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Combustion and Flame 156 (2009) 896–913

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Combustion and Flame

www.elsevier.com/locate/combustflame

A statistical approach to develop a detailed soot growth model usingPAH characteristics

Abhijeet Raj, Matthew Celnik, Raphael Shirley, Markus Sander, Robert Patterson, Richard West, Markus Kraft ∗

Department of Chemical Engineering, Cambridge University, New Museums Site, Pembroke Street, Cambridge CB2 3RA, UK

a r t i c l e i n f o a b s t r a c t

Article history:Received 4 July 2008Received in revised form 27 October 2008Accepted 15 January 2009Available online 11 February 2009

Keywords:PAHSootAromatic siteKinetic Monte CarloSimulationModelling

A detailed PAH growth model is developed, which is solved using a kinetic Monte Carlo algorithm. Themodel describes the structure and growth of planar PAH molecules, and is referred to as the kineticMonte Carlo–aromatic site (KMC-ARS) model. A detailed PAH growth mechanism based on reactionsat radical sites available in the literature, and additional reactions obtained from quantum chemistrycalculations are used to model the PAH growth processes. New rates for the reactions involved inthe cyclodehydrogenation process for the formation of 6-member rings on PAHs are calculated in thiswork based on density functional theory simulations. The KMC-ARS model is validated by comparingexperimentally observed ensembles on PAHs with the computed ensembles for a C2H2 and a C6H6 flameat different heights above the burner. The motivation for this model is the development of a detailedsoot particle population balance model which describes the evolution of an ensemble of soot particlesbased on their PAH structure. However, at present incorporating such a detailed model into a populationbalance is computationally unfeasible. Therefore, a simpler model referred to as the site-counting modelhas been developed, which replaces the structural information of the PAH molecules by their functionalgroups augmented with statistical closure expressions. This closure is obtained from the KMC-ARS model,which is used to develop correlations and statistics in different flame environments which describe suchPAH structural information. These correlations and statistics are implemented in the site-counting model,and results from the site-counting model and the KMC-ARS model are in good agreement. Additionallythe effect of steric hindrance in large PAH structures is investigated and correlations for sites unavailablefor reaction are presented.

© 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction

There is general consensus in the literature that soot parti-cles are formed from polycyclic aromatic hydrocarbons (PAHs), thegrowth of which is directly linked to the formation of soot parti-cles [1]. It was proposed in [2] that the smallest aromatic com-pound, benzene is formed in a flame by aliphatic molecules andradicals that are generated by the pyrolysis of fuel molecules [3].Recent studies on the flames of cyclic hydrocarbons such as cyclo-hexane [4], and aromatic and alkyl substituted aromatic hydrocar-bons [5,6] show that the formation of benzene is not restricted tothe presence of aliphatic molecules or radicals. For example, in acyclohexane flame, benzene can form by the dehydrogenation offuel molecules [4]. In the higher aromatic flames such as naph-thalene flame, the larger PAHs can form directly by the PAH addi-tion reactions and cyclodehydrogenation process [5]. A theoreticalstudy on the addition of two five-member rings to form naphtha-lene, indene and benzene conducted by Wang et al. [7] suggests

* Corresponding author.E-mail address: [email protected] (M. Kraft).

0010-2180/$ – see front matter © 2009 The Combustion Institute. Published by Elsevierdoi:10.1016/j.combustflame.2009.01.005

another mechanism for the PAH formation. These studies clearlyshow that the pathway for the formation of PAHs in a flame isstrongly dependent on the structure of the fuel molecules. Once aPAH molecule forms, it can grow by: (a) the hydrogen-abstraction–carbon-addition (HACA) mechanism [3,8], involving addition ofacetylene (C2H2) at aromatic radical sites; and (b) PAH additionreaction, involving addition of a PAH/PAH radical over another toform biaryls [5].

To determine the geometry of PAHs present in soot particles,a number of morphological studies have been conducted in thepast on soot generated from different sources, like flames and en-gines [9–11]. These studies reveal that soot particles mainly consistof planar PAH molecules. It is well known that a soot particle iscomposed of primary particles linked together to form a fractal ag-gregate [12,13]. High Resolution Transmission Electron Microscopy(HRTEM) images of soot particles show that the primary parti-cles are near-spherical in shape, and are crystalline near the outeredge. This crystallinity arises due to stacking of planar PAHs toform parallel atomic layers and their alignment along the periph-ery of primary particles [9–11,14]. However, the nuclei of primaryparticles are amorphous in nature. The presence of non-crystalline

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nuclei is evident from the experimental findings of Vander Walet al. [12]. In their work, HRTEM images of partially oxidised sootparticles were observed, which apparently had hollow centres, in-dicating the presence of amorphous and thus more reactive carbonsheets in the centre of the primaries. This amorphous nature arisesfrom the random orientation of PAHs inside the nuclei [12]. Braunet al. [15] observed soot particles from diesel engines in variousconditions and concluded that the amount of crystallites (stackedgraphene sheets or planar PAHs) always exceeds the amount ofrandomly oriented PAHs in diesel soot. A study on morphology ofsoot particles from different fuels was conducted by Vander Waland Tomasek [16]. It was concluded that the degree of crystallinitydepends on the fuel being pyrolysed to produce soot. The degreeof crystallinity was highest in soot from the pyrolysis of acety-lene as compared to soot from benzene and ethanol. The PAHs orgraphene sheets present in soot from acetylene and benzene wereplanar. However, the PAHs present in soot from ethanol were ob-served to have curvature.

There is experimental evidence that shows the presence of sub-structures in primary particles [13–15]. In [13], HRTEM imagesof soot particles in a propane flame were observed, and the au-thors believe that a soot particle is composed of three kinds ofnanostructures: primary particles forming chain-like aggregates;sub-primary particles having ellipsoidal shape and clustered toform small chains of less than 10 subunits; and elementary par-ticles having shell-like structures with outer edge composed ofamorphous carbon. Elementary particles were believed to form theinner graphitic nuclei of soot particles. In [14], the spherical sub-structures in the centre of primary particles were designated asinner core and fine particles. Several layers of PAHs having bend-ing structures were observed around the fine particles, indicatingthe involvement of non-planar PAHs having embedded 5-memberrings [17] in soot nucleation. In their work as well, the outer layerof soot was observed to have graphitic crystallites, which may un-dergo polycyclic growth due to molecules, ions or radicals withtwo to four carbon atoms [14]. In brief, the above experimentalfindings on soot morphology suggest that the outer edge of sootparticles mainly consists of crystalline stacks of planar PAHs, andtherefore, surface growth of a soot particle takes place through thegrowth of PAHs present in those crystalline stacks.

Current soot models consider the processes of inception, surfacegrowth and coagulation of soot particles. The model referred to asthe ABF model [8,18] is commonly used to describe soot formationin flames and other systems, though it must often be augmentedto match experimental observations [19,20]. Soot mass is gener-ated by the inception and surface growth processes, though it isbelieved that the majority of the soot mass comes from surfacegrowth reactions [21]. In the soot models, only the addition of oneC2H2 molecule contributes to surface growth, the rate of which isestimated for the rate of C2H2 addition at armchair sites (definedlater). Frenklach et al. [22,23] have presented a more completesoot surface growth model which includes C2H2 at different sur-face sites, as well as ring desorption, however, this model has notbeen implemented in population balances and has only been usedto simulate the growth of graphene sheets. Thus, it appears thatthe present soot models like the ABF model are incomplete andlacks a number of important reaction steps, by which a soot par-ticle can grow. It has been certainly observed in the past that theABF soot model underpredicts the soot volume fraction in flames[24–26].

A number of refinements to the PAH and soot growth modelhave been proposed [23,27], which allows a soot particle to berepresented by its constituent PAHs. An advantage of tracking theshape of constituent PAHs is that the reactive site density (andtypes) on soot particles can be exactly known, rather than esti-mated [8]. The PAH growth model presented by Violi et al. [27,28]

models large, non-planar PAH sheets in a flame-like environment.A large set of reactions, including 5-member ring formation, isused to describe the growth of a PAH molecule, using a kineticMonte Carlo (KMC) algorithm and molecular dynamics (MD) cal-culations [27]. An optimised PAH molecule geometry is generated.The study by Keller et al. [29] showed that fullerenes and PAHsthat exist independently in flames develop 3D structures, whichmotivated the KMC-MD model of Violi et al. [27]. This model de-scribes single-atomic layer PAHs, which are present towards thebeginning of the flame, and which are assumed to be involved insoot nucleation. Frenklach et al. [22,23] proposed a PAH growthmodel, by which a planar graphene structure is allowed to growonly in one particular direction. The assumption of planar struc-ture of PAHs is in agreement with the experimental findings aboutthe composition of soot particles (stated above) in the later stagesof soot growth.

