molecular dynamics simulations of laser-induced incandescence of soot using an extended reaxff...

Molecular Dynamics Simulations of Laser-Induced Incandescence of Soot Using anExtended ReaxFF Reactive Force Field

Amar M. Kamat,† Adri C. T. van Duin,*,† and Alexei Yakovlev‡

Department of Mechanical and Nuclear Engineering, PennsylVania State UniVersity, UniVersity Park,PennsylVania 16802, United States, and Scientific Computing & Modelling NV, Vrije UniVersiteit, TheoreticalChemistry, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands

ReceiVed: August 24, 2010

Laser-induced incandescence (LII) of soot has developed into a popular method for making in situ measurementsof soot volume fraction and primary particle sizes. However, there is still a lack of understanding regardingthe generation and interpretation of the cooling signals. To model heat transfer from the heated soot particlesto the surrounding gas, knowledge of the collision-based cooling as well as reactive events, including oxidation(exothermic) and evaporation (endothermic) is essential. We have simulated LII of soot using the ReaxFFreactive force field for hydrocarbon combustion. Soot was modeled as a stack of four graphene sheets linkedtogether using sp3 hybridized carbon atoms. To calculate the thermal accommodation coefficient of variousgases with soot, graphene sheets of diameter 40 Å were used to create a soot particle containing 2691 atoms,and these simulations were carried out using the ReaxFF version incorporated into the Amsterdam DensityFunctional program. The reactive force field enables us to simulate the effects of conduction, evaporation,and oxidation of the soot particle on the cooling signal. Simulations were carried out for both reactive andnonreactive gas species at various pressures, and the subsequent cooling signals of soot were compared andanalyzed. To correctly model N2-soot interactions, optimization of N-N and N-C-H force field parametersagainst DFT and experimental values was performed and is described in this paper. Subsequently, simulationswere performed in order to find the thermal accommodation coefficients of soot with various monatomic andpolyatomic gas molecules like He, Ne, Ar, N2, CO2, and CH4. For all these species we find good agreementbetween our ReaxFF results and previously published accommodation coefficients. We thus believe thatMolecular Dynamics using the ReaxFF reactive force field is a promising approach to simulate the physicaland chemical aspects of soot LII.

1. Introduction

Soot formation in combustion systems has been an importantproblem for combustion scientists for a long time. The formationof soot is a consequence of poor combustion conditions, and itaffects the efficiency of the combustion device. Soot emissionsinto the atmosphere are also known to have an adverse effecton human health.1,2 Laser-based diagnostics have had animportant role in studying the formation and destructionprocesses associated with soot in combustion systems. Laser-induced incandescence (LII) is developing into a popular methodfor making quantitative spatial and temporal measurements ofsoot.3 This method involves heating the soot particles by a high-energy pulsed laser to incandescent temperatures of about 3500K. Soot particles at these high temperatures radiate energy, andthe subsequent cooling signal is detected and analyzed. Heatedsoot particles lose energy by conduction (due to collisions withsurrounding gas molecules), radiation, and vaporization. Sootvolume fraction depends on the magnitude of the cooling signal,4

whereas the decay of the cooling signal is determined by thesurface area of the particles. The latter relation is important inquantifying the primary particle sizes, but this warrants a deepunderstanding of the processes that lead to the generation ofthe cooling signal.5

A number of theoretical models have been developed with aview to using LII for quantitative soot volume fraction andparticle size measurements. Theoretical models solve thetransient mass and heat balance equations of the soot particlesafter being hit by the laser pulse, to obtain its temperatureresponse with respect to time. Eckbreth6 in 1977 related thetime varying interference to laser heating, conduction, andsublimation of the soot particles. The earliest models of LIIwere developed by Melton7 and Dasch8 and further improve-ments were made by Case,9 Mewes,10 Liu,11 and Michelsen.12

Despite these modeling efforts, there are still questions regardingseveral heat transfer processes, particularly the collision-basedcooling which occurs in the free molecular regime and thevaporization of carbon fragments away from the soot particlewhich occurs in high fluence experiments.3 Forming a theoreticalmodel is thus challenging due to uncertainties in the choice ofphysical and optical properties of soot, as well as complicationsin modeling processes at the molecular level.

