4 August 2000

Ž .Chemical Physics Letters 325 2000 661–667www.elsevier.nlrlocatercplett

Effect of transiently bound collision on binary diffusioncoefficients of free radical species

Hai Wang)

Department of Mechanical Engineering, UniÕersity of Delaware, Newark, DE 19716-3140, USA

Received 24 February 2000; in final form 7 June 2000

Abstract

The influence of transiently bound collision on the diffusion coefficient of free radicals was examined using moleculardynamics simulations and the Green–Kubo formula. It was found that transiently bound collisions significantly increase thediffusion coefficients of free radicals at temperatures relevant to combustion. The present study suggests that a moleculartheory beyond the Chapman–Enskog equation is needed to evaluate the diffusion coefficients of free radicals in laminarflame and other high-temperature reacting flow simulations. q 2000 Elsevier Science B.V. All rights reserved.

1. Introduction

Diffusion is a factor in chemically reacting flow,notably combustion, where the propagation rate of aflame is determined by the rates of chemical reactionand heat release as well as diffusion. Because largespatial gradients exist in temperature and speciesconcentrations, the rate of chemical reaction, heatconduction, and diffusion of reactants, products, andintermediate species are strong coupled, especially inflames where fuel and oxidizer are not mixed priorto combustion. An accurate description of transportprocesses is as important in combustion simulationas quantitative knowledge of elementary reaction

w xkinetics 1 .Take a laminar premixed hydrogen–air flame as

an example. The computer laminar flame speed is assensitive to the value of the H-atom diffusion coeffi-cient as to the rate constant of the dominant chain

) Fax: q1-302-831-3619; e-mail: hwang@me.udel.edu

branching reaction H q O ™O q OH. Table 12

shows the ranked first-order sensitivity parameters,defined as E ln s 8rE ln x , where s 8 is the laminaru u

flame speed of hydrogen–air mixtures, and x is arate constant or the H-atom diffusion coefficient.Computations were performed for three equivalenceratios using a reaction mechanism of hydrogen com-

w x w xbustion 2 and the Sandia PREMIX 3 and TRAN-w xFIT 4 codes together with the transport database

accompanied with TRANFIT. It is seen that for allthree equivalence ratios the influence of the H-atomdiffusion coefficient is comparable to or larger thanthe influence of the rate constants. Assuming lineardependence of s 8 on the H-atom diffusion coeffi-u

cient, the sensitivity coefficient of 0.4 for a stoichio-metric mixture translates into a 20% increase inflame speed, about 40 cmrs, if the diffusion coeffi-cient of the H atom is increased by 50%.

Flame simulations are currently carried out withtransport models based on Chapman–Enskog theoryw x Ž .5 and the Lennard–Jones LJ 12-6 potential, origi-

0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0009-2614 00 00733-8

( )H. WangrChemical Physics Letters 325 2000 661–667662

Table 1Ranked first-order sensitivity parameters of diffusion coefficientsof H–M, where M is any chemical species, and reaction rateparameters, computed for the laminar flame speeds of hydrogen–air mixtures

Parameters Equivalence ratio, f

0.6 1.0 1.6aD 0.23 0.40 0.26H – M

OHqH ™HqH O 0.38 0.25 0.162 2

HqO ™OqOH 0.09 0.11 0.172

OqH ™HqOH 0.17 0.11 0.072

HqHO ™OHqOH 0.17 0.10 0.082Ž . Ž .HqO qM ™HO qM y0.14 0.00 0.042 2

HqHO ™H qO y0.11 y0.03 y0.062 2 2

HqOHqM™H OqM y0.07 y0.07 y0.062

HO qOH™H OqO y0.06 y0.03 y0.022 2 2

a Sensitivity parameters obtained by perturbing all binarydiffusion coefficients involving the H atom.

Žnating from the work of Mason and coworkers e.g.,w x. w x w xRef. 6 , Dixon-Lewis 7 , and Warnatz 8 . Based

w xon more recent work of Mason and coworkers 9–11 ,a modification to the repulsive part of the LJ poten-

w xtial was proposed 12 for small species like H andH . This modification makes the repulsive part of2

the potential energy softer and increases the H-atomdiffusion coefficient.

