Molecular Dynamics Study of Combustion Reactions in a Supercritical

Article Cite This: J. Phys. Chem. A XXXX, XXX, XXX−XXX

pubs.acs.org/JPCA

Molecular Dynamics Study of Combustion Reactions in a Supercritical Environment. Part 2: Boxed MD Study of CO + OH → CO2 + H Reaction Kinetics Sergey V. Panteleev,†,‡ Artem ̈ E. Masunov,*,†,§,∥,⊥ and Subith S. Vasu# †

NanoScienece Technology Center, University of Central Florida, 12424 Research Parkway, Suite 400, Orlando, Florida 32826, United States ‡ N. I. Lobachevsky State University of Nizhny Novgorod, Gagarin Av. 23, Nizhny Novgorod 603950, Russia § Department of Chemistry, and Department of Physics, University of Central Florida, 4111 Libra Drive, Orlando, Florida 32816, United States ∥ South Ural State University, Lenin pr. 76, Chelyabinsk 454080, Russia ⊥ National Research Nuclear University MEPhI, Kashirskoye shosse 31, Moscow, 115409, Russia # Center for Advanced Turbomachinery and Energy Research (CATER), Mechanical and Aerospace Engineering, University of Central Florida, Orlando, Florida 32816, United States S Supporting Information *

ABSTRACT: Oxy-fuel combustion technology holds a great promise in both increasing the efficiency of the energy conversion and reducing environmental impact. However, effects of the higher pressures and replacement of the nitrogen with carbon dioxide diluent are not well understood at present. The title reaction is one of the most important processes in combustion. Despite numerous studies, the effects of supercritical carbon dioxide environment did not receive much attention in the past. Here we report the results of boxed molecular dynamics simulations of these effects at QM/MM theory level with periodical boundary conditions. The free energy barriers for HOCO intermediate formation and decomposition were tabulated in a wide range of pressures (1−1000 atm) and temperatures (400−1600 K). Pressure dependence of calculated rate constants for these reaction steps and overall reaction were analyzed. We found that the CO2 environment may increase these rate constants up to a factor of 25, at near critical conditions. At higher temperatures, this effect weakens significantly. Numerical values for parameters of extended Arrhenius equation, suitable for combustion kinetic modeling are reported.

■

ics55−57 investigations were published. Recently, we used quantum chemistry methods to predict44 autocatalytic effects in reaction R1, where additional carbon dioxide molecule is involved in reaction intermediates, thus lowering the activation barriers. Despite considerable attention that the R1 reaction received, many aspects of this reaction remain controversial. Numerical parameters, describing this reaction, depend on the method used and vary widely. For instance, NIST Chemical Kinetics Database cites 150 different parameters sets.58 Early on, it was recognized59,60 that R1 involves HOCO radical intermediate, and thus proceeds in two steps (R2 and R3, the first of which is reversible):

INTRODUCTION Fossil and renewable fuel combustion remains predominant energy source in modern technology. One of proposed improvements to increase combustion efficiency is oxy-fuel combustion, where a high-pressure oxygen/carbon dioxide mixture is used instead of air.1,2 Design of turbines for use in oxy-combustion processes requires the accurate modeling of chemical processes at these conditions. One of the most important elementary processes in combustion,3,4 largely responsible for energy release is recombination of hydroxyl radical with carbon monoxide: •

OH + CO → CO2 + H•

(R1)

•

This process was investigated in details both experimentally5−20 and theoretically.20−32 In particular, rovibrational spectroscopy,33,34 shock tube ignition delay time measurements,5,35 laser-induced fluorimetry,5,36 photoelectron spectroscopy,37 NMR,38 and other experimental methods were used. Among theoretical methods, statistical mechanics studies of reaction rates,31,39−41 quantum chemistry predictions of potential energy surfaces,21,23,25−28,42−54 and direct dynam© XXXX American Chemical Society

OH + CO → HOCO•

(R2)

HOCO• → •OH + CO

(R2r)