In this paper, we have tried a computational approach byperforming molecular dynamics (MD) simulations of the LIIprocess. Daun et al.13,14 have previously used MD to find thethermal accommodation coefficients of various gas species withthe soot particle, which is an important coefficient to modelthe collision-based cooling of the heated soot particle. However,to date, no molecular simulation (MD or otherwise) has beencarried out to model the other heat transfer mechanisms, viz.,vaporization and oxidation. We have used the ReaxFF Reactive

* To whom correspondence should be addressed, [email protected].† Department of Mechanical and Nuclear Engineering, Pennsylvania State

University.‡ Scientific Computing & Modelling NV, Vrije Universiteit.

J. Phys. Chem. A 2010, 114, 12561–12572 12561

10.1021/jp1080302 2010 American Chemical SocietyPublished on Web 11/10/2010

Force Field15,16 to simulate the LII process and generate thecooling signal. Once the signal was generated, we carried outanalysis to link the cooling of the soot particle to bond breakageof the sp2/sp3 hybridized carbon atoms in the particle. The LIIsimulations were performed for a variety of pressures, and thesoot particle was placed in the presence of both inert and reactivegases to study the effect of oxidation on the cooling signal.

Simulations were also performed to find the thermal accom-modation coefficients of various monatomic and diatomic gaseswith soot. The accommodation coefficient R basically specifiesthe efficiency of energy transfer in a collision from a heatedsoot particle to a gas molecule. It was defined by Knudsen17 asthe ratio of the energy gained by the gas molecule during thecollision to the maximum possible energy transfer between thesoot particle and the molecule (corresponding to completeenergy accommodation of the gas with the soot particle)

Eout is the final energy of the molecule, Ein is the initial energyof the molecule, Es is the energy of the molecule if it iscompletely accommodated at the soot temperature, and the ⟨ ⟩signs denote time averaging of the above quantities. As collision-based cooling occurs in the free molecular regime, accurateknowledge of R is necessary to correctly model the conductionprocess. We have performed simulations in which gas-sootcollisions were allowed to take place and the values are reportedfor various monatomic and polyatomic gas species like He, Ne,Ar, N2, CO2, and CH4.

2. ReaxFF-Based Molecular Dynamics

A classical molecular dynamics simulation solves Newton’sequations of motion for all atoms in the system. The forcesexperienced by atoms are derived from space derivatives of theforce field used for the MD simulation, and the resultingaccelerations are integrated using a suitable scheme like theVelocity-Verlet algorithm, to find the velocities and displace-ments of atoms as a function of time. We have used the ReaxFFReactive Force Field for hydrocarbon oxidation developed byChenoweth et al.15 for all the simulations carried out. ReaxFFuses a bond order-bond distance relation in conjunction withthe bond order-bond energy relation, which enables it toproperly simulate the smooth dissociation of bonds.16 Coulomband Morse potentials are used for nonbonded interactions, andthese interactions are calculated for all atom pairs. The forcefield used in this paper has previously been extensivelyoptimized against quantum and experimental data.15 The majoradvantage of this force field is that it can be used to simulatebond formation and bond breakage, thus providing a powerfultool to simulate vaporization and oxidation of the heated sootparticle.

The accommodation coefficient simulations used a biggermodel of soot, and thus a new version of ReaxFF incorporatedinto the Amsterdam Density Functional (ADF)18 program wasused for this. The original ReaxFF program was modified tofacilitate calculations of large systems. The main modificationsare listed below:

1. Optimizations of the serial performance. In addition to theusual optimizations such as reorganization of loops for bettercache utilization, some algorithms have been changed. Forexample, a linearly scaling algorithm for solving a system ofthe EEM19 linear equations has been implemented, as well as a

faster algorithm for assigning atoms to molecules. Thus, anoverall linear scaling of the computational time with the numberof atoms has been achieved.

2. Parallelization using MPI. Atoms and the Verlet list pairsare distributed among MPI processes. The distribution isadjusted on the fly to minimize load imbalance.

3. Memory usage has been drastically reduced. Dynamicmemory allocation has been used where feasible. All arrays thatscale as O(N2) with the number of atoms have been eliminatedand replaced either with sparse matrices or with indexed arrays,thus resulting in an overall linear scaling.

