Conversion of methane to benzene in CVI by density functional theory study

A density functional theory (DFT) study was employed to explore the mechanism of the conversion of methane to benzene in chemical vapor infiltration (CVI) based on the concluded reaction pathways from C1-species to C6-species. The geometry optimization and vibrational frequency analysis of all the chemical species and transition states (TS) were performed with B3LYP along with a basis set of 6–311 +G(d, p), and Gaussian 09 software was used to perform the study. The rate constants were calculated by KiSThelP according to the conventional transition state theory (TST), and the Wigner method was applied to acquire the tunneling correction factors. Then the rate constants were fitted to the modified Arrhenius expression in the temperature range of 800–2000 K. As for the barrierless reactions calculated in this paper, the rate constants were selected from the relating references. Through the energetic and kinetic calculations, the most favorable reaction pathway for benzene formation from methane was determined, which were mainly made of the unimolecular dissociation. The conversion trend from C1-species to C4-species is mainly guided by a strong tendency to dehydrogenation and the pathways from C4-species to C6-species are all presumed to be able to produce C6H6 molecule.

simulated the reaction profile of CH 3 radicals with DFT and displayed a simplified reaction pathway from CH 3 to C 6 H 6 . Huang et al. 18 presented a DFT study on the sequential process of methane decomposition on iron oxides, which could provide a detailed explanation of the reaction mechanism at the atomistic for the design of more efficient oxygen carriers. Besides, in order to understand the activity of heterogeneous catalysis affected by nanostructure, Kozlov et al. 19 calculated the complete methane dehydrogenation on nanostructured palladium by DFT. Despite the plenty of theoretical studies of the decomposition of methane published already, there are few DFT researches concerning the detailed conversion pathways of methane to benzene in CVI, and it is of great necessity to figure out the mechanism from the molecular level.
During the decomposition of methane in CVI, benzene is a basic element of pyrolytic aromatic hydrocarbons which can further form the pyrocarbon, thus the present work is performed by limiting the reaction route only to the first cyclic ring formation. In this work, a DFT study was employed to explore the mechanism of the conversion of methane to benzene in CVI based on the concluded reaction pathways from C 1 -species to C 6 -species. The energy profiles of each reaction were calculated and the corresponding rate constants were obtained subsequently, then the most favorable reaction pathway was proposed.

computational Details
All the chemical species involved in the study were optimized with different spin multiplicities followed by vibrational frequency calculations to obtain the stable molecular structures that provided the lowest energy and correct geometry. In order to verify the structure of the transition states (TS) of all the chemical reactions, the frequency analysis was conducted after the geometry optimization, which turned out that each stable TS was equipped with only one imaginary frequency. Besides, the intrinsic reaction coordinate (IRC) 20 was performed to prove that each TS was connected to the desired reactant and product. To get the IRC results with high quality and complete curve, the max points of each IRC calculation were set to be 150 coupled with the default step size of 10. The relative energies of the species and TS were applied to figure out the corresponding reaction energies and activation energies which were used for the energy profiles. All the quantum calculations were performed with Becke three-parameter exchange and Lee, Yang and Parr correlation function (B3LYP) 21,22 along with a basis set of 6-311 +G(d, p), and Gaussian 09 23 software was used to study all the DFT calculations.
The rate constants as a function of temperature for the reactions involved in the study were calculated by using the kinetic and statistical thermodynamical package (KiSThelP) 24 according to the conventional transition state theory (TST) 25 . The parameters required by TST such as molecular energy, spin multiplicity, and vibrational frequencies were obtained from the Gaussian output files. As for the quantum tunneling effect, the Wigner method 26,27 was applied to acquire the tunneling correction factors for all the elementary reactions. Besides, the total reaction degeneracies of each case were taken into account by multiplying a symmetry number of each reaction by the number of equivalent abstractable hydrogen atoms 28 . Finally, the rate constants were fitted to the modified Arrhenius expression 26,28 in the temperature range of 800-2000 K. The formula is expressed as: where A is the Arrhenius prefactor, T is the temperature, E a is the barrier height, and n is the temperature exponent indicating the deviation from the standard Arrhenius equation. In addition, due to the fact that KiSThelP is only programmed for the reactions with TS, and for the barrierless reactions calculated in this paper, the rate constants were selected from the relative references published before.