Received: October 3, 2017 Revised: December 11, 2017

A

DOI: 10.1021/acs.jpca.7b09774 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A HOCO• → CO2 + H•

at high T. It was also reported17,40 to be significant only in 1−100 atm range, and remain flat below and above this interval. A similar sigmoidal shape of k(P) was observed in other studies of multistep reactions.10,41,69,77 More detailed investigation31 found R2 and R3 steps to have an opposite dependence on P and T. At low T and high P, reaction R2 plays the role of the rate limiting step, while at high T and low P step R3 becomes the rate limiting one. This contribution continues the series of studies investigating the effect of CO2 environment on combustion reactions. Earlier we examined how single molecule of carbon dioxide makes changes in reaction potential surfaces,44,90−92 and how force field parameters need to be tuned to describe transcritical conditions quantitatively.93 This work is aimed at computational investigation of CO2 diluent effects on R2 and R3 kinetics in the temperature range of 400−1600K and pressures between 0.3 and 1000 atm. We will analyze the results in terms of extended Arrhenius eq 1 and tabulate the empirical parameters for several pressure values.

(R3)

Experimental evidence of this intermediate were obtained in low temperature matrix studies, 61,62 as well as in the gas phase.12,62−64 It was later characterized by spectroscopic methods.15,33,34,36,64−66 and confirmed by quantum chemical modeling.21,25,54 More detailed investigations of R2 revealed that HOCO is formed in vibrationally excited state HOCO*.14,59,67−69 This hot intermediate may further deactivate in one of three channels:39 stabilization by a third particle M in R4, redissociation back to the reactants in R5, and product formation in R659 HOCO* + M → HOCO + M*

(R4)

HOCO* → CO + OH

(R5)

HOCO* → H + CO2

(R6)

70,55,5,68,71,39

Several studies focused on the vibrationally excited reactants: OH* and CO*. The rate constant for R1 was shown to decrease5,56,68 when CO* is involved but to increase with OH* involvement. Participation of the third particle in R4 results in the rate increase for the overall reaction R1 with increasing pressure.10,11,17,27,40,41,67,69,72−80 The nature of M is also important: while He diluent was found to be less efficient, H2 and SF6 increase reaction rate to a greater extent.72−74 At the same time, O2/Ar mixture has no effect.69 At low pressures, the rate increase by a third particle is relatively weak.75 Even in SF6 case the pressure increase from 1 to 800 Torr only doubles the R1 rate constant.40 According to another study,67 the relative pressure effect is increased from Ar < N2 < SF6, while a third study reports80 the relative effects to decrease in the series M = H2O: SF6: CF4: N2: He as 1.0:0.5:0.3:0.1:0.02. Surprisingly, water vapor was found to produce the greatest effect. This effect was attributed81 to the formation of polar radical/water complexes. Similar complexes were described earlier for the other systems82−86 and were found to play role only at room temperature and below, when concentration of these complexes is significant. However, DFT calculations87 seem to contradict this interpretation. They predict higher activation barriers when hydroxyl is hydrated. To summarize, rate acceleration effect of diluent pressure increases in the sequence He ≈ H2 < Ar < N2 < CF4 < SF6 < H2O. To the best of our knowledge, the effect of the third particle in case of M = CO2 was not studied experimentally, and it will be the focus of this study. Another aspect of our work is temperature dependence of the reaction rate constants. In order to analyze the temperature dependence, the extended Arrhenius equation is often used: k = AT n exp( −Ea /RT )

2. COMPUTATIONAL DETAILS We adopt mixed quantum mechanics/molecular mechanics (QM/MM) theory level and molecular dynamics (MD) approach, as implemented in CHARMM version c40b1.94 We selected a 0.2 fs time step in order to describe the hydrogen motion accurately. Verlet leapfrog integrator and NVT ensemble were combined with the periodical boundary condition and particle mesh Ewald method.95 The simulated system included single reactive intermediate HOCO, described by semiempirical MNDO Hamiltonian.96 The cubic unit cell with the side of 26 Å also included N molecules of CO2 (N = 0, 2, 4, 8, 16, 32, 77), described by classical force field with transferable potentials for phase equilibria (TraPPE),97 in flexible molecule version.98 Molden program99 was used for trajectory visualizations. The starting geometry for the reactive system was prepared by adding CO2 molecules to HOCO radical and optimizing the cluster at PM6 theory level with Gaussian 09 program,100 followed by 50000 steps QM/MM minimization with CHARMM. Next, the system was slowly (10 K/ps) heated up to the target temperature with MD setup described above. Seven target temperatures (400, 600, 800, 1000, 1200, 1400, and 1600 K) were used in simulation, which continued for 2−6 ns. The reaction coordinate was assigned the value of the bond distance CO reaction R2) and OH (reaction R3 less their equilibrium values in the range from 0 to 5 Å. In order to efficiently sample the reactive system at all values of the reaction coordinates, we employed accelerated MD approach, called boxed MD (BXD).101−104 In BXD method, the whole range of reaction coordinate is divided into series of intervals (boxes) 0.1 Å wide. The MD trajectory is propagated as usual, until reaction coordinate exceeds the limit of the given box. At this point the velocities of two atoms involved are inverted in the direction of the reaction coordinate. This returns the system back inside the box. After this reflection event occurred 500 times, the system is allowed to cross the boundary and move to the adjacent box. In order to collect sufficient statistics, we continue MD simulation until 10−30 passes from the first to the last box are made. For a typical example of reaction coordinate values, see Figure S1 in Supporting Information. The values of reaction coordinate at each time step, was produced by AXD module in CHARMM, and converted into potential of mean force (PMF), also known as the free energy profile G(ρ). PMF is a function of reaction