With current computing resources, the simulation of the de-tailed PAH growth models is either very computationally expen-sive [30], or involves restrictive assumptions [23]. The inclusionof such detailed models into particle population balances is alsoprohibitively computationally expensive, however, a method for in-clusion of such detailed PAH models into particle population bal-ances has been recently published [31]. The site-counting modelwas presented which does not store the structural information ofthe PAHs present in soot particles, but rather estimates the struc-ture by counting the number of each type of surface site and byusing correlations and statistics to describe their interaction.

The aim of this paper is to present a detailed PAH growthmodel, referred to as the kinetic Monte Carlo–aromatic site (KMC-ARS) model, which stores the structural information of a PAHmolecule as it grows in a flame-like environment. The model issolved using a kinetic Monte Carlo algorithm. A large set of sootsurface processes reported in the literature [23,28,31–33], includ-ing oxidation processes [31], has been used for the model devel-opment. A recently calculated rate for O2 oxidation of PAHs [34]has been employed in this study. New rates for the reactions in-volved in the cyclodehydrogenation process for the formation of6-member rings on PAHs have been calculated in this work basedon density functional theory simulations at HCTH407 level of the-ory. The KMC-ARS model has been validated by comparing thePAH ensembles observed experimentally for a C2H2 flame [35]and a C6H6 flame [29] with the computed ensembles at differentheights above the burner. A statistical method has been presentedwhich allows the structure of the resultant PAHs to be describedby simple, rapidly computed, expressions, which make it possi-ble to incorporate this advanced model into a particle populationbalance to account for surface growth of soot particles for thefirst time [31]: the site-counting model. The site-counting modelis computationally very cheap, and can be included easily in a sootparticle population balance. A detailed description of the methodsemployed to obtain statistics and correlations is provided. A studyon the variation of PAH statistics with flame properties like type offuel, cold gas velocity, C/O ratio and pressure has been conducted.The correlations and statistics are provided for commonly observedpremixed laminar flames to make their simulations possible usingthe site-counting model. A comparison between the KMC-ARS andsite-counting models is discussed.

2. ARS model

In a previous paper [31] a detailed PAH model was devel-oped, whereby PAHs are described by their reactive sites. A PAHmolecule is assumed to be fully described by three major char-acteristics: the number of carbon (C) atoms, number of hydrogen(H) atoms, and the number of reactive sites. A discussion of re-active sites has been given previously by Celnik et al. [31], and is

898 A. Raj et al. / Combustion and Flame 156 (2009) 896–913

Fig. 1. An example PAH showing the principal surface site types.

reviewed here. A site is a structure present on the outer edge ofPAHs formed by two consecutive surface carbon atoms (those car-bon atoms having bonded H atoms). The sites are differentiated bythe number of carbon atoms required for their formation. SimplePAH geometry requires that a site has exactly two surface carbonatoms, and zero to four bulk carbon atoms (those carbon atoms nothaving a bonded H atom). In this way four elementary sites can bedefined, which are shown in Fig. 1: free-edges, zig-zags, armchairsand bays. The zig-zag, armchair and bay sites can be easily dif-ferentiated from each other based on the number of bulk carbonatoms required in their formation. Additionally, a 5-member ringmay occupy a zig-zag site [23], therefore these have also been in-cluded in this model. PAH geometry also requires that, for a closedloop, each site must have two neighbouring sites. This is an impor-tant consideration as, in general, surface processes at one site willaffect at least two other sites.

In this paper, a number of site terminologies are used, whichare defined here:

• Elementary sites: The sites in Fig. 1 are called elementary sites.• Combined sites: These sites are formed by more than one con-

secutive elementary site.• Parent site: The site on which a chosen reaction takes place.

This site gets replaced in the reaction.• Neighbour sites: In a reaction on a parent site, the two sites

present on the either side of the parent site are called neigh-bour sites. After a reaction, neighbour sites change their typesin a defined way.

2.1. PAH reactions

In the literature, a large number of reactions by which a PAHmolecule can grow are available [23,28,31–33]. Celnik et al. [31]have collected an extensive set of PAH reactions along with theirrate constants at a pressure of 101.3 kPa, which are required todevelop the PAH growth mechanism. The present work involvesPAH growth simulations at two different pressures: 2.67 kPa and101.3 kPa. For the simulation at 101.3 kPa, the set of reactionsprovided in [31] was used. A literature survey was undertaken toobtain the rate constants for the PAH reactions at a pressure of2.67 kPa, and a list of elementary reactions along with their Arrhe-nius rate constants and references are provided in Table 1. Morethan one set of rate constants for some of the reactions at thesame pressure were available in the literature, which varied sig-nificantly from one another, and are listed in the same table. Forreactions 7, 9 and 10, rate constants were available for aromaticspecies with different number of rings, like benzene, naphthalene,phenanthrene and pyrene. For such reactions, the rate constantsevaluated for aromatic species with highest number of rings wereused in this work.

In Table 1, different symbols have been used to represent thesites present on the edge of a PAH molecule, and are explainedbelow:

C•s A radical site

Cs,feR6 6-member ring occupying a FE siteCs,zzR5 R5 next to ZZCs,acR6 6-member ring next to ACCs(eR5) Embedded R5 ring on PAH edgeCsR5 5-member ringCs,feR5 R5 next to FECs,acR5 R5 next to ACCs,i PAH with i number of rings

It can be noticed that a number of reactions involve adjacent ele-mentary sites (combined sites).

Table 1PAH surface reactions. The rate constants are of the form: AT n exp(−E/RT ), and at a pressure of 2.67 kPa.

No. Reaction Aa n Ea Ref.

Hydrogen abstraction from, and addition to free-edges, zig-zags and armchairs1 Cs − H + H � C•

s + H2 3.23 × 1007 2.095 15.84 [38–40]−1a C•

s + H2 � Cs + H 3.90 × 1012 46.00 [41]−1b C•

s + H2 � Cs + H 3.40 × 1009 0.880 7.871 [22]−1c C•

s + H2 � Cs + H 3.90 × 1012 9.330 [42]2a C•

s + H → Cs 3.49 × 1039 −7.770 13.37 [43]2b C•

s + H → Cs 8.02 × 1019 −2.011 1.968 [38,39,44]3 Cs − H + OH � C•

s + H2O 2.10 × 1013 19.10 [43]−3 C•

s + H2O � Cs − H + OH 3.68 × 1008 1.139 17.10 [43]

Hydrogen abstraction from, and addition to, 5-member rings4 CsR5 + H � CsR5• + H2 5.07 × 1007 1.930 12.95 [38,39,45]−4a CsR5• + H2 � CsR5 + H 3.01 × 1004 2.630 8.540 [46,47]−4b CsR5• + H2 � CsR5 + H 1.57 × 1011 5.260 [47,48]−4c CsR5• + H2 � CsR5 + H 9.46 × 1003 2.560 5.060 [45]5 CsR5• + H → CsR5 1.40 × 1030 −3.860 3.320 LP limit [37]b

6.08 × 1012 0.270 0.280 HP limit [37]b

6 CsR5 + H � CsR5H• 8.42 × 1008 1.490 0.990 [38,39,46]c

−6 CsR5H• � CsR5 + H 6.28 × 1037 −8.240 44.67 [43]d

Free-edge ring growth7a C•

s + C2H2 � CsC2H•2 7.70 × 1040 −9.190 13.40 [49]

7b C•s + C2H2 � CsC2H•

2 4.40 × 1049 −11.60 19.30 [49]8a CsC2H•

2 � CsC2H + H 2.74 × 1022 −4.061 37.04 [38,39]8b CsC2H•

2 � CsC2H + H 4.30 × 1051 −12.19 48.00 [49]9a C•

s + C2H2 � CsC2H + H 7.50 × 1026 −3.960 17.10 [49]9b C•

s + C2H2 � CsC2H + H 8.00 × 1017 −1.210 22.60 [49]


Table 1 (continued)


9c C•s + C2H2 � CsC2H + H 1.40 × 1022 −2.640 17.40 [49]

9d C•s + C2H2 � CsC2H + H 1.70 × 1030 −4.960 20.50 [49]

10a C•s C2H + C2H2 � Cs(C2H)C2H•

2 3.90 × 1048 −11.30 27.20 [49]10b C•

s C2H + C2H2 � Cs(C2H)C2H•2 8.56 × 1044 −10.50 13.22 [49]

11a Cs(C2H)C2H•2 → CsR6• 2.50 × 1012 −0.130 15.72 [31]

11b Cs(C2H)C2H•2 → CsR6• 2.93 × 1060 −14.99 24.40 [43]e

Free-edge ring desorption12 Cs,feR6• � Cs(C2H)C2H•

2 1.60 × 1005 [50], 1500 Kf

1.50 × 1007 [50], 2000 K1.20 × 1008 [50], 2500 K

13 Cs,fe(C2H)C2H•2 � C•

s C2H + C2H2 5.74 × 1056 −13.12 16.62 [49]g

14 C•s C2H + H → CsC2H same as reaction 2 [50]

15 CsC2H + H � CsC2H•2 1.80 × 1025 −4.500 4.200 [49]

16 CsC2H•2 � C•

s + C2H2 1.83 × 1046 −10.46 13.17 [49]g

5-member ring addition to zig-zags17 C•

s + C2H2 → CsR5 + H 4.00 × 1013 10.12 [43]

