↗
DOE OSTI2021

The Reaction of o-Benzyne with Vinylacetylene: An Unexplored Way to Produce Naphthalene

Herein, the mechanisms and kinetics of the reaction of ortho-benzyne with vinylactylene have been studied by ab initio and density functional CCSD(T)-F12/cc-pVTZ-f12//B3LYP/6-311G(d,p) calculations of the pertinent potential energy surface combined with Rice-Ramsperger-Kassel-Marcus - Master Equation calculations of reaction rate constants at various temperatures and pressures. Under prevailing combustion conditions, the reaction has been shown to predominantly proceed by the biradical acetylenic mechanism initiated by the addition of C4H4 to one of the C atoms of the triple bond in ortho-benzyne by the acetylenic end, with a significant contribution of the concerted addition mechanism. Following the initial reaction steps, an extra six-membered ring is produced and the rearrangement of H atoms in this new ring leads to the formation of naphthalene, which can further dissociate to 1- or 2-naphthyl radicals. The o-C 6 H 4 + C 4 H 4 reaction is highly exothermic, by ~143 kcal/mol to form naphthalene and by 31-32 kcal/mol to produce naphthyl radicals + H, but features relatively high entrance barriers of 9-11 kcal/mol. Although the reaction is rather slow, much slower than the reaction of phenyl radical with vinylacetylene, it forms naphthalene and 1- and 2-naphthyl radicals directly, with their relative yields controlled by the temperature and pressure, and thus represents a viable source of the naphthalene core under conditions where ortho-benzyne and vinylacetylene are available.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Formation of phenanthrenyl radicals via the reaction of acenaphthyl with acetylene

Ab initio G3(MP2,CC)/B3LYP/6–31G** calculations of the potential energy surface for the reaction of 3-acenaphthyl radical with acetylene were combined with Rice-Ramsperger-Kassel-Marcus Master Equation calculations of the temperature- and pressure-dependent reaction rate constants and product branching ratios with the goal to examine the transformation of an edge five-membered ring into a six-membered one. The reaction mechanism has been shown to contain two stages: (1) the formation of 3-ethynylacenaphthylene + H in the 3-acenaphthyl + C 2 H 2 reaction via acetylene addition to the radical site and an H loss, (2) the formation of the 1-phenanthrenyl radical in the 3-ethynylacenaphthylene + H reaction via H addition to the five-membered ring followed by the ring flip. The results of kinetic modeling show this pathway being faster than the pathway proposed earlier featuring the CCH chemisorption on the six-membered ring without involvement of the five-membered ring. A kinetic test demonstrated the dominance of the former pathway for the formation of pyrene from naphthalene. Here, it is expected that the new mechanism plays a major role in the transformation of five-membered rings on edges of PAH structures.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Formation of Phenanthrene via Recombination of Indenyl and Cyclopentadienyl Radicals: A Theoretical Study

This work presents quantum chemical G3(MP2,CC)//B2PLYPD3/6-311G(d,p) calculations of the potential energy surface for the indenyl (C 9 H 7 ) + cyclopentadienyl (C 5 H 5 ) reaction followed by unimolecular decomposition of the C 14 H 11 radicals formed as the primary products, as well as the Rice–Ramsperger–Kassel–Marcus master equation (RRKM-ME) calculations to predict temperature- and pressure-dependent reaction rate constants and product branching ratios. The reaction begins with the barrierless recombination of indenyl and cyclopentadienyl forming a C 14 H 12 molecule with a new C–C bond connecting two five-membered rings, which subsequently dissociates to C 14 H 11 radicals by H losses. The primary products of the C 9 H 7 + C 5 H 5 → C 14 H 11 + H reaction can directly decompose by another H loss to benzofulvalene, and this pathway is most favorable in terms of the entropy factor and hence is preferable at higher temperatures. Otherwise, the initial C 14 H 11 isomers can undergo significant structural rearrangements before eliminating an H atom and producing phenanthrene, anthracene, or benzoazulenes, among which the formation of phenanthrene via the “spiran” pathway is clearly preferred. The calculated barriers along the computed favorable dissociation pathways are relatively low, in the ~30–40 kcal/mol range, making the C 14 H 11 radicals unstable at temperatures above 1000–1250 K at 1 atm. The results of RRKM-ME calculations show that, under typical combustion conditions, the decomposition of the C 14 H 11 radicals predominantly leads to benzofulvalene. However, the latter can be rapidly converted to phenanthrene via H-assisted isomerization with the rate constant for the benzofulvalene + H → phenanthrene + H reaction being close to 10 –11 cm 3 molecule –1 s –1 at 1000–1500 K and 1 atm. Here, the results provide further support for the hypothesis that recombination of two π radicals containing five-membered rings can lead to a growth of PAH with the formation of two fused six-membered rings, but the reaction mechanism may not be direct and is likely to involve two consecutive H atom losses leading to a fulvalene-like product, with subsequent H-assisted isomerization of the latter to a benzenoid PAH.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Transformation of an Embedded Five-Membered Ring in Polycyclic Aromatic Hydrocarbons via the Hydrogen-Abstraction–Acetylene-Addition Mechanism: A Theoretical Study