(1)

where A is the preexponential factor, n is an empirical constant, and Ea is the phenomenological activation energy. R = universal gas constant, and T = absolute temperature. The low temperature studies often report non-Arrhenius behavior for R1.5,88,89 Somewhat surprisingly, the reducing the temperature below 300 K was found to increase the rate of R1,25 which was interpreted as tunneling effect. However, the importance of tunneling was estimated below 10% at room temperature.27 Other studies estimate the tunneling to have appreciable contribution only below 200 K.31 The high temperature experiments in the 600−1250 K interval found35 k(T) to be nearly Arrhenus,35 with different parameters at different pressures. This pressure dependence was found27 to be weaker B


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The Journal of Physical Chemistry A Table 1. Rate Constants k (cm3/mol·s), Predicted for Reaction R2 pressure of CO2, atm T, K

0.3

600 800 1000 1200 1400 1600

1.28 × 1011 4.68 × 1011 1.11 × 1012 2.07 × 1012 3.29 × 1012

1.0 1.28 × 1011 4.70 × 1011 1.12 × 1012 2.07 × 1012 3.29 × 1012

3.0 1.29 × 1011 4.73 × 1011 1.12 × 1012 2.08 × 1012 3.31 × 1012

10

30

100

300

1000

1.33 × 1011 4.82 × 1011 1.14 × 1012 2.11 × 1012 3.35 × 1012

3.0 × 1010 1.63 × 1011 5.42 × 1011 1.22 × 1012 2.21 × 1012 3.47 × 1012

8.0 × 1010 3.42 × 1011 9.14 × 1011 1.87 × 1012 3.16 × 1012 4.85 × 1012

1.96 × 1011 8.96 × 1011 2.26 × 1012 4.13 × 1012 6.21 × 1012 8.37 × 1012

4.70 × 1011 1.97 × 1012 4.70 × 1012 8.58 × 1012 1.31 × 1013 1.80 × 1013

Table 2. Rate Constants k (s−1), Predicted for Reaction R2r pressure of CO2, atm. T, K

0.3

1.0

3.0

10

30

100

300

1000

600 800 1000 1200 1400 1600

5.93 × 104 3.42 × 107 2.02 × 109 3.31 × 1010 2.48 × 1011

5.93 × 104 3.43 × 107 2.02 × 109 3.32 × 1010 2.49 × 1011

5.94 × 104 3.44 × 107 2.03 × 109 3.34 × 1010 2.50 × 1011

5.96 × 104 3.47 × 107 2.06 × 109 3.39 × 1010 2.54 × 1011

1.10 6.74 × 104 3.79 × 107 2.18 × 109 3.55 × 1010 2.65 × 1011

1.23 9.56 × 104 5.78 × 107 3.38 × 109 5.32 × 1010 3.81 × 1011

1.30 1.32 × 105 8.85 × 107 5.39 × 109 8.63 × 1010 6.20 × 1011

1.35 1.52 × 105 1.21 × 106 7.77 × 109 1.28 × 1011 9.32 × 1011

Table 3. Best Fit Parameters in the Extended Arrhenius Equation, eq 1, for R2 Reactions in the Temperature Range 600−1600 K fitting parameters in extended Arrhenius equation pressure Plog, atm

