Cluster and Nanoparticle Condensation and ... - ACS Publications

J. Phys. Chem. C 2008, 112, 11109–11121

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Cluster and Nanoparticle Condensation and Evaporation Reactions. Thermal Rate Constants and Equilibrium Constants of Alm + Aln-m T Aln with n ) 2-60 and m ) 1-8 Zhen Hua Li† and Donald G. Truhlar* Department of Chemistry and Supercomputing Institute, UniVersity of Minnesota, Minneapolis, Minnesota 55455-0431 ReceiVed: NoVember 30, 2007; ReVised Manuscript ReceiVed: February 18, 2008

The association reactions of Al atoms with Aln clusters and nanoparticles and the unimolecular dissociation reactions of Aln clusters and nanoparticles have been studied using classical molecular dynamics trajectory simulations. Thermal reaction rate constants of the association rate constant with m ) 1 and the dissociation rate constant with m ) 1-8 have been simulated, and subsequently, the association rate constants for m > 1 can be determined indirectly. It was found that the monomer association rate constants depend weakly on temperature. For the unimolecular dissociation reactions, the rate constants depend strongly on temperature, and the temperature dependences can be fitted using the Arrhenius equation. The results indicate that the unimolecular dissociation reaction has a high activation barrier that tends to increase with particle size and furthermore that the preferred dissociation process is always monomer emission. With both the monomer association and monomer emission rate constants, the standard Gibbs free energy changes of the Alm + Aln-m T Aln reactions on the ground-state potential energy surface with n ) 2-20, 30, 40, 50, and 60 have been determined. These standard Gibbs free energy changes determined by the molecular dynamics trajectory simulations agree fairly well with the corresponding values determined by previous Monte Carlo equilibrium simulations. The rate constants determined in this study can be used to model the formation and growth of metal nanoparticles under a wide range of conditions from 1100 to 3300 K. M + Mn-1 T Mn

1. Introduction The unique physical and chemical properties of metal clusters and nanoparticles (particles) and nanomaterials fabricated from them have been subjected to extensive experimental and theoretical research.1 Many experimental techniques have been used to synthesize nanoparticles,1–5 such as mechanical attrition, chemical synthesis, laser vaporization, and plasma expansion. Since the properties of clusters and nanoparticles are different from those of atoms and bulk materials and depend strongly on size, it is important to control the size of clusters and nanoparticles in order to tailor their properties. To achieve this goal, it is critical to understand the growth mechanism of particle formation and the size distribution it yields. Thus, detailed knowledge of the equilibrium properties of the particles and the kinetics of the formation and growth reactions is required. In the homogeneous (i.e., gas phase) nucleation of metal particles from a supersaturated gas of metal atoms, both condensation (association) and evaporation (dissociation) reactions must be considered:

Mm + Mn-m T Mn

(R1)

The two processes, condensation and evaporation, compete with each other. In classical nucleation theory (CNT)6–12 the mechanism is further simplified in that the condensation and evaporation are assumed to occur through the addition or evaporation of monomers, i.e., reactions with m ) 1 in reaction R1: * Corresponding author. E-mail: [email protected]. † Present address: Department of Chemistry, Fudan University, Handan Road 220, Shanghai 200433, People’s Republic of China.

(R2)

This is a reasonable assumption since at the initial stage of the condensation the majority of the species in the supersaturated gas phase are monomers. (One may need to consider more complicated mechanisms for heteronuclear nucleation,13 but here we consider only the homonuclear case.) In CNT, a critical particle is defined as the one for which the condensation and evaporation rates are the same. In dynamic nucleation theory (DNT)14,15 the condensation and evaporation are viewed as gasphase association and dissociation reactions, respectively, and the rate constants are calculated using variational transition state theory (VTST).16,17 In DNT, to calculate the evaporation rate constant, which is proportional to the derivative of the Helmholtz free energy for particle formation with respect to the radius of the spherical dividing surface,14 the Helmholtz free energy of particle formation needs to be determined. In a recent classical trajectory study to estimate the effective reactive cross section of water cluster formation in order to correct the VTST rate constants, Schenter et al. found that the correction factor for the VTST calculated rate constants using a spherical dividing surface is approximately a factor of 2.18 Trajectory simulations have also been used to simulate the association and dissociation of clusters and nanoparticles.19 Several studies have used trajectory calculations to study various association and dissociation rate constants, for example, the collision of argon monomers with argon clusters,20 the formation of water clusters,18 collisions between nickel clusters21 and between silver clusters,22 and the unimolecular dissociation of nickel clusters.23,24 Usually, fixed-energy rate constants are determined in these simulations. In general, the rate constants for bimolecular association and unimolecular dissociation reactions are pressure dependent. In

10.1021/jp711349v CCC: $40.75  2008 American Chemical Society Published on Web 07/02/2008

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Li and Truhlar

TABLE 1: Rate Constants (cm3 molecule-1 s-1) of Al + Aln-1 f Aln Association Reactions n 2

T (K) 1100 3000

8

1100 3000

a

Ntpa

run 1

3 5 7 3 5 7 3 5 7 3 5 7

1.11 × 10 1.11 × 10-9 1.11 × 10-9 1.74 × 10-9 1.72 × 10-9 1.71 × 10-9 1.62 × 10-9 1.61 × 10-9 1.64 × 10-9 2.64 × 10-9 2.73 × 10-9 2.67 × 10-9 -9

run 2

run 3

run 4

run 5

average

1.11 × 1.11 × 10-9 1.11 × 10-9 1.73 × 10-9 1.72 × 10-9 1.72 × 10-9 1.60 × 10-9 1.67 × 10-9 1.59 × 10-9 2.60 × 10-9 2.68 × 10-9 2.61 × 10-9

1.11 × 1.11 × 10-9 1.11 × 10-9 1.74 × 10-9 1.73 × 10-9 1.72 × 10-9 1.54 × 10-9 1.66 × 10-9 1.61 × 10-9 2.55 × 10-9 2.59 × 10-9 2.63 × 10-9

1.11 × 1.11 × 10-9 1.11 × 10-9 1.74 × 10-9 1.73 × 10-9 1.72 × 10-9 1.58 × 10-9 1.56 × 10-9 1.59 × 10-9 2.63 × 10-9 2.66 × 10-9 2.59 × 10-9

1.11 × 1.11 × 10-9 1.11 × 10-9 1.74 × 10-9 1.73 × 10-9 1.72 × 10-9 1.71 × 10-9 1.58 × 10-9 1.66 × 10-9 2.63 × 10-9 2.65 × 10-9 2.67 × 10-9

1.11 × 10-9 1.11 × 10-9 1.11 × 10-9 1.74 × 10-9 1.72 × 10-9 1.72 × 10-9 1.61 × 10-9 1.62 × 10-9 1.62 × 10-9 2.61 × 10-9 2.66 × 10-9 2.64 × 10-9

10-9

10-9

10-9

10-9

A collision is considered associative if the number of turning points is greater than or equal to Ntp.

the high-pressure limit, nascent association complexes are stabilized before they redissociate, and dissociation reactions proceed out of a Boltzmann distribution of internal states for canonical ensembles or a microcanonical distribution for fixedenergy ensembles (this is the “local-equilibrium” assumption, that reactant-state populations are related to other reactant-state populations by Boltzmann statistics, and the same for product states, even when reactants are not in equilibrium with products). At lower pressures, nonequilibrium effects become large, and one says that the rates fall off, eventually becoming proportional to the concentrations of third bodies; in applications one should include such falloff effects, which are especially large for smaller clusters. The dissociation of diatomics is the extreme case where the dissociation is essentially always bimolecular. In general, one needs to use a master equation formalism to find the nonequilibrium distributions and effective rate constants consistent with a set of state-to-state energy transfer rate constants. Further discussion of such effects may be found elsewhere.25,26 In the present paper we discuss the local equilibrium rate constants, which are unimolecular for dissociation and bimolecular for association. These may be used as input or constraints on a simulation where reactions and energy transfer processes compete. In the rest of this article, we will simply use the label “rate constants” for the local-equilibrium rate constants that we are discussing, without a qualifier, but the reader should keep in mind that these are the well-defined local-equilibrium rate constants, not the phenomenological rate constants that depend on total pressure, the concentrations of all species in the system, and sometimes even the initial conditions. Classical trajectory calculations27 (also called molecular dynamics simulations) may be used to calculate the rate constants, and in the present article, we present a classical trajectory simulation method to determine the association (forward) and dissociation (backward) thermal (i.e., canonical ensemble) rate constants of reaction R2 for aluminum systems. With these thermal rate constants, the equilibrium constant of reaction R2 in concentration units can be determined by detailed balance,28 which yields