Results and Discussion
The primary goal of this section is to obtain the main reaction pathways of the decomposition of methane during CVI by distinguishing the pathways collected from the previous researches 11,12,[29][30][31][32][33][34][35][36][37][38][39][40] . By studying and summarizing the related researches published previously, detailed reaction schemes for benzene to be prepared using methane as precursor are summarized, as shown in Fig. 1. Since there are some species produced during the decomposition of methane that make little impact on the mechanism of the formation of pyrolytic carbon, it is generally accepted that only the important molecules and radicals which determine the mechanism of the pyrolysis are supposed to be focused on. As a matter of fact, it is widely agreed that the decomposition and formation of hydrocarbon are fundamentally a free-radical mechanism 11 , so most of the concluded reactions are radical ones. As depicted in Fig. 1, from inside to outside, it shows the conversion pathway from methane to C 2 -species, C 4 -species and C 6 -species, respectively. The molecular structures of all the reactants and products studied in this work are displayed in Fig. 2 ranging from hydrogen molecule to benzene. As can be seen from this figure, apart from the common species with only one specific structure, there are some isomers in this paper such as C 6 H 6 that has four different chemical structures. The reaction pathways proposed in the following are all involved with the dominant species mentioned above.
Reaction pathways from c 1 -species to c 2 -species. For the discussion on the mechanism of reaction pathways from C 1 -species to C 2 -species, the reaction profiles containing every reaction pathway and their potential energies are displayed in Fig. 3, where the energies of each species and TS have been plotted relative to the total energy of the reactants that are all set to be 0 kJ/mol 41,42 . The reactions contributing to the same product have been divided into the same column, which are arranged from left to right in substantially the order of the formation of hydrocarbon shown in Fig. 1.
The initial step of the whole reaction system is the dehydrogenation of CH 4 molecule, which decomposes directly into CH 3 radical and H atom with the endoergic energy of 463.7 kJ/mol, and this is consistent with the www.nature.com/scientificreports www.nature.com/scientificreports/ results calculated before 17,43,44 , which proves the accuracy of this calculation method. With H atom generated, it can combine with CH 4 molecule to produce CH 3 radical and H 2 molecule through TS1 with the activation energy of 40.1 kJ/mol. At the second column, two methyl can bond together to produce an ethane molecule, which is a barrierless process with the exothermic energy of 383.9 kJ/mol. The methyl can also react with methane to yield  ethane and hydrogen atom by overcoming the energy barrier of 212.1 kJ/mol, and the total energy of the products is 79.7 kJ/mol higher than that of the reactants. At the column of C 2 H 5 , C 2 H 6 molecule can be abstracted by an external H atom to form C 2 H 5 radical and H 2 molecule by conquering the energy barrier of 27.8 kJ/mol corresponding to TS3, and the energy of products lies 15.8 kJ/mol below the reactants. C 2 H 6 molecule can also undergo H-elimination reaction forming C 2 H 5 radical and H atom with the barrierless endothermic energy of 444.7 kJ/mol. As for the formation of C 2 H 4 molecule, it can come from both C 2 H 6 molecule and C 2 H 5 radical. After overcoming the high energy barrier of 541.2 kJ/mol, C 2 H 4 molecule and H 2 molecule can be produced from C 2 H 6 molecule with the reaction energy being 161.4 kJ/mol, indicating an endothermic reaction. C 2 H 5 radical can decompose into C 2 H 4 molecule and H atom by the dehydrogenation reaction, of which the activation energy and reaction energy are 178.5 kJ/mol and 176.8 kJ/mol, respectively. This column also involves the abstraction of H atom from C 2 H 5 radical by an external H atom to form C 2 H 4 molecule and H 2 molecule with the barrierless reaction energy lying 283.3 kJ/mol below the reactants. The fifth column shows two endothermic reactions to the formation of C 2 H 3 radical. By losing one H atom of C 2 H 4 molecule, C 2 H 3 radical can be produced with the high barrierless reaction energy being 484.1 kJ/mol. And with an extraneous H atom attacking C 2 H 4 molecule, the corresponding products are C 2 H 3 radical and H 2 molecule with the energy barrier and reaction energy being 48.4 kJ/mol and 23.7 kJ/mol, respectively. Finally, the rightmost column of Fig. 3 shows the energy process of the formation of C 2 H 2 molecule. The most complicated pathway is the formation of C 2 H 2 molecule and H 2 molecule from the dehydrogenation of C 2 H 4 molecule, which gets the reaction energy of 208.7 kJ/mol after overcoming the energy barriers of 326.3 kJ/mol and 472 kJ/mol corresponding to TS8 and TS9, respectively. Another endothermic reaction is the H atom elimination of C 2 H 3 radical, which overcomes the energy barrier of 191.5 kJ/ mol and results in the energy of products being 184.2 kJ/mol. The last reaction is exothermic and barrierless, C 2 H 3 + H → C 2 H 2 + H 2 , and the energy of products is 275.9 kJ/mol below than that of reactants.