3

A, cm /mol·s

n

Ea, cal/mol

0.3 1.0 3.0 10 30 100 300 1000

(5.948 ± 0.118) × 1013 (1.166 ± 0.136) × 1014 (1.036 ± 0.107) × 1014 (5.427 ± 0.140) × 1013 (2.815 ± 0.097) × 1012 (1.001 ± 0.119) × 1013 (9.122 ± 0.077) × 1015 (6.913 ± 0.111) × 1014

0.0435 ± 0.0017 −0.0397 ± 0.0011 −0.0250 ± 0.0010 0.0548 ± 0.0021 0.4046 ± 0.0032 0.3810 ± 0.0029 −0.5894 ± 0.0044 −0.1729 ± 0.0023

(1.022 ± 0.029) × 104 (1.041 ± 0.026) × 104 (1.036 ± 0.030) × 104 (1.013 ± 0.021) × 104 (8.829 ± 0.143) × 103 (8.664 ± 0.140) × 103 (8.406 ± 0.136) × 103 (7.545 ± 0.121) × 103

approximated by eq 1 using the best fit for parameters A, n, and E a.

coordinate ρ; it determines the probability p(ρ) to find the system at a point ρ along reaction pathway: p (ρ ) =

exp( −G(ρ)/kbT )

∫ exp(−G(ρ)/kbT ) dρ

3. RESULTS AND DISCUSSION The force field parameters TraPPE were explicitly optimized97 to accurately reproduce experimentally obtained data on densitypressure isotherms for transcritical CO2 fluid over long simulation times. Indeed, our simulations produced averaged internal pressure values (shown in Table S1) in good agreement with the NIST Chemistry Webbook.105 The PMF obtained as result of BXD simulations are presented on Figure S2a−f (see Supporting Information). The values for the activation free energies and rate constants predicted for each system composition are shown Tables S2 and S3. Since simulations were performed in series of fixed densities, the rate constants presented in Tables S2 and S3 correspond to the varied pressures. These values were used to interpolate rate constants in isobaric series (P = 0.3, 1.0, 3, 10, 30, 100, 300, and 1000 atm) and discussed in the following. 3.1. Reactions R2 and R2r. The rate constant values for reactions R2 and R2r are shown in Tables 1 and 2 respectively. One can see from these tables, that temperature increase accelerates R2r to a larger degree, than the R2. This is expected, as inverse reaction is endothermic with high activation barrier. Indeed, free activation energy for R2r ranges from 11 to 38 kcal/ mol, depending on the temperature (Table S2). Free activation energy of direct reaction R2 ranges from 2 to 8 kcal/mol.

(2)

We divided each box into 0.05 Å intervals and used the BXDanalysis.py script to plot the histogram of probability p(ρ) and convert the resulting distribution into PMF. These PMFs were plotted for all the systems studied at each temperature and used to calculate free energy barriers for reactions R2, R2r, and R3. The Eyring−Polany equation, eq 2, was employed to predict rate constant: k=κ

kbT −ΔG≠ / RT e h

(3)

Here kb is the Boltzmann constant, h is the Plank constant, and κ is the transmission constant (assumed to be unity). Tabulated values for the rate constant were interpolated for seven pressure values, and fitted by the extended Arrhenius equation, eq 1. An in house software program was used for fitting purposes. The quality of the fit was accessed by RMSD value Δ: N

Δ=

∑i = 1 (ki − k 0i)2 N

(4)

Here N is number of data points, k0i is the rate constant obtained from MD simulation using eq 3, and ki is the rate constant, C


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The Journal of Physical Chemistry A Table 4. Best fit parameters in extended Arrhenius eq. 1 for R2r reactions in the temperature range 600 K - 1600 K fitted parameters in extended Arrhenius equation pressure Plog, atm

A, s−1

n

Ea, cal/mol

0.3 1.0 3.0 10 30 100 300 1000

(5.252 ± 0.262) × 1034 (1.326 ± 0.289) × 1035 (1.058 ± 0.196) × 1035 (7.163 ± 0.177) × 1034 (1.549 ± 0.240) × 1036 (1.721 ± 0.279) × 1039 (9.963 ± 0.260) × 1039 (9.668 ± 0.286) × 1040