Kc(1, n) )

Ass k1,n Frag k1,n

standard Gibbs free energy change of reaction R2 can be determined from the equilibrium constant by

∆G1,n ° ) -RT ln Kp(1, n)

(2)

where Kp(1,n) is the equilibrium constant in pressure units. Finally, the standard Gibbs free energy (Gf°) of the particles (also known as the free energy of formation) can be determined via a recursion relationship

Gf ° (n) ) ∆G1,n ° + Gf °(n - 1) + Gf °(1)

(3)

(Gf°(1))30

if the standard free energy of the monomer is known. The borderline between nanoparticles and clusters is not uniquely defined, but we use the definition that particles with a diameter of 1 nm or larger are nanoparticles and particles (molecules) with smaller diameters are clusters. With this definition and a reasonable estimate of the typical radius, Al19 and smaller particles are clusters, and Al20 and larger particles are nanoparticles.29,30 The heat of formation Al2O3(s) is more than 400 kcal/mol at 298.15 K,31 and thus aluminum nanoparticles are a potential high-energy fuel.32–34 The physical and chemical properties of the aluminum clusters and nanoparticles have been extensively studied, both experimentally35–40 and theoretically.41–53 Recently, economical and accurate analytic potentials for aluminum systems have been developed by fitting to highly accurate electronic-structure data54 for Aln clusters and nanoparticles as well as to experimental bulk properties.29,55 The potentials named NP-A and NP-B are probably the two best available analytic potentials for the aluminum systems.29 NP-B is slightly less accurate than NP-A but is at least an order of magnitude faster to evaluate. In recent work this potential was used to study the vapor-liquid coexistence properties of Al up to the critical temperature56,57 and to study the free energy of formation of Aln nanoparticles up to n ) 60.58 In the present study, we use the NP-B potential to study the rate constants, equilibrium constants, and free energy changes of reaction R2 for aluminum systems with n ) 2-60. 2. Theory and Simulation Details

(1)

Frag where kAss 1,n and k1,n are the thermal rate constant of the association and dissociation reactions, respectively, of process R2, where the subscript “1,n” denotes monomer emission (that is, m ) 1). The equilibrium constants of reaction R1 with m > 1 are then calculable, and thus the particle equilibrium size distributions are known. Furthermore, the

The cutoff distance (Rcut) of the NP-B potential is 5.38 Å (that is, there is no interaction between atoms separated by more than this distance). We use Stillinger’s definition of a particle by which any two atoms with a distance less than Rcut are members of a particle.59 This definition of particle is good enough to judge if an isolated particle is dissociated: Once a bigger particle dissociates into fragments (smaller particles by the Stillinger definition, and thus they are separated by at least

Alm + Aln-m T Aln Reactions

J. Phys. Chem. C, Vol. 112, No. 30, 2008 11111

a distance of Rcut), these fragments will separate with conservation of total linear momentum, and they cannot re-form the bigger particle because there are no forces to stop their separation. All the trajectory simulations reported here were carried out using the Liouville formulation60 of the velocity Verlet integrator61 with a time step of 2.0 fs. We used the ANT 0762 molecular dynamics program. All the simulations are performed at a constant temperature controlled by a two-chain Nose´-Hoover thermostat.63–66 The temperature of the system is calculated using60

p2

T)

∑ 2mi i i

(4)

1 k N 2 B f

where Nf is the number of degrees of freedom of the system. The use of a thermostat is very important for reactions involving small particles. In extreme cases, for example for the association of two Al atoms, if no thermostat is used, no stable Al2 cluster can be formed because there are no other bath molecules to remove excess energy, while for dissociation reactions, if no thermostat is used, molecules with a total energy below the dissociation limit can never dissociate. For reactions involving large particles, other internal motions not directly involved in the breaking or forming of bonds (transition modes) can serve as the bath to remove or provide extra energy. The thermostat here thus mimics bath molecules that remove or provide extra energy.67 As suggested by Nose´,64 the “mass” (with units of mass times distance squared) of the two additional thermostat degrees of freedom in the thermostat is calculated using

Q)

( )

2Nf kBT t0 〈s〉2 2π

Figure 1. Plots of ln(1 - PrFrag(t)) vs t for the dissociation of Al16 cluster at 2100, 2300, 2500, and 2700 K. PrFrag(t) includes all the dissociation channels.

2

(5)

where s is the unitless additional variable, 〈s〉2 is set to 1, and t0 is the typical vibrational period of the system. In the current study, we use 20 fs for t0/2π, corresponding to a vibration frequency of about 265 cm-1, which is a typical vibrational frequency for aluminum clusters. For both association and dissociation simulations, we start from the minimum-energy structure of the nonmonomer reactant, and we randomly assign initial coordinates and momenta to the atoms according to the classical phase space distribution of separable harmonic oscillators.43 The translation motion of the center of mass is then zeroed, and angular momentum is assigned to the particle according to a Boltzmann distribution. The nonmonomer reactant is then equilibrated for 20 ps at a constant temperature controlled by a two-chain Nose´-Hoover thermostat.63–66 After equilibration, a total of Ntraj sets of coordinates and momenta are saved as the initial configurations for later collision or dissociation simulations (one out of ten is randomly saved). At high temperatures, using the Stillinger criterion for dissociation, the particle may dissociate into fragments (small particles) before all Ntraj configurations are saved. In such cases, the previously saved configurations are kept, another 20 ps equilibration is performed with different initial conditions, and the previous procedure is repeated until all Ntraj configurations are saved. For the simulation of the association reactions, Ntraj is 5000, while for the dissociation reactions, Ntraj is 10 000 for Aln particles with n e 20 and 5000 for Alnparticles with n ) 30, 40, 50, and 60. The results for the Ntraj (5000 or 10 000) trajectories are then used to determine

Frag Figure 2. Plots of ln(1 - Pm,n ) vs t for the fragmentation-specific dissociation of Al16 cluster at 3300 K.

association or dissociation rate constants. Since the starting configurations of the Ntraj trajectories in one simulation are not completely uncorrelated, we carry out the above procedure with completely different starting configurations five times for each temperature T and each n, and the final results are an average over the five rate constants with each obtained from a different set of Ntraj trajectory simulations. The rate constant of the monomer association reaction, i.e., Al + Aln-1 f Aln, is simulated by a trajectory calculation27,68,69 in which the thermal bimolecular rate constant is obtained by calculating the collisions of an Al atom with a rotating Aln-1 particle (target) placed initially at the origin of the coordinate system, and the thermal rate constant is obtained by a Monte Carlo sampling over all initial conditions including the relative

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Frag Figure 3. Plots of ln km,n vs 1000/T for the fragmentation-specific Frag dissociation of Al16 cluster, where km,n is in s-1. The straight lines are obtained by linear-least-squares fitting.

odd.) In the current work, a collision in which VPrel changes sign Ntp times or more is deemed to be associative. Table 1 shows that the reaction rate constant is not very sensitive to Ntp, and so we set Ntp equal to 3. A particle is considered to be fragmented when the minimum distance between any two atoms belonging to different fragments is larger than the cutoff distance (5.38 Å) of the potential (Stillinger definition of a particle). At very high temperature, according to the Stillinger definition, the particle may dissociate into fragments before the atom collides with it. When this happens, the trajectory is not counted as associative (i.e., not counted in NAss r(1,n)) and is thrown away (also not counted in Ntraj). Otherwise the rate constant obtained will depend on the choice of bmax, and the larger bmax is, the smaller the rate constant will be. Another possibility at very high temperature is that a larger cluster is formed according to the Stillinger criterion, but this larger cluster dissociates before the criterion for an associative collision is satisfied (Ntp g 3). When this happens, the trajectory is not counted as associative (i.e., not counted in NAss r(1,n)) but is not thrown away (i.e., is still counted in Ntraj); these trajectories correspond to collision-induced dissociation. To simulate the rate constant of the evaporation reaction, i.e., Aln f Alm + Aln-m, the unimolecular dissociation of an Aln cluster is monitored, again using the Stillinger criterion. The probability that dissociation has occurred by time t is

translational energy between the two reactants. Then the association rate constant is given by Ass k1,n )

