Theoretical Study of the Reactions of Ethanol with Aluminum and

Article pubs.acs.org/JPCA

Theoretical Study of the Reactions of Ethanol with Aluminum and Aluminum Oxide Alexander S. Sharipov and Alexander M. Starik* Central Institute of Aviation Motors, 111116 Moscow, Russia Scientific Educational Centre “Physical-Chemical Kinetics and Combustion”, 111116 Moscow, Russia S Supporting Information *

ABSTRACT: Quantum chemical calculations with the use of B2PLYP method were carried out to study the reactions of Al and AlO with the C2H5OH molecule. The values of energy barriers were estimated by means of extrapolation to the basis set limit. Examination of the potential energy surface revealed the energetically favorable reaction pathways. It has been found that for the Al + C2H5OH reaction, the OH-abstraction process leading to the formation of AlOH and C2H5 prevails. During investigation of the AlO + C2H5OH reaction it has been found that resulting products of this reaction were AlOH and C2H5O in different isomeric forms: hydroxyethyl and ethoxyl radicals. Appropriate rate constants for revealed channels have been estimated by using a canonical variational theory and capture model. The Arrhenius approximations for these processes have been proposed for the temperature range T = 400−4000 K.

1. INTRODUCTION

Recent studies, addressing the analysis of kinetic processes in the CH4−Al−O2 mixture, have showed that one of the crucial reactions, responsible for the appearance of novel channels of chain mechanism development upon oxidation of CH4−Al composite fuel, is the reaction of CH4 with AlO.15 Therefore, it can be assumed that analogous process C2H5OH + AlO → products plays an important role in the oxidation of composite fuel composed of ethanol and aluminum. The other reaction, which can be responsible for the chain mechanism development in the C2H5OH−Al−O2 system, is the reaction of C2H5OH with Al atom. Note that Al atoms form during the oxidation of Al nanoparticles that are substituted in the primary fuel. The point is that for nonoxidized nanoparticles with diameter d < 50 nm, the time of heating of the Al particle caused by energy release in the course of aluminum oxidation is much higher than the characteristic time of the phase change. Therefore, the Al core of small particles undergoes phase transition. This results in the appearance of high temperature gradient in a thin oxide surface shell of particle and its rapid destruction16 or deformation.17 As a result, small aluminum clusters or even Al atoms come into the environment. Further, Al atoms can react with O2 that leads to the AlO formation.8,18 Thus, at the initial stage, Al atoms and AlO radicals are abundant in the reacting fuel−air mixture and can react with C2H5OH. However, the kinetic data on these reactions in the gas phase are absent. Moreover, even the reaction products are unknown. The goal of the present work is theoretical analysis of these

For the past years great attention of researchers has been focused on the analysis of ignition and combustion of Al particles in various environments (O2, CO2, HCl, H2O, D2O).1−9 This interest is caused by the fact that aluminum possesses rather high calorific power and high heat of combustion. Nowadays, aluminum is widely used to increase the energy of reaction of various explosives.10,11 Today, a hot topic of combustion science is the studies of ignition and combustion of composite fuels composed of hydrocarbons and Al nanoparticles.12−15 As was demonstrated in these works, the addition of Al nanoparticles to hydrocarbon fuels increases their reactivity and energetic output. Especially, this strategy is promising for the biofuels such as methanol, ethanol, dimethyl ether, and others, because they have smaller calorific power than methane and other alkanes and lower reactivity. Experiments conducted with the usage of aerosol shock tube technique demonstrated that addition of 2% aluminum particles of 50 nm diameter to ethanol shortened the ignition delay by 32%.13 However, up to now there is no explanation of this effect. It can be supposed that the presence of Al nanoparticles in ethanol and hydrocarbons accelerates the chain processes during oxidation of such fuels. To gain an insight in the chain mechanism development in the combustible mixtures comprising aluminum, one needs to develop adequate reaction mechanisms. Such mechanisms are based on the kinetic data on elementary processes obtained both experimentally and theoretically with the use of ab initio calculations of potential energy surfaces (PESs). Therefore, there exists a strong need to obtain such kinetic data on the reactions of ethanol with Al−containing species. © 2015 American Chemical Society

Received: February 20, 2015 Revised: April 7, 2015 Published: April 20, 2015 3897

DOI: 10.1021/acs.jpca.5b01718 J. Phys. Chem. A 2015, 119, 3897−3904

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The Journal of Physical Chemistry A reactions. The estimates of temperature-dependent rate constants for possible reaction pathways are also presented.