5-member ring desorption18 CsR5• → CsC2H• 1.60 × 1014 177.5 [51]

Armchair ring growth19 C•

s + C2H2 � CsC2H•2 6.70 × 1045 −10.55 21.2 [49]

One step armchair ring growth (ABF model)20a C•

s + C2H2 → CsR6 + H 1.87 × 1007 1.787 3.262 [38,39]20b C•

s + C2H2 → CsR6 + H 3.98 × 1013 10.10 [52,53]20c C•

s + C2H2 → CsR6 + H 4.00 × 1023 −3.20 14.40 [49]

5-member ring migration21 Cs,zzR5H• → CsR5 + H 5.63 × 1011 [23], 1500 K

1.02 × 1012 [23], 2000 K1.13 × 1012 [23], 2300 K

5- to 6-member ring conversion at free-edges [23]22 C•

s,fe(R5H•) + C2H2 � Cs(R5H•)C2H•2 same as reaction 7 [22,23]

23 C•s,fe(R5H•) + C2H2 � Cs(R5H•)C2H + H same as reaction 9 [22,23]

24 Cs,fe(R5H•)C2H → CsR6• same as reaction 11 [23]

6- to 5-member ring conversion at armchairs25 Cs,acR6• → CsR5• + C2H2 same as reaction 12 [23]

5- to 6-member ring conversion at armchairs26 Cs,acR5H• → CsR6 + H 5.63 × 1011 [23], 1500 K

1.02 × 1012 [23], 2000 K1.13 × 1012 [23], 2300 K

Free-edge oxidation27a CsR6• + O2 → CsC2H + HCO + CO 2.20 × 1012 31.40 [18]27b CsR6• + O2 → CsC2H + HCO + CO 2.10 × 1012 31.30 [54]27c CsR6• + O2 → CsC2H + HCO + CO 9.70 × 1003 2.42 38.48 [34]28 CsR6 + OH → CsC2H + CH2CO 1.30 × 1013 44.40 [54]29 CsC2H + O → C•

s + HCCO 2.04 × 1007 2.00 7.90 [54]

Armchair oxidation30a C•

s,fe + O2 → C•s + 2CO 2.20 × 1012 31.40 [18]

30b C•s,fe + O2 → C•

s + 2CO 2.10 × 1012 31.30 [54]30c C•

s,fe + O2 → C•s + 2CO 9.70 × 1003 2.42 38.48 [34]

31 Cs,fe + OH → C•s + CHO + CH 1.30 × 1013 44.40 [18]

Benzene addition32a C•

s + C6H6 � CsC6H5 + H 1.90 × 1076 −18.90 39.47 [38,39]32b C•

s + C6H6 � CsC6H5 + H 1.27 × 1005 [33], 300 K1.60 × 1008 [33], 500 K7.83 × 1009 [33], 700 K5.37 × 1010 [33], 1000 K8.43 × 1010 [33], 1500 K8.01 × 1009 [33], 2000 K1.62 × 1008 [33], 2500 K

5-member bay closure [32]33 C•

s,i → Cs,i + 1 + H 3.86 × 1011 0.210 17.70 [32]

6-member bay closure [55]34 Cs,i + H → C•

s,i + H2 9.24 × 1007 1.50 9.646 HCTH407h

5.60 × 1007 1.78 10.60 B3LYPh

−34 C•s,i + H2 → Cs,i + H 9.60 × 1004 1.96 9.021 HCTH407h

35 C•s,i → C•

s,i + 1H 1.11 × 1011 0.658 23.99 HCTH407h

(continued on next page)


Table 1 (continued)


36 C•s,i + 1H → Cs,i + 1 + H 1.44 × 1013 −0.57 3.662 HCTH407h

37 C•s,i → Cs,i + 1 + H 3.49 × 1012 −0.39 2.440 HCTH407h

Embedded 5-member ring migration to zig-zags [32]38 C•

s (eR5) → C•s (R5) 4.96 × 1011 0.755 50.0 [32]

39 C•s (R5) + H → Cs(R5) 2.33 × 1008 1.390 −1.950 [32]

a The units are kcal, mol, cm and s.b Analogous to C2H3 + H → C2H4 [23]. The rate constant for this reaction was calculated using the high-pressure and the low-pressure limit rate constants, and was

expressed in Troe falloff form [36] as suggested in [37]. The parameters required to express the rate constant in Troe form were obtained from [37].c Analogous to C2H4 + H → C2H5 [23].d Analogous to C2H5 → C2H4 + H [23].e Analogous to n-C6H5 � A1−.f Assumed to be same as decyclisation of a phenyl radical.g Calculated using equilibrium constant.h Level of theory used for DFT calculations in the present work. The details about the theoretical calculations required to obtain these rate constants can be found in the

supplementary material.

2.2. PAH processes

Table 2 shows a list of PAH processes involving 5-memberand 6-member ring addition and removal, and oxidation of re-active sites. This list is based on the reactions provided in [31]amended with the reaction for bay closure studied in this workand some other reactions from the literature [23,32,33,56]. Thereaction mechanisms are provided in Table 2 for only the newlyadded processes. For each process listed in this table, several inter-mediate species are involved in the reaction mechanism. If all theintermediate species present in the reaction mechanisms are con-sidered, the simulation of a PAH molecule would become compu-tationally very expensive. Therefore the intermediate species wereassumed to be in steady state, and that the ring addition and des-orption steps are irreversible [23,31]. A detailed explanation forthese assumptions can be found in [31]. The PAH processes were

then represented as jump processes (see Table 2). The rate ex-pressions for the jump processes obtained after the steady-stateanalysis are presented in the same table. In the rate expressions,the term [CS] represents the concentration of parent site S. Appelet al. [18] argued that the hydrogen atom on armchair and bayradical sites can easily jump from one side to another [57]. Thisprocess will double the rate of PAH growth processes involvingthese sites, like armchair growth reaction and bay closure reac-tions. In the light of above argument, PAH growth simulations werecarried out by doubling the rates of armchair growth and bay clo-sure reactions.

In the literature, two different routes for the free-edge growthreaction are available, as shown in Fig. 2: Route I involves an in-termediate species with a free-radical after the addition of C2H2in step 2 as suggested in [22]; Route II involves direct additionof C2H2 with the removal of an H atom. It is for the first time

Table 2Monte Carlo jump processes.

Process [Ref.] Parent site

S1 Free-edge ring growth [22,56] Free-edge

Jump process: Rate:(k7b + k9b)

( k1[H]+k3[OH]k−1b [H2]+k2a [H]+k−3[H2O]+(k7b+k9b)[C2H2]

)[C2H2][Cfe]

S2 Armchair ring growth [18,31] Armchair

Jump process: Rate:k20a

( k1[H]+k3[OH]k−1b [H2]+k2a [H]+k−3[H2O]+k20a [C2H2]

)[C2H2][Cac]

S3 Free-edge ring desorption [22,31] 6-member ring

Jump process: Rate:k12

( k1[H]+k3[OH]k−1b [H2]+k2a [H]+k−3[H2O]+k12

)[CfeR6]


Table 2 (continued)


S4 6- to 5-member ring conversion at armchair [23,31] 6-member ring next to armchair



)[CacCfeR6]

S5 5-member ring addition [31,51] Zig-zag


( k1[H]+k3[OH]k−1b [H2]+k2a [H]+k−3[H2O]+k17[C2H2]

)[C2H2][Czz]

S6 5-member ring desorption [31,51] 5-member ring

Jump process: Rate:

k18( k4[H]

k−4c [H2]+k5[H]+k18

)[CzzR5]

S7 5- to 6-member ring conversion at free edge [23] 5-member ring next to free-edge

Jump process: Rate:(k7b + k9b)

( k6[H]k−6+ f

)f [C2H2][CfeR5],

where f = ( k1[H]+k3[OH]k−1b [H2]+k2a [H]+k−3[H2O]+(k7b+k9b)[C2H2]

)

S8 5- to 6-member ring conversion at armchair [23,31] 5-member ring next to armchair


( k6[H]k−6+k26

)[CacR5]

S9 Free-edge oxidation by O2 [31,54] 6-member ring

Jump process: Rate:k27c

( k1[H]+k3[OH]k−1b [H2]+k2a [H]+k−3[H2O]+k27c [O2]

)[O2][CfeR6]

S10 Free-edge oxidation by OH [31,54] 6-member ring

Jump process: Rate:k28[OH][CfeR6]

S11 Armchair oxidation by O2 [31,54] Free-edge

Jump process: Rate:k30c

( k1[H]+k3[OH]k−1b [H2]+k2a [H]+k−3[H2O]+k30c [O2]