Concerning the diffusion coefficient of free radi-cals, a hitherto neglected issue is the elastic collisionassumption of Chapman–Enskog theory. Taking theH–CO pair as an example, the interaction is gov-erned by the long-range van der Waals forces andalso the short-range binding forces resulting fromtransient formation of the complex HCOU ; the dy-namics of this excited species inevitably affect thetrajectories of H and CO upon dissociation andthereby the diffusion coefficient. These effects areprevalent as long as the association barrier is small.Pairs like H–O , H–CO, H–CO , O–CO, H–C H ,2 2 2 2

O–C H , H–C H and O–C H all fall into this2 2 2 4 2 4

category, whose energy barriers to association aremerely 1 or a few kcalrmol. At combustion temper-atures, most collisions are inelastic because of thetransient formation of ro-vibrationally excited com-plexes. The effects of such collisions have not beenconsidered so far in the treatment of free radicaldiffusion.

The objective of the present work is phenomeno-logical examination of the influence of transientbinding forces on the diffusion coefficient. Our focusis not to obtain quantitative information, becauseeach binary system has its own binding interaction,but to obtain an order-of-magnitude estimate for this

Ž .influence. Molecular dynamics MD simulationswere performed for this purpose using a potentialenergy function appropriate to radical–molecule in-teractions. Diffusion coefficients were calculated forvarious association barriers. It was found that tran-sient binding forces indeed have to be considered incomputing diffusion coefficients of certain free-radi-cal-molecule pairs.

2. Theoretical background and simulation re-marks

We are interested in the isothermal and isobaricŽ .system. For a K-component system let r r,t bek

the mass density of component k at coordinate r andŽ .time t. The rate of change in r r,t is related to thek

Ž .diffusion flux, J r,t , by the conservation of masskw x13 ,

Er r ,tŽ .kq=PJ r ,t s0 , 1Ž . Ž .k

Et

where ks1, PPP Ky1. An additional equation isobtained from the definition of the barycentric refer-ence frame, i.e., ÝK J s0.ks1 k

For a system at local equilibrium in pressure andŽ .temperature, the diffusion flux in Eq. 1 is related to

w xdensity gradients by 14 ,

Ky1

J r ,t sy D =r r ,t , 2Ž . Ž . Ž .Ýk k j jjs1

where the D are the binary diffusion coefficients.k jŽ . Ž .Combining Eqs. 1 and 2 , one obtains Fick’s law

of diffusion in the form of

Ky1Er r ,tŽ .k 2s D = r r ,t . 3Ž . Ž .Ý k j jEt js1

( )H. WangrChemical Physics Letters 325 2000 661–667 663

For convenience, we define the flux of species krelative to the center-of-mass velocity, z s

K Ž . KÝ r Õ t rÝ r , asks1 k k ks1 k

J sr z yz , 4Ž .Ž .k k k

where z is the velocity of species k.k

MD simulation is performed in the microcanoni-Ž .cal NVE ensemble. The center-of-mass reference

frame is defined by setting zs0. The velocity ofthe k th species is given as the average of the individ-ual particle velocity:

Nk1z t s z t , 5Ž . Ž . Ž .Ýk aNk as1

where N is the number of particles of species k andk

z is the velocity of particle a of species k.a

Ž .Combining the phenomenological Eq. 2 with theequation of mass conservation and solving theseequations in the frequency domain leads to the

w xGreen–Kubo formula 15 , which relates the self-dif-fusion coefficient, D , to the velocity autocorrela-11

tion functions of single particle motions:

1 t™`

D s c t d t 6Ž . Ž .H11 3 0

Ž .where c t is the velocity autocorrelation function,an ensemble average over all particles,

² :c t s z tqt Pz t 7Ž . Ž . Ž . Ž .where the brackets denote an average over differenttime origins t . Expressions relating the binary diffu-sion coefficient to velocity autocorrelation functions

w xare also available 14,16 . The diffusion coefficientwas also evaluated with the Einstein–Smoluchowski

w xexpression 15 , which relates diffusivity to meansquared displacements,

12² < < :D s lim r tqt yr t . 8Ž . Ž . Ž .11 6 tt™`

Ž .The limitation of Eq. 8 is that for spatially finitesystems D first approaches a plateau, but later11

drops to zero as t™`. Here the diffusion coefficientis taken as the computed plateau value.

