Computational Prediction of Rate Constants for Reactions Involved in

Subscriber access provided by UNIV OF TASMANIA

Article

Computational Prediction of Rate Constant for Reactions Involved in Al Clustering Ning Ning, Lénaïc Couedel, Cécile Arnas, and Sergey Khrapak J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b06557 • Publication Date (Web): 12 Oct 2017 Downloaded from http://pubs.acs.org on October 16, 2017

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry A is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Computational Prediction of Rate Constant for Reactions Involved in Al Clustering Ning Ning,

∗

Lénaïc Couedel, Cécile Arnas, and Sergey Khrapak

Aix-Marseille-Université, CNRS, PIIM, 13397 Marseille, France

E-mail: [email protected] Phone: +33 (0)491288173

1

ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 25

Abstract Aluminium (Al) clustering processes via three types of association reactions are herein studied using classical molecular dynamics trajectory calculations. The simulations were carried out under realistic experimental conditions. The dependence of rate constants on temperature and cluster size was obtained. The association reactions have a very small activation barrier, and the activation energy increases with increasing temperatures. Our prediction of reaction rate constants can be of interest for the study of

Al

nanoparticle growth using kinetic models.

Introduction Aluminium, the most abundant metallic element in the earth's crust after silicon, is a nonprecious metal widely employed in commercial and industrial application due to its interesting physical and chemical properties. During recent years, aluminium nanostructures have attracted attention for a variety of applications.

With high enthalpy of combustion and

rapid kinetics, aluminium nanoparticles are very reactive due to their large specic surface area. The incorporation of aluminium powder into materials can improve their performance by altering the physical properties of the material; for example, increased reaction energies, ame temperatures and blast rates can be desirable properties obtained through aluminium doping.

This property makes aluminium nanoparticles desirable in high-energy fuel, pyro

techniques and explosive industries.

1

The plasmon resonance of aluminium nanostructures

makes it an interesting candidate for plasmonic material to detect organic and biological molecules that exhibit strong UV absorptions. includes optical antennas,

5

2, 3, 4

The application of this property also

plasmon-enhanced spectroscopy

6

and display technology.

Aluminium nanoparticles can be produced via various chemical synthesis, evaporation technique , nique.

15

13

pulsed laser ablative deposition,

14

7, 8

9 , 10 , 11 , 12

gas

and magnetron sputtering tech-

Nanoparticle dimensions are an important factor for a given application as the

mechanical and electrical properties of a material are related to the size of the nanoparticles.

2


Page 3 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Since the real-time, direct observation of the growing process is normally challenging, theoretical studies are essential to provide a better understanding of the particle growth kinetics and mechanisms. In this paper, we rst show an example of laboratory produced aluminium nanoparticles using magnetron sputtering discharge in the next section. We then present in the following section the computational method employed to investigate

Al

clustering, followed by the

validation of our methods. In the last sections we investigated the rst steps of nanoparticle growth:

Al clustering process under similar experimental conditions by using Molecular

Dynamic (MD) simulations based on dierent empirical potentials.

We focus on reaction

kinetics, especially, the prediction of the reaction rate constant. It is of great interests for kinetic modeling of the macroscopic processes involved in experiments and provides atomscale insight of

Al

nanoparticle growth.

Laboratory produced Al naoparticle Aluminium nanoparticles were grown in an argon (30 Pa) direct-current magnetron discharge consisting of an aluminium cathode, biased to

∼-250 V ,

and a ground anode. The discharge

system was contained in a cylindrical vacuum chamber. A similar experimental apparatus was employed to study tungsten nanoparticle growth in reference 16.

Under the chosen

operating conditions, the cathode was sputtered and aluminium nanoparticles started to grow. After a 200 s discharge duration, nanoparticles were collected from the chamber. Fig. 1 shows the Scanning electron microscopy (SEM) image of the produced

Al nanopar-

ticles. We can see that there is a mixture of icosahedra, trigonal bipyramids, octahedral and irregular nanoparticles.