Five-membered rings are constituents of many polycyclic aromatic hydrocarbons (PAHs) and their presence on the edges of large PAHs has been repeatedly observed experimentally. However, modern kinetic combustion models often do not consider the growth of PAHs through the transformation of the five-membered rings. Here, in connection with the above, we carried out a theoretical study of the mechanism of Hydrogen-Abstraction-aCetylene-Addition (HACA) transformation of an embedded five-membered ring on the armchair PAH edge to a six-membered ring, considering cyclopenta[d,e,f]phenanthrene (4,5-methylenephenanthrene) as a prototype system for this process. The potential energy surface for the reactions of cyclopenta[d,e,f]phenanthrenyl radicals produced by direct H abstractions from cyclopenta[d,e,f]phenanthrene with acetylene has been compiled at the G3(MP2,CC)//B3LYP/6-311G(d,p) level of theory including zero-point vibrational energy corrections. The computed energies and molecular parameters were then used to solve the Rice-Ramsperger-Kassel-Marcus Master Equation (RRKM-ME) in order to calculate reaction rate at various pressures and temperatures, which were fitted to modified Arrhenius equation for further kinetic modeling. The results show that the HACA transformation of the embedded five-membered ring to a six-membered ring is possible, albeit slow. The most viable reaction mechanism involves the R2 + C 2 H 2 reaction, where the acetylene molecules adds to a σ-radical in the six-membered ring adjacent to the five-membered ring via a low entrance barrier. The predominant product of R2 + C 2 H 2 is predicted to be 3 ethynyl 4H cyclopenta[def]phenanthrene Pr5 via immediate H elimination from the initial addition complex. Next, Pr5 undergoes H-assisted isomerization to 4aH-pentaleno[4,3,2,1-cdef]phenanthrene Pr4, and the latter adds an H atom eventually forming the 1-pyrenylmethyl radical Pr3: R2 + C 2 H 2 ⇌ 3 ethynyl 4H cyclopenta[def]phenanthrene (Pr5) + H ⇌ 4aH-pentaleno[4,3,2,1-cdef]phenanthrene (Pr4) + H Pr5 + H ⇌ Pr4 + H Pr4 + H → 1-pyrenylmethyl (Pr3) This HACA sequence may be competitive with the methyl radical addition to the R1 radical formed by H abstraction from the CH 2 group in the five-membered ring of cyclopenta[d,e,f]phenanthrene, which provides a pathway to pyrene following two H atom losses. Relative contributions of the two mechanisms of the five- to six-membered ring transformation would strongly depend on the branching ratios of the R1 and R2 radicals produced by the H abstractions and the available concentration of C 2 H 2 vs. CH 3 and hence differ in different flames.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Theoretical Study of the Phenoxy Radical Recombination with the O( 3 P) Atom, Phenyl plus Molecular Oxygen Revisited

Quantum chemical calculations of the C 6 H 5 O 2 potential energy surface (PES) were carried out to study the mechanism of the phenoxy + O( 3 P) and phenyl + O 2 reactions. CASPT2(15e,13o)/CBS//CASSCF(15e,13o)/DZP multi-reference calculations were utilized to map out the minimum energy path for the entrance channels of the phenoxy + O( 3 P) reaction. Stationary points on the C 6 H 5 O 2 PES were explored at the CCSD(T)-F12/cc-pVTZ-f12//B3LYP/6-311++G** level for the species with a single-reference character of the wave function and at the CASPT2(15e,13o)/CBS//B3LYP/6-311++G** level of theory for the species with a multi-reference character of the wave function. Conventional, variational, and variable reaction coordinate transition state theories were employed in Rice-Ramsperger-Kassel-Marcus Master Equation calculations to assess temperature- and pressure-dependent phenomenological rate constants and product branching ratios. Here, the main bimolecular product channels of the phenoxy + O( 3 P) reaction are concluded to be para/ortho-benzoquinone + H, 2,4-cyclopentadienone + HCO, and, at high temperatures, also phenyl + O 2 . The main bimolecular product channels of the phenyl + O 2 reaction include 2,4-cyclopentadienone + HCO at lower temperatures, and phenoxy + O( 3 P) at higher temperatures. Both for the phenoxy + O( 3 P) and phenyl + O 2 reactions, the collisional stabilization of peroxybenzene at low temperatures and high pressures competes with the bimolecular product channels.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Theoretical study of the reaction mechanism and kinetics of the phenyl + propargyl association