The particle trajectory and velocity information inŽ . Ž .Eqs. 7 and 8 was obtained from MD simulations

employing periodic boundary conditions. Particleswere confined within a cubic box, initially at FCCsites and with equal velocity at directions generated

by a random number generator. Integration of theequations of motion was carried using Beeman’s

w xLeapfrog algorithm 17 . The system was allowed toequilibrate initially until no systematic drifts in pres-sure or temperature were evident. Velocity autocorre-lation functions were then sampled by averagingover 105 time origins. To suppress force calculationsat large particle–particle separations, the potentialenergy calculation was truncated at a cutoff distanceusually 5 times the LJ collision diameter.

3. Results and discussion

To determine the accuracy of the numerical im-plementation, we performed two series of test runs:one for liquid argon at a density of 1.43 grcm3, andthe other for gaseous argon at atmospheric pressure.

˚We assume the LJ collision diameter, ss3.4 A andw xwell-depth ´rk s120K 4 . In both series, theB

integration time step size was sufficiently small suchthat further reduction did not affect the results. Thenumber of particles was 216 for both series ofcomputations.

Fig. 1 shows the radial distribution function com-puted for an equilibrium pressure of 5.15"1.02

Fig. 1. Radial distribution function computed for a 216-particlesystem at the density of 1.43 grcm3, with a pressure 5.15"1.02dynercm2 and temperature 100"4 K, averaged over 5=105

configurations.

( )H. WangrChemical Physics Letters 325 2000 661–667664

dynercm2 and temperature 100"4 K. It is seen thatw xthe known fluid behavior 15 is well reproduced.

ŽThe normalized velocity autocorrelation function not.shown obtained for the same run exhibited negative

correlations at times shorter than ;0.3 ps, whichw xare characteristic of high-density fluids 16 . Integra-Ž .tion of the autocorrelation function, Eq. 6 , gave a

diffusion coefficient of 1.65=10y5 cm2rs, whichcompares well with the 1.75=10y5 cm2rs value

Ž .obtained from the Einstein expression 8 . Thesevalues are also consistent with the result of Hoheiselw x y5 218 , who found Ds1.65=10 cm rs at the samedensity and temperature. The difference in the D

Ž . Ž .values from Eqs. 6 and 8 is attributable to thelimited ensemble size.

In the second series of tests we compared thegaseous self-diffusion coefficient of argon with the

w xChapman–Enskog expression 5 for a pressure of 1Ž .atm Fig. 2 . It is seen that the MD results agree with

the Chapman–Enskog theory. The discrepancy athigh temperatures is again caused by the limitednumber of particles in the simulation.

To investigate the influence of transiently boundcollisions on the diffusion coefficient, we performed

Fig. 2. Variation of the self-diffusion coefficient of argon at 1 atmŽ .with temperature, comparing MD results symbols and the predic-Ž .tion of the Chapman–Enskog equation line .

Fig. 3. Particle–particle potential energies employed in the MDŽ .simulation. The numbers denote test cases see text .

several simulations using the potential energy func-tion

E rŽ .u

612 6 12s s 6 s1 1 2smin 4´ y , ´1 25ž / ž / ž /½ r r r5

10s2)y q´ 9Ž .ž / 5r

where the first term is the traditional 12-6 potential,and the second one represents the transiently bindingforces. This representation of the potential energyfunction, while arbitrary, is certainly adequate for thepresent purpose.

Table 2Ž Ž .. aPotential parameters Eq. 9 used in MD simulations and the

effective association barrier, E0

UCase ´ rk ´ rk E2 B B 0Ž . Ž . Ž .K K kcalrmol

1 – – `

2 40000 20000 23.03 30000 10000 11.54 25000 5 000 5.55 21000 1 000 0.7

a ˚Other parameter values are fixed at s s2.5 A, ´ rk s1001 1 B˚K, and s s1.3 A.2

( )H. WangrChemical Physics Letters 325 2000 661–667 665

Fig. 4. Variation of diffusivity ratio, DrD , where D is the` `

Ž .diffusivity at the infinite classical energy barrier Case 5 , as afunction of the classical energy barrier, E . Symbols: MD results,0

Ž .line: fit to symbols. The numbers denote test cases see text .