Reaction pathways from c 2 -species to c 4 -species. The aim of this section is to deal with the different reaction pathways from C 2 -species to C 4 -species, and the corresponding potential energies of the possible and reasonable pathways are shown in Fig. 4. As is discussed in section 3.1, the same means of expression is applied to Fig. 4, which displays the formation from C 4 H 7 radical to C 4 H 2 molecule.  www.nature.com/scientificreports www.nature.com/scientificreports/ As is depicted at the first column of Fig. 4, there is only one feasible pathway leading to C 4 H 7 radical, which comes from the combination of C 2 H 3 radical and C 2 H 4 molecule, and the reaction energy of this exothermic process is 140.3 kJ/mol with the energy barrier of 13.6 kJ/mol. At the second column of C 4 H 6 molecule, three distinct reactions are calculated. The bonding of C 2 H 2 molecule and C 2 H 4 molecule results to the formation of C 4 H 6 molecule with releasing 176.9 kJ/mol, after undergoing TS11 with the energy barrier of 296.3 kJ/mol. C 4 H 6 molecule can also be formed by the dehydrogenation of C 4 H 7 radical, which requires the activation energy of 154.3 kJ/mol, and the energy of products is 143.3 kJ/mol higher than that of reactant. Another reaction leading to the appearance of C 4 H 6 molecule is the radical association of two C 2 H 3 radical, which is observed to be a barrierless process with the exothermic energy being 480.1 kJ/mol. With C 4 H 6 molecule brought into being, the C 4 H 5 radical can be generated consequently. As displayed at the third column of Fig. 4, on the one hand, by breaking one C-H bond of C 4 H 6 molecule, the C 4 H 5 radical is raised with the energy of the products being 488.6 kJ/mol. On the other hand, by attacking C 4 H 6 molecule with an external H atom, the C 4 H 5 radical can be formed as well with another product of H 2 molecule. This step is slightly endoergic (28.2 kJ/mol) and presents the energy barrier of 51.9 kJ/mol corresponding to TS13. Besides, C 4 H 5 radical can also be brought out by the chemical integration of C 2 H 3 radical and C 2 H 2 radical, which loses 172.6 kJ/mol after conquering the energy barrier of 24 kJ/mol corresponding to TS14. At the fourth column demonstrated in Fig. 4, the formation of C 4 H 4 molecule is analyzed by two reactions. Like the H-abstraction of C 4 H 7 radical, C 4 H 5 radical can decompose into C 4 H 4 molecule and H atom by dehydrogenation, which is calculated to be endoergic (176 kJ/mol) and needs to overcome the energy barrier of 189.3 kJ/mol. Another reaction is the combination of two acetylene molecules, which goes through one intermediate of C 4 H 4 b radical and two transition states of TS16 and TS17. This complicated process needs to overcome a small energy barrier of TS16 to form C 4 H 4 b , the energy of which is 228.7 kJ/mol lower than that of the reactants, and then C 4 H 4 b radical is transformed into the more stable structure of C 4 H 4 a molecule after overcoming the barrier energy of TS17. As can be observed from the structures of C 4 H 4 a molecule, C 4 H 4 b radical and TS17, which are illustrated in Figs. 2 and 4, C 4 H 4 a molecule is transformed from C 4 H 4 b radical by moving one H atom to the end of the carbon chain. At the fifth column of Fig. 4, the interaction of two acetylene molecules contributes to the presence of C 4 H 3 radical and H atom, which is calculated to be barrierless and endothermic. Another endothermic reaction is the interaction of C 4 H 4 a molecule and H forming C 4 H 3 radical and H 2 molecule with the energy of 22.6 kJ/mol, which undergoes TS18 of 56.4 kJ/mol. The last part of this diagram is the formation of C 4 H 2 molecule, which is produced by abstracting the H atom of C 4 H 3 radical, as can be seen from the structure of TS19. The energies of TS19 and the products are 178.5 kJ/mol and 157.2 kJ/mol, respectively.