4. Writing the system snapshots and the information aboutcalculation progress has been optimized to reduce the disk I/Ooverhead, which was significant in the earlier version.

5. The program has been integrated into the ADF graphicaluser interface (GUI). The GUI support for ReaxFF includes botha model editor and modules for visualization of the results.

6. The ReaxFF program has been integrated with the ADFprogram to enable geometry optimizations and IR frequencycalculations by ADF using ReaxFF forces.

3. Force Field Development

The credibility of any MD simulation depends strongly onthe quality of the force field used; hence force field developmentis an important step before starting an MD simulation. ReaxFFis an empirical reactive force field, which consists of a generalfunctional form and parameters which vary according to thetypes of elements considered. These parameters are optimizedagainst a training set containing quantum chemical and/orexperimental data, using a successive one-parameter searchalgorithm20 to minimize the error shown in eq 2 (σ is the weightattached to each data point according to its relative importancein the optimization15 and x is a data point in the training set)

The quantum chemical calculations performed in this paperused the B3LYP21,22 hybrid DFT functional and the 6-311G**23

basis set, implemented using Jaguar.24 We used the existing forcefield for hydrocarbons, described first by van Duin et al.16 andextended to hydrocarbon oxidation by Chenoweth et al.15 Thisforce field was optimized to correctly predict the chemical andphysical interactions between hydrocarbons as described byDFT. However, since our accommodation coefficient studiesalso included nitrogen gas, the N-C-H and N-N parametershad to be optimized before using them in the MD simulations.

3.1. N-N Parameter Optimization. The training set for theN-N parametrization consisted of quantum mechanical (QM)data describing heats of formation of various nitrogen-containingcompounds, N2-N2 dimer complexes data taken from Shaw etal.,25,26 and bond dissociation energies of single, double, andtriple N-N bonds. Also included in the training set were cellparameters and the cohesive energy of the face center cubiclattice of R-nitrogen crystal, with the values taken from Bolz etal.27 The N-N parameters in the ReaxFF force field were thusoptimized to correctly predict the various physical and chemicalinteractions between nitrogen atoms. The optimization wasallowed to run until ReaxFF correctly predicted, within accept-able error bars, the heats of formation of various nitrogencompounds, the bond energies of N-N bonds, and the crystalparameters of R-nitrogen as described in ref 27. Figure 1 showsa comparison between ReaxFF and QM energies for the

R )⟨Eout-Ein⟩⟨Es-Ein⟩

(1)

Error ) ∑i)1

n [(xi,QM - xi,ReaxFF)

σ ]2

(2)

12562 J. Phys. Chem. A, Vol. 114, No. 48, 2010 Kamat et al.

dissociation of a single, double, and triple bond of N-N, forthe N-N-N angle bending case and for the heats of formation

of some all-nitrogen compounds. Figure 2 shows a similarcomparison for N2 dimer interactions.

Figure 1. Comparison of ReaxFF and QM for (a) N-N bond, (b) heats of formation of all nitrogen compounds, and (c) angle energies.

Laser-Induced Incandescence of Soot J. Phys. Chem. A, Vol. 114, No. 48, 2010 12563

To validate these newly optimized parameters, a heat-up MDsimulation on a R-nitrogen crystal was performed to checkwhether the new force field predicted the boiling and meltingpoints of the crystal correctly or not. The crystal consisted of256 N2 molecules with periodic boundary conditions in all threedirections, thus allowing us to simulate a bulk R-nitrogen crystal.The system was minimized with respect to energy and thenequilibrated at 5 K in a NPT ensemble to allow the crystal torelax and acquire its most energy-favorable configuration. Anincrease of about 3.2% of its original volume was observedduring this equilibration.

After this, the crystal was heated up gradually from 5 to 100K at constant pressure, at a rate of 0.4 K/ps. This was done inan NPT ensemble, with Berendsen29 temperature and pressuredamping constants of 100 and 250 fs, respectively. The radialdistribution function (RDF) of the crystal (calculated betweenthe centers of the N2 molecules) was monitored throughout thesimulation, to observe the phase change of the crystal. Thetemperature-volume curve for the crystal is shown in Fig-ure 3.

