Detailed Kinetic Monte Carlo Simulations of Graphene-Edge Growth

Detailed Kinetic Monte Carlo Simulations of Graphene-Edge Growth

(Parte 1 de 4)

Detailed Kinetic Monte Carlo Simulations of Graphene-Edge Growth

Russell Whitesides and Michael Frenklach*

Department of Mechanical Engineering, UniVersity of California, and EnVironmental Energy Technologies DiVision, Lawrence Berkeley National Laboratory, Berkeley, California 94720

ReceiVed: July 10, 2009; ReVised Manuscript ReceiVed: NoVember 18, 2009

A new detailedchemical-kineticMonte Carlo model of graphene-edgegrowth is presented.The model employs a fine-grained approach to chemically resolved species, allows for incorporation of five-member rings into growingstructures,and links the stochastickineticsteps to a geometryoptimization,therebyproperlyaccounting for curving of molecular structures. The evolving morphology is greatly affected by the rates of key reactions and hence by surface-site steric environment and gas-phase species concentrations. The evolving graphene morphology and growth rates seemingly reach “asymptotic” behavior, independent of the initial substrate. Most noteworthy, growing layers become significantly curved. The curvature occurs regardless of initial substrate at both 1500 and 2000 K with higher curvature occurring at the lower temperature. More intriguing is the observation that, at 2000 K, transition from planar to curved growth does not commence immediately but occurs at some later time, seemingly when the growing graphene reaches a size significantly larger than coronene. No curvature is produced in numerical simulations at 2500 K, indicating that high-energy environments cause the five-member-ring to be less stable, thus preventing them from forming.

I. Introduction

Graphene. Graphene, a single sheet of graphite, has recently gained substantial interest in the areas of condensed-matter physics and materials science due its unique properties such as high electrical conductivity,1-3 superior thermal conductivity,4 and intrinsic strength.5 The envisioned applications of freestanding graphene sheets are broad6 and include composites,7 electronic devices,8 sensors,9 photodetectors,10 batteries,1,12 ultracapacitors,13 and imaging substrates.14 The current methods of producing graphene sheets are mechanical exfoliation of graphite15,16 (i.e., peeling off layers with scotch tape), chemical reduction of exfoliated graphite oxide,17-19 vacuum graphitization of silicon carbide substrates,20 chemical vapor deposition on substrates,21-29 and substrate-free gas-phase synthesis.30-3

Graphenesubstrateshave also become an importantsurrogate for theoretical studies of carbonaceous (soot and interstellardust) particle surface34-39 because such particles are composed of graphitic elements.40-43 Understanding the chemical mechanisms underlying the growth of soot in high-temperature environments, such as flames, has progressed substantially in recent years. Our objective here is to summarize the current knowledge and present an emerging model of graphene-edge growth, resolved at elementary-reaction level. In addition to approaching the goal of developing predictive models of soot formation in combustion, the gained knowledge may also assist in seeking mass-production methods of graphene synthesis.

Mechanism of Graphene-Edge Growth. It has been established in experimental studies that acetylene is the principal gaseous growth species that reacts at the soot particle surface, and that this carbon deposition process follows first-order kinetics.4-46 Theoretical treatment of surface reactions was initiatedby introductionof the hypothesisof chemicalsimilarity, which postulated that chemical reactions taking place on soot particle surface are analogous to those of large PAHs.47-49 In other words, the surface of soot particles was assumed to look chemically like an edge of a large PAH molecule, covered with C-H bonds. The gaseous environment forming soot is usually dominated by acetylene for molecular growth species and by hydrogen atoms for radicals, and the growth of carbon mass was suggestedto follow the hydrogen-abstraction-C2H2-addition (HACA) mechanism,49,50 a repetitive sequence in which a surface radical site is created by hydrogen abstraction and subsequent acetylene addition to the created surface radical forms an aromatic ring,

The initial HACA mechanism for surface growth was based on the armchair edge of aromatics,48,50 as depicted above. However, there is no experimental or theoretical evidence indicating the growing surface has to be exclusively armchair. Furthermore, it was later noted that applying the HACA mechanism to a graphene armchair edge of a finite size leads to the formation and persistence of zigzag edges.34 It was also noted34 that application of the HACA mechanism to zigzag edges results in the formation of five-member rings. A followup study51 corroborated that five-member ring adsorption can proceed via a similar mechanism and with a similar rate to those of the “canonical” HACA reaction mechanism. The same study presented a mechanism for five-member ring migration, which included a triradical species as an intermediate.