−4.7652 ± 0.0082 −4.8817 ± 0.0091 −4.8519 ± 0.0085 −4.7983 ± 0.0069 −5.1504 ± 0.0077 −6.0067 ± 0.0095 −6.1461 ± 0.0088 −6.3707 ± 0.0101

(5.898 ± 0.226) × 104 (5.921 ± 0.247) × 104 (5.917 ± 0.231) × 104 (5.911 ± 0.189) × 104 (5.996 ± 0.239) × 104 (6.156 ± 0.263) × 104 (6.262 ± 0.240) × 104 (6.362 ± 0.280) × 104

Figure 1. Temperature dependence of rate constants on linear (a, c), and logarithmic (b, d) scales for reactions R2 (a, b) and R2r (c, d).

It is worth noting that, at 400 K, the MD trajectory did not move to the next box during the simulated time. Apparently, the kinetic energy was insufficient to overcome the steep potential energy slope for breaking the covalent bond. At 600 K, the system moved from one box to the other for N = 16 to N = 77 only. This illustrates thermal activation effect at higher pressures. Apparently, CO2 molecules accelerate the chemical process and supply reactants with excess kinetic energy by means of inelastic collisions. Part of this excess energy is transferred into vibrational excitations of the reactants. There are experimental evidence

supporting the notion that reaction R1 proceeds faster when reactants are vibrationally excited.39,55,70 Our calculations predict that CO2 environment accelerates reaction R2 by a factor of 2− 15, while in section 3.2, we will demonstrate that reaction R3 is accelerated by a factor of 2−2000. The rate constant values, shown in Tables 1 and 2, were used to fit the extended Arrhenius parameters for each pressure value. Results of this fitting are collected in Tables 3 and 4. One can see from these tables that at all pressure values the calculated rate constants demonstrate nearly Arrhenius behavior. One can see D


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The Journal of Physical Chemistry A

Figure 2. Pressure dependence of rate constants for R2 (a) and R2r (b) reactions on the pressure exerted by CO2 molecules at different temperatures.

Figure 3. Pressure dependence in terms of acceleration ratio k(P)/k(0) for reaction R2 (a), and reaction R2r (b).

Table 5. Calculated Values of the Rate Constants for Reaction R3 in the Temperature Range 400−1600 K, Obtained from the Results of Calculations, s−1 pressure of CO2, atm T, K

0.3

1.0

3.0

10

30

100

300

1000

400 600 800 1000 1200 1400 1600

1.32 × 106 3.18 × 109 5.49 × 1010 3.12 × 1011 7.77 × 1011 1.95 × 1012 4.07 × 1012

1.44 × 106 3.36 × 109 5.69 × 1010 3.19 × 1011 7.98 × 1011 1.96 × 1012 4.08 × 1012

1.67 × 106 3.82 × 109 6.17 × 1010 3.34 × 1011 8.45 × 1011 2.01 × 1012 4.11 × 1012

3.50 × 106 5.40 × 109 7.59 × 1010 3.83 × 1011 1.00 × 1012 2.17 × 1012 4.19 × 1012

2.68 × 107 1.89 × 1010 2.28 × 1011 6.97 × 1011 1.52 × 1012 2.66 × 1012 4.44 × 1012

1.45 × 108 5.72 × 1010 4.08 × 1011 1.41 × 1012 3.20 × 1012 4.63 × 1012 5.61 × 1012

8.00 × 109 1.54 × 1011 8.30 × 1011 2.43 × 1012 4.45 × 1012 6.14 × 1012 6.91 × 1012

3.10 × 1010 3.44 × 1011 2.02 × 1012 3.82 × 1012 5.89 × 1012 7.20 × 1012 7.63 × 1012

that the quality of the fit, as described by RMSD value Δ (given by eq 4) remain below 15%. Temperature dependences of rate constants for reactions R2 and R2r are shown in Figure 1. The pressure dependence of the rate constants of R2 and R2r reactions is plotted on Figure 2. One can see that increasing pressure accelerates R2, and this dependence saturates at higher P. Inverse reaction R2r is nearly pressure-independent, while it is more temperature-dependent.

It would be instructive to also presented our results in the form of acceleration ratio AR: AR(P) = k(P)/k(0)

(4a)

where k(0) is rate constant at low pressure, calculated with no CO2 environment, and k(P) is rate constant at particular pressure (see Figure 3). One can see from Figure 3 that the pressure dependence nearly levels off above 200 atm. Similar situation was frequently E


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Figure 4. Temperature dependence for the rate constants in linear (a) and logarithmic (b) scale for reaction R3.