It can be seen from Figure 3 that phase change from liquidto gas begins to occur at approximately 78 K. The solid to liquidphase change is less obvious, and as shown in Figure 4, it isdifficult to predict when the crystal actually melts from the RDFplots. One reason for this is the small difference (around 13 K)between the melting and boiling points of R-nitrogen. Adiffusivity analysis was then carried out, and the diffusivity ofthe crystal was calculated as a function of temperature. However,no significant change in the diffusivity was observed before 80K which could signal a solid to liquid phase change. Thediffusivity analysis thus proved inconclusive in determining themelting point. The RDFs shown in Figure 4 can be used toprovide a rough estimate of the predicted melting point. It can

be seen that in the 50-60 K temperature range, the g(r) showstwo peaks at around 4 and 6 Å. In the 60-70 K range however,the peak at 6 Å flattens out, suggesting some diffusion of theN2 molecules in the crystal. As a rough estimate then, the solidto liquid transition can be guessed to be in the 55-65 K range.A comparison of the predicted melting/boiling points of thecrystal with literature values is shown in Table 1.

The agreement between ReaxFF and experimental values forthe melting and boiling points of R-nitrogen is found to be good,thus validating the new N-N force field.

3.2. N-C-H Parameter Optimization. The N-C param-eters needed to be optimized to correctly describe the physicalinteractions between nitrogen molecules and the soot particleduring the accommodation coefficient simulations. Since thenitrogen gas-soot collision is nonreactive under the conditions

Figure 2. Comparison of ReaxFF and QM for N2 dimer interactions. QM data were provided by Shaw et al.25,26 who used FreeON28 (formerlyMondoSCF) for the calculations.

Figure 3. The temperature-volume curve during the heat up simula-tion. The “kink” in the curve is circled, signaling a phase change fromliquid to gas occurring at a temperature of around 78 K.


considered (nitrogen at 300 K and soot at 2000 K), more weightwas placed on getting the van der Waal’s interaction betweennitrogen and carbon atoms right in the optimization, while thereactive data such as bond dissociation of C-N was includedin the training set, albeit with a lower weight σ placed on it asdescribed in eq 1. The training set thus included C-N single,double, and triple bond dissociation data acquired from DFTcalculations, the energies for various angles involving N, C,andHatoms(N-C-N,C-N-C,C-C-N,H-C-N,H-N-C),and the heats of formation of compounds involving C, N, andH (hydrogen cyanide, methanamine, methylamine, and am-monia). Comparisons between the improved ReaxFF force fieldand QM calculations are shown in Figure 5.

To get reliable estimates of the accommodation coefficient,it is necessary to train the force field to correctly describe thephysical interaction between nitrogen and soot as mentionedearlier. To get these van der Waals interaction energies right,nitrogen was brought toward acetylene through a bond restraintapplied between the two molecules, which kept them at thedesired distance from each other. It can be seen in Figure 6that there is good agreement between ReaxFF and QM until N2

gets within 2 Å of C2H2, which is where the repulsive van derWaal’s forces dominate. For distances less than 2 Å, reactiveevents start taking place between nitrogen and acetylene. Thisis not captured accurately by this force field because this reactiveregion was not needed for the accommodation coefficient studiesperformed in this paper, and hence these data points were notweighted strongly in the force field training set. Thus, the newoptimized force field was able to describe the physical interac-tions between nitrogen and carbon atoms correctly and was usedfor simulations involving soot in the presence of nitrogen gas.A full description of the ReaxFF force field parametersdeveloped in this work can be found in the SupportingInformation.

4. Molecular Dynamics Simulations of LII

Simulations were setup in a cubic periodic box of length 80Å. As graphite properties are normally used to determinephysical properties of soot in theoretical models,3 soot wasmodeled as a graphite-like structure consisting of a stack offour graphene sheets containing 44 atoms each, connected bysp3 hybridized carbon atoms (Figure 7), to form a particle thatwas about 20 Å in diameter. Two different particles wereconstructed based on varying the sp2:sp3 hybridized carbon atomratios in the model. This was done to include the effect of thisratio on the cooling signal. Seven simulation runs wereconducted for the initial configurations detailed in Table 2.

The structures were first minimized with respect to energy.Thereafter, the whole system (soot and gas molecules) wasequilibrated at 2000 K in a NVT ensemble, i.e., the system wasmaintained at 2000 K using Berendsen29 thermostats until thefluctuations in the total energy became minimal and the averageenergy became constant.