Shown in Table 1 are the atomization enthalpies for reactants and possible products in the Al−C2H5OH and AlO−C2H5OH

2. METHODOLOGY A series of quantum chemical computations with the use of density functional theory (DFT) were performed to investigate the PESs for the Al + C2H5OH and AlO + C2H5OH systems. Special attention was paid to the choice of theoretical method suitable for the problem under study. It is believed that the Becke’s three-parameter hybrid functional (B3LYP) can be an efficient approach for the calculations of thermochemical properties of Al-containing species due to its acceptable accuracy and low computational costs.19−21 However, the B3LYP level of theory as well as the other DFT methods has a tendency to underestimate the activation barriers for exchange reactions22 including reactions with atomic aluminum.23,24 It is worth noting that for the past years a few hybrid density functionals with perturbative second-order correlation (for example, B2PLYP and mPW2PLYP) were developed.22,25,26 Today, the B2PLYP method is considered as the best general purpose density functional to determine accurately not only the reaction enthalpy but also the activation barriers of chemical reactions.22,25,27 In the present work, the PESs of Al + C2H5OH and AlO + C2H5OH systems were explored by using the UB2PLYP method and Dunning’s correlation consistent basis set with diffuse functions.28 So, the geometry of reactants, transition states, and possible products of reaction pathways were optimized at the UB2PLYP/aug-cc-pvDZ level of theory. For each stationary point, to characterize it either as a minimum (all vibrational frequencies are real) or as a first-order saddle point (there exists a single imaginary frequency), a vibrational frequency analysis at the same level of theory was conducted. In line with the recommendations of Kesharwani et al.,29 the calculated values of frequency were scaled by a factor of 0.9648 to take into account an incomplete treatment of electron correlation as well as the use of truncated basis set and ignoring the anharmonicity of vibrations in the frequency analysis. The fact that reactants and products are connected by the transition state, was confirmed by following minimum-energy paths (MEPs) of the reactions by using the Gonzalez−Schlegel intrinsic reaction coordinate (IRC) algorithm30 in both directions. To refine the energy of revealed critical points, we performed pointwise electronic energy calculations with the use of extended basis sets. In this case, the Dunning’s aug-cc-pvnZ basis set28 with maximal angular momentum quantum numbers n = 3, 4 were applied. Then, the obtained energy values were extrapolated to the basis set limit via the simple exponentially decaying function31

Table 1. Atomization Enthalpies (kJ/mol) for the Species Involved in the Kinetics of Al−C2H5OH and AlO−C2H5OH Reactive Systems Calculated by Using the Present Methodology for Different Values of Maximum Angular Momentum (n) in the Basis Set and Corresponding Reference Data31,32

E(n) = E(∞) + A e−an

molecule

n=2

n=3

n=4

n=∞

ref data

C2H5OH C2H5 CH2OH AlOH AlO AlCH3 C2H5O mean dev, %

3124.0 2331.5 1586.7 932.7 467.8 1462.5 2707.3 4.1

3193.0 2387.1 1626.4 964.5 496.4 1505.7 2769.1 1.2

3207.9 2397.2 1635.6 974.2 506.3 1512.7 2781.8 0.5

3210.4 2397.6 1637.6 978.2 511.2 1511.9 2783.4 0.3

3225.4 2403.6 1636.8 977.07 512.3 1511.5 2786.1

reactive systems calculated by using the methodology described above, as well as their values reported in the reference databases.34,35 The magnitudes of mean deviation of the calculated values of atomization enthalpy from the reference ones are also presented there. If the aug-cc-pvDZ (n = 2) basis set is used, the mean deviation amounts to 4%. However, applying the extrapolated basis set allows one to improve the accuracy until 0.3%. Thus, the applied methodology provides the reliable energy values for the systems under study. The values of electronic energy and enthalpy correction required for these thermochemical calculations as well as the values of formation enthalpy of corresponding Al-containing species obtained in the present work are summarized as Supporting Information. The strategy for estimating the reaction rate constant for each reaction path depends on the type of elementary process. For the barrierless reaction, the rate coefficient is controlled by long-range forces exerting between reactants.36,37 For a pair of interacting polar molecules (A and B), one can take into account an attractive part of Stockmayer (SM) potential including dispersion, dipole−dipole, and dipole−induced dipole interaction terms. This potential is expressed in the form ⎛ ⎛ σ ⎞6 SM * ζd − id(ωAB)]⎜ AB ⎟ φAB (r ) = −4εAB⎜[1 + ξAB ⎝ r ⎠ ⎝ ⎛ σ ⎞3 ⎞ * ζd − d(ωAB)⎜ AB ⎟ ⎟ + δAB ⎝ r ⎠⎠