It is worth noting that the icosahedron corresponds to the magic

number cluster conguration. We will therefore focus our theoretical studies on the pathway to such nanocrystals. For this purpose, the simulation conditions were chosen to be close to that of the experi-

3



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 1: Scanning electron microscopy image of

Al

Page 4 of 25

nanoparticles obtained by using a mag-

netron sputtering source

ments. Wolter et al.

17

determined the

Al

atom density and temperature in a DC magnetron

sputtering set-up similar to ours. The Al atom temperature was measured to be between 300 and 400 K Li et al.

18

17

(varies according to discharge pressures and magnetron power). Besides,

studied the kinetics of reactions involving aluminium clusters from 1100 to 3300

K. In our simulation, the temperature is consequently set in the range of 300 to 1100 K in order to extend over a wide range of experimental conditions and procedures which have not yet been studied. In our experimental conditions, the proposed by Wolter et al.,

17

is

Al

5 × 1016 m−3

atom density was calculated using the function or

5 × 10−11 nm−3 .

Such a low density indicates

that clustering in such conditions is undergone mainly through successive binary collisions. In magnetron sputtering discharge, both ionic and neutral species contribute to aluminium clustering. In the present study, we focus on the clustering kinetics via neutral association reactions, aluminium ion clustering is beyond the scope of the present study. The latter rate constant can be estimated using ratio between ion and neutral rate constants derived by Luo et. al.

15

4


Page 5 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Computational Methods There is a lack of experimental and rst principle data in the reaction kinetics of

Al clustering

at 300-1100 K. Therefore, two dierent analytical potential models were used for study of Al clustering process in order to have verication of our results. The selection of the potentials was based on the study of A. W. Jasper,

19 , 20 , 21

where the authors performed a thorough

study to dene the most accurate method to describe small Al systems.

Schultz et al.

19

explored several quantum mechanical methods in describing small Al systems, and found that PBE0 hybrid density functional theory with MG3 basis set is the best performed candidate. Jasperet al.

20

then tested over 19 dierent analytical potentials including most of the

typical potentials used in describing metals such as: pairwise additive PEFs, explicit many body potentials, embedded-atom methods, eective medium theory, and nonpairwise additive potentials with two-and three-body terms. After calculating 224 Al clusters and FCC crystal energies and geometries, it was found that two methods, namely Embedded-Atom Method (EAM) model developed by Mei and Davenport, al.,

21

and explicit many-body potential (EMB),

20

22

reparameterized by Jasper et

provide the smallest mean unsigned errors

while comparing results with PBE0/MG3 method. These two analytical methods were then reparameterized against PBE0/MG3 methods in describing Al clusters and nanoparticles from 3 to 177 atoms. The validation of those two methods was performed by calculating energetic and structural properties of aluminium clusters,

23

thermal properties of aluminium,

and kinetics of reactions involving aluminium clusters.

24

thermodynamics

18 , 25

Both methods describe two-body and many-body interactions of aluminium system. The main dierence between these two methods is the way many-body eects are modeled. EMB employs a more complicated formalism to include many-body eects, which makes it much more computationally time consuming (7 times slower) than the EAM method. A detailed comparison of these two methods can be found in the reference.

5


20


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 25

Validation of the methods in describing Al system The two analytical potential were implemented into the chemical dynamics program VENUS

26 27

Then the simulation methods were validated by comparing the calculated cohesive energy (Ecoh ) of aluminium clusters with experimental and ab initio calculations results. The results are plotted in Fig. 2. The opened green diamonds and black stars are the values calculated by EMB for values for

Aln

Aln

(n=2-64) and EAM for

Aln

(n=2-20), respectively. Solid red triangles are

(n=2-9) obtained using density functional theory with the PBE0 exchange-

correlation functional.