8kBT Ass πbmax2Pr(1,n) πµ

(6)

where kB is the Boltzmann constant, µ is reduced mass, bmax is Ass is the fraction of the maximum impact parameter, and Pr(1,n) trajectories that are reactive: Ass Pr(1,n) )

Ass Nr(1,n)

Ntraj

(7)

where is the number of reactive trajectories, and Ntraj is the total number of trajectories. The choice of bmax was made by first calculating the maximum distance (dmax) between the atoms in the target particle (if the target particle is an atom, this distance is set to zero) and bmax is determined by

(8)

For association reactions, the Stillinger definition alone is not enough to judge if a bigger particle is formed because the two reactants may just pass each other or they may just reflect nonreactively (rebound). Therefore, for association reactions, one needs a criterion other than Rcut to judge if a bigger particle is formed. A reactive collision of Al with Aln-1 may be either dissociative or associative: our goal here is to calculate the association rate constant. The method used in the current study to judge whether a collision is associative or not is to P ) of the relative velocity V bemonitor the projection (Vrel rel tween the two reactants on the vector Rrel of their center-of-mass separation: P Vrel ) Vrel · Rrel

NFrag (t) r Ntraj

(10)

where NrFrag(t) is the number of dissociative trajectories by time Frag is the rate constant for processes leading to the t. If km,n dissociation channel Alm + Aln-m, the total dissociation probability is related to the rate constants by

ln(1 - PFrag (t)) ) -t r

Frag ∑ km,n

(11)

m

Ass Nr(1,n)

bmax ) 2.2(dmax/2 + Rcut)

PFrag (t) ) r

(9)

If the collision is nonassociative, the incoming atom will pass P will change sign just the target particle or rebound, and Vrel once. If the collision is associative, the Al atom will stick to P will change sign the Aln-1 cluster for a period of time, and Vrel more than once. (The number of sign changes is always finally

Fitting ln(1 - PrFrag(t)) to t by linear-least-squares fitting, one can obtain the sum of the rate constants of all the dissociation channels (Figure 1). The rate constant of an individual channel can then be obtained by23 Frag Frag km,n ) f m,n

Frag ∑ km,n

(12)

m

Frag is the fragmentation-channel probability, which is where fm,n the ratio of the number of trajectories dissociating into one specific channel to the total number of dissociative trajectories.25 The above method requires that all the dissociation channels be monitored to obtain the total dissociation rate constant. One can monitor just one specific dissociation channel, for example Frag is the the evaporation of a single atom from a particle. If Pm,n probability of dissociation channel Alm + Aln-m, and if all PFrag m,n are small, then

(

ln 1 -

Frag Frag (t) ≈ ∑ ln[1 - Pm,n (t)] ∑ Pm,n )

(13)

Frag Frag ln[1 - Pm,n (t)] ≈ -tkm,n

(14)

m

m

Therefore,

As shown in Figure 2, when the simulation time is long and Frag PFrag m,n (t) is large, the linear relationship between ln [1 - Pm,n (t)] and t no longer holds. To obtain reliable rate constants with this more economical method, we decrease t when fitting with eq 14 until the absolute value of the correlation coefficient is Frag vs 1000/T for Al larger than 0.999. Figure 3 shows ln km,n 16



3 -1 -1 Ass Figure 4. Plots of ln kAss s . Temperatures range from 1900 to 2900 K. (a) n ) 2-9; (b) n ) 10-17; 1,n vs 1000/T, where k1,n is in cm molecule (c) n ) 18-20, 30, 40, 50, 60.

with m ) 1, 2, 3, and 4. There is a good linear relationship Frag and 1/T. between ln km,n 3. Results and Discussion Association Reactions. The logarithm of the forward rate constant is plotted in Figure 4 for temperatures from 1900 to 2900 K, where rate constants are in cm3 molecule-1 s-1. In this temperature range, for most n, ln kAss 1,n shows a linear relationship70 with 1/T. From the plot, it can be seen that the association rate constants depend weakly on temperatures: the rate constants at 2900 K are just roughly 1.2-1.3 times larger than the rate constants at 1900 K. In addition, the plots show that the larger the target particle, the larger the association rate constants. This is easy to understand because larger particles have larger radii and thus have larger cross sections. Since the geometric cross section is proportional to n2/3, the rate constant is plotted in Figure 5, as a function of n2/3 for temperatures of 1900, 2300, and 2900 K. The plots show a good linear relationship between 2/3 kAss 1,n and n . The straight lines in Figure 5 are linear-leastsquares fits to all the data points for each temperature; the parameters of these fits are given in Table S2 of Supporting Information. Although for a narrow temperature range the plots of ln kAss 1,n show a linear relationship between ln kAss 1,n and 1/T, for a larger temperature range the relationship no longer holds. Figure 6 shows ln kAss 1,n vs 1000/T for the association reactions of Al atom with Al, Al5, Al10, Al15, and Al29 for a temperature range of 1100-3300 K. The curves are far from straight. Collisioninduced dissociation trajectories are not counted as associative trajectories but are still counted in Ntraj. Due to the short time period between forming the larger cluster and fulfilling the Ntp g 3 criterion, the number of these trajectories is small, for example, an average of 53 out of 5000 for the collision between Al and Al10 at 3300 K. The data points between 1100 and 3300 K can be well fitted by the popular form Ass Ass Ass ln k1,n ) A1,n + B1,n ln T -

Ass ∆E1,n RT

(15)

Ass Ass where AAss 1,n , B1,n , and ∆E1,n are fitting parameters. The lines in Figure 6 are fitted to eq 15, and the fits represent the data quite

2/3 3 -1 -1 Ass Figure 5. Plots of kAss s for 1,n vs n , where k1,n is in cm molecule 1900, 2300, and 2900 K.

well. All the fitting parameters are available in Table S3 of the Supporting Information. The Arrhenius activation energy of the association reaction is Ass Ass Ass ∆Ea(1,n) ) ∆E1,n + B1,n RT

AAss 1,n ,

BAss 1,n ,

∆EAss 1,n

(16)

In Figure 7, and are plotted as functions of n. All three plots show zigzag behavior. If one averages out the Ass oscillations, AAss 1,n and ∆E1,n both become smaller as n increases, and BAss becomes larger. At high n BAss 1,n 1,n is close to 1, which indicates a linear dependence71 of Gibbs free energy of activation on temperature for large n. It seems that there is a compensating Ass effect between AAss 1,n and ∆E1,n (Figure 8). The rate constants for the association reactions of Alm with Aln-m for m > 1 are not simulated directly in the present study, but they can be calculated from the equilibrium constants of reaction R1 and the dissociation rate constants of Aln to Alm

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Li and Truhlar using eq 14 are presented for the dissociation of Al14, Al15, Al16, and Al17. Most of ratios are larger than unity, indicating that the fitted rate constant is smaller than the one obtained by eq 12. This is easy to understand from Figure 2. Table 2 shows though that the two methods generally agree well with each other. As previously mentioned, the fitting method has the advantage of not requiring one to monitor all the dissociation channels. However, this method has the shortcoming that, due Frag may not exactly to the nature of the fitting, the sum of km,n equal the total rate constant. Since, in this study, we have monitored all the dissociation channels, the results presented afterward are all obtained using rate constants determined by eq 12. Results for Al16 are shown in Figures 1, 2, and 3, and these results are typical for the unimolecular dissociation of Aln particles in that monomer emission is always the dominant channel. This is reasonable since the monomer emission channel is the one that requires the least energy. In Figure 9, the dissociation energies for the channels producing Alm + Aln-m