Table 6. Fitted Parameters in Extended Arrhenius Equation for R3 Reaction in the Temperature Range 400−1600 K fitting parameters in extended Arrhenius equation pressure Plog, atm

A, s−1

n

Ea, cal/mol

0.3 1.0 3.0 10 30 100 300 1000

(8.391 ± 0.098) × 1022 (7.275 ± 0.134) × 1022 (1.001 ± 0.124) × 1021 (6.156 ± 0.142) × 1022 (3.315 ± 0.140) × 1024 (3.515 ± 0.113) × 1026 (6.714 ± 0.151) × 1028 (1.594 ± 0.157) × 1030

−2.5024 ± 0.0136 −2.0250 ± 0.0174 −1.6132 ± 0.0166 −2.3962 ± 0.0184 −3.0622 ± 0.0169 −3.4413 ± 0.0142 −4.2102 ± 0.0180 −4.8920 ± 0.0182

(1.701 ± 0.034) × 104 (1.912 ± 0.058) × 104 (1.753 ± 0.051) × 104 (1.649 ± 0.052) × 104 (1.596 ± 0.046) × 104 (1.543 ± 0.039) × 104 (1.458 ± 0.040) × 104 (1.460 ± 0.043) × 104

Figure 5. Pressure dependence for R3 rate constants at different temperatures.

observed in experiments.39,55,70 For instance, the rising pressure results in nonlinear increase of the rate constant in R1.40 The rate constant pressure dependence k(P) increases rather sharply below 200 atm, then platoes at a higher pressure. Increasing the

temperature delays this transition to asymptotic limit. In another study,69 the pressure dependence k(P)T=const was linear at low pressures, then gradually shifted to the flat one. We will discuss the nature of these trends in Section 3.3. F


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The Journal of Physical Chemistry A 3.2. Reaction R3. The reaction R3 represents the second step of overall process R1: decomposition of the intermediate by means of OH bond breaking. The respective activation free energies are collected in Table S2. As one can see, these activation free energies vary from 3.1 to 10.5 kcal/mol, depending on the pressure and temperature. This range of activation free energies is very similar to the reaction R2. The density specific (isopycnic) rate constants are shown in Table S3, while the interpolated rate constants in isobaric series are collected in Table 5, and plotted in Figure 4. As one can see from Figure 4, the reaction constant for R3 is sensitive to both temperature and pressure, similar to R2. It also demonstrates nearly Arrhenius behavior at all the temperatures studied. Best fit for parameters in eq 1 at particular pressures, are reported in Table 6. Pressure dependence for R3 rate constants at different temperatures is shown in Figure 5. Similar to R2, increasing pressure accelerates the R3 rate constant. Compared to the R2 case, the asymptotic platoes on k(P) curves is more pronounced. We also plotted AR, defined in eq 4a on Figure 6. One can see from this figure that at lower T the

one may apply the formal kinetics approach (see Supporting Information for details), and arrive at the following expression: k1 =

k 2k 3 k −2 + k 3

(6)

The rate constants for the reactions R2, R2r, and R3 were predicted as described in sections 3.1 and 3.2 and used in eq 6 to calculate the rate constant for the overall reaction. The results are shown in Table 7 and plotted in Figure 7. Similar to reactions R2, R2r, and R3, the temperature dependence for the overall reaction is nearly Arrhenius (Figure 7b). Best fit for parameters in eq 1 at various pressures, are reported in Table 8. The pressure dependence of the reaction rate is plotted in Figure 8, and acceleration ratio is shown on Figure 9. One can see from the Figure 8, that acceleration of the reaction rate by pressure is more pronounced at lower temperatures. Unfortunately, our simulations for T = 400 were only partially successful. However, we were able to use Figure 9b to extrapolate R1 reaction rates based on the data available. The results of this extrapolation are reported in Table 9. One can see that at low pressures (30 atm. and below) the reaction rate constant is nearly pressure-independent. In the entire pressure range, overall reaction R1 appears to be less pressure dependent, than any of the constituent elementary steps R2, R2r, and R3. One can compare our predictions to experimental observations obtained with other diluents.32,40,67,67,72−74 Similar to what was observed for M = He, Ar, H2, N2, O2, CF4, SF6, H2O, our simulations predict that the concentration of the third particle M will accelerate the overall reaction R1. Following works by Troe et al.,10,41,69,77 this can be rationalized as following. Extending the classical Lindemann mechanism106 to bimolecular reactions, one can introduce the first step of this reaction as collision of the third particle M with one of reactants. Only if the reactant was activated in such a way, it has sufficient kinetic energy to undergo R2 and cross the activation barrier. At low pressure, this activation step is rate limiting, and overall reaction rate law for R1 appears to have third order. Alternatively, it may be described by second order rate law with rate constant linearly proportional to the pressure. At the high pressure limit, deactivation process R4 reduces the concentration of activated intermediates HOCO*, so that the overall reaction R1 becomes second order reaction which pressure-independent rate constant. This rate law change from third to second order is responsible for the k(P) shape, similar to one on Figure 8. Apparently, our simulations capture these effects successfully.