)[O2][Cfe]

S12 Armchair oxidation by OH [31,54] Free-edge

Jump process: Rate:k31[OH][Cfe]

(continued on next page)


Table 2 (continued)


S13 Benzene addition [33] All site types

Jump process: Rate:k32a

( k1[H]+k3[OH]k−1b [H2]+k2a [H]+k−3[H2O]+k32a[C6H6]

)[C6H6][Cs]

S14 5-member ring migration [23] 5-member ring next to zig-zag


( k6[H]k−6+k21

)[CzzR5]

S15 6-member bay closure [58] 6-member bay


( k34[H]+k3[OH]k−34[H2]+k2a [H]+k−3[H2O]+k36

)[Cbay]

S16 5-member bay closure [32] 5-member bay



)[Cbay]

S17 Embedded 5-member ring migration to zig-zags [32] Embedded 5-member ring next to two adjacent free-edges



)[Cfe,feeR5]

that both the routes are included in a PAH growth mechanism.There can be two different ways to include the two routes inthe PAH growth mechanism: (a) Considering free-edge growth viathe two routes as two different growth processes. In this case,the two resulting jump processes will be identical, but will have

different rates. (b) Combining the two routes to form a single re-action mechanism (as shown in Table 2 for process S1). In thiswork, the second method was used to incorporate the two routesin the PAH growth processes as they were very similar to eachother.


Fig. 2. Different routes for free-edge growth.

3. KMC-ARS model

In this paper a KMC algorithm has been developed to simu-late the growth of single PAH molecules. KMC allows the structureof the growing PAH molecule to be tracked explicitly, hence noassumptions are required about the shape of the structure. How-ever, this method is computationally expensive, and can currentlyonly be used to simulate single molecules in reasonable compu-tation times. The model is formulated mathematically in terms ofthe state-space and the jump processes (and their rates). In orderto create an efficient numerical algorithm a higher level data struc-ture is also defined, on which the jump processes are more easilydescribed.

3.1. State space

Tracking the structure of a PAH requires knowledge about theposition of the C atoms and the bonds between them. As a PAHhas a similar structure to graphite, it is clear that the bulk C atomsmust bond with three others (not differentiating between singleand double bonds), while surface C atoms form chemical bondswith two other C atoms and possibly a H atom. In other words,sp2 hybridisation for all the C atoms is assumed. Thus, the simpleststate space required to track the structure of the PAH moleculeincludes the following information: the positions of the C atoms,and the information about the bonds associated with the C atoms.The latter information is stored in terms of the positions of theother C atoms to which a C atom is bonded. For 2D representationof PAHs with planar or near-planar geometry [59], the state spaceE can be represented as:

E = (i, j, i1, j1, i2, j2, i3, j3) ∈ Z8, (1)

where (i, j) are the coordinates of a C atom, and (in, jn)n=1,2,3are the coordinates of the surrounding C atoms bonded to the Catom at (i, j). For a surface C atom, (i3, j3) is not defined, and isset to a predefined value, thus providing a method to differentiatea surface C atom from a bulk C atom. This state space includesall the possible states/structures adoptable by a PAH molecule. Itwould also be possible to generalise this state space further to 3D,by including additional coordinates and defining them in the set ofreal numbers.

While this state space is sufficient to fully define a PAH struc-ture, definition of the Monte Carlo jump processes on it is verydifficult. Therefore, in order to simplify the problem, a higher or-der data structure is defined using the above state space, and thejumps are defined using this data structure, which are describedin the next section.

Fig. 3. An example grid showing a pyrene molecule.

3.2. Data structure

In this study, to track the PAH structure, a 2D grid is generatedand the starting PAH molecule is placed on it such that each Catom is assigned a grid point. Fig. 3 shows an example grid with apyrene molecule. This 2D grid does not store the actual C–C bondlengths, however, this is acceptable as the bonds are implicitly rep-resented in the rate expressions by the surface sites. Therefore, ifthe PAH molecule represented on a 2D grid is not present in anystack, it may develop a non-planar geometry.

Two vectors are defined using the above state space to describethis problem: a carbon atom vector c, and a site vector s. The vec-tor c stores the following information: the two site types of whichthe C atom is a part, S1 and S2, the site indices (each site of aparticular type is differentiated from others based on its index),Sin1 and Sin2 , the carbon atom type (bulk or surface), Ctype , andits spatial coordinates, i and j. Thus, c can be represented as:

c = (S1, S2, Sin1 , Sin2 , Ctype, i, j) ∈ Z7.

All the required information about a site can be obtained from itssurface C atoms. However, the derivation of site information fromthe carbon atoms allows the jump processes to be easily defined,but requires some duplication of information. Therefore, a site vec-tor s was employed to store the following information: site type,Stype , site index (explained above), Sin , and the coordinates of thetwo surface C atoms, (i1, j1) and (i2, j2). Therefore, s can be rep-resented as:

s = (Stype, Sin, i1, j1, i2, j2) ∈ Z6.

The PAH structure is then fully described by the data structure e:

e = (c, s). (2)


3.3. Jump processes

The jump processes listed in Table 2 are defined on sites of aparticular type (parent sites). The information stored in vector s isused to define the rate (Ri ) of jump process i using the equation:

Ri = ki × f i × C × Nsite(e), (3)

where ki is the Arrhenius rate constant of the form AT ne−Ea/RT

(listed in Table 1), f i is the fraction of radical sites of the par-ent type, found using the steady-state assumption [31], C is theconcentration of the gas-phase species involved in the reactionand Nsite(e) is the number of parent sites on the structure, andis obtained from the data structure e. The expressions for the rateconstants of intermediate reactions and the fraction of radical sitesof the parent type can be obtained from Tables 1 and 2.

On simulation of a jump the vectors c and s must be updatedto reflect the changes to the PAH structure. Processes may add orremove C atoms, which must be reflected in the c vector, and theparent site will usually be destroyed. Additionally, the neighboursites must also be updated in the s vector.

3.4. KMC algorithm

The KMC algorithm used to track the growing structure is de-scribed below. This algorithm updates the PAH structure by remov-ing the used sites and carbon atoms and adding the newly formedones after each reaction.

(1) Set t ← 0.(2) Initialise 2D starting structure by providing all the required

information (Fig. 3): e ← e0, where e0 ∈ E .(3) Calculate the jump rates:

Ri = ki × f i × C × Nsite, i = 1,2, . . . , I,

where I is the number of processes in Table 2.(4) Generate an exponentially distributed waiting time τ with pa-

rameter λ, where λ is the sum of all the jump rates:

λ =I∑

i=1

Ri .

(5) Update t ← t + τ .(6) If t � tstop , then END.(7) Choose a reaction based on the probability calculated using

its reaction rate.(8) Uniformly select a site of correct type (Table 2) and determine

the positions of the two surface C atoms.(9) If ring addition reaction, determine the positions of the C

atoms to be added and go to step 11.(10) If ring desorption reaction, go to step 3.(11) If the site is hindered by surrounding C atoms then go to

step 3.(12) Update e ∈ E according to reaction (Table 2).(13) Update neighbour sites:

(a) 6-member ring: For an addition reaction, change theneighbour site type by moving it ahead in the one ofthe following lists (depending on whether 0 or 1 five-member ring is involved): {F E → Z Z → AC → BY } or{R5F E → R5Z Z → R5AC → R5BY }. For a desorption re-action, move back in one of the above lists.

(b) 5-member ring: For addition reaction, change the neigh-bour site type in the following way: For site S ∈ {F E, Z Z ,

AC, BY }, S → R5S and R5S → R5S R5. For desorption re-action, R5S → S , and R5S R5 → R5S .

(14) Go to step 3.

3.5. Error estimation

The random variation in the values of a variable with the simu-lation runs is an inherent property of a stochastic algorithm. Toestimate the statistical fluctuation in the simulated result of avariable, L trial runs are generated. The empirical mean of therealisations of the random variable Y : [y1(t), y2(t), . . . , yL(t)] iscalculated, and is represented by y(t):

y(t) = 1

L

L∑l=1

yl(t). (4)

An example of a random variable Y used in this work can be thenumber of 6-member rings (NR6). A method to obtain this vari-able is explained in Section 6.1.1. The empirical variance, η is thencalculated using Eq. (5):

η(t) = 1

L

L∑l=1

y2l (t) − [

y(t)]2

. (5)

The half-width of the error bound about the average value is cal-culated using the central limit theorem (Eq. (6)) with a = 3.29 fora confidence level of 99.9%.