(1)

(2)

Here εAB is the dispersion potential well depth, σAB is the collision diameter of interacting particles, ζd−id and ζd−d are the angle-dependent functions for dipole−induced dipole and * and δAB * = μAμB/ dipole−dipole types of interaction, ξAB 2εABσAB3 are the mean reduced polarizability and mean reduced dipole moment of interacting particles. In accordance with the method of dipole reduced formalism (DRFM) developed by Paul and Warnatz,38 the attractive part of SM potential subjected to orientation-averaged procedure can be reduced to the spherically symmetrical potential with the effective C6eff coefficient

where E(n) is the energy value calculated with aug-cc-pvnZ basis set at given n, E(∞) is the energy estimated at the basis set limit, A and a are the parameters specified on the basis of extrapolation procedure. The scaled vibrational frequencies were also applied for zero point energy (ZPE) correction of energy values obtained at the UB2PLYP/aug-cc-pv∞Z level of theory (sign ∞ indicates an extrapolation of aug-cc-pvnZ basis set to the complete basis set). Note that all quantum chemical calculations have been performed by using the Firefly QC program package,32 which is partially based on the GAMESS(US) source code.33 3898


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The Journal of Physical Chemistry A φeff (r ) = −

C6eff r6

2 ⎛ C6ind C6el ⎞ ⎟ + C6eff = C6disp⎜⎜1 + disp ⎟ C6 4C6disp ⎠ ⎝

In accordance with CTST, the transition state is located at the saddle point on the surface of Born−Oppenheimer potential energy, and the rate constant is calculated at the coordinate s = 0:

(3)

C6disp,

C6el,

kCTST(T )=η(s = 0,T )

C6ind

where and terms specify the dispersion interaction, electrostatic interaction of dipoles, and polarization interaction. These terms were determined in the same manner as in our recent work.39 The values of polarizability and dipole el moment of molecules, required for the evaluation of Cdisp 6 , C6 , ind and C6 terms, were calculated in line with the procedure proposed there.39 So, the estimations of dipole moments μ and static isotropic polarizabilities α were performed by using the UB3LYP/6-311+G(2d) level of theory for the geometries optimized with the 6-31+G(d) basis set (UB3LYP/6-311+G(2d)//UB3LYP/6-31+G(d)). The values of α were additionally scaled by a factor of 1.11 to compensate empirically the finiteness of basis set and contribution of zero point vibrational motion. In accordance with the known capture model,40 the capture occurs when the reactive system moves inside a sphere with critical distance δ between molecules. The value of δ is specified by the location of the maximum of the centrifugal barrier. After capture, the formation of products occurs with the efficiency equal to unity. Thus, for the attractive potential (3), the rate constant of bimolecular barrierless reaction can be expressed as41,42 1/6 Q ePES ⎛ (C6eff )2 k bT ⎞ ⎜ ⎟ k(T ) ≅ 1.706 A B ⎜ ⎟ M3 Qe Qe ⎝ ⎠

When CVT is used,43 the expression for η(s,T) (eq 5) is minimized with respect to the s coordinate, and the rate constant kCVT is defined as kCVT(T ) = min(η(s ,T ))

(7)

s

At first, the rate constant of the process with nonzero activation barrier was estimated with the use of CTST, applying the VMEP value calculated at the UB2PLYP/aug-cc-pv∞Z level of theory (kCTST ∞Z ). To evaluate the variational effects on the CTST calculations, the projected vibrational frequency analysis along the reaction path45 at the UB2PLYP/aug-cc-pvDZ level of theory (precisely this level of theory was applied during IRC calculations) was performed. The reaction rate constant estimated by CVT for the UB2PLYP/aug-cc-pvDZ calculated reaction profile (kCVT DZ ) was compared with the rate constant kCTST calculated by using CTST with the same activation DZ barrier, and the total rate constant for the process with energy barrier was estimated as following CTST k(T ) = k∞ Z (T ) × r ( T )

r (T ) =

CVT kDZ (T ) CTST kDZ (T )

(8)

where the factor r(T) shows the extent of overprediction in the value of rate constant caused by the use of nonvariational transition-state theory against the variational one.