Frenklach, Schuetz and Ping (FSP)36 found a more plausible five-member ring migration mechanism with only monoradical

* To whom correspondence should be addressed. Phone: 510-643-1676. E-mail:

J. Phys. Chem. A X, x, 0 A

10.1021/jp906541a X American Chemical Society

intermediates and lower barriers. The authors performed sterically resolved kinetic Monte Carlo (KMC) simulations by employingthe lone-ringmigrationreaction,shown above, along with adsorption, desorption, and growth steps. The KMC model made use of the steady-state assumption for all radical intermediates to reduce the computational expense of the calculations. The model was able to match initial soot growth rates observed in premixed ethylene-oxygen flame studies of Harris and Weiner.4 The authors concluded that competition between migration of five-member rings along the zigzag edge and nucleation of six-member rings was a key mechanistic feature dictatinggrowth rate and morphology.An importantimplication of the migration phenomenon is that while five-member rings are constantly being formed on the growing edge, they do not accumulate; rather, they are converted to six-member rings.

FSP also noted the possibilityof five-memberrings migrating toward each other and “colliding”,

Whitesides et al.52 investigated the energetics and kinetics of this ring-collision reaction and found that its rate was comparable to the rate of lone-ring migration.36 They reexamined the lone-ring migration, including additional reaction products, and found that ring desorption is coupled to ring migration and that radical intermediates on both the migration and collision pathways had lifetimes exceeding microseconds at flame conditions. Both of these conclusions indicated a need for a more detailed description of surface species and processes than was used in the FSP simulations.

Further theoretical analysis by Whitesides et al.53 revealed that the product of the ring collision can undergo isomerization to reverse its orientation, or “flip”. The viability of the flip reaction motivated study54 that identified the phenomenon we call embedded-ring migration, in which a five-member ring moves through the zigzag edge of a graphene layer. Theoretical rate coefficients were found to be comparable to the previously investigated flip reaction and other competitive zigzag-edge reactions. The fast kinetics indicated that the embedded ring moves essentially freely within the zigzag edge. On larger substrates, the reaction has a weak thermodynamic preference for configurations with the five-member ring in the interior of the edge as opposed to at the corner, causing embedded rings to be found more often away from the corner of zigzag edges. In spite of this slight thermodynamic tendency, the occurrence of the embedded-ring migration reaction gives embedded rings ample access to the edge corner where they may interact with migrating rings or with gas-phase species. The high mobility of embedded rings enables the layer to minimize the inclusion of five-member rings, and thus should contribute significantly to annealing and smoothing of growing surfaces.

Kraft and co-workers have developed a KMC model37-39 that follows the form of the FSP model. In their initial formulation37 the model included the reactions of the FSP model and added edge oxidation. The KMC model was then linked to a soot populationbalance through statisticalinformationgathered from simulations of single layers and used in simulations of a soot particle formation in a plug-flow reactor55 and laminar premixed flames.56 The most recent study39 included aromatic bay closure reactions57,58 along with embedded-ringmigration58 and benzene addition to radical edge sites.59 That study showed agreement between simulation results and experimental measurements for H-to-C ratios of PAH forming in acetylene and benzene flames.


Their work has focused on soot particle inception with small (2-3 ring) aromatic molecules used as growth species, employing the numerical technique60 that couples KMC with molecular dynamics (MD).68-73 The reaction steps in the initial model60,61

Russell Whitesides earned his B.S. and Ph.D. degrees in Mechanical Engineering from Virginia Commonwealth University and the University of California at Berkeley, respectively. His dissertation research focused on the chemistry of aromatic edges. He is now a post-doctoral researcher at Lawrence Livermore National Laboratory performing fluid-dynamic modeling of internal combustion engines.

MichaelFrenklachis Professorin the Departmentof MechanicalEngineering of the University of California at Berkeley. He received his Diploma in Chemical Technology from the Mendeleyev Chemical-Technological University,Moscow,Russia,and his Ph.D. in PhysicalChemistryat Hebrew University, Jerusalem, Israel. His faculty appointments began in 1979 in the Department of Chemical Engineering at Louisiana State University. In 1985 he joined the Materials Science Department of the Pennsylvania State University, and in 1995 he accepted his current position at Berkeley. ProfessorFrenklach’sresearchinterestsare primarilyin the area of chemicalkinetic modeling and elucidation of reaction mechanisms of complex reaction networks, including combustion chemistry, inception and growth of particulatematter, and chemical-vapordepositionof diamond.His current activities focus on global system approach to predictive modeling and uncertainty quantification.