∆E(m,n) ) Ee(Alm) + Ee(Aln-m) - Ee(Aln) e 3 -1 Ass Figure 6. Plots of ln kAss 1,n vs 1000/T, where k1,n is in cm molecule s-1 with n ) 2, 6, 11, 16, and 30 for a temperature range between 1100 and 3300 K. The lines are fitted using all data points with eq 15.

with Aln-m (see Equilibrium Constants and Standard Gibbs Free Energy Change). Although our association rate constants were obtained from full simulations, there may be some interest in comparing them to the gas-kinetic collision rate constant of (8kBT/πµ)1/2πσ2, where σ is a temperature-dependent effective collision diameter equal (in this model) to the maximum impact parameter that leads to a reactive collision. One may in fact calculate such σ values from our results. Consider, for example, the case of n ) 30 at T ) 2500 K; the association rate is 4.1 × 10-9 cm3 molecule-1s-1 (see Figure 4c, orsfor higher precisionsTable S1 in the Supporting Information) which leads to an effective collision diameter σ of 0.96 nm. If we assume that σ for Al30 + Al is (σAl + σAl30)/2, and if we assume that σAl/2 is a standard atomic radius72 of 0.126 nm for Al, then we obtain an effective radius σAl30/2 of 0.84 nm for Al30, which is a reasonable result for this size of particle. Dissociation Reactions. For the unimolecular dissociation reaction, it is important to allow the particle to rotate, especially for the small particles such as Al2 and Al3. Nevertheless, in some simulations, the angular rotation motion of the particle has been removed.26 Here we compare the dissociation rate constants of Al3, Al11, and Al17 at 3000 K with and without including rotation. For the dissociation of Al3, the total dissociation rate constant (kFrag n ) simulated with rotation is larger than the one simulated without rotation by a factor of 2.0. For Al11, kFrag simulated with rotation is 1.6 times larger than the n one simulated without rotation, while for Al17 this ratio decreases to 1.35. A factor of 2 in the rate constant corresponds to a difference of about 4 kcal/mol in the calculated free energy change of reaction R2 at 3000 K, whereas a factor of 1.35 corresponds to only a 1.8 kcal/mol difference. The effect of rotation is smaller for the larger particles because, with the same rotational energy, larger particles rotate slower since they have larger moments of inertia. In section 2, we presented two methods for the determination of the rate constant of an individual dissociation channel. In Table 2, the ratios between kFrag 1,n obtained by eq 12 and by fitting

(17)

where Ee is the equilibrium potential energy given by the NP-B analytical potential, are plotted as a function of particle size, where the particle geometries used are all global minima geometries located in a previous study.73 It can be seen that monomer emission requires the least energy. The results for Al16 shown in Figure 3 are typical of the plots Frag vs 1000/T for other n in that they show a good linear of ln km,n Frag and 1/T in the temperature range relationship between ln km,n studied (2000-3300 K). Lower temperatures are not simulated because the dissociations are rare events at these temperatures; thus much longer simulation time is needed and the simulation can only be done for small particles. For the dissociation of Al2, simulations were done in a wider temperature range from 1500 to 3300 K. The plot of ln kFrag 1,2 vs 1000/T for Al2 in this temperature range, unlike that of the association reaction, shows a good linear relationship between ln kFrag 1,2 and 1/T (Figure 10). Therefore, it is possible to obtain the rate constants for other temperatures by linear extrapolation of ln kFrag m,n vs 1/T, or by using the Arrhenius fitted preexponential factors and activation energy for the dissociation rate constants in a wider temperature range. Although the rate constants of channels other than monomer emission have also been determined in the present study, in Figure 11, only ln kFrag 1,n as a function of 1000/T is plotted. A Frag and 1/T, and good linear relationship is found between ln km,n they can be fitted using the Arrhenius equation70

ln

Frag Frag km,n ) Am,n -

Frag ∆Ea(m,n)

RT

(18)

Frag and ∆EFrag are fitting parameters corresponding where Am,n a(m,n) to the logarithm of preexponential factor and activation energy for the Alm cluster emission reaction, respectively. AFrag 1,n and Frag as a function of size are plotted in Figure 12. Their ∆Ea(1,n) values as well as the values for other dissociation channels with m ) 2-8 can be found in the Supporting Information. Results for larger m values are not available because of the small sample of trajectories or too few fitting data (less than three temperatures) for that dissociation channel: in the present study, rate constants are not calculated for channels with less than 50 Frag dissociative trajectories. As n increases, both AFrag 1,n and ∆Ea(1,n) increase. For n between 13 and 20, the plots show a moderate oscillation. This is different from that of the association, where zigzag behavior with respect to n is observed for both AAss 1,n and



Ass Ass Figure 7. Plots of (a) AAss 1,n , (b) B1,n , and (c) ∆E1,n in eq 15 as a function of size n for the association reactions of Al atom with Aln particles.

Frag Obtained by eq 12 and by TABLE 2: Ratio between k1,n Fitting Using eq 14

n Al14

Al15

Al16

Al17

2000 2100 2300 2500 2700 2900 3000 3100 3300

1.02 0.92 1.04 1.06 1.07 1.13 1.15 1.10 1.03

0.98 1.06 1.08 1.05 1.12 1.11 1.11 1.15 0.85

1.03 1.03 1.01 1.04 1.12 1.14 1.15 1.13 1.11

0.96 1.07 1.01 1.04 1.10 1.15 1.14 1.13 1.10

average

1.06

1.06

1.09

1.08

T (K)

Ass Figure 8. Plot of AAss 1,n vs ∆E1,n for the association reactions of Al Ass atom with Aln particles. AAss 1,n and ∆E1,n are obtained by fitting using eq 15.

∆EAss 1,n . The plots also indicate that the dissociation rate constant has much stronger temperature dependence than the association reaction rate constant. Comparison of the plot of ∆E(1,n) vs n in Figure 9 to the plot e Frag vs n in Figure 12b shows similar trends in the two of ∆Ea(1,n) plots, although the plot in Figure 9 shows much larger oscillation amplitude for n between 10 and 20. Since the association is barrierless, the activation energy of the dissociation reaction might be expected to have an approximately linear dependence on ∆E(1,n) e . It should be noted that the activation energy includes thermal contributions from both the transition state and the reactant, and it also includes contributions from other isomer geometries, whereas ∆E(1,n) is calculated solely based on the e potential energies of the global minima structures.

Similar to the association reaction, a compensating effect Frag between AFrag 1,n and ∆Ea(1,n) is observed (Figure 13). Equilibrium Constants and Standard Gibbs Free Energy Change. Having calculated both the forward and backward reaction rate constants for reaction R2, we can now calculate the equilibrium constant from eq 1 and the standard Gibbs free energy change of reaction R2 can be determined using eq 2 from the equilibrium constant of reaction R2 in pressure units, which is related to the equilibrium constant in concentration units by

Kp(1, n) ) Kc(1, n)/(RT/P°)

(19)

where P° is 1 atm. Before presenting numerical values of ∆G1,n°, it is important to stress the interpretation of this free energy change. In general, the free energy change of a chemical reaction may be written as

∆G◦1,n ) ∆G◦ GPES + ∆GElec

(20)

where ∆GGPES° corresponds to the value calculated from the ground-state potential energy surface (GPES) by ignoring the underlying electronic degrees of freedom, and ∆GElec is the contribution from the electronic partition functions not being Frag unity. Since we calculate kAss 1,n and k1,n for the GPES by Born-Oppenheimer trajectories, calculating ∆G1,n° by eqs 1

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Li and Truhlar

Figure 9. Plots of dissociation energy (as defined by eq 17) vs n for the dissociation of Aln into Alm + Aln-m, with m ) 1, 2, 3, 4, and 5.