(4)

where M is the reduced molecular mass of colliding molecules, kb is the Boltzmann constant, QeA, QeB ,and QePES are the electronic partition functions of reactants (A and B) and the electronic degeneracy of reactive PES, respectively. For the processes with nonzero activation barrier, when there exists a single potential maximum, which is associated with the transition-state structure, separating the reactant and product PES regions, the temperature-dependent rate constant can be estimated with both canonical variational theory (CVT) and nonvariational conventional transition-state theory (CTST).43 The generalized expression for the rate constant at the temperature T is specified as a function of reaction coordinate along the minimum energy reaction path43 k T Q TS(T ,s) ⎛ VMEP(s) + VZP(s) ⎞ η(s ,T ) = Γ b exp⎜ − ⎟ h Q R (T ) k bT ⎠ ⎝

(6)

3. POTENTIAL ENERGY SURFACES AND REACTION PATHWAYS During the analysis of the PES for the reaction system composed of ground-state aluminum atom and ethanol molecule the following reaction pathways were revealed: Al + C2H5OH → AlH + C2H5O (H‐abstraction, ethoxyl radical formation)

(a)

Al + C2H5OH → AlH + C2H4OH (H‐abstraction, hydroxyethyl radical formation) Al + C2H5OH → AlOH + C2H5

(5)

(b)

(OH‐abstraction) (c)

Here Γ is the tunneling factor, h is the Planck constant, QTS(T,s) is the partition function of the transition state for the bound degrees of freedom orthogonal to the reaction path at the coordinate s, QR(T) is the partition function of reactant, VMEP(s) is the value of the Born−Oppenheimer potential energy at the coordinate s and VZP(s) is the value of zero point vibrational energy along the reaction path at the coordinate s. Note that VMEP(s) and VZP(s) values are given with respect to the energies for reactants. The partition function for the degrees of freedom, specified as internal rotation, was calculated in line with the expression of Truhlar,44 which gave a smooth approximation for the partition function from free rotator to harmonic oscillator. Tunneling factor Γ was evaluated by using the semiclassical transmission probability for asymmetric Eckart potential in the same manner as in our previous work.23

Al + C2H5OH → AlCH3 + CH 2OH

(CH3‐abstraction) (d)

The investigation of the doublet PES for the Al + C2H5OH system at the UB2PLYP/aug-cc-pv∞Z level of theory with appropriate ZPE correction showed that the activation barriers for reaction pathways a−d were equal to +146.7, +150.3, −5.3, and +203.0 kJ/mol, respectively (structure, the frequencies of normal vibrations and rotational constants for the corresponding transitions states are given as Supporting Information). From these data, one can conclude that channel c corresponding to OH-abstraction can occur with considerable rate in the reactive system comprising atomic Al in the ground state and ethanol. The rate constants for the rest reaction paths must be substantially smaller. The energy diagram for reaction 3899


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∼4 kJ/mol with respect to the reactants, and the difference between the energies of SPc and reactants with an allowance for the dispersion correction was found equal to −10.4 kJ/mol. Thus, the calculations with dispersion correction verified the barrierless character of the reaction path c. Additional calculations with the use of composite computational scheme CBS-QB3, developed to achieve high target accuracy,46 also proved this fact (Table 2). It should be emphasized that composite CBS-QB3 method predicts notably smaller activation barriers for all channels of the reaction Al + C2H5OH → products. During the investigation of the PES of AlO(X 2Σ+) + C2H5OH system, the following reaction pathways were revealed:

path c is shown in Figure 1. Depicted in Figure 2 is the corresponding structure of the saddle point. One can observe

Figure 1. Relative energies of critical points on the doublet PES correlating with Al + C2H5OH system at the UB2PLYP/aug-cc-pv∞Z (closed symbols) and UB2PLYP-D/aug-cc-pv∞Z (open symbols) levels of theory augmented with ZPE correction.