After equilibration a production run was carried out, duringwhich the simulation was allowed to run for about 10 millioniterations on a single processor. Because of the high tempera-tures attained by the soot particle, a time step analysis wascarried out to determine the highest permissible time step thatwould conserve the total energy of the system; this time stepturned out to be 0.02 fs. In the production run, a strongBerendsen thermostat of about 10 fs was applied to the sootparticle, to raise its temperature from 2000 to 4000 K in 1200time steps. This temperature ramping was done to simulate a

Figure 4. The radial distribution function g(r) of the crystal is shownduring heat up. The three phases of R-nitrogen can be clearlydistinguished in the respective temperature intervals shown in the figure.It should be noted that this RDF was plotted between the centers of N2

molecules; hence it does not show a peak at 1.1 Å which is the typicalN-N triple bond length.

TABLE 1: Comparison of ReaxFF Values of Melting andBoiling Points (in K) of r-Nitrogen with Literature Values

ReaxFF literature30,31

melting point 55-65 63.6boiling point 78 77.4

Figure 5. Comparison of ReaxFF and QM for (a) C-N bond and (b)angle energies.


high-energy laser pulse, thus giving a temperature spike to thesoot particle similar to that in LII experiments. At the end ofthis spike, the thermostat on the soot particle was turned off,thus enabling the development of a velocity distributionunaffected by the thermostat effects. The gas molecules (if

present) were maintained at 2000 K using a mild thermostatwith a damping constant of 2500 fs at all times during thesimulation, to simulate a typical in situ measurement environ-ment. The temperature of the soot particle was monitoredthroughout the run and was plotted as a function of time.

For the thermal accommodation coefficient runs, it wasnecessary to capture the long phonon wavelength physics inthe soot particle during the gas-soot collisions. For this reason,a bigger soot model was constructed along the same lines. Fivegraphene sheets were linked together using methyl and ethylgroups, each sheet being 40 Å in diameter, creating a particlecontaining 2691 atoms in all. The top and side views of thesoot particle are shown in Figure 8. As mentioned earlier, thesesimulations were performed using the ReaxFF version integratedinto the ADF program.

The soot particle was maintained at a temperature of 2000 Kusing a thermostat of time constant 100 fs. This temperaturewas chosen to keep the soot particle together without carbonfragments vaporizing away, as this is essential for an accom-modation coefficient calculation. The gas molecules weremaintained at a room temperature of 300 K using a mildthermostat of 1000 fs. The pressure of the gas was maintainedat low enough values to ensure that the mean free path was atleast twice the diameter of the soot particle. Only collisions

Figure 6. The four approaches of nitrogen toward acetylene which were included in the training set. N2 was brought toward C2H2 through a bondrestraint applied between the two molecules, which allowed the distance between the two molecules to be held constant.

Figure 7. Soot model consisting of a stack of four graphene sheetsconnected by sp3 hybridized carbon atoms (sp3:sp2 C atom ratio ) 0.66).Carbon atoms are colored brown, whereas hydrogen atoms are white.

TABLE 2: Initial Configurations Used for Simulations

sp3:sp2 carbon atomratio in soot surrounding gas pressure (atm)

case 1 0.25 none 0case 2 0.66 none 0case 3 0.66 argon 108case 4 0.66 argon 54case 5 0.66 argon 27case 6 0.66 argon 13.5case 7 0.66 OH radicals 54


occurring between the gas molecules and the upper and lowersheets of the soot particles were considered. After equilibration,the system was allowed to run until there were enough gas-sootcollisions for the accommodation coefficient value to stabilizenear some constant value. An analysis code was written to trackall the gas-soot collisions that occurred during the simulation.Whenever a gas molecule got within 9 Å of the upper/lowersheets of the soot particle, its incoming velocity was logged;similarly after colliding with the soot particle its outgoingvelocity was logged. After many of these collisions haveoccurred, the average energy transfer between the soot particleand the gas molecules stabilizes, and by eq 1 the accommodationcoefficient R can be computed.

