A Steady-State Approximation to the Two ... - ACS Publications

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A Steady-State Approximation to the Two-Dimensional Master Equation for Chemical Kinetics Calculations Thanh Lam Nguyen, and John F. Stanton J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.5b00997 • Publication Date (Web): 27 Mar 2015 Downloaded from http://pubs.acs.org on April 5, 2015

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The Journal of Physical Chemistry

A Steady-State Approximation to the Two-Dimensional Master Equation for Chemical Kinetics Calculations Thanh Lam Ngu yen and John F. Stanton* Dep a rt men t o f Ch em is t r y, Th e Un iv er s ity o f Tex a s a t A u st in , Ma i l S to p A 5 3 0 0 , T exa s 7 8 7 1 2 - 0 1 6 5 , Un i ted S ta t es.

*E-mail: [email protected] Abstract: In the field of chemical kinetics, the solution of a two-dimensional master equation that depends explicitl y on both total internal energy (E) and total angular momentum (J) is a challenging problem. In this work, a weakE/fixed-J

collisional

model

(i.e.

weak-collisional

internal

energy

relaxation/free-collisional angular momentum relaxation) is used along with the stead y-state approach to solve the resulting (simplified) twodimensional (E,J)-grained master equation. The corresponding solutions give thermal rate constants and product branching ratios as functions of both temperature and pressure. We also have developed a program that can be used to predict and anal yze experimental chemical kinetics results. This expedient technique, when combined with highl y accurate potential energy surfaces, is cable of providing results that may be meaningfull y compared to experiments. The reaction of singlet oxygen with methane proceeding

through

vibrationall y

ex cited

methanol

is

used

as

an

illustrative example. Keywords: master-equation, 2DME, SCTST, stead y-state, O+CH 4 ,

ACS Paragon Plus Environment

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Introduction Chemical reactions occurring in combustion, atmospheric environments, and space are complicated processes, which often proceed through multiple intermediates (also called wells) to produce various products. In these

s ystems,

there

can

be

considerable

competition

between

unimolecular reactions (i.e. dissociation/isomerization) and collisional energy transfer processes of vibrationally ex cited intermediates; thermal reaction rate constants and product yields can depend strongl y on both temperature and pressure. 1 - 6 To deal with this issue, one can construct and then solve an energy-grained master equation. 1 - 11 Apart from two extreme conditions (i.e. at the high– and low–pressure limits where anal ytical solutions of the master-equation can be obtained), 9 , 1 2 - 1 4 one must solve an energy–grained

master

equation

numericall y. 1 - 11

The

solutions

(phenomenological (thermal) rate constants and product yields) can vary with both temperature and pressure; and they are frequently calculated within the stead y-state approach. In addition, solutions that can vary as a function of reaction time under non–equilibrium conditions (e.g. time evolution for concentrations of intermediates) can also be computed. There are at least two techniques that are commonl y used to solve the master

equation,

including

stochastic

simulation

and

deterministic

methods. 1 0 Each method has its own advantages and drawbacks. The advantages

of

the

stead y–state

approach

(which

belongs

to

the

deterministic method) used here include a rapid solution of the equation


2

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that is independent of pressure (unlike the stochastic approach); and that it is well-suited to complicated reaction mechanisms. However, the stead y–state method cannot provide chemical kinetics properties that vary as a function of reaction time. Such time–dependent properties can onl y be computed using either stochastic simulation method or the eigenvalue– eigenvector (matrix) technique. 1 0 There are a number of chemical kinetics software programs available, including MULT IWE LL 1 5 and USESAM 1 6 software packages that use (exact) stochastic simulation methods; matrix techniques have been applied in MESMER, 1 7 VAR IFLEX, 1 8 UNIMOL, 1 9 CHEMRATE, 2 0 and other programs. All these packages have been designed to work with a one–dimensional energy–grained mater equation (1DME), in which onl y the internal energy (E) is considered. The roles of angular momentum are either ignored or just partl y included. However, the effects of angular momentum 2 1 - 2 6 (e.g. centrifugal effects) are found to play an important role in some reactions where there are significant changes

of

rotational

constants

amongst

the

transition

states

and

intermediates along the reaction path. Therefore, such effects should sometimes (or often) be taken into account in computing microcanonical rate constants. Various approximate models have been suggested to capture these effects. 2 7 - 2 9 The most common assumes that the stationar y points on potential energy surface are described b y a s ym metric top in which the rotations are completel y separable from the vibrations and ma y