AlO + C2H5OH → AlOH + C2H4OH (H‐abstraction, hydroxyethyl radical formation)

(e)

AlO + C2H5OH → AlOH + C2H5O (H‐abstraction, ethoxyl radical formation)

Analysis of the doublet PES for the AlO + C2H5OH system at the UB2PLYP/aug-cc-pv∞Z level of theory showed that the activation barriers for reaction pathways e and f are relatively small and equal to 28.0 and 8.2 kJ/mol, respectively. The geometries of corresponding saddle points are presented in Figure 3. The calculated structures, frequencies of normal vibrations, and rotational constants for the corresponding transitions states are also given as Supporting Information. In accordance with the obtained values of energy barriers, one can conclude that both channels e and f can potentially occur with comparable rates in the reactive system comprising groundstate AlO(X 2Σ+) and an ethanol molecule. The corresponding energy diagrams for these reaction paths are depicted in Figure 4. One can see that the H-abstraction process via reaction path f is energetically preferred due to lthe ower value of the activation barrier (8.2 kJ/mol). Note that if one applies the empirical dispersion correction,26 the values of these barriers become smaller (Table 2). It should be emphasized that earlier the possible products of the reaction of ground-state aluminum atom with ethanol in adamantine matrix at T = 77 K were identified by Joly et al.47 So, the following products of insertion reactions were revealed: Al(OCH2CH3)2 and HOAlOCH2CH3. These experimental results cannot be compared directly with the results of theoretical analysis of the present work, as far as at the conditions of experiment47 more than one ethanol molecule could be involved in the reactions with Al. However, the possibility of the formation of HOAlOCH2CH3 product in the course of the insertion reaction AlO + C2H5OH was additionally explored at the UB2PLYP/aug-cc-pvDZ level of theory. The corresponding activation barrier was found to be very high (∼250 kJ/mol) compared to those for the abstraction channels, and therefore, this insertion process is not important for gas-phase kinetics.

Figure 2. SPc configuration with geometric parameters (angstroms, degrees) calculated at the UB2PLYP/aug-cc-pvDZ level of theory.

that the reaction of Al with C2H5OH leads to the formation of AlOH and C2H5 products through the saddle point SPc, which lies slightly lower (by ∼5 kJ/mol) than the reactants on the doublet PES. Therefore, in fact, this reaction is a barrierless one. In addition, B2PLYP calculations revealed the existence of a weakly bound van der Waals Al···C2H5OH complex (IMc) on the reactant side of the transition state. In view of the rather small energy difference between SPc and reactants, it would be desirable to verify the fact that the reaction of Al with C2H5OH proceeds, in whole, without activation barrier. So, the energy values of critical points were augmented with empirical dispersion correction that can be essential for the long-range interaction26 (Figure 1). Herein and hereafter such calculations are denoted as UB2PLYP-D/aug-ccpv∞Z. The obtained values of activation barriers are summarized in Table 2. As a result, the complex Al··· C2H5OH after dispersion correction became to lie deeper by Table 2. Values of Energy Barriers for Processes a−f Calculated at Different Levels of Theory activation energy, kJ/mol reacn path

UB2PLYP/ aug-cc-pv∞Z

UB2PLYP-D/ aug-cc-pv∞Z

CBS-QB3

a b c d e f

146.7 150.3 −5.3 203.0 28.0 8.2

153.7 157.2 −10.4 210.0 24.7 5.3

137.5 143.3 −2.2 191.2 21.8 19.2

(f)

4. REACTION KINETICS IN THE Al + C2H5OH AND AlO + C2H5OH SYSTEMS Because the process Al + C2H5OH→ AlOH + C2H5 was proved to be barrierless, its rate constant can be correlated, in a first approximation, with the rate constant of IMc formation process Al + C2H5OH → IMc (k1). The temperature3900


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Figure 3. SPe (a) and SPf (b) configurations with geometric parameters (angstroms, degrees) calculated at the UB2PLYP/aug-cc-pvDZ level of theory.

Figure 4. Relative energies of critical points on the doublet PES correlating with the AlO + C2H5OH system at the UB2PLYP/aug-ccpv∞Z (closed symbols) and UB2PLYP-D/aug-cc-pv∞Z (open symbols) levels of theory augmented with ZPE correction.