Figure 6. Pressure dependence of the acceleration ratio (AR) for the reaction R3 at different temperatures.

pressure effect on the rate constant is found to be greater than that at higher T. For example, from 0.3 to 300 atm, AR is nearly 600 at 400 K, is 48 at 600 K, and drops to 1.7 at 1600 K. Despite this relatively large acceleration, the absolute values for the rates remain low and become comparable to R2 rates only above 800 K, where the pressure effects are modest. 3.3. Reaction R1. In order to obtain the overall rate k1 for a two-step process, where the first step is reversible

Table 7. Values of the Rate Constants for the Overall Reaction R1 in the Temperature Range 600−1600 K, Obtained from Eq 4a, cm3/mol·s pressure of CO2, atm T, K 600 800 1000 1200 1400 1600

0.3 1.28 × 1011 4.68 × 1011 1.11 × 1012 1.99 × 1012 2.92 × 1012

1.0 1.28 × 1011 4.69 × 1011 1.11 × 1012 1.99 × 1012 2.92 × 1012

3.0 1.29 × 1011 4.72 × 1011 1.12 × 1012 2.00 × 1012 2.94 × 1012

10

30

100

300

1000

1.33 × 1011 4.82 × 1011 1.13 × 1012 2.04 × 1012 3.00 × 1012

3.02 × 1010 1.63 × 1011 5.42 × 1011 1.22 × 1012 2.16 × 1012 3.16 × 1012

8.15 × 1010 3.42 × 1011 9.14 × 1011 1.86 × 1012 3.11 × 1012 4.48 × 1012

1.96 × 1011 8.96 × 1011 2.26 × 1012 4.12 × 1012 6.09 × 1012 7.60 × 1012

4.70 × 1011 1.97 × 1012 4.70 × 1012 8.57 × 1012 1.29 × 1013 1.59 × 1013

G


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Figure 7. Temperature dependence for the rate constants in linear (a) and logarithmic (b) scale for reaction R1.

Table 8. Fitted Parameters in Extended Arrhenius Equation for R1 Reaction in the Temperature Range 400−1600 K fitting parameters in extended Arrhenius equation pressure Plog, atm

3

A, cm /mol·s

n

Ea, cal/mol

0.3 1.0 3.0 10 30 100 300 1000

(4.398 ± 0.085) × 1016 (6.089 ± 0.089) × 1016 (5.348 ± 0.091) × 1016 (2.395 ± 0.079) × 1016 (5.568 ± 0.077) × 1014 (3.951 ± 0.081) × 1014 (2.587 ± 0.086) × 1018 (3.136 ± 0.096) × 1016

−0.7998 ± 0.0131 −0.8405 ± 0.0126 −0.8235 ± 0.0134 −0.7234 ± 0.0130 −0.2699 ± 0.0125 −0.2238 ± 0.0119 −1.3096 ± 0.0145 −0.6752 ± 0.0137

(4.919 ± 0.120) × 104 (4.955 ± 0.104) × 104 (4.939 ± 0.187) × 104 (4.829 ± 0.143) × 104 (4.208 ± 0.161) × 104 (3.760 ± 0.124) × 104 (4.068 ± 0.111) × 104 (3.427 ± 0.186) × 104

Figure 8. Pressure dependence of the rate constants for reaction R1 on total pressure.