Figure 10. Plot of ln kFrag 1,n vs 1000/T for the dissociation of Al2, where Frag k1,n is in s-1. The straight line is obtained by a linear-least-squares fitting.

and 2 corresponds to approximating it as ∆GGPES°. The neglected contribution may be written74

∆GElec ) -RT ln

QElec(Aln) QElec(Al) QElec(Aln-1)

(21)

where QElec(X) is the electronic partition function of X. If Al and Aln-1 were doublets and Aln were a singlet, and if

none of these species had any low-lying excited electronic states, ∆GElec° would be RT ln 4. In contrast, if Aln also had a low-lying triplet whose separation from the ground singlet was much less than RT, then ∆GElec would be zero. However, one must consider all excited electronic states, especially for metals at the high temperatures considered here. (Even Al and Al2 have low-lying electronic states, with their lowest electronic excitations at 11275 and 170 cm-1,76 respectively.) Adding the electronic excitation contribution to free energy changes simulated by the ground-state potential is even more complicated for particles with multiple lowenergy isomers, which is the case for most metal particles, where an isomer with higher potential energy may have comparable and even larger population than the global minimum structure at moderate temperatures.73 In fact, ∆GElec for n ) 2-60 has been estimated in a previous study,58 and we will not consider it here. We will, however, compare the present values of ∆GGPES° to the values of ∆GGPES° calculated previously58 by version L of the aggregation-volume-bias Monte Carlo (AVBMC-L) method for potential NP-B for n ) 2-60. With this understanding, we now drop the subscript GPES; this comparison is primarily intended as a consistency check on the methodology because the present ∆G1,n° are derived from dynamic calculations of rate constants whereas AVBMC-L is a nondynamic equilibrium simulation. (The final ∆G1,n° values of our previous study58 begin with NP-A for n ) 2-10 and with NP-B for n ) 11-60 and also include a high-level electronic structure correction for n ) 2-60 and a high-level isomeric-rovibrational correction for n ) 2-5 and 11-60, where “high-level” denotes any level higher than NP-B, but these corrections, along with ∆GElec are not included in the present comparison since the present results are all for NP-B. Thus we compare to NP-B results for ∆GGPES for all n without either electronic contributions or high-level corrections.) In Figure 14, ∆G1,n° of reaction R2 is plotted as a function of n for temperatures 2100, 2500, and 3000 K. ∆G1,n° changes greatly from n ) 2 to n ) 3, and then it gradually decreases as n increases further. As mentioned in the previous paragraph, we have directly simulated the equilibrium constant of reaction R2 using Monte Carlo (MC) simulations for n ) 2-60 in a previous study.58 In Figure 14, the NP-B MC results for ∆G1,n° (without high-level corrections or electronic contributions) at 2500 and 3000 K are also plotted for comparison. The previous MC method for n ) 2 is well validated by comparing to other theoretical approaches (the direct configurational integral method and the Monte Carlo configurational integral method) for a given potential and (in conjunction with high-level corrections and the electronic contribution) to experimental results.58 The comparison in Figure 14 indicates that the present simulation greatly underestimates ∆G1,n° for dimer formation. At 2500 K, ∆G1,n° is underestimated by about 4 kcal/mol, while at 3000 K, it is underestimated by about 7 kcal/mol; these differences correspond to an overestimate of the equilibrium constant for dimer formation by factors of 2.3 and 3.1 at 2500 and 3000 K, respectively. Since the equilibrium constant is related to both the forward and backward reaction rate constants, this implies that either the forward reaction rate constant is overestimated, or the backward reaction rate constant is underestimated, or both. We are not sure of the source of errors. It is well-known that a dimer cannot be stabilized without third-party collision, and dissociation of a dimer cannot occur without collisions by other bath molecules. Here



-1 Frag Figure 11. Plot of ln kFrag 1,n vs 1000/T for the dissociation of Aln into Al + Aln-1, where k1,n is in s . (a) n ) 2-8; (b) n ) 9-16; (c) n ) 17-20, 30, 40, 50, 60.

Frag Figure 12. Plots of (a) AFrag 1,n and (b) ∆Ea(1,n) vs 1000/T for the dissociation of Aln particles into Al + Aln-1.

we use a thermostat to serve as a bath to remove or provide extra energy. It is well-known that a one-dimensional harmonic oscillator, especially for systems away from equilibrium, is difficult for the Nose´-Hoover thermostat.77 However, we should not focus excessively on the dimer, which is included here only for completeness. It is a wellknown special case with large nonequilibrium effects because of cluster redissociation in a single vibrational period.25 At 2500 K, except for dimer formation, the two methods agree very well with each other, all within 2 kcal/mol. At 3000 K, the kinetic method and the equilibrium method agree well with each other (within 1.5 kcal/mol) for small n, but for the larger n, the difference is large and the difference increases as n increases with the largest difference being more than 4 kcal/mol. Another difference is that ∆G1,n° obtained by the Monte Carlo simulations slowly increases when n > 30, whereas ∆G1,n° given by the current MD simulations

always decreases for large n. From the plots, one can see that, for large n, ∆G1,n° is almost a smooth function of n. Therefore, we expect that ∆G1,n° for n not studied can be extrapolated from the available ∆G1,n° values without introducing a large error in ∆G1,n°. In Figure 15, ∆G1,n° is plotted as a function of temperature. The straight lines in the plots are linear-least-squares fitting to all the data points for a specific n. One can see that very good linear relationship between ∆G1,n° and temperature is observed for n ) 20, but the curves for n ) 5 and 10 deviate slightly from a straight line. The curvature of the curve is consistent with the prediction of using classical rigid-rotor harmonic-oscillator approximation to calculate the equilibrium constant.58 All the free energy changes are listed in Table 3. In our previous work, we have thoroughly discussed the high-level (any level higher than the analytical potential used

11118


Frag Figure 13. Plot of AFrag 1,n vs ∆Ea(1,n) for the dissociation of Aln particles into Al + Aln-1.

Figure 14. Plot of ∆G1,n° vs n for the Al + Aln-1 T Aln reactions. Open symbols represent results from previous Monte Carlo simulations for the NP-B potential without high-level corrections or electronic contributions.58

Li and Truhlar

Figure 15. Plot of ∆G1,n° vs T for the Al + Aln-1 T Aln reactions with n ) 5, 10, and 20. The straight line is obtained by least-squares fitting.

Information, the monomer evaporation rate constants calculated using the previously determined accurate ∆G1,n° values and the association rate constants obtained from current MD simulations are presented. It should be noted that, in the previous study, accurate ∆G1,n° values are only available for four temperatures, 1500, 2000, 2500, and 3000 K, except for n e 4. Equilibrium constants and rate constants for other temperatures can be obtained by extrapolation assuming that ∆G1,n° is a linear function of temperature58 (see also Figure 15). With the equilibrium constants of process R2, the equilibrium constants, and also the standard Gibbs free energy changes of process R1 with m > 1, can also be determined. In the present study, we have obtained the rate constants for the Aln f Aln + Aln-m reactions with m > 1. With the equilibrium constants of reaction R1 and these dissociation rate constants, the association rate constants of reaction R1 with m > 1 can also be obtained indirectly. For example, when m ) 2 the association rate constant can be obtained by the equation Ass Frag k2,n ) Kc(2, n)k2,n

(22)

where Kc(2,n) is calculated by in the molecular simulations) corrections for the ∆G1,n° values predicted by molecular simulations and accurate ∆G1,n° values have been determined.58 These corrections are far from negligible. With these accurate ∆G1,n° values accurate equilibrium constants for reaction R2 can then be determined. If the associate rate constants of reaction R2 are deemed to be accurately determined, high-level corrected rate constants for the evaporation reactions can then be determined, or vice versa. It is more likely that the evaporation rate constants are affected more by these high-level corrections because the association reactions are almost barrierless and their rate constants are mainly determined by the size of the colliding particles (see Figure 5) and the sticking probability, while for the dissociation reactions, the fitted Arrhenius activation energies show a strong correlation (see the discussion above) with the ground-state dissociation energies (as defined by eq 17) which are greatly affected by the high-level electronic structure corrections. In Table S15 of the Supporting

Kc(2, n) ) Kc(1, n) Kc(1, n-1)/Kc(1, 2)

(23)

In the Supporting Information, we have collected the fitted Arrhenius parameters for the dissociation reactions Aln f Alm + Aln-m with m up to 8. Interested readers may do the math to calculate the equilibrium constants Kc(m,n) and association rate Ass with m > 1. It should be noted that K (1,2) constants km,n c obtained by the current trajectory simulations is not very accurate (see above discussions and Figure 14). Instead, one may use the more accurate value (the value without high-level corrections) determined in the previous study.58 4. Concluding Remarks In the present work, the association reactions of Al atom with Aln particles and the unimolecular dissociation of Aln particles have been studied using classical molecular dynamics trajectory simulations. Thermal reaction rate constants of both reactions have been determined. The results indicate