Figure 5. Temperature-dependent rate constants for the Al + C2H5OH → AlOH + C2H5 process: k1 (dotted curve) and effective rate constant kc calculated with UB2PLYP-D/aug-cc-pv∞Z level of theory (solid curve) and CBS-QB3 method (dashed curve). Experimental data on the rate constants for the reactions of Al with O2, DME, and DEE are depicted by symbols.

dependent rate constant k1(T) was estimated with the use of capture model (eqs 2−4). For the calculations, the following values of reactant electric properties were taken: α(Al) = 7.22 Å3, α(C2H5OH) = 5.15 Å3, μ(C2H5OH) = 1.72 D. These values of static polarizability and dipole moment were estimated in accordance with the procedure reported in our recent work.39 As the ground-state doublet aluminum atom has 2 P1/2 and 2P3/2 low-lying spin−orbit states, and the 2P3/2 state lies 161.2 K higher than the 2P1/2 state,48 the electronic partition function for Al(2P) can be expressed as Qe = 2 + 4 exp(−161.2/T). The electronic weights for the C2H5OH molecule and for the reactive PES were taken equal to 1 and 2, respectively. The resulting theoretical dependence k1(T) is depicted in Figure 5. Over the temperature range T = 250− 4000 K it can be approximated by the single Arrhenius dependence:

where k−1 is the rate constant of the decomposition process IMc → Al + C2H5OH, and k2 is the rate constant for the process of the conversion of complex IMc to the reaction products. The function f(T) defines the influence of the existence of saddle point SPc on the effective rate constant of barrierless process c. The rate constant k−1(T) was estimated via k1(T) one by using the detailed balancing principle. The rate constant k2(T) was calculated with the use of eqs 5−8. Note that, for these estimations, the energy values obtained at the UB2PLYP-D/ aug-cc-pv∞Z level of theory were applied. The results of the calculations of k1, k−1, and k2 rate constants as well as the values of Γ(T), r(T), and f(T) factors and the effective rate constant kc for reaction path c in the temperature range T = 250−4000 K are presented in Table 3. One can see that, at low temperatures (T < 400 K), the inequality k−1 ≪ k2 is valid due to the difference in the activation energy values. At such temperatures, we have f(T) ∼ 1, and the rate constant of reaction path c is governed mainly by the capture rate coefficient k1. At higher temperatures (T > 400 K), the magnitudes of k−1 and k2 become comparable, and the effective rate constant kc slightly decreases with temperature growth (up to a factor of 1.6 at T = 4000 K) owing to the existence of saddle point SPc. As regards the tunneling factor, the method applied in the present work is based on the assumption that the reaction and tunneling paths coincide. This leads to underestimation of the Γ value as far as the corner cutting effects are ignored. Therefore, we do not recommend using the rate constants obtained in the present work at low temperatures, where Γ is far from unity (T < 400 K). Certainly, more advanced tunneling methods based on the inclusion of deviations between the tunneling path and the reaction path37 are desirable for using at low temperatures.

k1(T ) = 1.256 × 1013T 0.157 exp(88.5/T ) cm 3 mol−1 s−1

This rate coefficient possesses slightly positive temperature dependence at T > 300 K that is due to the principal ∼T 1/6 temperature behavior of barrierless reaction rate coefficient (eq 4). The negative temperature dependence at T < 300 K is ascribed to the population of the fine structure levels of Al(2P) atom. However, the existence of saddle point SPc, which separates IMc and products PES regions, can contribute to the effective rate constant of channel c. To estimate the magnitude of this effect, a simple version of two-transition-state model49 was applied. When a steady-state assumption for the concentration of IMc complex was implemented, the effective rate constant for channel c could be expressed as kc(T ) = k1(T ) f (T ) = k1(T )

k 2(T ) k −1(T ) + k 2(T )

(9) 3901


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Table 3. Values of Rate Coefficients k1, k−1, k2, and kc as Well as the Values of Γ(T), r(T), and f(T) Factors at Different Temperatures T, K

k1, cm3 mol−1s−1

k−1, s−1

250 400 700 1000 1300 1600 1900 2200 2500 2800 3100 3400 3700 4000

× × × × × × × × × × × × × ×

× × × × × × × × × × × × × ×

4.25 4.03 4.00 4.06 4.14 4.23 4.31 4.38 4.45 4.51 4.57 4.63 4.68 4.74

13

10 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013

1.16 8.74 9.99 6.66 1.85 3.51 5.43 7.46 9.51 1.15 1.34 1.52 1.69 1.85

k2, s−1 5

10 107 109 1010 1011 1011 1011 1011 1011 1012 1012 1012 1012 1012

2.13 1.48 4.64 2.06 4.73 7.98 1.14 1.48 1.80 2.10 2.37 2.62 2.84 3.04

× × × × × × × × × × × × × ×

7

10 109 1010 1011 1011 1011 1012 1012 1012 1012 1012 1012 1012 1012

Γ

r

f(T)