Reaction pathways from c 4 -species to c 6 -species. This part discusses the reaction pathways leading to benzene, which is originated from C 4 H 6 molecule, C 4 H 5 radical, C 4 H 4 molecule, and C 4 H 3 radical, respectively. All the reactions analyzed in the following are multistep which are more complicated than the reactions from C 1 -species to C 4 -species, and they are divided into five columns for the convenience of discussion, as shown in Fig. 5. Unlike the other two figures discussed above, due to the fact that all these series of reactions result in the same product, benzene, Fig. 5 puts the reactions with the same main reactants into one column, such as the second column of C 4 H 5 → C 6 H 6 .
As depicted in the left column of Fig. 5, there is just one reasonable pathway resulting in benzene from C 4 H 6 molecule, and four transition structures can be found. This process is initiated by the chemical combination of C 4 H 6 molecule and C 2 H 3 radical that brings C 6 H 9 a radical into existence after overcoming the small energy barrier of 20.5 kJ/mol. Then C 6 H 9 a radical undergoes a ring closure reaction after overcoming the activation energy of TS21 to form C 6 H 9 b molecule, which lies 203.2 kJ/mol below the reactants. And for the reaction to proceed further, benzene can arise from the dehydrogenation of C 6 H 9 b molecule followed by subsequent decomposition of C 6 H 8 c molecule into C 6 H 6 molecule and H 2 molecule, which goes through two TS configurations of TS22 and TS23. At the second column, the interactions between C 4 H 5 radical and C 2 H 2 molecule or C 2 H 3 radical are www.nature.com/scientificreports www.nature.com/scientificreports/ calculated separately. The addition of C 2 H 2 molecule takes place on C 4 H 5 radical to form C 6 H 7 a radical, which turns out to be a barrierless and exothermic process with the energy below than that of the reactants by 148.7 kJ/ mol. Then C 6 H 7 a radical undergoes a ring closure step after overcoming a small energy barrier of TS24 to produce a more stable structure of C 6 H 7 b radical, which lies 388.5 kJ/mol below the group of C 4 H 5 radical and C 2 H 2 molecule. After that, the C-H bond scission happens to C 6 H 7 b radical to generate benzene molecule and H atom, and TS25 is observed during this endothermic step with the energy of the final products being 273.3 kJ/mol lower when compared to the initial reactants. As for the addition of C 2 H 3 radical to C 4 H 5 radical, the whole process is very similar to the addition from C 2 H 2 radical to C 4 H 5 radical. At first, C 6 H 8 a molecule is produced with its energy lower than the reactants by 437.3 kJ/mol after the chemical integration of C 2 H 3 radical and C 4 H 5 radical, which is proved to be barrierless and exothermic. And C 6 H 8 a molecule is then transformed into C 6 H 8 b molecule by the ring closure step, which needs to conquer the energy barrier of TS26 and release energy to achieve the stable structure of C 6 H 8 b molecule with its energy lying 521.7 kJ/mol below the reactants. Further, in the third step of this route, two neighboring hydrogen atoms of C 6 H 8 b molecule ruptured from C−H bonds tend to form a dihydrogen molecule leaving a stable benzene molecule, which needs to overcome the high activation energy of TS27.