4. Results and Discussion

4.1. Cooling Signal. The cooling signal from the soot particlewas plotted as a function of time for the various cases detailedin Table 1. For the stand-alone soot case, cooling is entirelydue to vaporization (Figure 9 shows the initial and finalsnapshots of the simulation box for the stand-alone case),whereas when the soot particle is surrounded by gas molecules,cooling is also due to collisions with cooler gas molecules.

Figure 10 shows the cooling signals for all seven casesconsidered in this study. As shown, the cooling rate increaseswith increasing pressure. This is expected due to the increasednumber of collisions with the gas molecules. When the sootparticle is surrounded by OH radicals, cooling is seen to beslower than the corresponding case with the same number ofargon atoms. This is because the OH radicals oxidize the sootparticle, which is an exothermic reaction and hence negates thecooling effect to a certain extent. Slight oxidation of the soot

particle is observed at the end of the simulation (Figure 11),with three oxygen atoms being incorporated into the heavierfragments of the soot particle and many smaller carbonfragments being oxidized to form aldehydes and carbonylradicals. The slower cooling may also be attributed to thedifferences in the accommodation coefficient of OH as comparedto that of argon; this will be the subject of future studies.

For the two cases in which there are no gas moleculessurrounding the soot particle, cooling is faster when there aremore sp3 hybridized carbon atoms per sp2 hybridized carbonatom. In these soot models, the carbon atoms that link thegraphene sheets are all in the sp3 hybridized state, whereas thecarbon atoms in the graphene sheets are sp2 hybridized. Thus,variation in this sp3:sp2 carbon atom ratio was brought aboutby introducing more of these “linking” carbon atoms. Thecarbon atoms in the graphene sheets are more stable than thelinking carbon atoms due to the aromaticity of graphene. It wasconjectured that the cooling rate difference in the two caseswas due to the difference in the number of sp3 hybridized carbonbonds which could be broken more easily than the carbon bondspresent in the aromatic ring.

To test this hypothesis, an analysis code was written whichwould track the carbon atoms present in all the molecules forall iterations. The code determines whether the carbon atom ina given molecule was originally sp3 or sp2 hybridized based onthe original connectivity of the atoms and thus gives the numberof broken C-C bonds for both the saturated and aromatic carbonatoms in the soot particle as a function of time. We wereinterested in knowing which of the sp3 or sp2 hybridized carbonatoms were more likely to defragment and become smallermolecules (typically molecules weighing less than 70 amu) andultimately contribute more toward the cooling. The results forthe two different soot models are plotted in Figure 12.

From the plots, it can be seen that about 50% of the carbonbonds which were sp3 hybridized earlier (i.e., carbon atomswhich were linking the graphene sheets), break off from theoriginal particle to form smaller fragments, whereas about 5%of those which were sp2 hybridized initially (i.e., carbon atomsin the graphene sheets) break their bonds. Thus it can be inferredthat sp3 hybridized carbon atoms contribute extensively towardthe cooling signal by vaporizing and forming smaller molecules,whereas the much more stable carbon atoms inside the graphenesheets tend to remain inside the graphene sheets. This alsoexplains why cooling is faster in case 2, because there are moresp3 hybridized carbon sites to break off and vaporize.

4.2. Thermal Accommodation Coefficients. The thermalaccommodation coefficient simulations were carried out for the

Figure 8. Top and side views of the soot particle (surrounded by argon atoms) used for accommodation coefficient calculations. This soot particleis 40 Å in diameter, and contains 2691 atoms. Gas-soot collisions occurring with the upper and lower sheets were taken into account to calculatethe coefficient.

Figure 9. Initial (left) and final (right) snapshots of the simulationbox with only the soot particle inside (case 2 from Table 1). The initialframe shows the soot particle before being hit by the laser, while thefinal frame shows the soot particle broken up into four separate aromaticrings, along with many other smaller carbon fragments.


following gases: helium, neon, argon, nitrogen, carbon dioxide,and methane. The denominator of eq 1 corresponds to the

maximum energy exchange that can occur between the gasmolecule and the soot particle during a collision, and hence

Figure 10. Cooling signals for the seven cases discussed in Table 1. The effect of varying parameters such as (a) pressure, (b) type of gas, and(c) sp3:sp2 C atom ratio in the soot particle can be seen on the cooling signal.