Simulations were performed with the five poten-tials shown in Fig. 3. The LJ part of the potential

˚was fixed with ´ rk s100 K and s s2.5 A1 B 1

throughout. Case 1 is the ordinary LJ 12-6 potentialwithout any transiently binding interaction. It repre-sents the current method of diffusion-coefficientevaluation. Cases 2–5 describe transiently bindinginteractions. Table 2 shows the values of potential

Ž .parameters of Eq. 9 . The barrier, E , varied from0

23 kcalrmol in Case 2 to 0.7 kcalrmol in Case 5.Lower values of E increase the fraction of colli-0

sions which can access the bound part of the poten-tial energy at a given temperature. In all cases thewell-depth value of the binding potential energy was40 kcalrmol, representing the typical exothermicityof radical–molecule association.

Simulations were carried out for 125 particles ofidentical mass of 10 amu, at an average temperatureof 1500 K and a pressure of 1 atm. Diffusion coeffi-cients were computed from the velocity autocorrela-tion functions. Fig. 4 shows the variation of the ratio,

Fig. 5. Trajectories of particle collisions in the relative coordinate system computed at four impact parameter values. Solid lines, Case 5;shaded lines, Case 1. s

U is the critical impact parameter above which the intramolecular potential-energy well is not accessible duringŽ .collision, u is the deflection angle, and parameters s and s are as defined in Eq. 9 .1 2

( )H. WangrChemical Physics Letters 325 2000 661–667666

DrD , as a function of the association barrier, where`

D is the diffusion coefficient of Case 1, represent-`

ing an infinite association barrier. It is seen thatDrD increases with a decrease in E . The DrD` 0 `

value is finite as E approaches zero. For large0

values of E , DrD approaches unity. D was0 ` `

computed to be 10.0 cm2rs.The dependence of D upon E is intuitively0

obvious. The increased deviation of D from D is`

caused by the fact that at a given kinetic energy alarger fraction of collisions access the binding part ofthe potential, when the association barrier becomessmaller, and thus the diffusivity is affected to agreater extent by the binding part of the potentialenergy. It is also reasonable to expect that for a fixedenergy barrier, this influence is larger at highertemperatures, where again a larger population ofcollisions can overcome the association barrier.

The extent of diffusivity increase due to transientbinding forces is significant. At 1 kcalrmol barrier,the diffusion coefficient increases by 50% at 1500 Kand 1 atm. The diffusivity enhancement should beeven more notable at higher temperatures. Followingthis analysis and the flame sensitivity results dis-cussed in Section 1, we conclude that the approachcurrently adopted in combustion simulations for esti-mating radical diffusion coefficients is inadequate.The uncertainty in diffusion coefficients caused byneglecting binding interactions is sufficiently large toaffect flame simulations significantly.

Lastly we discuss the cause of diffusivity en-hancement in the presence of transiently bindingforces. Fig. 5 shows the trajectories of particle colli-sion in the relative coordinate system at four differ-

w xent impact parameters 15 . The solid curves wereobtained from the Case 5 potential energy curve

Ž .shown in Fig. 3 E s0.7 kcalrmol , the shaded0

ones from the potential energy curve of Case 1Ž . UE ™` . We use s to represent the critical im-0

pact parameter above which the binding potential-en-ergy well is not accessible during collision. In thiscase, shown in Fig. 5a, the trajectories are identicalfor Cases 1 and 5. For impact parameter equal to3s

Ur4, however, the angle of deflection, u , is muchlarger in Case 5 than in Case 1. For smaller impact

Ž .parameters Fig. 5c,d the differences in u aresmaller, but the qualitative difference persists; thedeflection angles in Case 5 are always larger than in

Case 1. This increased duration of collision is re-sponsible for the diffusivity enhancement.

4. Conclusion

It was shown by MD simulations that transientlybound collisions have a notable influence on thebinary diffusion coefficients of free radicals. Thisinfluence should be adequately studied in order toobtain a reliable free-radical diffusion model forflame simulations.

Acknowledgements

This work was supported by the Air Force OfficeŽ .of Scientific Research AFOSR under the technical

monitoring of Dr. Julian Tishkoff. Computationswere performed at the Facility for ComputationalChemistry at the University of Delaware, which is

Ž .funded by the NSF CTS-9724404 .

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