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be treated approximatel y as either active or inactive (i.e. adiabatic) motions or used in a h ybrid model that is a mixture of the two. 2 7 - 2 9 In 1996, Jeffrey et al. 3 0 were the first to report full iterative solutions of the two-dimensional master equation for thermal unimolecular reactions using the Nesbet algorithm. 3 1 In the same year, Robertson et al. 3 2 reformulated the 2DME problem in terms of a simplified diffusion equation and solved it using inverse iteration. One year later, Venkatesh et al 3 3 carried out a full theoretical anal ysis of the two-dimensional master equation formulation of the problem of intermolecular energy transfer in the

context

of multiple-well,

multiple-channel

chemically activated

reactions. In 1999, the same group 3 4 applied efficient numerical methods to anal yze the internal structure of the master equation kernel (that describes intramolecular and intermolecular energy transfer in multiplewell, multiple-channel s ystems) to the chemicall y activated reaction C 2 H 5 + O 2 . They suggested the use of a Krylov-subspace method to determine the uppermost portions of the internal spectrum of the master equation kernel. In 2002, Miller and Klippenstein 3 5 reported solution of a twodimensional master equation for thermal dissociation of methane in the low-pressure limit. In addition, these authors also suggested three different collisional models to reduce the two–dimensional problem to one dimension using a technique of Smith and Gilbert. 2 1 , 2 2 Note that in the current version of the Multiwell software suite, 1 5 Marcus’s approach 1 5 that reduced the two–dimensional to a one–dimension problem b y


4

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invoking “centrifugal corrections” has been fairl y widely used. This approach is exact at the high–pressure limit, but onl y approximate at lower pressures. For reviews of master equation methods, the interested reader is referred to the literature. 1 - 11 In this work, we use the matrix technique to solve 2D master-equation in parallel, simplified b y assuming the stead y-state approach. The paper is organized as follows: the second section briefl y presents the theoretical background of 1D- and 2D-master equation techniques; the third section will appl y the 2DME method advocated here to the reaction of O( 1 D) with CH 4 ; the fourth and final section concludes.

Theoretical Background 1. One-Dimensional Master-Equation (1DME): Solution of the 1DME, which depends onl y on internal energy (or vibrational energy), is well documented in textbooks. 1 - 11 , 3 6 - 4 1 It is briefl y summarized below for chemicall y activated reactions, although solutions for thermall y activated reactions or other scenarios can be obtained in a similar manner. The one–dimensional energy–grained master equation that describes the competing processes of population loss and gain of hot intermediates b y unimolecular reactions (i.e. dissociation/isomerization) and collisional energy transfer processes of vibrationally ex cited intermediates with bath gases is expressed by 1 - 11


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[ ] = ∙ − [ ] − [ ] + , . 1

where [C i ] is the population of the vibrationall y excited intermediate in state i. ω is the collisional frequency of an energized intermediate with a third

bod y,

and

k(E)

is

the

microcanonical

rate

constant

for

a

dissociation/isomerization step. Note that the dissociation of energized intermediates leading to fragmented products is assumed to be irreversible while the isomerization from one well to another well is treated reversibl y. P i j represents the probabilit y of energy transfer from state j to state i. R•f i is the source term; R is the total rate of the entrance flux leading to the formation of the energized adduct. f represents the initial energy distribution function of the energized adduct just formed from a chemical association reaction and is given b y 7 - 11 exp − # . 2

= ' $( exp %− #&

where G is the sum of vibrational states of the entrance transition structure. T is the temperature and R is the gas constant. In a matrix form, eq. 1 can be rewritten as:

ℂ = + − ,ℂ − -ℂ + ℙℂ . 30

or


6

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ℂ = + − 1ℂ,

23ℎ 1 = , + - − ℙ, . 35

viz. + − , −9, 8 … 7 1=7 … 7 −;, −, 6

−,9 9 + − 9,9 … … −;,9 −,9

… … … … … …

… −,; … −9,; … … … … … ; + − ;,; … −,;

−, −9, > … = = … = −;, + − ,
… = … = 0 9←

A Steady-State Approximation to the Two ... - ACS Publications

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