3.17 1.53 1.15 1.07 1.04 1.03 1.02 1.02 1.01 1.01 1.01 1.01 1.01 1.01

0.73 0.75 0.74 0.72 0.69 0.67 0.65 0.63 0.62 0.60 0.59 0.58 0.57 0.56

0.99 0.94 0.82 0.76 0.72 0.69 0.68 0.67 0.65 0.65 0.64 0.63 0.63 0.62

kc, cm3 mol−1s−1 4.23 3.81 3.29 3.07 2.98 2.94 2.92 2.91 2.91 2.92 2.92 2.93 2.94 2.94

× × × × × × × × × × × × × ×

1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013

temperature range T = 50−1000 K using a supersonic molecular beam technique: k(T) = 4.0 × 1013 T 0.2/(1 + T/ 2100)1.7 cm3 mol−1 s−1. So, it would be helpful to compare the calculated values of k1 and kc with these measurements. Shown in Figure 5 are the measured rate constants for the reactions of Al with O2, DME, and DEE. One can see that the rate constant of barrierless reaction of Al with O2 has a slightly negative temperature dependence. A similar dependence was obtained for the Al + C2H5OH → AlOH + C2H5 reaction channel in the present work. Note that at T = 1000 K the rate coefficient of the Al + O2AlO + O reaction is by a factor of 3 greater than the kc value. The values of rate constants for the reactions of Al with DME and DEE (these organic compounds are structurally similar to ethanol) at T = 296 K are close to the obtained value of kc. This fact suggests that the rate constant predicted for reaction path c in the present work displays the same temperature behavior and the same order of magnitude as the other barrierless reactions with atomic aluminum. Therefore, the estimated rate constant kc(T) can be treated as a reliable one. It should be emphasized that the product of the reaction path c, C2H5, is a highly reactive radical. Thus, this reaction path can be considered an effective channel of chain initiation in the Al−C2H5OH−O2 combustible mixture. The pathways of the reaction AlO(X 2Σ+) + C2H5OH, in contrast to the reaction Al + C2H5OH, have small but nonzero activation barriers (Table 2). Therefore, the rate constants of the processes e and f were estimated with the use of eqs 5−8 and energies of critical points obtained at the UB2PLYP-D/ aug-cc-pv∞Z level of theory. For the estimations, the following electronic weights for the reactants and reactive PES were used: Qe(C2H5OH) = 1, Qe(AlO(X 2Σ+)) = 2,53 Qe(TS) = 2. The resulting temperature-dependent rate constants for channels e and f are depicted in Figure 6. Over the temperature range T = 400−4000 K, these rate coefficients can be approximated in the form

However, the temperature range T < 400 K is not important for the conditions of combustion of the C2H5OH + Al composite fuel. The temperature dependences of k1 and kc are depicted in Figure 5. In the temperature range T = 400−4000 K, the resulting reaction rate coefficient kc can be approximated by single Arrhenius expression: kcB2PLYP(T ) = 1.49 × 1013T 0.075 exp(198.9/T ) cm 3 mol−1 s−1

It is also interesting to estimate the uncertainty in the values of rate constant associated with the inaccuracy in the calculations of activation barriers. As far as the UB2PLYP/ aug-cc-pv∞Z and CBS-QB3 levels of theory predict that the formation of IMc is barrierless process (Table 2), the rate constant k1 does not depend on the accuracy of PES calculations. However, k−1 and k2 rate constants are sensitive to the uncertainty in the predictions of corresponding activation barriers. To estimate this effect, the rate constant kc was also evaluated with the usage of energy barriers obtained by CBS-QB3 method. The resulting version of kc(T) dependence defined by the expression kcCBSQB3(T ) = 4.85 × 1012T 0.203 exp(111.0/T ) cm 3 mol−1 s−1

is also shown in Figure 5. One can see that at low temperatures (T ≤ 1000 K), the uncertainty in the kc value can achieve a factor of 1.8. However, at higher temperatures (T > 1000 K), it is substantially smaller. Though the CBS-QB3 method is considered as a slightly more accurate technique than the B2PLYP one for the predictions of activation barriers for certain types of reactions,27,50 we believe that the choice of a more accurate level of theory for the reaction under study is somewhat problematic in view of the absence of experimental kinetic data. Therefore, both versions of temperature-dependent rate constants are reported here and hereafter. Today, there is not any experimental data on the rate constant for the reaction of Al atom with ethanol molecule in the available literature. However, the results of the measurements of the rate constants for barrierless reactions of atomic aluminum with O2, dimethyl ether (DME), and diethyl ether (DEE) were reported.51,52 So, Parnis et al.51 reported the lowtemperature measurements (at T = 296 K) of rate constant for reactions of Al with DME (1.6 × 1013 cm3 mol−1s−1) and DEE (5.4 × 1013 cm3 mol−1s−1). Later Naulin and Costes52 derived the k(T) temperature dependence for the reaction of aluminum with molecular oxygen from cross section measurements in the