ε(t) = a

√η(t)

L. (6)

The confidence interval for a variable, y(t), is then represented by:

I = [y(t) − ε(t), y(t) + ε(t)

]. (7)

4. Site-counting model

The site-counting model presented by Celnik et al. [31] deter-mines the soot particle composition and PAH sites information,assuming soot surface to be formed by stacking together of planarPAHs. It preserves the information about reactive sites, but neglectsthe spatial structure of PAHs to make this model computationallyvery cheap. This model abridges the gap between computation-ally expensive kinetic Monte Carlo or molecular dynamics modelsand the simple spherical carbon soot particle models [26,60,61].The motivation of this model was to produce as much informa-tion about the PAH structure of soot as possible, while keeping thecomputational expense low enough to simulate particle popula-tions, rather than just single particles [31]. The state space requiredfor this model is nine-dimensional, and is given by [31]

E = (C, H, N F E , N Z Z , N AC , NBY , NR5, Sa, N P AH ),

where Nx is the number of site type x, C and H are the number ofcarbon and hydrogen atoms respectively, Sa is the surface area ofthe particle and N P AH is the number of PAHs which make up theparticle. It is worth mentioning that the size of the PAHs in a sootparticle can be different from each other. Based on this state space,particle processes like, inception, jump processes representing sur-face growth, and coagulation, on each particle of type x ∈ E can bedefined by ordinary differential equations (ODEs) of the followingform:

dx

dt= F (x, NR6_AC , NR5F E , NR5AC , NR5Z Z , NR6, P S,m), (8)

where N y is the number of combined-site type y in the particle(see Fig. 10 for combined sites on an example PAH), P S,m is theprobability of a site type S ∈ {E D, Z Z , AC, BY } to act as a neigh-bour site in a process with index m ∈ {1, . . . ,number of processesin Table 2}. The terms N y and P S,m in the function F are requiredto completely close the ODEs. It is clear that these terms requirestructural information about PAHs, and cannot be exactly deter-mined if the PAH structure is not tracked. Therefore, a statistical


Fig. 4. An example of jump process S2. The site counts for both the molecules areprovided. The affected sites are shown in bold face.

method is developed using the KMC-ARS model for the equationclosure, in which N y and P S,m are expressed as functions of x, andare briefly explained below:

N y : The combined sites (N y) are involved in a number of reac-tion steps (S3, S4, S7–S10, S14 and S17) in Table 2. In thesite-counting model, since the relative positions of the sitesare not tracked, the counts of combined sites, required tocalculate the reaction rates, cannot be determined. There-fore, the site-counting model needs an approximation to thestructure to be calculated. This structural information is ob-tained using the KMC-ARS model to find correlations for thecombined sites (N y) in terms of elementary sites involvedin their formation. A method to obtain the correlations forthe major combined sites is detailed in Section 6.1.1.

P S,m: As mentioned before, a reaction on a parent site affectsmore than one site at a time. Fig. 4 shows how an armchairgrowth reaction on phenanthrene to form pyrene affectstwo free-edges (neighbour sites) along with an armchair site(parent site).

In this example, the types of the neighbour sites were changedfrom free-edges to zig-zags. Thus, unless the structure of PAHs istracked, it is not possible to exactly know the neighbour sites to beupdated after a reaction. The KMC-ARS model provides the infor-mation about the neighbour sites in different reactions in terms ofweights to the counts of elementary sites. These weights are usedto calculate P S,m . A detailed explanation of the method used toobtain the site-weights and neighbour probabilities is provided inSection 6.1.2.

A major advantage of site-counting model is that it is compu-tationally very fast compared to the KMC-ARS model. A real-timeKMC simulation of 2.4 ms for a PAH molecule takes about 1.5 s ofcomputation time with the site-counting model and about 30 minwith the KMC-ARS model in the same gas-phase environment.

5. Experimental validation

In this work, PAH growth simulations were carried out in sev-eral laminar premixed flames of C2H2, C2H4 and C6H6 to developcorrelations revealing structural information of PAHs in differentflame environments. Their variation with flame properties like typeof fuel, cold gas velocity, C/O ratio and pressure was also studied.The operating conditions for these flames are provided in Table 3.In the literature, no experimental study on a C2H4–O2 flame with-out any diluent at a pressure of 2.67 kPa was available. Therefore,a hypothetical C2H4–O2 flame (flame 7) was created for simula-tion, which uses the operating conditions of flame 4. All the flameswere simulated using the ABF chemical mechanism [18] along withCHEMKIN package [62] and PREMIX [63] to obtain the chemi-cal species profiles. Fig. 5 shows the computed profiles of majorchemical species for the flames 1, 2 and 4. In the literature, ex-perimentally observed species profiles were available for flame 1

Table 3Flame initial conditions.

Flame Pressure(kPa)

Fuel Composition (mol%) Cold gasvelocity(cm/s)

[C]/[O] Ref.

Fuel O2 Ar

1 2.67 C2H4 19.4 30.6 50 62.5 0.65 [67]2 2.67 C6H6 20.1 78.9 0 42 0.8 [29]3 2.66 C2H2 44.4 55.6 0 42 0.8 [35]4 2.66 C2H2 50 50 0 42 1.0 [35]5 2.66 C2H2 50 50 0 50 1.0 [35]6 2.66 C2H2 51.4 48.6 0 42 1.06 [35]7 2.67 C2H4 50 50 0 42 1.0 see text8 101.3 C2H4 16.3 23.7 60 13 0.69 [68]9 101.3 C2H4 16.3 23.7 60 10 0.69 [68]

10 101.3 C2H4 16.3 23.7 60 8 0.69 [68]11 101.3 C2H4 16.3 23.7 60 6.53 0.69 [68]

only and its comparison with the computed profiles is shown inFig. 5(a). An experimental validation of the chemical mechanismin the diluted flame environments of C2H2 and C6H6 (not presentin Table 3) is shown in the supplementary material. The growthof PAH molecules in all the flame environments was studied usingthe computed species profiles.

PAHs present in a low-pressure C6H6 flame (flame 2) and C2H2flames (flames 3–6) have been observed experimentally in [29,35]using Resonance Enhanced Multi-Photon Ionisation (REMPI) andtime of flight–mass spectrometry (TOF-MS). The experimental de-tails are briefly reviewed here. Ensembles of PAHs present in thegaseous phase at different heights above the burner (HAB) are col-lected. Neutral PAHs are ionised using REMPI, and the time of flightinside a flight tube is used to determine the mass of PAHs [17].The observed ensemble of PAHs at a particular HAB is then rep-resented on a C–H diagram, where each data point denotes a PAHwith a fixed number of C and H atoms. In this work, the KMC-ARSmodel was validated by comparing the PAH ensembles obtainedcomputationally with the experimentally observed ensembles rep-resented on C–H diagrams for the two flames. For the simulationsusing the KMC-ARS model, a seed molecule (or starting structure)is required. To study its effect on simulation results, computedC–H diagrams for flame 2 were obtained with two different seedmolecules: pyrene and benzene (not shown here). No significantvariation in the C–H diagrams with the seed molecule was ob-tained. In this work, pyrene was chosen as the seed molecule, asit is considered to be important for soot nucleation [18]. Fig. 6shows C–H diagrams for flame 2 at three different HABs. The dot-ted lines in the figures represent the positions of peri-condensedPAHs on C–H diagrams [29]. This has been provided as a guide-line to observe the development of PAH ensembles with HAB. Thedata points above this line represent H-rich PAHs and below H-poor PAHs with respect to the most peri-condensed six-memberring structures [29]. The C–H diagrams show that the observedPAH ensembles at the three HABs were very well predicted by theKMC-ARS model. In this flame environment, benzene addition onPAHs was found to dominate over other reactions for the lowerHABs (less than 7 mm), and the reactions involving acetylene dom-inated at higher HABs. This result is in agreement with the predic-tion of [28]. Fig. 7 shows a C–H diagram for flame 4 containingthe simulated along with the experimentally observed ensembleof PAHs with up to 70 carbon atoms. It is evident from the C–Hdiagrams of the two flames that the concentration of data pointsnear the peri-condensed line in case of flame 4 is higher than thatof flame 2 indicating the presence of H-poorer PAHs in flame 4 ascompared to PAHs in flame 2 with same C atom count. It was in-teresting to note that in both the flame environments, free-edgegrowth through route 2 (Fig. 2) dominated at lower flame temper-atures (<1000 K), and route 1 dominated at higher temperatures(or higher HABs). In the environment of flame 4, benzene addition


(a)

(b)

(c)

Fig. 5. Major chemical species profiles for the flames 1, 2 and 4, respectively (Ta-ble 3). (a) C2H4 flame. Filled markers in the figure denote the computed speciesprofiles and hollow markers denote the experimentally observed profiles. (b) C6H6

flame (simulated profiles only). (c) C2H2 flame (simulated profiles only).

on PAHs did not play a significant role due to its low concentra-tion.