TABLE 3: ∆G1,n° (kcal/mol) for the Al + Aln-1 T Aln Reactions n T (K)

2

3

4

5

6

7

8

9

2000 2100 2300 2500 2700 2900 3000 3100 3300

-8.1 -6.5 -3.2 -0.5 2.1 5.0 6.7 8.4 11.8

-5.3 -3.2 0.4 4.2 8.0 11.6 13.3 15.2 19.2

-7.2 -5.3 -1.3 2.6 6.6 10.5 12.2 14.2 17.9

-8.6 -6.6 -2.6 1.6 5.4 9.2 11.3 13.1 16.8

-9.3 -7.1 -3.1 0.7 4.7 8.6 10.4 12.3 16.0

-9.7 -7.8 -3.7 0.2 4.2 7.9 9.9 11.5 15.7

-10.2 -8.1 -4.1 -0.2 3.8 7.5 9.4 11.2 15.1

-10.5 -8.4 -4.6 0.0 3.3 7.5 9.3 10.8 14.6

T (K)

10

11

12

13

14

15

16

17

2000 2100 2300 2500 2700 2900 3000 3100 3300

-11.1 -8.9 -4.5 -0.7 3.2 7.0 8.9 10.6 13.7

-10.8 -9.0 -5.1 -1.4 3.5 6.9 8.9 10.7 15.1

-11.5 -9.4 -4.9 -1.1 3.3 7.2 8.9 11.1 15.0

-11.4 -9.2 -5.6 -1.7 2.8 6.7 8.1 10.7 14.6

-11.8 -10.0 -5.9 -1.3 2.6 6.6 8.6 10.6 13.9

-12.2 -10.5 -5.7 -2.0 1.6 6.6 8.4 10.2 14.0

-12.3 -10.4 -6.1 -1.8 2.5 5.8 7.9 8.2 11.2

-11.8 -10.5 -5.7 -1.8 1.6 5.5 7.9 10.0 14.1

n

n T (K)

18

19

20

30

40

50

60

2000 2100 2300 2500 2700 2900 3000 3100 3300

-12.1 -10.2 -6.4 -2.9 1.5 4.6 7.0 8.5 12.4

-12.7 -10.7 -7.4 -2.8 0.8 4.9 7.0 8.6 12.6

-13.2 -11.3 -7.2 -3.2 0.8 5.0 7.0 8.8 13.1

-12.3 -7.5 -3.5 0.4 4.0 5.9 7.7

-12.6 -8.9 -4.2 0.0 4.0 5.3 6.7

-12.9 -8.6 -4.7 -0.2 3.1 5.2 7.4

-13.4 -9.9 -4.9 -1.6 2.3 4.7 7.1

that, for the association reaction, the logarithm of the rate constant does not depended linearly on 1/T in a wide temperature range, and can be fitted by eq 15. The results indicate that the association rate constants depend weakly on temperature and the reaction is almost barrierless. For the unimolecular dissociation reactions, the simulation indicates that the logarithm of the rate constant depends linearly on 1/T and can be fitted using the Arrhenius equation. The simulation indicates that the unimolecular dissociation rate constant depends strongly on temperature and the reaction has a high activation barrier that has a trend to increase with particle size. The unimolecular dissociation rate constants for other dissociation channels have also been determined. The results indicate that monomer emission is the one with largest rate constant, which is consistent with the fact that monomer emission requires the least energy. With both the forward (monomer association) and backward (monomer emission) rate constants, the equilibrium constants and standard Gibbs free energy changes of the Al + Aln-1 T Aln reactions with n ) 2-20, 30, 40, 50, and 60 have been determined. Subsequently, the equilibrium constants and Gibbs free energy changes of the Alm + Aln-m T Aln reactions, and in conjunction with the dissociation rate constants of Aln f Alm + Aln-m, the association rate constants of Alm + Aln-m f Aln reactions with m > 1 can also be determined. With the free energy changes of Al + Aln-1 T Aln reactions, as well as with the experimental standard Gibbs free energy of formation for the Al atom, the standard Gibbs free energies of Aln particles with n > 1

can be determined. The comparison between the present simulated free energy changes and those simulated earlier by a Monte Carlo method indicates that the equilibrium constant for the dimer formation is overestimated. Alternatively, one can use the experimental standard Gibbs free energy of formation of Al monomer and dimer58 to determine the standard Gibbs free energy of formation of larger particles using the free energy changes of Al + Aln-1 T Aln determined in the present work. Acknowledgment. The authors are grateful for helpful assistance from Steven Girshick, Andreas Heyden, Ahren Jasper, and Nathan E. Schultz. This work was supported by the National Science Foundation (Grant CHE07-04974) and by the DefenseUniversity Research Initiative in Nanotechnology (DURINT) through a grant managed by the Army Research Office. Supporting Information Available: A Microsoft Word document containing rate constants for both the forward and backward reactions and the fitting parameters for ln k with respect to 1/T using eqs 15 and 18. This material is available free of charge via the Internet at http://pubs.acs.org. Note Added after Print Publication. The Supporting Information was incorrect in the version of the manuscript published on the Web on July 2, 2008 (ASAP) and in print (J. Phys. Chem. C 2008, 112, 11109-11121). The correct version was published on June 3, 2009 and an Addition and Correction will be published (DOI: 10.1021/jp904971d).