Potential energy surface for the phenyl + propargyl radical recombination reaction has been studied at the CCSD(T)-F12/cc-pVTZ-f12//B3LYP/6-311G** level of theory for the closed-shell singlet species and at the triplet–singlet gap CASPT2/cc-pVTZ-CCSD(T)-F12/cc-pVTZ-f12//CASSCF/cc-pVTZ level of theory for the diradical species. High-pressure limit rate constants for the barrierless channels were evaluated with variable reaction coordinate transition state theory (VRC-TST). Rice–Ramsperger–Kassel–Marcus Master Equation (RRKM-ME) calculations have been performed to assess temperature- and pressure-dependent phenomenological rate constants and product branching ratios. Here, the entrance channels of the radical association reaction produce 3-phenyl-1-propyne and phenylallene which can further dissociate/isomerize into a variety of unimolecular and bimolecular products. Theoretical evidence is presented that, at combustion relevant conditions, the phenyl + propargyl recombination provides a feasible mechanism for the addition of a second five-member ring to the first six-member aromatic ring producing the prototype two-ring species indene and indenyl. Rate expressions for all important reaction channels in a broad range of temperatures and pressures have been generated for kinetic modeling.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Mechanism and kinetics of the oxidation of 1,3-butadien-1-yl (n-C 4 H 5 ): a theoretical study

Here, ab initio CCSD(T)-F12/cc-pVTZ-f12//B3LYP/6-311G(d,p) calculations of the C 4 H 5 O 2 potential energy surface have been combined with Rice–Ramsperger–Kassel–Marcus Master Equation (RRKM-ME) calculations of temperature- and pressure-dependent rate constants and product branching ratios to unravel the mechanism and kinetics of the n-C 4 H 5 + O 2 reaction. The results indicate that the reaction is fast, with the total rate constant being in the range of 3.4–5.6 × 10 –11 cm 3 molecule –1 s –1 . The main products include 1-oxo-n-butadienyl + O and acrolein + HCO, with their cumulative yield exceeding 90% at temperatures above 1500 K. Two conformers of 1-oxo-n-butadienyl + O are formed via a simple mechanism of O 2 addition to the radical site of n-C 4 H 5 followed by the cleavage of the O–O bond proceeding via a van der Waals C 4 H 5 O∙∙∙O complex. Alternatively, the pathways leading to acrolein + HCO involve significant reorganization of the heavy-atom skeleton either via formal migration of one O atom to the opposite end of the molecule or its insertion into the C1–C2 bond. Not counting thermal stabilization of the initial peroxy adducts, which prevails at low temperatures and high pressures, all other products share a minor yield of under 5%. Rate constants for the significant reaction channels have been fitted to modified Arrhenius expressions and are proposed for kinetic modeling of the oxidation of aromatic molecules and 1,3-butadiene. As a secondary reaction, n-C 4 H 5 + O 2 can be a source for the formation of acrolein observed experimentally in oxidation of the phenyl radical at low combustion temperatures, whereas another significant (secondary) product of the C 6 H 5 + O 2 reaction, furan, could be formed through unimolecular decomposition of 1-oxo-n-butadienyl. Both the n-C 4 H 5 + O 2 reaction and unimolecular decomposition of its 1-oxo-n-butadienyl primary product are shown not to be a substantial source of ketene.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Charge transfer rate constants for the carotenoid-porphyrin-C 60 molecular triad dissolved in tetrahydrofuran: The spin-boson model vs the linearized semiclassical approximation