Figs. 8(a) and 8(b) show computed PAHs from the two flameshaving the same number of C atoms. It is well known that thePAHs having 5-member rings bordered by not more than two 6-member rings are planar in geometry (the term “planar” refers tothe planar arrangement of C atoms) [17]. PAHs with up to 70 Catoms from flame 5 mostly have 5-member rings on zig-zags, thusbordered by two six-member rings only, and a very few embed-

(a)

(b)

(c)

Fig. 6. C–H diagrams for the C6H6 flame at different HABs. (a) HAB = 10 mm.(b) HAB = 7 mm. (c) HAB = 5 mm.

ded 5-member rings. Hence, the simulation predicts almost planarPAHs for the C2H2 flame, as was observed in [35]. The embed-ded five-member rings occur due to the closure of 5-member bayswith the present reaction mechanism. Bay sites on PAHs were ob-served to occur most commonly due to benzene addition reactionat lower HABs and due to free-edge growth reaction at higherHABs, and hence, bay closure reaction was more important in theenvironment of flame 2. Embedded 5-member rings formed due toclosure of 5-member bays were common in computed PAHs fromflame 2. Therefore, the simulation predicts PAHs with some geo-metrical distortions for this flame. These results are in concordancewith the predictions of Griesheimer et al. [5] for an aromatic flame.In [5], a naphthalene flame was observed experimentally, and was


Fig. 7. C–H diagram for the C2H2 flame at HAB = 7 mm.

(a)

(b)

Fig. 8. Example computed PAHs for the flames 2 and 4, respectively. (a) C6H6 flame.(b) C2H2 flame.

concluded that the PAHs dominating the gas phase were formedby the addition of one PAH/PAH radical over another (biaryl forma-tion). The biaryls thus formed were considered to be highly reac-tive, and can form 5- or 6-member rings by cyclodehydrogenation,if they have bay sites. It is clear that the addition of PAHs/PAH rad-icals can play an important role in the PAH growth mechanism forthe flames of aromatic hydrocarbons and cyclic compounds such ascyclohexane flame [4], where the concentration of PAHs is higheras compared to aliphatic flames. In this work, only the benzeneaddition reaction was considered, which was found to be impor-tant for the benzene flame. In the future, the reactions involvingthe addition of higher aromatic compounds such as naphthalenewill also be included.

The C–H diagrams presented in this paper contain data pointscorresponding to PAHs with even number of C and H atoms. PAHs

Fig. 9. Number fraction of elementary sites for different C atom counts and HABs.

with odd number of C atoms can arise due to: presence of aliphat-ics with odd number of C atoms like C1 and C3 hydrocarbons,presence of 5-member rings on free-edges (for example, indene) oron armchairs (for example, fluorene), and presence of 6-memberrings on zig-zags (for example, phenalene). Reactions leading toPAHs with odd number of C atoms were not included in the re-action mechanism, and therefore, such PAHs were not obtained.These reactions will be explored in future studies.

The H content of PAHs with the same number of C atoms notonly varies with the type of the fuel generating them, but alsowith HAB. The C–H diagrams for flame 2 in Fig. 6 show that PAHsat lower HABs are H-richer than those at higher HABs. It was sug-gested by Weilmünster et al. [35] that for the same number ofC atoms, H-rich PAHs involve more bays and armchairs and less5-member rings than H-poor PAHs. To verify this, ensembles ofPAHs with the same number of C atoms were generated for dif-ferent HABs, and the variation in the concentration of elementarysites on the PAHs with HAB was studied. Fig. 9 shows the concen-tration of elementary sites on PAHs, represented in terms of theirnumber fractions for different HABs and C atom counts. For thecomputed PAHs with n number of C atoms, the empirical mean ofthe number fraction of a site S ∈ {R5, E D, Z Z , AC, BY }, xS,n wasobtained using the following expression: xS,n = N S,n/

∑i∈S Ni,n ,

where N S,n is the average number of sites of type S present onthe PAHs. It is clear from Fig. 9 that for the PAHs with the samenumber of C atoms, the number fractions of armchairs and baysites decrease and that of 5-member rings increases with HAB,thus making PAHs H-poorer with HAB. This result is in line withthe predictions of Weilmünster et al. [35].

The comparison of the computed results with experimentalfindings presented here suggests that the model is capable of pre-dicting the experimental observations very well. However it is lim-ited by the assumption of 2D growth of PAHs, and therefore, needsfurther improvement in the future.

6. Results and discussion

The validation of the KMC-ARS model against some key exper-imental findings in the previous section warrants its use to getthe statistical closure expressions for the site-counting model, asindicated in Section 4. In order to obtain this closure, a detailedstatistical analysis of the results based on the study of growth ofPAHs in the flame environments listed in Table 3 is presented.

6.1. Statistical analysis

This section details the methodology adopted to obtain the un-closed terms in the site-counting model: N y and P S,m (definedin Section 4). Additionally, the effect of steric hindrance in larger


Fig. 10. An example PAH molecule showing combined sites.

PAH structures is investigated. For the results presented in this sec-tion, PAH growth simulations were carried out a large number oftimes in order to obtain average variations in the required randomvariables and confidence intervals over them (see Section 3.5 fordetails).

6.1.1. Combined-site correlationsThe site-counting model requires some approximations to de-

scribe the combined sites because it does not store the PAH struc-tural information. Fig. 10 shows four types of combined sites on anexample PAH structure. The KMC-ARS model can be used to pro-vide this information by developing correlations. The particle statespace is multidimensional, therefore the number of combined siteswill be functions of many variables, particularly the number of car-bon atoms, and the number of all elementary sites. However, itwas assumed to be sufficient to describe the combined site countsby just one particle property. The correlations for the combinedsites are based on the variation in their counts with one of theirconstituent elementary sites. To obtain the correlations, the sim-ulations described above were run 500 times and the combinedsite counts were stored along with the elementary site counts.Fig. 11(a) shows the number of R6 rings as a function of free-edgecount, and Fig. 11(b) shows the number of R6 rings adjacent toarmchairs as a function of armchair count for flame 8. A nearlylinear relationship was observed in all cases. The correlations wereobtained by using a least-squares algorithm to fit a linear func-tion to the data, averaged over all simulations. The correlations forthe combined site, R5ZZ involved in process S14 (Table 2) were notobtained as this reaction does not affect the site counts in the site-counting model [31]. Meaningful correlations for the combined siteinvolving two FE’s next to an embedded R5 (process S17, Table 2)could not be obtained as it occurred very infrequently on the com-puted PAHs.

As an R6 ring consists of three consecutive free-edges, they can-not exist when there are just two free-edges, hence a correlationof the form NR6 = a(N F E − 2) was obtained. The armchair countwas chosen as the fitting variable for the AC_R6 site count as thearmchair count was always found to be lower than the free-edgecount, hence the combined site count should be more sensitiveto the number of armchairs. For the combined sites involving R5,the R5 count was chosen for the same reason. The correlations forthe combined sites required by the site-counting model (N y) wereevaluated for all the flames listed in Table 3 to study their variationwith flame properties like fuel, cold gas velocity, pressure and C/Oratio. In this paper, detailed statistics are provided only for atmo-spheric pressure flames 8–11. It is because such correlations havebeen used in [34] in the site counting model to predict the sootobservables like soot volume fraction, number density and parti-

(a)

(b)

Fig. 11. Correlations for combined sites. Solid lines show linear approximations,dashed lines show confidence intervals. (a) R6 ring count as a function of free-edgecount. (b) AC_R6 site count (R6 ring next to an armchair) as a function of armchaircount.

cle size distribution functions for flames 8–11. Given below are thecorrelations for these flames:

NR6 =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0.169 × (N F E − 2) if N F E > 2 (flame 8),

0.168 × (N F E − 2) if N F E > 2 (flame 9),

0.169 × (N F E − 2) if N F E > 2 (flame 10),

0.166 × (N F E − 2) if N F E > 2 (flame 11),

0 if N F E � 2;

(9)

N AC_R6 =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0.36 × N AC if N F E > 2, N AC > 0 (flame 8),




0 if N F E � 2, N AC � 0;

(10)

NR5F E =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

1.3 × NR5 if N F E > 0, NR5 > 0 (flame 8),

1.25 × NR5 if N F E > 0, NR5 > 0 (flame 9),

1.2 × NR5 if N F E > 0, NR5 > 0 (flame 10),

1.22 × NR5 if N F E > 0, NR5 > 0 (flame 11),

0 if N F E � 0, NR5 � 0;

(11)

NR5AC =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0.066 × NR5 if N AC > 0, NR5 > 0 (flame 8),

0.024 × NR5 if N AC > 0, NR5 > 0 (flame 9),

0.031 × NR5 if N AC > 0, NR5 > 0 (flame 10),

0.025 × NR5 if N AC > 0, NR5 > 0 (flame 11),

0 if N AC � 0, NR5 � 0.

(12)

It can be noticed that these correlations do not vary much forflames 8–11, and can be concluded that the change in cold gasvelocity does not affect the correlations. In a similar fashion, cor-relations for flames 3–6 were evaluated. These low-pressure flamesdiffered in C/O ratio and/or cold gas velocity. For these flames aswell, correlations were found to be varying negligibly indicating noeffect of C/O ratio on the correlations. Average values of the cor-


Table 4Correlation coefficients for the simulated flames.