28

Purple solid squares are the results for

using the density functional theory with the BPW91 functional. of aluminium cluster measurements tion between model:

30

Aln

Aln 29

(n=2-15) obtained by

Experiential prediction

(n=2-17) cohesive energies were obtained from photodissociation

(blue open circle in Fig. 2). Dashed line shows the prediction of the rela-

Ecoh

and the fraction of surface atoms in a cluster

n−1/3

described by a simple

30 bulk Ecoh = Ecoh − BN −1/3

(1)

This model assumes spherical shapes of clusters, which does not apply to small clusters (n 12

gives

bulk Ecoh

= 3.438 eV, which

is in good agreement with the experimental value of bulk cohesive energy 3.43 eV.

31

We can see from Fig. 2 that EMB agrees better with experimental results and DFTPEB0 method for clusters with

n < 7.

For clusters with

n > 6,

the two methods both

give adequate predictions. Both methods show a better energetic description of

Aln

clusters

than DFT-BPW91 method. In order to quantitatively demonstrate the accuracy of the two methods, we calculated the mean absolute error (MAE) of cohesive energies by comparing with experimental values (see Table 1). The results conrm our previous conclusions. Validation was performed not only by description of energetic properties of

Aln

clusters,

but also by their structural properties. We plotted in Fig. 3a coordination number (CN), a count of each atom's nearest neighbors, and in Fig. 3b average nearest-neighbor distance in

6


.

Page 7 of 25

8

n 4

100 50

20

10

5

Bulk Value

3.5

Cohesive energy, eV/atom

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2

1

EMB EAM DFT-PBE0 (a)(b) DFT-BPW91 EXP (c)

3 2.5 2 1.5 1 0.5 0

0

0.2

0.4

n-1/3

0.6

0.8

1

Figure 2: Cohesive energy per atoms of aluminium cluster as a function of by EMB (for

Aln

(n=2-20)) and EAM (for

the comparison with (a): values of of

Aln

from.

Aln

Aln

calculated

(n=2-64)) methods in this work, as well as

(n=2-9) obtained by DFT-PBE0 from;

(n=2-15) obtained by DFT-BPW91 from;

30

n−1/3

29

28

(b): values

Aln (n=2-17) n−1/3 (n > 12).

(c): experiential values of

Dashed line shows the prediction of the relation 1 between

Ecoh

and

Table 1: Mean absolute errors of calculated cohesive energies for Aln (n=2-17) when compared with experimental values. Unit is in eV/atom

Aln n7

EAM

EMB

0.122

0.047

0.049

0.043

7



2.9

9 8

2.8

7 6 EAM EMB

5 4

R, Angstrom

Coordination number

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 25

3

EMB EAM DFT-PBE0*

2.7 2.6 2.5

2 2.4

1 0

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 n

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 (b) Average nearest-neighbor distance as a funcn tion of cluster size. Red dash line represents (a) Coordination number as a function of cluster bulk value. 29 Red triangle represents values of size. Aln (n=2-9) obtained by DFT-PBE0 from. 28 Figure 3: Structural properties of

Aln

cluster presented by coordination number (a) and

average nearest-neighbor distance (b), respectively.

Aln

clusters. . Demonstrated by Fig. 3a, there are no signicant dierences if the coordination numbers

are small (CN

< 3)

where the geometries are planar ( n surface. 29

the coordination number is 12 in bulk aluminium, and 9 for atoms at the

The clusters investigated in this study have a large proportion of surface atoms and have not yet reached the bulk limit. The average nearest-neighbor distance is shown in Fig.3b. The dierences between the EAM and EMB methods are not negligible especially for clusters with

n < 8.

For

n > 4,

both analytical methods give better predictions of the average nearest-neighbor distance when compared to DFT-PBE0 method. We can see from Fig.3b that the average nearestneighbor distance increases with the increasing cluster size.

For cluster with

average nearest-neighbor distances are reaching the bulk value of 2.86 Å.