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References and Notes (1) (a) Metal Nanoparticles: Synthesis, Characterization, and Applications; Feldheim, D. L., Foss, C. A., Eds.; Marcel Dekker; New York, 2002. (b) Nanoparticles: from Theory to Applications; Schmid, G., Ed.; WileyVCH: Weinheim, 2004. (c) Structure and Properties of Atomic Nanoclusters; Alonso, J. A., Ed.; Imperial College Press: London, 2005. (2) Gates, B. C. Chem. ReV. 1995, 95, 511. (3) Rao, N.; Girshick, S.; Haberlein, J.; McMurry, P.; Jones, S.; Hansen, D.; Michael, B. Plasma Chem. Plasma Process. 1995, 15, 581. (4) Cushing, B. L.; Kolesnichenko, V. L.; O’Connor, C. J. Chem. ReV. 2004, 104, 3893. (5) Wilcoxon, J. P.; Abrams, B. L. Chem. Soc. ReV. 2006, 35, 1162. (6) (a) Becker, R.; Do¨ring, W. Ann. Phys. 1935, 24, 719. (b) Frenkel, J. J. Chem. Phys. 1939, 7, 200. (c) Zeldovich, Ya. B. Zh. Eksp. Teor. Fiz. 1942, 12, 525. (d) Abraham, F. F. Homogeneous Nucleation Theory; Academic: New York, 1974. (7) Hoare, M. R.; Pal, P.; Wegener, P. P. J. Colloid Interface Sci. 1980, 75, 126. (8) (a) Hagena, O. F. Surf. Sci. 1981, 106, 101. (b) Kulmala, M.; Olin, M.; Raunemaa, T. J. Aerosol Sci. 1987, 18, 615. (c) Schweitzer, F.; Schimansky-Gieger, L.; Ebeling, W.; Ulbrecht, H. Physica A 1988, 150, 261. (d) Oxtoby, D. W. J. Phys.: Condens. Matter 1992, 4, 7627. (9) (a) Jokiniemi, J. K.; Lazaridis, M.; Lehtinen, K. E. J.; Kauppinen, E. I. J. Aerosol Sci. 1994, 25, 429. (b) Kobraei, H. R. Atmos. Res. 1994, 31, 3. (c) Schaaf, P.; Senger, B.; Reiss, H. J. Phys. Chem. B 1997, 101, 8740. (d) Schmelzer, J. W. P.; Schmelzer, J., Jr Colloid Interface Sci. 1999, 215, 345. (e) Athanasopoulos, D. C. Surf. Sci. 2000, 462, 55. (f) Vehkamaki, H.; Ford, I. J. J. Chem. Phys. 2000, 112, 4193. (g) Toxvaerd, S. J. Chem. Phys. 2003, 119, 10764. (10) (a) Suh, S.-M.; Girshick, S. L.; Zachariah, M. R. J. Chem. Phys. 2003, 118, 736. (b) Prakash, A.; Bapat, A. P.; Zachariah, M. R. Aerosol Sci. Technol. 2003, 37, 892. (c) Mukherjee, D.; Prakash, A.; Zachariah, M. R. Aerosol Sci. 2006, 37, 1388. (11) (a) Krasnochtchekov, P.; Albe, K.; Ashkenazy, Y.; Averback, R. S. J. Chem. Phys. 2005, 123, 154314. (b) Neimark, A. V.; Vishnyakov, A. J. Chem. Phys. 2005, 122, 174508. (c) Krasnochtchekov, P.; Averback, R. S. J. Chem. Phys. 2005, 122, 044319. (d) Tanaka, K. K.; Kawamura, K.; Tanaka, H.; Nakazawa, K. J. Chem. Phys. 2005, 122, 184514. (e) Kraska, T. J. Chem. Phys. 2006, 124, 054507. (f) Fladerer, A.; Strey, R. J. Chem. Phys. 2006, 124, 164710. (g) Saika-Voivod, I.; Poole, P. H.; Bowles, R. K. J. Chem. Phys. 2006, 124, 224709. (h) Schmelzer, J. W. P.; Boltachev, G. S.; Baidakov, V. G. J. Chem. Phys. 2006, 124, 194503. (i) Lauri, A.; Zapadinsky, E.; Vehkamaki, H.; Kulmala, M. J. Chem. Phys. 2006, 125, 164712. (j) Wedekind, J.; Reguera, D.; Strey, R. J. Chem. Phys. 2006, 125, 214505. (12) Seinfeld, J. H.; Pandis, S. N. Atmospheric Chemistry and Physics; Wiley: Hoboken, NJ, 2006; Chapter 11. (13) (a) Girshick, S. L. J. Chem. Phys. 1997, 107, 1948. (b) Suh, S.M.; Girshick, S. L.; Kortshagen, U. R.; Zachariah, M. R. J. Vac. Sci. Technol., A 2003, 21, 251. (14) (a) Schenter, G. K.; Kathmann, S. M.; Garrett, B. C. J. Chem. Phys. 1999, 110, 7951. (b) Schenter, G. K.; Kathmann, S. M.; Garrett, B. C. Phys. ReV. Lett. 1999, 82, 3484. (c) Schenter, G. K.; Kathmann, S. M.; Garrett, B. C. J. Chem. Phys. 1999, 111, 4688. (15) Kathmann, S. M. Theor. Chem. Acc. 2006, 116, 169. (16) Truhlar, D. G.; Garrett, B. C. Annu. ReV. Phys. Chem. 1984, 35, 159. (17) Truhlar, D. G.; Garrett, B. C. Variational Transition State Theory in Theory and Applications of Computational Chemistry: The First Forty Years; Dykstra, C. E., Frenking, G., Kim, K. S., Scuseria, G. E., Eds.; Elsevier: Amsterdam, 2005; pp 67-87. (18) Schenter, G. K.; Kathmann, S. M.; Garrett, B. C. J. Chem. Phys. 2002, 116, 4275. (19) (a) Zhukhovitskii, D. I. J. Chem. Phys. 1995, 103, 9401. (b) Venkatesh, R.; Marlow, W. H.; Lucchese, R. R.; Schulte, J. J. Chem. Phys. 1996, 104, 9016. (c) Yasuoka, K.; Matsumoto, M. J. Chem. Phys. 1998, 109, 8451. (d) Laasonen, K.; Wonczak, S.; Strey, R.; Laasonen, A. J. Chem. Phys. 2000, 113, 9741. (e) Krasnochtchekov, P.; Averback, A. S. J. Chem. Phys. 2005, 122, 044319. (f) Calvo, F.; Parneix, P.; Gadeá, F. X. J. Phys. Chem. A 2006, 110, 1561. (20) (a) Brady, J. W.; Doll, J. D.; Thompson, D. L. J. Chem. Phys. 1979, 71, 2467. (b) Brady, J. W.; Doll, J. D.; Thompson, D. L. J. Chem. Phys. 1980, 73, 2767. (c) Napari, I.; Vehkama¨ki, H.; Laasonen, K. J. Chem. Phys. 2004, 120, 165. (21) Venkatesh, R.; Lucchese, R. R.; Marlow, W. H.; Schulte, J. J. Chem. Phys. 1995, 102, 7683. (22) Yukna, J.; Wang, L. J. Phys. Chem. C 2007, 111, 13337–13347. (23) Lo´pez, M. J.; Jellinek, J. Phys. ReV. A 1994, 50, 1445. (24) Avci, H.; Çivi, M.; Guvenc, Z. B.; Jellinek, J. J. Phys. B: At. Mol. Opt. Phys. 2003, 36, 3487. (25) (a) Johnston, H.; Birks, J. Acc. Chem. Res. 1972, 5, 327. (b) Burns, G.; Wong, W. H. J. Am. Chem. Soc. 1975, 97, 710. (c) Snider, N. S. J. Chem.