Flame Correlation coefficients

R6a AC_R6b R5F Ec R5ACd unavail′ F Ee unavail′ AC f

a b c d e1 e1 f1 f2

1 0.052 0.250 1.064 0.006 0.476 0.446 0.579 0.1422 0.062 0.163 0.774 0.028 0.38 0.39 0.972 0.9853–6 0.040 0.123 0.718 0.022 0.183 0.026 0.155 0.6317 0.074 0.206 0.786 0.018 0.41 0.58 0.99 0.0798–11 0.168 0.355 1.243 0.036 0.573 0.024 0.599 0.044

a NR6 = a × (N F E − 2).b N AC_R6 = b × N AC .c NR5F E = c × NR5.d NR5AC = d × NR5.e Nunavail′ F E = tanh(e1 × log(e2 × N F E + 1)).f Nunavail′ AC = tanh( f1 × log( f2 × N AC + 1)).

relation coefficients for these flames along with the other flamessimulated in this work are provided in Table 4. It can be seen inTable 4 that the correlations for the flames having different fuelsbut same pressure (flames 2–7) differ from each other. Similarly,comparing the correlations for flame 1 and flames 8–11 in thesame table shows that the correlations are dependent on flame op-erating pressure as well. Since the correlations reflect the resultingPAH structure in a flame environment, the changes in correlationswith the type of fuel and pressure of the flames may be due to thechange in the dominant PAH growth processes (discussed in Sec-tion 5 for C6H6 and C2H2 flames), which can significantly changethe PAH structures [64]. The domination of a PAH growth pro-cess in a flame environment depends on its rate, which in turndepends on species concentration and rate constants of the PAHreactions (Eq. (3)). Chemical species profiles vary significantly withthe type of fuel in the flame (see Fig. 5), and the rate constantsof PAH reactions depend on pressure [54]. Therefore, the dominantPAH processes in flames with varying pressure and/or fuels maybe different, leading to a change in PAH structures and thus thecorrelations. The correlations and the PAH statistics discussed fur-ther in this paper have been provided for all the flames listed inTable 3. This is to make the simulation of these flames along withthe flames observed under similar conditions possible using thesite-counting model to predict substantial details of soot particleslike soot volume fraction, number density, particle size distributionfunctions, C/H ratio and average PAH sizes [31,34].

6.1.2. Neighbour statisticsWhen a surface reaction occurs it is necessary to update the

neighbour site types, however, information about the neighboursites is not available in the site-counting model. As outlined inSection 4, P S,m is required to obtain this information. Only thejump processes involving six-member rings, and oxidation of re-active sites in Table 2 are considered here, as those involvingonly 5-member rings do not change neighbour sites, and pro-cesses S7 and S8 were observed to occur so infrequently thatmeaningful statistics could not be obtained. It was initially as-sumed that neighbour sites of type S ∈ {E D, Z Z , AC, BY } are se-lected for process m with the probability P S,m(t) = N S(t)/Ntot(t),where N S(t) is the number of sites of type S at time t , Ntot(t)is the total number of sites (ignoring 5-member rings), and m ∈{1, . . . ,number of processes in Table 5} is the process index. How-ever, it was found that when using these probabilities the site-counting model did not agree with the KMC-ARS model, whichsuggests that some sites are more likely than others to be neigh-bours for certain processes. Therefore, site count weights were in-troduced for each process such that the weighted counts are givenby: N ′

S,m(t) = W S,m × N S (t), hence the probability of a site of typeS being selected at time t for process m becomes:

Table 5Neighbour site-weights for the principal reactions for flames 8–11.

No. Process Reaction Flame W F E W Z Z W AC W BY

1 S1 FE growth C1 0.24 0.24 0.3 0.22C2 0.23 0.26 0.31 0.2C3 0.22 0.26 0.31 0.21C4 0.22 0.26 0.31 0.21

2 S2 AC growth C1 0.38 0.26 0.19 0.17C2 0.38 0.27 0.19 0.16C3 0.38 0.28 0.19 0.15C4 0.38 0.27 0.19 0.16

3 S3 R6 desorption C1 0 0.19 0.38 0.43C2 0 0.21 0.37 0.43C3 0 0.21 0.37 0.42C4 0 0.21 0.36 0.43

4 S4 R6 to R5 at AC C1 0 0.19 0.39 0.42C2 0 0.2 0.37 0.43C3 0 0.23 0.39 0.38C4 0 0.20 0.36 0.44

5 S9 FE oxidation by O2 C1 0 0.19 0.37 0.44C2 0 0.21 0.37 0.43C3 0 0.21 0.36 0.43C4 0 0.21 0.39 0.4

6 S10 FE oxidation by OH C1 0 0.19 0.37 0.44C2 0 0.21 0.37 0.43C3 0 0.21 0.36 0.43C4 0 0.21 0.39 0.4

7 S11 AC oxidation by O2 C1 0 0.37 0.4 0.23C2 0 0.35 0.36 0.29C3 0 0.35 0.36 0.29C4 0 0.37 0.37 0.26

8 S12 AC oxidation by OH C1 0 0.37 0.4 0.23C2 0 0.35 0.36 0.29C3 0 0.35 0.36 0.29C4 0 0.37 0.37 0.26

9 S15 Bay closure C1 0.41 0.29 0.30 0C2 0.41 0.29 0.30 0C3 0.42 0.29 0.29 0C4 0.43 0.29 0.28 0

P S,m(t) = N ′S,m(t)

N ′tot,m(t)

, (13)

where N ′tot,m(t) is the sum of the weighted site counts and de-

pends on the process m. It is assumed that the site weights donot depend strongly on PAH size, therefore they are considered tobe constant for each process. This assumption is supported by theKMC-ARS simulations conducted for this study.

In order to calculate the site-weights, several KMC-ARS simula-tions were performed. For each simulation there are K events, andKm denotes the number of times event m occurred. On selection ofthe tth jump process, where t ∈ {1,2, . . . , K }, the following infor-mation was stored: the time point t , the jump process index, mt ,the types of both neighbour sites, T1(t) and T2(t) and the countsof all site types; NE D(t), N Z Z (t), N AC (t), NBY (t). L S,m shall denotethe number of times a site of type S acted as a neighbour for pro-cess m. In the limit of large Km , the probability of a site acting asa neighbour is assumed to approach P S,m = L S,m/Km , therefore, bysumming over each jump process and solving the following equa-tion, the site-weights can be obtained:

Km∑t=1

W S,m N S (t)W F E,m N F E (t)+W Z Z ,m N Z Z (t)+W AC,m N AC (t)+W BY ,m NBY (t) = L S,m, (14)

where S ∈ {E D, Z Z , AC, BY } and m ∈ {1, . . . ,9} (processes listedin Table 5). As there are four possible site types which may actas neighbours; free-edges (F E), zig-zags (Z Z ), armchairs (AC ) and


Fig. 12. Neighbour site-weights of elementary sites for different reactions for theC2H2 flames (flames 3–6). The reactions corresponding to the reaction numbers (onx-axis) are listed in Table 5. For each reaction, the four consecutive parallel stacksrepresent flames 3 to 6 from left to right.

bays (BY ), Eq. (14) gives a system of four linear equations for eachprocess m, which can be solved using a standard numerical tech-nique such as a Newton method. Equation (14) was solved for thefour elementary sites, with the additional constraints of W S,m � 0and

∑S W S,m = 1. Table 5 shows the site-weights for the princi-

pal reactions for flames 8–11. These weights are used to calculateP S,m = f (W S,m) in the site-counting model using Eq. (13).

It is clear from Table 5 that assuming equal weights for allthe sites would lead to an unrealistic PAH molecule using thesite-counting model. For the reactions involving R6 desorption andconversion of a 6-member ring to a 5-member ring, the free-edgesite-weight (W F E(3–6)) is zero, as it is not possible to desorb asix-member ring having more than three consecutive free-edgestogether with the current reaction mechanism. Thus, a free-edgecannot exist as the neighbour site of a R6 ring. Similarly, forarmchair oxidation reactions (reactions S11 and S12 in Table 2),a free-edge cannot exist as a neighbour site. Therefore, the free-edge site-weight (W F E(7,8)) is zero. For a 6-member bay closurereaction, a bay cannot be present as a neighbour, as it is geo-metrically prohibited. Therefore, for this reaction, W BY is zero.A study on the variation in neighbour site-weights with flameproperties was also conducted. It can be seen in Table 5 thatthese weights for flames 8–11 do not vary significantly indicatingno effect of cold gas velocity on them. Variation in the neigh-bour site-weights with C/O ratio, operating pressure and type offuel of the flames has been shown graphically in Figs. 12, 13and 14 respectively. It can be easily concluded from these fig-ures that the site-weights do not vary significantly with C/O ra-tio (variation within 8%), but varies with the type of fuel andpressure, as was observed in the case of combined sites correla-tions.