8


29

n > 13,

the

Page 9 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


In general, both analytical potentials give an adequate description of

Aln

clusters in

terms of energetic and structural properties. It seems, however, that EMB can provide more accurate values when compared to experiments and DFT-PBE0 calculations. Nevertheless, EMB method is much more time consuming compared to the EAM method.

Taking this

into account, we use the EAM method in the present work as the primary method to study all the reactions.

The EMB method is only employed to calculated part of the reactions

involving small molecules (where EMB gives more accurate results than EAM).

Collision Simulations Details Trajectory calculations using classical molecular dynamic simulations were performed to study the aluminium clustering process. G. H. Perslherbe et al.

32

studied the dissociation of

aluminium clusters using an analytic potential energy function. According to their results, dissociation is negligible in our temperature range. Therefore, in this study, the aluminium clustering involved only three types of association reactions: successive

cluster growth by adding a

Al atom, clustering by adding a successive Al2 , and growth by adding a successive

Al3 .

Al + Aln−1 −→ Aln (n = 2 − 20)

(R1)

Al2 + Aln−2 −→ Aln (n = 4 − 20)

(R2)

Al3 + Aln−3 −→ Aln (n = 6 − 20)

(R3)

9



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 25

In total, there are 54 reactions to consider. We employed EAM to study all 54 reactions, and EMB method to study

Al + Aln (n = 1 − 19) −→ Aln+1 ,Al2 + Aln (n = 2 − 7) −→ Aln+2 ,

Al3 + Aln (n = 2 − 7) −→ Aln+3 .

The kinetic temperature of the incoming atoms and

molecules was limited to the translational kinetic energy, which leads to energy transfer in direct collisional transfer. The two reactants were initially placed with a distance much larger than the cuto distance of the analytical potentials. Both reactants were initially oriented randomly. The smallest reactant with a given translational energy was running towards the bigger one which was rotating with energies randomly selected from a thermal distribution function. All the reactants were chosen to be initially in vibrational ground state. For each reaction, 5000 trajectories per each kinetic temperature were performed for each run using velocity Verlet integrator and a time step of 0.1

f s.

5 runs were performed in parallel. Each

of the reactant molecules reached its equilibrium state at the target temperature. After the atoms of two reactants had been interacting with each other for a given time, a Berendsen thermostat was employed during 2

ps

to cool down the product to the target

temperature. The cooling process was then followed by a relaxation process (microcanonical ensemble) for 1

ps.

Given our experimental condition of 30 Pa of Ar pressure in the plasma,

−6 5 the estimated Ar density is approximately 7.2 ×10 , which is about 10 time higher than atom density.

Al

Therefore, the involved association reactions are predominantly three-body

reactions:

Alm + Aln−m + Ar −→ Aln + Ar As Ar is suciently abundant in comparison with

Al atoms, the stabilization of produced

excited clusters by Ar can be ensured. The cooling process has therefore been incorporated into our simulation by thermostat cooling. The rate of the overall reaction is then determined by the production rate of excited

Aln ,

and rate constant of association reaction here refers

to the high-pressure limit rate constant. In order to verify our method, we calculated the kinetic energies of the produced molecules/clusters

10


Page 11 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


before cooling, and found that only a small proportion of the produced molecules/clusters (see Table 2) contains excessive energy above the energy threshold of thermal dissociation proposed by Peslherbe and Hase

33 34 .

It is worthy noting that the energy threshold of ther-

mal dissociation is a bit lower than that of dissociation induced by rare gas impact proposed by Sainte Claire and Hase.

35

Hence, the thermostat cooling method is well justied and

necessary in this study.

Table 2: Percentage of produced molecules/clusters which possess more energy than the energy threshold 33 to dissociate before thermostat cooling (1100 K case).