Charge transfer rate constants were calculated for the carotenoid-porphyrin-C 60 (CPC 60 ) molecular triad dissolved in explicit tetrahydrofuran. The calculation was based on mapping the all-atom anharmonic Hamiltonian of this system onto the spin-boson Hamiltonian. The mapping was based on discretizing the spectral density from the time correlation function of the donor–acceptor potential energy gap, as obtained from all-atom molecular dynamics simulations. Different spin-boson Hamiltonians were constructed for each of the possible transitions between the three excited electronic states in two different triad conformations. The rate constants of three possible transitions were calculated via the quantum-mechanically exact Fermi’s golden rule (FGR), as well as a progression of more approximate expressions that lead to the classical Marcus expression. The advantage of the spin-boson approach is that once the mapping is established, the quantum-mechanically exact FGR and the hierarchy of approximations are known in closed form. The classical Marcus charge transfer rate constants obtained with the spin-boson Hamiltonians were found to reproduce those obtained from all-atom simulations with the linearized semiclassical approximation, thereby confirming the equivalence of the two approaches for this system. Furthermore, within the spin-boson Hamiltonian, we also found that the quantum-mechanically exact FGR rate constants were significantly enhanced compared to the classical Marcus theory rate constants for two out of three transitions in one of the two conformations under consideration. The results confirm that mapping to the spin-boson model can yield accurate predictions for charge transfer rate constants in a system as complex as CPC 60 dissolved in tetrahydrofuran.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Three-state harmonic models for photoinduced charge transfer

A widely used strategy for simulating the charge transfer between donor and acceptor electronic states in an all-atom anharmonic condensedphase system is based on invoking linear response theory to describe the system in terms of an effective spin-boson model Hamiltonian. Extending this strategy to photoinduced charge transfer processes requires also taking into consideration the ground electronic state in addition to the excited donor and acceptor electronic states. In this paper, we revisit the problem of describing such nonequilibrium processes in terms of an effective three-state harmonic model. We do so within the framework of nonequilibrium Fermi’s golden rule (NE-FGR) in the context of photoinduced charge transfer in the carotenoid–porphyrin–C 60 (CPC 60 ) molecular triad dissolved in explicit tetrahydrofuran (THF). To this end, we consider different ways for obtaining a three-state harmonic model from the equilibrium autocorrelation functions of the donor–acceptor, donor–ground, and acceptor–ground energy gaps, as obtained from all-atom molecular dynamics simulations of the CPC 60 /THF system. The quantum-mechanically exact time-dependent NE-FGR rate coefficients for two different charge transfer processes in two different triad conformations are then calculated using the effective three-state model Hamiltonians as well as a hierarchy of more approximate expressions that lead to the instantaneous Marcus theory limit. Our results show that the photoinduced charge transfer in CPC 60 /THF can be described accurately by the effective harmonic three-state models and that nuclear quantum effects are small in this system.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Modified Fermi’s golden rule rate expressions

Fermi’s golden rule (FGR) serves as the basis for many expressions of spectroscopic observables and quantum transition rates. The utility of FGR has been demonstrated through decades of experimental confirmation. However, there still remain important cases where the evaluation of a FGR rate is ambiguous or ill-defined. Examples are cases where the rate has divergent terms due to the sparsity in the density of final states or time dependent fluctuations of system Hamiltonians. Strictly speaking, assumptions of FGR are no longer valid for such cases. However, it is still possible to define modified FGR rate expressions that are useful as effective rates. The resulting modified FGR rate expressions resolve a long standing ambiguity often encountered in using FGR and offer more reliable ways to model general rate processes. Finally, simple model calculations illustrate the utility and implications of new rate expressions.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS2007

Pressure and Temperature Dependence of the Reaction of Vinyl Radical with Ethylene

This work reports measurements of absolute rate coefficients and Rice-Ramsperger-Kassel-Marcus (RRKM) master equation simulations of the C2H3 + C2H4 reaction. Direct kinetic studies were performed over a temperature range of 300-700 K and pressures of 20 and 133 mbar. Vinyl radicals (H2C=CH) were generated by laser photolysis of vinyl iodide (C2H31) at 266 nm, and time-resolved absorption spectroscopy was used to probe vinyl radicals through absorption at 423.2 nm. Measurements at 20 mbar are in good agreement with previous determinations at higher temperature. A weighted three-parameter Arrhenius fit to the experimental rate constant at 133 mbar, with the temperature exponent fixed, gives k = (7 +/- 1) x 10(exp -l4) cu cm/molecule/s (T/298 K)(exp 2) exp[-(1430 +/- 70) K/T]. RRKM master equation simulations, based on G3 calculations of stationary points on the C4H7 potential energy surface, were carried out to predict rate coefficients and product branching fractions. The predicted branching to 1-methylallyl product is relatively small under the conditions of the present experiments but increases as the pressure is lowered. Analysis of end products of 248 nm photolysis of vinyl iodide/ethylene mixtures at total pressures between 27 and 933 mbar provides no direct evidence for participation of I -methylallyl.