As displayed in the middle column of Fig. 5, the coupling of C 4 H 4 molecule and C 2 H 2 molecule resulting in benzene is calculated, which undergoes three transition structures TS28, TS29 and TS30, respectively. The first step of this route is exothermic (−102.7 kJ/mol) and presents the activation energy of 136.3 kJ/mol, which brings the C 6 H 6 b species out. Then there are two steps of H atom migration on C 6 H 6 b species leading to benzene molecule eventually after overcoming the energy barriers of TS29 and TS30, respectively. And the energy of the benzene molecule is 446.5 kJ/mol lower than that of the beginning reactants showing the exothermicity of this process. As for the fourth column of Fig. 5, at first glance, the chemical mechanism for this route seems as complicated as that described for the conversion from C 4 H 4 molecule to benzene molecule. It begins with the addition of C 2 H 3 radical to C 4 H 3 radical forming C 6 H 6 a species by releasing 476.6 kJ/mol without overcoming the energy barrier. Then C 6 H 6 a species is converted to C 6 H 6 b species by conquering the activation energy of TS31. From the next step on, the following process is exactly the same as the conversion from C 6 H 6 b species to benzene molecule described in the third column except for the absolute values marked in Fig. 5. Finally, in the last column of Fig. 5, it demonstrates the conversion from C 4 H 3 radical to the benzene molecule. Actually, this process is composed of two separate reactions, which is connected by the C 6 H 5 b radical. At first, the chemical bonding of species C 4 H 3 radical and C 2 H 2 molecule takes place resulting in the presence of C 6 H 5 a radical, which releases 177.2 kJ/mol without any activation energy. Then C 6 H 5 a radical is converted to a phenyl radical after overcoming the slight energy barrier of TS32. Due to the instability of phenyl radical, it is very possible for it to bond with an external H atom to form benzene molecule, which is calculated to be barrierless and releases the energy of 490.3 kJ/mol. Because the external H atom is added during this process rather than in the beginning, it makes sense to draw the energy profiles of these two reactions separately to illustrate this process more clearly and correctly.

Reaction kinetics.
To figure out the reactions among those discussed above that play important roles during the conversion of methane to benzene, reaction kinetics of the elementary reactions were calculated and used for kinetic simulations, as illustrated below. Table 1 provides the kinetic parameters for elementary reactions with TS among the reactions of C 1 -C 6 in the temperature range of 800-2000 K which covers the possible formation temperature of the species studied in this work, and for the benefit of the analysis of the mechanisms of these reactions the rate constants of the corresponding reactions in the temperature of 1353 K are listed individually because this temperature is very special during the decomposition of methane by using CVI method 9 . As is shown in Table 1, the kinetic parameters for the 32 reactions differ greatly with each other with the difference of the class and progress of the reactions. For example, the lowest reaction rate in 1353 K is 4.30 × 10 −22 cm −3 mol −1 s −1 of C 2 H 2 + C 2 H 4 → C 4 H 6 , and the highest reaction rate in 1353 K is 1.52 × 10 12 cm −3 mol −1 s −1 of C 6 H 6 c → C 6 H 6 d , which shows a difference of 34 orders of magnitude between these two reaction rates. Table 2 displays the reaction rates for 13 elementary reactions without TS among the reactions of C 1 -C 6 . Because the KiSThelP is unable to calculate the rate constants of the barrierless reactions, the corresponding rate values are taken from the literatures by searching the NIST chemical kinetics database 45 . To the best of our knowledge, there are no relevant theoretical or experimental results about the rate constants for the reaction C 4 H 3 + C 2 H 3 → C 6 H 6 a , and as a result, this specific reaction is disappeared in Table 2. As can be seen, almost all the rate constants are selected at the temperature of 1400 K which is close to 1353 K and can be valuable as well, except for the reaction C 2 H 3 + C 2 H 3 → C 4 H 6 whose rate constant is elected at 298 K because of the lack of its kinetic study at high temperatures. Like Table 1, the rate constants of these barrierless reactions dispread in a wide range. The smallest value is 5.00 × 10 −21 cm −3 mol −1 s −1 of reaction CH 4 → CH 3 + H, and the largest value is 13.70 cm −3 mol −1 s −1 of reaction C 2 H 6 → C 2 H 5 + H, which indicates that all the rate constants of the barrierless reactions are commonly low.