%

Al + Al2 −→ Al3 0.63±0.04

Al + Al5 −→ Al6 2.1±0.1

Al + Al12 −→ Al13 5.8±0.3

In this study, we sampled over all initial conditions which included relative translational energies between the two reactants to obtain the rate constant:

Nreac k = πb2max Ntot where

bmax

is max impact parameter given by

s

18

8kB T πµ

(2)

18

bmax = 2.2(dmax /2 + Rcut )

dmax

(3)

is the maximum distance between two atoms in the reactant, and

Rcut

is the cuto

distance set in the empirical potential. As our reactants were not all spherical, this choice of

bmax

allowed inclusion of all the possible eective collision cases.

For the values of

b

superior to

bmax , it has been tested to conrm that there is no collision occurring.The impact

parameter

b

was then selected randomly between 0 and

bmax .

Please note that the calculated rate constants refer to local equilibrium rate constants, which are bimolecular associative reactions.

11



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 25

Results and discussion In this section, we discuss separately our results for all three types of

Al

clustering paths.

The reaction rate constants of the involved association reactions and their inuencing factors are deduced.

Clustering through association reaction R1 Using a connected component labeling method,

36

the product of each association reaction

was identied in order to calculate the reaction probability. The resulting rate constants of the association reactions are tabulated in the Supporting Information Table S1-2. The factors that inuence the rate constant of the association reactions are discussed. Firstly, the calculated rate constants as a function of temperature are plotted. For example: the logarithm of calculated rate constant of the reaction

1000/T

Al + Al13 −→ Al14

as a function of

is shown in Fig. 4.

As our temperature range (300-1100 K) includes a high temperature gas-phase kinetic system, the temperature dependence is usually non-linear and is described by the modied Arrhenius equation:

37

k = BT l e−C/RT

(4)

where the physical meaning of B and C do not represent the pre-exponential factor and activation energy respectively. Eq. 4 was used to t the calculated rate constant with respect to temperature. The results of this tting procedure are the parameters

B, l

and

C,

which

are listed in the Supporting Information Table S3. We then plot the resulting tting function for the dierent cluster size

Al + Aln (n = 1 − 19) −→ Aln+1 Aln , k

n

for reaction

(see Fig. 5a and Fig. 5b). We found that for all reactants

increases with the increasing temperature.

Rate constants calculated by the EMB

method (Fig. 5a) are in general larger than those calculated by the EAM method (Fig. 5b).

12


Page 13 of 25

-20

MD Fit

-20.2

lnk

-20.4

-20.6

-20.8

-21

1

2

1.5

3

2.5

3.5

1000/T, 1/K Figure 4: Calculated rate constant of reaction

Al + Al13 −→ Al14

with varing temperatures,

and the tting by a modied Arrhenius equation (eq. 4).

-19.8

-19.8

-20

-20

-20.2

-20.2

-20.4

ln(k)

ln(k)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


19 16 13 10 7 4 1

-20.6 -20.8 -21 -21.2 0.9

1.2

19 16 13 10 7 4 1

-20.4 -20.6 -20.8 -21 -21.2

1.5

1.8 2.1 2.4 1000/T, 1/K

2.7

3

3.3

0.9

1.2

(a) EMB

1.5

1.8 2.1 2.4 1000/T, 1/K

2.7

3

3.3

(b) EAM

Figure 5: Rate constant of association reaction

Al + Aln (n = 1 − 19) −→ Aln+1

temperatures calculated by EMB (a) and EAM (b) methods, respectively.

13


with varing


As the rate constants are described by a modied Arrhenius function, the activation energy changes with temperature. We can calculate the activation energy at a given temperature by the derivatives of modied Arrhenius equation:

Ea (T ) = −R

37

d(ln k(T )) = lRT + C d(1/T )

0.4

2.4 EMB EAM

Activation energy, kcal/mol

Activation energy, kcal/mol

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 25

0.35

0.3

0.25

(5)

n=1 n=7 n=13 n=19 k BT

2.1 1.8 1.5 1.2 0.9 0.6 0.3

0.2

2

4

6

8

10 n

12

14

16

18

400

20

600 800 Temperature, K

1000

(a) Activation energy as a function of cluster (b) Activation energy as a function of the temsize n for the 300 K case predicted by EMB and perature predicted by EMB method. Dash line EAM method. represents energies equal to k T . B

Figure 6: Activation energy as a function of cluster size (a) and temperature (b), respectively.