Ismail, Huzeifa↗
DOE OSTI2020

Formation of phenanthrene via H–assisted isomerization of 2–ethynylbiphenyl produced in the reaction of phenyl with phenylacetylene

Model chemistry G3(MP2,CC)//B3LYP/6–311G(d,p) calculations of the potential energy surface for the reaction of phenyl radical (C 6 H 5 ) with phenylacetylene (C 8 H 6 ) have been carried out and combined with Rice–Ramsperger–Kassel–Marcus/Master Equation calculations of temperature– and pressure–dependent rate constants. The results showed that the reaction can serve as a viable source for the formation of phenanthrene via an indirect route involving a primary reaction of phenyl addition to the ortho carbon in the ring of phenylacetylene and H elimination producing 2–ethynylbiphenyl followed by secondary H–assisted isomerization of 2–ethynylbiphenyl to phenanthrene. In the secondary reaction, the H atom adds to the α carbon of the ethynyl side chain, then a six–member ring closure takes place followed by aromatization via an H loss. The channel of H addition to the side chain of 2–ethynylbiphenyl appears to be much faster than H addition to the ortho carbon in the ethynyl–substituted ring leading back to the initial C 6 H 5 + C 8 H 6 reactants. Rate constants for the primary C 6 H 5 + C 8 H 6 ⇌2–ethynylbiphenyl (p1) + H and secondary p1 + H⇌phenanthrene (p2) + H reactions have been computed in the temperature range of 500–2500 K at pressures of 30 Torr, 1, 10, and 100 atm and fitted to modified Arrhenius expressions. Here, the suggested kinetic scheme and rate constants are proposed as a prototype for the modeling of the growth of polycyclic aromatic hydrocarbons via the phenyl addition–dehydrocyclization (PAC) mechanism involving an addition of a PAH radical to an ethynyl–substituted PAH molecule.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Automated reaction kinetics and network exploration (Arkane): A statistical mechanics, thermodynamics, transition state theory, and master equation software

The open-source statistical mechanics software described here, Arkane–Automated Reaction Kinetics and Network Exploration–facilitates computations of thermodynamic properties of chemical species, high-pressure limit reaction rate coefficients, and pressure-dependent rate coefficient over multi-well molecular potential energy surfaces (PES) including the effects of collisional energy transfer on phenomenological kinetics. Arkane can use estimates to fill in information for molecules or reactions where quantum chemistry information is missing. The software solves the internal energy master equation for complex unimolecular reaction systems. Inputs to the software include converged electronic structure computations performed by the user using a variety of supported software packages (Gaussian, Molpro, Orca, TeraChem, Q-Chem, Psi4). The software outputs high-pressure limit rate coefficients and pressure-dependent phenomenological rate coefficients, as well as computed thermodynamic properties (enthalpy, entropy, and constant pressure heat capacity) with added energy corrections. Some of the key features of Arkane include treatment of 1D, 2D or ND hindered internal rotation modes, treatment of free internal rotation modes, quantum tunneling effect consideration, transition state theory (TST) and Rice-Ramsperger-Kassel-Marcus (RRKM) rate coefficient computations, master equation solution with four implemented methods, inverse-Laplace transform of high-pressure limit rate coefficients into the energy domain, energy corrections based on bond-additivity or isodesmic reactions, automated and efficient PES exploration, and PES sensitivity analysis. The present work describes the design of Arkane, how it should be used, and refers to the theory that it employs. Arkane is distributed via the RMG-Py software suite (https://github.com/ReactionMechanismGenerator/RMG-Py).

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS2017

Reachability Subspace Exploration Using Continuation Methods

Reachability manifold computation suffers from the curse of dimensionality and for large state spaces is computationally intractable. This paper examines the use of continuation methods to address this issue by formulating the reachability sub-space manifold calculation into a number of initial valued problems. As a result of computing the reachability manifold for a subspace of interest, an exponential improvement in computational cost occurs. This concept is applied to a position subspace reachability problem of a spacecraft in a Keplerian orbit under maximum thrust constraints. Future work includes a comparison of the proposed method with computing reachability manifolds using viscosity solutions of the Hamilton Jacobi Bellman partial differential equation.

orbital mechanics↗
NASA NTRS1973

Space shuttle GN and C equation document: Entry and transition guidance

The entry-guidance routine presented is designed to take the orbiter vehicle from entry interface through the critical heating phase of entry down to the start of the approach phase. The material includes: (1) a functional flow diagram, (2) input and output variables, (3) a description of equations, and (4) detailed flow diagrams.

Marcus, F. J.↗