6.1.3. Unavailable sitesAs PAH molecules grow, their structure can become complex

and it is possible for some sites to be unavailable for reactionbecause they are hindered by nearby aromatic rings. This hasbeen referred to as steric hindrance. Fig. 15 shows how free-edges and armchairs can become unavailable on a computed PAHmolecule from flame 8. This generally occurs when the structuregrows around on itself. This effect should also be observed if PAHmolecules are stacked close together, and is probably one of themajor contributing factors of the alpha correlation for inactive sitesin the ABF model [18]. Such contorted structures, as observed inFig. 15 would most likely give rise to a 3D structure as the nearbyhydrogen atoms interact, however, if the molecule was constrainedin a stack this may be less energetically favourable, as the bending

Fig. 13. Variation in neighbour site-weights of elementary sites for different reac-tions with pressure for the C2H4 flame. The reactions corresponding to the reactionnumbers (on x-axis) are listed in Table 5. For each reaction, first stack represents av-erage neighbour site-weights for flames 8–11, and second stack represents flame 1.

Fig. 14. Variation in neighbour site-weights of elementary sites for different reac-tions with the type of fuel. The reactions corresponding to the reaction numbers(on x-axis) are listed in Table 5. For each reaction, the three consecutive parallelstacks represent three different flames at the same pressure (2.67 kPa): first stack—C6H6 flame (flame 2), second stack—C2H4 flame (flame 7), third stack—C2H2 flame(flames 3–6).

Fig. 15. A computed PAH molecule from flame 8 at HAB = 5 mm. Filled circles andstars denote unavailable free-edges and armchairs respectively. Hollow symbols de-note available sites.

of a PAH may not be allowed due to the presence of nearby PAHs.It is more likely in this case that the hydrogen are abstracted anda C–C bond is formed through bay closure reactions later on in theflame.

The KMC-ARS model allows unavailable sites to be counted ex-plicitly. Figs. 16(a) and 16(b) show the fraction of unavailable free-edges and armchairs respectively as functions of the free-edge andarmchair counts. The fraction of unavailable sites firstly increaseswith the site counts before reaching an asymptotic limit at largersite counts. A function of the form Y = tanh(a × log(b X + 1)) wasfound to describe this asymptotic behaviour well. The curves inFig. 16 were fitted by trial and error, and the equations for flames8–11 are given below:


(a)

(b)

Fig. 16. Unavailable sites. Solid lines show a tanh fit. Dashed lines show confi-dence intervals. (a) Fraction of unavailable free-edges. (b) Fraction of unavailablearmchairs.

Nunavail′ F E =

⎧⎪⎨⎪⎩

tanh(0.573 × log(0.024N F E + 1)) flame 8,



tanh(0.984 × log(0.014N F E + 1)) flame 11;(15)

Nunavail′ AC =

⎧⎪⎨⎪⎩

tanh(0.599 × log(0.044N AC + 1)) flame 8,



tanh(0.934 × log(0.024N AC + 1)) flame 11.

(16)

The above equations predict very similar variation of unavail-able free-edges with free-edges count for flames 8–11. The equa-tions for unavailable armchairs show a similar trend. Therefore, thecorrelation coefficient for only flame 8 is presented in Table 4.A similar trend was observed for flames 4–7. Therefore correla-tion coefficients for only flame 4 is provided in the same tablealong with the coefficients for other flames listed in Table 3. Theequations of the above form are remarkably similar to the alphacorrelation [18], used in the ABF soot model to determine em-pirically the number of active sites on soot particles. The alphacorrelation is used to model particle “aging,” which is the observedphenomenon that soot particle become less reactive with age. Thecorrelations presented here have a similar role, as they effectivelydecrease the number of active sites present on the PAHs with thesimulation time.

6.2. Site-counting model validation

The statistics and correlations described above were imple-mented in the site-counting model, and identical simulations were

(a)

(b)

Fig. 17. Comparison of the results from the KMC-ARS model and the site-countingmodel for flame 11. Dashed lines show the results from the site-counting model andsolid lines show the result from the KMC-ARS model. (a) C atom count as a functionof time. (b) Site count as a function of time.

performed using the site-counting model and the KMC-ARS modelfor flames 8–11. Fig. 17 shows the comparison of the PAH charac-teristics: carbon atom count and number of elementary sites onthe PAHs in flame 11. For flames 8–10, a similar comparison hasbeen shown in the supplementary material.

Fig. 17(a) shows that the site-counting model and the KMC-ARS model predict similar carbon atom counts at all observedflow times. Fig. 17(b) shows that the agreement for the numberof free-edges, zig-zags and armchairs is reasonable also. The closeagreement of the number of elementary sites is important becausethe site counts are used to calculate the process rates. Also, thissignifies that the site-counting model predicts PAHs very similarto those computed by the KMC-ARS model. A comparison betweenthe two models has also been shown in our previous work [31]in a flame-like environment. The extent of agreement of the com-puted PAH characteristics by the two models is very encouraging.It shows that the detailed PAH growth model can be confidentlyaccommodated into a soot particle population balance using thesite-counting model.

As the PAH molecules present in flames are very small in size(PAHs with less than 100 C atoms are observed in flames [9,10]),a comparison between the two models was carried out over thisrange with and without the correlation for unavailable sites, to testits importance in the experimental size range. Fig. 18 shows thevariation in the number of PAH sites with the number of C atomsfor this case. It can be concluded from this figure that the corre-lations for unavailable sites are not very significant for small PAHmolecules.


Fig. 18. Comparison of the results: Dashed lines show the results from the site-counting model and solid lines from the KMC-ARS model. Filled symbols show theresults from site-counting model without correlations for unavailable sites.

7. Conclusion

A new detailed PAH growth model (KMC-ARS model) has beendeveloped. This model considers growth of a PAH molecule in alldirections, irrespective of the orientation of reactive sites. A largeset of PAH processes along with the reactions involved in the PAHgrowth has been presented. DFT simulations to evaluate the ratesof the reactions involved in the cyclodehydrogenation process toform a 6-member ring on PAHs have been carried out. An algo-rithm has been developed to track the structure of a PAH molecule,as it grows in a flame-like environment. This model allows theexact estimation of the distribution of sites on a PAH molecule.However, the computational time required to track the growth ofa single PAH molecule is very high, therefore it is computationallyunfeasible to incorporate this detailed PAH growth model into asoot particle population balance.

The site-counting model proposed by Celnik et al. [31] allowsthis detailed PAH growth model to be coupled into soot parti-cle population balances by neglecting PAH structure. This modelis computationally very fast (computational time required to trackthe growth of a single PAH molecule is about 1.5 s with the site-counting model, and is about 30 min with the KMC-ARS modelfor a KMC simulation of 2.4 ms). Since the site-counting modelneglects the PAH structure, the structural information about thePAH molecule is provided by the KMC-ARS model. Correlationshave been developed for different flame environments which de-scribe the number of combined sites (more than one adjacent site),which are required to calculate the rates of some reactions. Sitetype weights have been found which allow neighbour sites to beselected appropriately without prior knowledge of the structure.A method to calculate the site weights using results obtained us-ing the KMC-ARS model has been described.

It has also been shown that as a PAH molecule grows, reactivesites can become unavailable due to steric hindrance. The fractionof unavailable sites increases rapidly with the size of the PAHs ini-tially, but shows an asymptotic behaviour afterwards. Correlationsfor the fraction of unavailable sites have also been developed. Allthe PAH statistics detailed in this paper have been shown to beindependent of cold gas velocity and C/O ratio of the flames, butdepend on the type of fuel and pressure. The statistics have beenobtained for the commonly observed premixed laminar flames,which can be used in the site-counting model along with sootparticle population balance to predict soot observables like sootvolume fraction, number density, particle size distribution func-tions, C/H ratio and average PAH sizes [31,34]. The results of theKMC-ARS model have been compared to the site-counting model.

The agreement between the carbon atom counts and the princi-pal site counts is reasonable, which suggests that the detailed PAHgrowth model can be implemented in the soot particle populationbalances through the site-counting model without much loss of in-formation.

Acknowledgments

A.R. is grateful to Cambridge Commonwealth Trusts (CCT) andClare College, Cambridge for their financial support. The authorshighly acknowledge the support of EPSRC under EP/C547241/1 andEP/E01724X/1.

Supplementary material

The online version of this article contains additional supple-mentary material. Section 1 provides the details of the theoreticalcalculations required to obtain the rate constants for the elemen-tary reactions (listed in Table 1) involved in the cyclodehydrogena-tion process. Section 2 consists of figures showing the experimen-tal validation of the chemical mechanism used in this work bycomparing the experimentally observed species profiles for a C2H2and a C6H6 flame with the computed ones. It should be noted thatthese two flames have Ar as a diluent, and have not been used inthis work for generating PAH statistics.

Please visit DOI: 10.1016/j.combustflame.2009.01.005.

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