The resulting activation energies are tabulated in the Supporting Information Table S4. After plotting activation energy with respect to cluster size

n in Fig. 6a, we found that there is

no correlation between activation energy and cluster size. In order to conrm our observation, we calculated the Pearson's correlation coecient of activation energy versus cluster size obtained by the EMB and EAM methods. We found that the correlation coecients are -0.18 and -0.11, respectively, which means that there is a negligible relationship between activation energy and cluster size. Figure 6b demonstrates that the activation energy increases with increasing temperature, which can be attributed to the relative energy dependence of the cross section.

38

We also deduced that even with the biggest cluster

energy of the association reaction is smaller than the thermal energy association reactions have a very small barrier.

39

14


n = 19, kB T ,

the activation

meaning that the

Page 15 of 25

From Fig. 5a and Fig. 5b, we can see that the rate constant size: with increasing

n,

the value of

k

k is related to the reactant Aln

increases. The dierence in

k

becomes less important

while reactant molecule gets bigger. We found that some clusters share very similar values of

k,

e.g.

we plot

k

Al10 − Al13 , Al18

Al19 .

and

In order to see the eect of cluster size quantitatively,

as a function of cluster size in Fig. 7a (obtained by using the EMB method) and

Fig. 7b(obtained by using the EAM method) .

2.6

2.6

2.4

2.4

900 K 600 K 300 K

2.2

900 K 600 K 300 K

2.2

2

k (x10-9), cm3 s-1

k (x10 -9), cm 3 s -1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.8 1.6 1.4 1.2

2 1.8 1.6 1.4 1.2

1

1

0.8

0.8

0.6

0.6 1

2

3

4 n2/3

5

6

7

1

2

(a) EMB

3

4 n2/3

5

7

6

(b) EAM

Figure 7: Rate constant of association reaction as a function of cluster size by EMB (a) and EAM (b) methods, respectively. f (n) = An2/3 where A is a constant.

n

calculated

Black dash lines represent function of

We can see from Fig. 7a and Fig. 7b that the rate constant is proportional to

n

for both

methods. This dependence relates to the cross section of the cluster size; the probability that an

Al

atom strikes the

Aln

cluster increases with the radius of the

that the geometric cross section of the cluster is proportional to

Aln

n2/3

cluster. It is known

for a spherical or cubic

cluster if the density of the cluster is not changing with the cluster size.

40

For bigger clusters

the geometrical cross section can be used in dening the reaction rates. Using the present simulations, we found this reasonable for

n > 13,

but less accurate for smaller cluster. The

results for larger clusters will be reported elsewhere. We found that both the EMB and EAM methods predict the same dependence.

The

values predicted by EMB are in general larger than those obtained using EAM. This dierence can be caused by the dierence in handling many-body eects. In the EAM method,

15



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 25

many-body eects are incorporated by a term called embedding energy, which is the energy needed to embed an atom into a lattice.

This embedding energy is a functional of local

electron density, which is approximated by the superposition of contributions from atom's neighbors.

In the EMB method, the many-body eects consist of screening and coordi-

nation number eects.

The coordination number term considers the weakening eect of

large-coordinated structures to the interatomic bonding. The screening function simulates the electronic screening eects in solids, where the interatomic strength gets weaker if an intervening atom resides between the two bonding atoms.

Moreover, a linear scaling has

been introduced to the many-body functions by a Stillinger and Weber cuto function

41

in order to smoothly switch from many-body function coupling to just interpair potential. This complex way of introducing many-body eects makes the EMB method more accurate in describing smaller systems, at the price of a larger computing time.