this expression differs for gases depending on the number ofinternal degrees of freedom (rotational and vibrational). Forgas-surface interactions, the denominator reduces to 2kb(Ts -Tg) + 0.5n(Ts - Tg),32 where n is the number of internal degreesof freedom of the gas molecule. Thus, eq 1 now becomes

Monatomic gases have no internal degrees of freedom, andhence n ) 0. Nitrogen has two rotational and two vibrationaldegrees of freedom; however at 300 K the vibrational modesare not populated and so n ) 2. Similarly, “n” is 2 for carbondioxide (since it is a linear molecule, it has only two rotationaldegrees of freedom) and 3 for methane. Along similar lines,partial accommodation coefficients33 for various degrees offreedom can also be defined as follows

It should be noted here that since these are classical MDsimulations, population of the vibrational energy levels is alwaysexaggerated at low temperatures. Simulations carried out in thisstudy showed that the average vibrational contribution to thetotal energy of the molecule is negligible for nitrogen (one bond)and substantial for carbon dioxide (two bonds) and methane(four bonds). The average distribution of the total energy ofgas molecules into translational, vibrational and rotational modesfor the three polyatomic gases is shown in Figure 13 as afunction of time. It can be seen that the vibrational contributionto the total energy is negligible for nitrogen, whereas it increases

for carbon dioxide and methane. The large fluctuations observedin the case of carbon dioxide can be attributed to the relativestiffness of the CdO double bond as compared to the CsHsingle bonds. As mentioned earlier, however, little should beread into the vibrational energy levels because quantum me-chanical theory predicts little or no population of vibrationallevels at 300 K, although some of these modes may get excitedwhen a gas molecule collides with the hot soot particle andleaves with high temperatures. The accommodation coefficientvalues found from the analysis are listed in Table 3.

Although the values predicted by ReaxFF are not in completeagreement with those available in literature, the trend capturedis correct, i.e., that the accommodation coefficient increases withthe increase in the weight of the monatomic gas molecule. It iswell-known through previous experimental gas-surface scat-tering data for monatomic gases that the accommodationcoefficient increases with atomic weight36 and the depth of thepotential well between the gas-surface interactions.33 Ereminet al.34 attribute this to the fact that since lighter molecules travelfaster (at a given temperature) and hence have lower collisiontimes with the soot particle, atoms like He have a much lowerprobability of equilibrating with the soot particle, as comparedto a heavier atom like argon. This can be easily verified throughMD simulations, and the average collision times for He, Ne,Ar, and Xe were found to be 40, 48, 58, and 78 ps, respectively.Despite the relative differences in the average collision timesof Ne and Ar, their accommodation coefficients are almostsimilar, suggesting that the collision time explanation is notsufficient to describe the difference in accommodation coef-ficients for various monatomic gases.

The picture is less clear for polyatomic gases, with no obviouscorelation between the accommodation coefficient and molecularweight observed. Using MD simulations, Daun13 showed thatin general the higher the internal degrees of freedom of themolecule, the lower is its accommodation coefficient. This isbecause to be completely accommodated at the soot temperature,all of the internal modes have to be accommodated. Anotherimportant factor to be considered is the force field used for thegas-soot interaction. The accommodation coefficient is prima-rily a function of the nature of the potential function betweenthe gas molecule and soot, and hence the accuracy of theevaluated coefficient depends mainly on how rigorously theempirical potential used in the MD simulation is trained againstthe “real” potential (obtained, for example, from ab initiomethods).

The extent to which the internal modes of energy of amolecule accommodate to the soot temperature can be obtainedfrom eqs 4-6. The partial accommodation coefficients for thepolyatomic gases are shown in Table 3.

It can be seen that vibrational accommodation coefficientsare relatively small compared to the other two partial coef-ficients. The highest is for methane, because it has four C-Hbonds which can carry vibrational energy after interaction withthe soot particle. Methane is also found to have the highesttranslational coefficient and is interestingly the lightest moleculeamong all three.

5. Conclusion

A thorough understanding of the processes that lead to thegeneration of the LII cooling signal is required to make accuratespatial and temporal measurements of soot morphology. Sub-limation and oxidation processes of the heated soot particlesare still not properly understood and quantified. Also, there isa lot of uncertainty in the LII community regarding the values

Figure 11. Oxidation of one of the heavier fragments of the sootparticle. Carbon atoms are colored brown, hydrogen atoms are white,and oxygen atoms are red.