3.4. Comparison with Available Experimental Data. In order to estimate reliability of the method used in this work, it seems beneficial to compare our predictions with the published experimental data, where available. Unfortunately, no measurements were available for CO2 diluent. In Table 10 we included R1 reaction rate constants, measured in He, Ar and N2 diluents at

800 K and four pressure ranges in. Our predictions for CO2 diluent and pressures closely matching experimental conditions are also listed in Table 10. They were obtained with the protocol described in section 3.3. As one can see from this table, despite the differences in diluent gas the values of the rate constant k1 are in close H


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Figure 9. Pressure (a) and temperature (b) dependence of the acceleration ratio (AR) for the rates of reaction R3.

Table 9. Acceleration Ratios (AR) for the R1 Reaction Rate, Obtained by Extrapolating the Data from Figure 9b pressure, atm T, K

1

3

10

30

100

300

1000

800 700 600 500 400

1.00 1.00 1.00 1.01 1.02

1.01 1.01 1.01 1.02 1.04

1.04 1.05 1.07 1.11 1.19

1.27 1.36 1.52 1.84 2.12

2.67 3.12 3.81 4.73 5.92

7.01 7.69 9.58 12.96 17.54

15.39 21.84 30.36 45.17 73.12

Table 10. R1 Reaction Rate Constants for 800 K and 0.3,1,3 10, 100, and 300 atma rate constant, 1011 cm3/mol·s pressure, atm.

0.3

1.0

3.0

10.0

100

300

experiment

0.8b

1.42d

5.12g

1.28

1.41f 1.81g 1.33

3.67g

prediction

1.13c 1.72e 1.28

3.42

8.96

1.29

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1 Both pressure and temperature accelerate all the processes considered. However, the pressure dependence of k2 and k3 is much more pronounced at low temperatures, while the k−2 pressure dependence is weak in the whole temperature interval. 2 Extended Arrhenius equation describes rate constants R2, R2r, and R3 with 15% RSMD or better. Fitted parameter values were reported. 3 The pressure dependence becomes weaker at high pressure. For k3 it saturates already at 100 atm, while k2 and k−2 apparently saturate above 1000 atm. 4 Comparison with available experimental data shows that CO2 molecules are among the most efficient to accelerate k1 with pressure, approaching that of SF6.

ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.7b09774. Calculated pressures, potentials of mean force, free energies of activation, rate constants, and eq 4a derivation (PDF)

a

Predicted values are calculated from BXD molecular dynamics simulation results and compared with experimental values under the same conditions. bAr diluent, pressure range: 0.19−0.82 atm.101−104 c Ar diluent, pressure range: 1.2−1.6 bar.101−104 dN2 diluent.107 eAr diluent.101−104 fAr diluent, pressure range: 9.6−9.8 bar.101−104 gHe diluent.101−104

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AUTHOR INFORMATION

Corresponding Author

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

agreement with experiment, except the lowest pressure of 0.3 atm. We did not expect our predictions to be accurate for the low pressure conditions, taking in account the limitations of our computational model. The simulation system for this pressure included only reactive system in a periodic box with no surrounding CO2 molecules. This indicates that computational protocol adopted in this work, based on Boxed MD at QM/MM theory level is appropriate to describe the reactive system studied in this work.

ORCID

Sergey V. Panteleev: 0000-0001-8780-5642 Artëm E. Masunov: 0000-0003-4924-3380 Subith S. Vasu: 0000-0002-4164-3163 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are grateful to Dr. Glowacki for his assistance with AXD program module. This work was supported in part by the Department of Energy (Grant Number: DE-FE0025260). The authors also acknowledge the National Energy Research Scientific Computing Center (NERSC), and the University of Central Florida Advanced Research Computing Center (https://arcc.ist.ucf.edu) for providing computational resources

4. CONCLUSIONS We presented results of QM/MM boxed molecular dynamics predictions of the rate constants k2, k−2, and k3 for reactions R2, R2r, and R3 in supercritical carbon dioxide environment. The expression for the overall R1 reaction rate is used to predict k1(T,P). On the basis of the pressure and temperature analysis, the following conclusions were made: I


Article


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and support. A.E.M. acknowledges support by Act 211, Government of the Russian Federation (Contract No. 02.A03.21.0011). Disclaimer: This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

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