keB2PLYP(T ) = 1.91 × 102T 3.254 exp(− 2261.2/T ) cm 3 mol−1 s−1

k fB2PLYP(T ) = 9.04 × 102T 3.011 exp(− 8.6/T ) cm3 mol−1 s−1

if B2PLYP values of energy were used. One can see that the rate constant of process f is considerably greater (by 2 orders of magnitude at T = 500 K) than that of reaction path e. This is mainly due to smaller activation barrier for channel f. However, with the temperature increase the rate constants of pathways e 3902


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AlO + C2H5OH → AlOH + C2H5O up to the temperature T ∼ 2000 K.

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ASSOCIATED CONTENT

S Supporting Information *

Tables containing the values of electronic energy, enthalpy correction, formation enthalpy of certain species, the geometrical structure and properties of the computed transition states, and Cartesian coordinates. These materials are available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*A. M. Starik. E-mail: [email protected].

Figure 6. Temperature-dependent rate constants for (e) ke and (f) kf reaction pathways calculated with the use of the UB2PLYP-D/aug-ccpv∞Z level of theory (solid curve) and CBS-QB3 method (dashed curve). Arrows show the possible uncertainties in the rate constants caused by using different quantum-chemical methods.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research (grants nos. 14-01-00464 and 14-08-90034). The part concerning the PES calculations and rate constant evaluation in the AlO + C2H5OH system was supported by the grant of Russian Science Foundation (project 14-19-01128). The authors thank Dr. Boris I. Loukhovitski for his help and assistance with software implementation of CVT calculations.

and f become closer to each other. Therefore, both reaction channels resulting in the formation of hydroxyethyl and ethoxyl radicals must be taken into account during kinetic modeling in combustible mixtures comprising aluminum and ethanol. However, the values of these rate constants become rather high (greater than 1013 cm3 mol−1 s−1) only at T > 2000 K. Therefore, one can conclude that reaction pathways e and f will contribute to the chain mechanism development in the Al− C2H5OH−O2 system much less than reaction path c. To estimate the uncertainty arising from errors in the calculated activation barriers, the rate constants ke and kf were also evaluated by assuming the energy barriers obtained by CBS-QB3 method. The resulting ke(T) and kf(T) dependences shown in Figure 6 can be expressed as

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REFERENCES

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keCBSQB3(T ) = 2.41 × 102T 3.226 exp(− 1957.8/T ) cm 3 mol−1 s−1

k fCBSQB3(T ) = 3.32 × 101T 3.392 exp(− 722.1/T ) cm 3 mol−1 s−1

It is seen that the rate constant of the process f, occurring with a lower activation barrier, is more sensitive to the inaccuracy in the calculation of activation energy.

5. CONCLUSIONS Quantum-chemical calculations with the use of a hybrid density functional with perturbative second-order correlation (B2PLYP) were conducted to examine the potential energy surfaces and to study the kinetics in Al(2P) + C2H5OH and AlO(X 2Σ+) + C2H5OH systems. The most likely reaction channels important for the combustion in the Al−C2H5OH− O2(air) system were identified. The critical points on PESs were determined. This allowed us to calculate the values of energy barrier for each reaction path configuration. It was revealed that, in the case of interaction of atomic aluminum with ethanol molecule, the barrierless OH-abstraction pathway leading to formation of AlOH and C2H5 dominates. For the reaction of AlO(X 2Σ+) with an ethanol molecule, the reaction paths leading to the formation of different C2H5O isomers (hydroxyethyl and ethoxyl radicals) occur with comparable rates. The appropriate rate constants for Al(2P) + C2H5OH and AlO(X 2Σ+) + C2H5OH reactions were determined on the basis of capture model and variational transition-state theory. It turned out that the rate constant of the reaction path Al + C2H5OHAlOH + C2H5 was much higher than those for the reaction channels AlO + C2H5OH → AlOH + C2H4OH and 3903


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Theoretical Study of the Reactions of Ethanol with Aluminum and

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