R )⟨Eout - Ein⟩

2kB(Ts - Tg) + 0.5n(Ts - Tg)(3)

Rtr )⟨Etr,out - Etr,in⟩2kB(Ts - Tg)

(4)

Rrot )⟨Erot,out - Erot,in⟩

0.5nrotkB(Ts - Tg)(5)

Rvib )⟨Evib,out - Evib,in⟩

0.5nvibkB(Ts - Tg)(6)


of the thermal accommodation coefficients of soot with variousgases, and their variation with experimental conditions. Weperformed molecular dynamics simulations of the LII processunder varying conditions of pressure and surrounding gasspecies. The ReaxFF force field for hydrocarbon oxidation wasused for simulating processes such as vaporization and oxidationwhich occur in high fluence experiments. The cooling rate of

the soot particle was found to be faster when subjected to ahigher pressure, and also when there were more sp3 hybridizedcarbon atoms per sp2 hybridized carbon atom in the soot particle.Vaporization was linked to the breaking of bonds of the carbonatoms in the soot particle, which were primarily in a sp3

hybridized state. The presence of a reactive gas like OH radicalswas found to slow down the cooling as compared to the argonsurrounding, mainly due to the oxidation events releasing energyand the high accommodation coefficient of argon.

To enable accommodation coefficient studies, development offorce field parameters for N-N and N-C-H was carried out.ReaxFF was able to correctly predict the melting and boiling pointsof R-nitrogen, as well as reproduce the physical and chemicalinteractions between nitrogen-carbon and nitrogen-nitrogen.

Simulations were also performed to find the thermal accom-modation coefficients of various monatomic and polyatomicgases with soot. For monatomic gases, the dependence on themolecular weights was captured correctly and we found good

Figure 12. Graphs showing the temperature decay of the soot particle along with the breaking of the sp2 and sp3 hybridized carbon bonds for (a)model I with lower sp3:sp2 carbon atom ratio (0.25) and (b) model II with higher sp3:sp2 carbon atom ratio (0.66).

TABLE 3: Thermal Accommodation Coefficient ValuesPredicted by ReaxFF Compared to Values Available inLiterature

gas ReaxFFliterature (previousMD simulations13) literature (experiments)

He 0.22 ( 0.03 0.2 0.1,34 0.13 ( 0.00514

Ne 0.34 ( 0.05 0.35 0.26 ( 0.0114

Ar 0.36 ( 0.06 0.43 0.44,34 0.36 ( 0.0114

Xe 0.41 ( 0.13 NA NAN2 0.12 ( 0.03 0.26 0.18,14 0.43 ( 0.0435

CO2 0.2 ( 0.03 0.19 0.1814

CH4 0.16 ( 0.02 0.11 0.0914


qualitative agreement with literature. We also found transla-tional, rotational, and vibrational accommodation coefficientsfor polyatomic gases with soot, which can be used to betterunderstand the physics of gas-soot interactions.

In conclusion, we believe that molecular dynamics can be auseful tool to simulate LII and understand the fundamentalprocesses that are associated with it. ReaxFF based MDsimulations allow us, for the first time, to combine the chemicaland physical aspects of LII, which will enable us to form bettertheoretical models for LII, thus improving its reliability as asoot diagnostic method.

Acknowledgment. We gratefully acknowledge funding fromKISK Startup Grant #C000032472 and from Illinois Coal GrantICCI 10/7B-3. We also acknowledge financial support fromNWO through Grant #OND1338770 for speeding up andparallelizing ReaxFF through integration into the ADF program.We also thank M. Sam Shaw from Los Alamos NationalLaboratory for providing QM data on N2 dimer interactions.

Supporting Information Available: A full description ofthe ReaxFF force field parameters. This material is availablefree of charge via the Internet at http://pubs.acs.org.

References and Notes

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TABLE 4: Partial Accommodation Coefficient Values ofPolyatomic Gases Predicted by ReaxFF

gas Rtrans Rvib Rrot

N2 0.1 0.01 0.14CO2 0.18 0.04 0.17CH4 0.36 0.06 0.19


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molecular dynamics simulations of laser-induced incandescence of soot using an extended reaxff...

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