Therefore, in this

study we only employed partially the EMB method to the studied reactions R2 and R3 .

Cluster through association reaction R2 Under our simulation conditions, dissociative attachment was negligible as a result of the collisions. The association reactions are the dominant reactions for the collisions between

Al2

and

Aln

under our simulation conditions.

the simulation results of reaction parameters for non-linear tting of

We performed the same data analysis to

R2, of which the resulting values, such as rate constants,

ln(k) versus 1000/T plot, activation energy are tabulated

in the Supporting Information Table S5-9. We found that the plot of rate constants of association reaction as a function of temperatures (see Fig. 8a) follows the same trend as the rate constant

k

Al + Aln

case. Nevertheless, the reaction

only weakly depends on cluster size (see Fig. 8b).

16


Page 17 of 25

-20.2

2

17 14 11 8 5 2

-20.6

900 K 600 K 300 K

1.8

k (x10-9), cm 3 s -1

-20.4

ln(k)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-20.8 -21

1.6 1.4 1.2 1

-21.2

0.8 0.9

1.2

1.5

1.8 2.1 2.4 1000/T, 1/K

2.7

3

(a) Rate constants of association reaction

3.3

0.6

k with

varying temperatures.

1

2

3

(b) Rate constants

Figure 8: Rate constants and their inuencing factors for

4 2/3 n

k

5

7

6

as a function of

n.

Al2 + Aln (n = 2 − 18) −→ Aln+2

calculated by using EAM method.

Clustering through association reaction R3 Under our simulation conditions, dissociative attachment was negligible as a result of the collisions between

Al3

and

Aln .

Association reactions are still the dominant reactions under

our simulation conditions. The same data analysis was applied to the simulation results of reaction R3, of which the resulting values, such as rate constants, parameters for non-linear tting of

ln(k)

versus 1000/T plot, and activation energy are tabulated in the Supporting

Information Table S10-14. We found that the plot of rate constants of association reaction as a function of temperatures can be also described by the modied Arrhenius function (see Fig. 9a). The dependence of

k

on cluster size becomes even weaker (see Fig. 9b).

The resulting rate constants of these three types of clustering path have the same order of magnitude. Therefore, they should all be taken into account for understanding and accurate description of the clustering process. It can be of interests for kinetic modeling to study the selectivity and yields of dierent reactions.

17



15 12 9 6 3

-20.6 -20.8 -21 -21.2 0.9

900 K 600 K 300 K

1.4

k (x10-9), cm 3 s -1

-20.4

ln(k)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 25

1.2 1 0.8

1.2

1.5

1.8

2.1 2.4 1000/T, 1/K

2.7

3

(a) Rate constants of association reaction

3.3

0.6 2

k with

3

4

(b) Rate constants

varying temperatures.

Figure 9: Rate constants and their inuencing factors for

n2/3

k

5

7

6

as a function of

n.

Al3 + Aln (n = 2 − 17) −→ Aln+3

calculated using the EAM method.

Conclusions In this paper, we studied three types of

Al

clustering paths and determined the reaction

rate constants for the involved association reactions using molecular dynamics trajectory calculations based on two dierent analytical potentials under realistic experimental conditions. The reaction rate constants predicted by the EMB method are in general larger than those acquired by the EAM method. Both methods demonstrate a similar dependence of rate constants on the temperature and the size of the reactant. Activation energies of the involved reactions increases with increasing temperature.

Association reactions discussed

here all have a small barrier. Under the studied conditions, for collisions between and

Aln (n = 2 − 19),

Alm (2 − 3)

the association reaction is the dominant one when compared with the

dissociative attachment.

All three association types should be taken into account for the

clustering process in the preliminary calculation, where the concentrations of are not negligible compared to

Al

Al2

and

Al3

atom. The EMB method provides a better description in

smaller systems (n

Computational Prediction of Rate Constants for Reactions Involved in

Recommend Documents