B J. Phys. Chem. A, Vol. x, No. x, X Whitesides and Frenklach included radical combination reactions of naphthalene and acenaphthylene and aromatic ring closure. Later work62,63 includedC1 throughC4 speciesas growthcomponentsand found

C2H2 to be the most prevalentnonaromaticgrowth species.Violi also investigated the following reaction58

which is the smallest analog of the embedded-ring migration, and considered oxidation reactions.64 More recent studies have focused on clustering of aromatic molecules formed from the KMC/MD simulations.65-67 However, surface processes aside from ring closure were not considered.

Here we present a new kinetic Monte Carlo model for graphene-edge growth, built on accumulated knowledge of elementary reaction processes. The model employs a more detailed description of surface reactions and sites and includes many more reactions creating five- and six-member ring complexes.Incorporationof five-memberringsleadsto graphene sheet curvature and so the KMC model is linked to a molecular mechanics (M) geometry optimization to account for the resulting structures. Our focus remains on growth of graphene and so oxidation is not included.

I. Methodology

The KMC simulations tracked a single graphene “molecule” evolving in a flame environment. At each time step, a reaction event was selected stochastically and then applied. The methodology for specifying these reactions followed Frenklach et al.36 The simulationwas treated as a series of Markov processes, i.e., each subsequent simulation step was only dependent on the current simulation state and not on the previous states. The selection of the reaction event and specific graphene-edge site was done by application of the Gillespie algorithm74 adapted for surface processes.75 In short, given a current state at time tn, the time of the next reaction event at surface site i, tn+1,i, was evaluated as tn+1,i ) tn - ln u ktotal,i where u is a random number distributeduniformlybetween zero and one and ktotal,i is the sum of the rates of the j reactions possible at site i, kj,i, ktotal,i ) ∑j kj,i

The smallest of the tn+1,i determined the time instant, tn+1, and hence the site, i, of the next KMC step. The selection of the particular reaction that occurs at that time was done by comparing the rate-constant ratios computed for site i, pj,i ) kj,i ktotal,i with another random number u, again distributed uniformly between zero and one. The selected reaction was applied to the selected site, the time was advanced to tn+1, and the process was repeated until the simulation ended.

The first modification of the FSP model36 was inclusion of more individually resolved surface species. The steady-state assumptionused for intermediateelementaryreactionsof “single step” transformation, applied to speed up the numerical simulation, was found to be inadequate in many cases due to longlived intermediates.52,53 Removal of these assumptions necessitated a more “fine-grained” description of surface processes and hence of surface species.

In the current model, nine surface species were defined and classified as either active (bonded to two other surface carbon atoms) or solid (bonded to three other surface carbon atoms). There are six active carbon atom species: hydrogen terminated six-memberring carbon atoms (a1H), hydrogen terminatedfivemember ring carbon atoms (a2H), radical six-member ring carbon atoms (a1-), radical five-member ring carbon atoms (a2-), five-member ring carbon atoms bonded to two hydrogen atoms (a2H2), and six-member ring carbon atoms bonded to a

C2H moiety (a1C2H). There are two classifications for solid carbon species (i.e., bonded to three other carbon atoms), those that are part of six member rings (s1) and those that are part of five and six-memberrings (s2). Figure 1 illustratesthe six active carbon atoms and two solid carbon atoms. Note that a lower case “a” is used in these designations for carbon atoms to

differentiate them from the notation A1,A 2,used to describe

free aromatic molecules of varying sizes.76,7 In addition to the site types shown in Figure 1, one more type was included, “a1solid”. It designates a nonreactive site, the type occurring at the bottom edge of a graphene substrate, used in the present study to simulate larger substrates, as shown in Figure 2.

A total of 42 surface transformations were included in the present model and are depicted in Table 1. Reactions forming five- and six-member ring complexes (14, 3, 34, and 39) and those capping embedded five-member rings (35-38) are of particular interest. Rate coefficients for the reactions were taken from experimental data, quantum chemical calculations, or assigned on the basis of analogy to other reactions when data were not available. A detailed account of assignments and sources for the reaction rate coefficients is provided in the Supporting Information.

(Parte 1 de 4)