To better understand the variation tendency of the rate constants of the reactions listed in Table 1, Fig. 6 presents a comparison of those reactions, which are divided into four parts due to the different reaction classes, dehydrogenation, H-abstraction by H atom, isomerization and combination. As demonstrated in Fig. 6, through the detailed comparisons of the rate constants within the same reaction class, it is concluded that the ranges of the rate constants for dehydrogenation reactions and H-abstraction reactions are both relatively small in the temperature range of 800-2000 K when compared with the isomerization reactions and combination reactions whose rate constants vary greatly with the increase of temperature. The reason may be due to that the rate constants for similar reactions are mainly dependent on the molecular structure. In general, the rate constants of all those reactions increase with the increase of temperature, except for the reaction C 2 H 2 + C 2 H 3 → C 4 H 5 whose rate constants decrease gently with the increase of temperature, illustrated in Fig. 6d. Further, there are a few reactions showing high rate constants all the time, such as the reaction C 6 H 6 c → C 6 H 6 d in Fig. 6c, which is attributed to the fact that www.nature.com/scientificreports www.nature.com/scientificreports/ the less stable radical has changed into the reasonably stabilized molecule with low energy. In addition, there are another 4 reactions that are not included in Fig. 6 Obviously, the dissociation of these 4 reactions is quite different from the previous four reaction classes, and the difference of the orders of magnitude among these four reaction rates are too large to be plotted in one figure. However, the rate constants of these reactions still increase with the increase of temperature.

General Discussion
It is widely accepted that all the reactions proposed above are possible pathways for the overall process of the conversion of methane to benzene in CVI. Furthermore, based on the above analysis, the most preferred pathway can be identified by comparing the potential energies and rate constants of the reactions resulting in the same products, concluded in Table 3. On the one hand, the exothermic reactions are more preferred than the endothermic reactions. For example, for the production of C 2 H 6 molecule in Fig. 3, the reaction CH 3 + CH 3 → C 2 H 6 has an advantage over the reaction CH 3 + CH 4 → C 2 H 6 + H because of its exothermicity. On the other hand, the reactions with higher rate constants are more favorable. For instance, between the two reactions forming C 2 H 5 radical in Fig. 3, the reaction C 2 H 6 → C 2 H 5 + H is preferential rather than the reaction C 2 H 6 + H → C 2 H 5 + H 2 because the rate constant of the former reaction is 11 orders of magnitude higher than that of the latter one. And by adopting the two rules into all the reactions, the concluded main pathway is drawn in Fig. 7. Meanwhile, the high energy inputs required for some of the reactions are provided by the high-temperature environment. As for the pathways from C 4 -species to C 6 -species, each reaction route is composed of at least three steps and exhibits similar potential energies and rate constants, which makes it quite difficult to distinguish the most favorable route among these reactions. Consequently, it is inferred that C 6 H 6 molecule can be possibly produced through all these routes shown in Fig. 5.
As depicted in Fig. 7, the most fundamental and initial step of this route is the dehydrogenation of CH 4 molecule which is the original material of the process, even though this step takes a lot of energy to proceed and has a low rate constant. Besides, the dehydrogenation effect plays a significant role in the whole process. The C 2 H 4 molecule, C 2 H 3 radical, and C 2 H 2 molecule produced during the dehydrogenation can help the reactions of C 4 -species to perform further, and C 2 H 3 radical and C 2 H 2 molecule can also take part in the production of C 6 H 6  www.nature.com/scientificreports www.nature.com/scientificreports/ molecule. The multistep reactions forming C 6 H 6 molecule are mainly attributed to the participation of C 4 H 3 radical, C 4 H 4 molecule, C 4 H 5 radical, and C 4 H 6 molecule. From Fig. 7, it can be seen that the main reaction routes are made of the unimolecular dissociation.

conclusions
Based on the concluded pathways from methane to benzene, the DFT study was performed to explain the reaction mechanisms and corresponding kinetics, which presented the calculation results of the potential energy and Arrhenius parameters involved in the process. The reaction profiles and rate constants of each pathway from C 1 -species to C 6 -species were obtained and the most favorable route was proposed. The conversion trend from C 1 -species to C 4 -species is mainly guided by a strong tendency to dehydrogenation, which is found to play an important role in this route. And the main reaction routes are made of the unimolecular dissociation. In addition, the pathways from C 4 -species to C 6 -species are all presumed to be able to produce C 6 H 6 molecule.  Table 3. Comparisons of the potential energies and rate constants of the reactions.