Li and Truhlar Phys. 1976, 65, 1800. (d) Dove, J. E.; Raynor, S. J. Phys. Chem. 1979, 83, 127. (e) Truhlar, D. G.; Blais, N. C.; Hajduk, C. J.; Kiefer, J. H. Chem. Phys. Lett. 1979, 63, 337. (f) Lim, C.; Truhlar, D. G. J. Phys. Chem. 1983, 87, 2683. (g) Burns, G.; Cohen, L. J. Phys. Chem. 1985, 89, 4161. (h) Haug, K.; Truhlar, D. G.; Blais, N. C. J. Chem. Phys. 1987, 86, 2697. (i) Haug, K.; Truhlar, D. G.; Blais, N. C. J. Chem. Phys. 1992, 96, 5556(E).. (26) (a) Forst, W. Theory of Unimolecular Reactions; Physical Chemistry A Series of Monographs Series 30; Academic: New York, 1973. (b) Troe, J. In Kinetics of Gas Reactions; Jost, W., Ed.; Physical Chemistry An Advanced Treatise Series 6B; Academic: New York, 1975; p 865. (c) Boyd, R. K. Chem. ReV. 1977, 77, 93. (d) Tardy, D. C.; Rabinovitch, B. S. Chem. ReV. 1977, 77, 369. (e) Breshears, W. D. J. Phys. Chem. 1995, 99, 12529. (f) Dunn, S. M.; Anderson, J. B. J. Chem. Phys. 1995, 102, 2812. (g) Lifshitz, C. Chem. Soc. ReV. 2001, 30, 186. (h) Zhil’tsova, I. V.; Zaslonlko, I. S.; Karasevich, Y. K. Kinet. Catal. 2001, 42, 435. (i) Fernańdez-Ramos, A.; Miller, J. A.; Klippenstein, S. J.; Truhlar, D. G. Chem. ReV. 2006, 106, 4518. (j) Liu, Y.; Lohr, L. L.; Barker, J. R. J. Phys. Chem. A 2006, 110, 1267. (k) Weston, R. E., Jr.; Barker, J. R. J. Phys. Chem. A 2006, 110, 7888. (27) Truhlar, D. G.; Muckerman, J. T. In Atom-Molecule Collision Theory: A Guide for the Experimentalist; Bernstein, R. B., Ed.; Plenum Press: New York, 1979, pp 505-566. (28) Smith, I. W. M. Kinetics and Dynamics of Elementary Gas Reactions; Butterworths: London, 1980; p 6. (29) Jasper, A. W.; Schultz, N. E.; Truhlar, D. G. J. Phys. Chem. B 2005, 109, 3915. (30) Schultz, N. E.; Jasper, A. W.; Bhatt, D.; Siepmann, J. I.; Truhlar, D. G. In Multiscale Simulation Methods for Materials; Ross, R. B., Mohanty, S., Eds.; Wiley-Interscience: Hoboken, NJ, 2008, 169–188. (31) NIST-JANAF Themochemical Tables, 4th ed. Chase, M. W. J. Phys. Chem. Ref. Data, Monogr. 1998, No. 9, 1-1951. (32) (a) Ramaswamy, A. L.; Kaste, P.; Trevino, S. F. J. Energ. Mater. 2004, 21, 1. (b) Ramaswamy, A. L.; Kaste, P. J. Energ. Mater. 2005, 23, 1. (33) Galfetti, L.; De Luca, L. T.; Severini, F.; Meda, L.; Marra, G.; Marchetti, M.; Regi, M.; Bellucci, S. J. Phys.: Condens. Matter 2006, 18, S1991. (34) Ivanov, Y. F.; Osmonoliev, M. N.; Sedoi, V. S.; Arkhipov, V. A.; Bondarchuk, S. S.; Vorozhtsov, A. B.; Korotkikh, A. G.; Kuznetsov, V. T. Propellants, Explos., Pyrotech. 2006, 28, 319. (35) (a) Lerme´, J.; Pellarin, M.; Vialle, J. L.; Baguenard, B.; Broyer, M. Phys. ReV. Lett. 1992, 68, 2818. (b) Baguenard, B.; Pellarin, M.; Lerme´, J.; Vialle, J. L.; Broyer, M. J. Chem. Phys. 1994, 100, 754. (36) Martin, T. P.; Na¨her, U.; Schaber, H. Chem. Phys. Lett. 1992, 199, 470. (37) (a) Cha, C. Y.; Gantefo¨r, G.; Eberhardt, W. J. Chem. Phys. 1994, 100, 995. (b) Gantefo¨r, G.; Eberhardt, W. Chem. Phys. Lett. 1994, 217, 600. (38) (a) Li, X.; Wu, H.; Wang, X. B.; Wang, L. S. Phys. ReV. Lett. 1998, 81, 1909. (b) Akola, J.; Manninen, M.; Hakkinen, H.; Landman, U.; Li, X.; Wang, L. S. Phys. ReV. B 1999, 60, 11297. (c) Akola, J.; Manninen, M.; Hakkinen, H.; Landman, U.; Li, X.; Wang, L. S. Phys. ReV. B 2000, 62, 13216. (39) Schnepf, A.; Schno¨ckel, H. Angew. Chem., Int. Ed. 2002, 41, 3532. (40) (a) Breaux, G. A.; Neal, C. M.; Cao, B.; Jarrold, M. F. Phys. ReV. Lett. 2005, 94, 173401. (b) Neal, C. M.; Atarace, A. K.; Jarrold, M. F. J. Am. Soc. Mass Spectrom. 2007, 18, 74. (41) (a) Jones, R. O. Phys. ReV. Lett. 1991, 67, 224. (b) Jones, R. O. J. Chem. Phys. 1993, 99, 1194. (42) Elbayyariz, Z.; Erkoc, S. Phys. Status Solidi B: Basic Res. 1992, 170, 103. (43) Peslherbe, G. H.; Hase, W. L. J. Chem. Phys. 1994, 101, 8535. (44) (a) Claire, P. d. S.; Peslherbe, G. H.; Hase, W. L. J. Phys. Chem. 1995, 99, 8147. (b) Peslherbe, G. H.; Hase, W. L. J. Chem. Phys. 1996, 104, 9445. (c) Claire, P. d. S.; Hase, W. L. J. Phys. Chem. 1996, 100, 8190. (d) Peslherbe, G. H.; Hase, W. L. J. Chem. Phys. 1996, 105, 7432. (e) Peslherbe, G. H.; Hase, W. L. J. Phys. Chem. A 2000, 104, 10566. (45) (a) Lloyd, L. D.; Johnston, R. L. Chem. Phys. 1998, 236, 107. (b) Lloyd, L. D.; Johnston, R. L.; Roberts, C.; Mortimer-Jones, T. V. ChemPhysChem 2002, 3, 408. (46) Ahlrichs, R.; Elliott, S. D. Phys. Chem. Chem. Phys. 1999, 1, 13. (47) Rao, B. K.; Jena, P. J. Chem. Phys. 1999, 111, 1890. (48) Turner, G. W.; Johnston, R. L.; Wilson, N. T. J. Chem. Phys. 2000, 112, 4773. (49) Akola, J.; Manninen, M. Phys. ReV. B 2001, 63, 193410. (50) Joswig, J.-O.; Springborg, M. Phys. ReV. B 2003, 68, 085408. (51) Sebetci, A.; Gu¨venç, Z. B. Model. Simul. Mater. Sci. Eng. 2005, 13, 683. (52) Noya, E. G.; Doye, J. P. K.; Calvo, F. Phys. ReV. B 2006, 73, 125407. (53) Zhang, W.; Zhang, F. S.; Zhu, Z. Y. Phys. ReV. B 2006, 74, 033412. (54) Schultz, N. E.; Staszewska, G.; Staszewski, P.; Truhlar, D. G. J. Phys. Chem. B 2004, 108, 4850.

Alm + Aln-m T Aln Reactions (55) Jasper, A. W.; Staszewski, P.; Staszewski, G.; Schultz, N. E.; Truhlar, D. G. J. Phys. Chem. B 2004, 108, 8996. (56) Bhatt, D.; Jasper, A. W.; Schultz, N. E.; Siepmann, J. I.; Truhlar, D. G. J. Am. Chem. Soc. 2006, 128, 4224. (57) Bhatt, D.; Schultz, N. E.; Jasper, A. W.; Siepmann, J. I.; Truhlar, D. G. J. Phys. Chem. B 2006, 110, 26135. (58) Li, Z. H.; Bhatt, D.; Schultz, N. E.; Siepmann, J. I.; Truhlar, D. G. J. Phys. Chem. C 2007, 111, 16227. (59) Stillinger, F. H. J. Chem. Phys. 1963, 38, 1486. (60) Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applications; Academic Press: San Diego, 2002. (61) Swope, W. C.; Andersen, H. C.; Berens, P. H.; Wilson, K. R. J. Chem. Phys. 1982, 76, 637. (62) Li, Z. H.; Jasper, A. W.; Bonhommeau, D. A.; Truhlar, D. G. ANT 07; University of Minnesota, Minneapolis, 2007. (63) Nose´, S. J. Chem. Phys. 1984, 81, 511. (64) Nose´, S. Mol. Phys. 1984, 52, 255. (65) Hoover, W. G. Phys. ReV. A 1985, 31, 1695. (66) Martyna, G. J.; Klein, M. L.; Tuckerman, M. E. J. Chem. Phys. 1992, 97, 2635.

J. Phys. Chem. C, Vol. 112, No. 30, 2008 11121 (67) Gro¨nbeck, H.; Tomańek, D.; Kim, S. G.; Roseń, A. Chem. Phys. Lett. 1997, 264, 39. (68) Raff, L. M.; Thompson, D. L. In Theory of Chemical Reaction Dynamics; Baer, M., Ed.; CRC Press: Boca Raton, 1985; Vol. 3, p 1. (69) Levine, R. D. Molecular Reaction Dynamics; Cambridge University Press: Cambridge, 2005. (70) Arrhenius, S. Z. Phys. Chem. 1889, 4, 226. (71) Claire, P. d. S.; Peslherbe, G. H.; Wang, H.; Hase, W. L. J. Am. Chem. Soc. 1997, 119, 5007. (72) Kittel, C. Introduction to Solid State Physics, 4th ed.; Wiley: New York, 1971; p 129. (73) Li, Z. H.; Jasper, A. W.; Truhlar, D. G. J. Am. Chem. Soc. 2007, 129, 14899. (74) Fay, J. A. Molecular Thermodynamics; Addison-Wesley: Reading, MA, 1965; pp 181-206. (75) Martin, W. C.; Zalubas, R. J. Phys. Chem. Ref. Data 1979, 8, 817. (76) (a) Fu, Z. W.; Lemire, G. W.; Bishea, G. A.; Morse, M. D. J. Chem. Phys. 1990, 93, 8420. (b) Woon, D. E.; Dunning, T. H. J. Chem. Phys. 1994, 101, 8877. (77) Bran´ka, A. C.; Wojciechowski, K. W. Phys. ReV. E 2000, 62, 3281.

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