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Open AccessCCS ChemistryRESEARCH ARTICLE1 Dec 2020

Steric Hindrance Effect in High-Temperature Reactions

    CCS Chemistry. 2020, 2, 460–467
    https://doi.org/10.31635/ccschem.020.202000155

    High-temperature reactions widely exist in nature. However, they are difficult to characterize either experimentally or computationally. The minimum energy path (MEP) model routinely used in computational modeling of chemical reactions is not justified to describe high-temperature reactions since high-energy structures are actively involved at high temperatures. In this study, we used methane (CH4) decomposition on Cu(111) surface as an example to compare systematically results obtained from the MEP model with those obtained from an explicit sampling of all relevant structures via ab initio molecular dynamics (AIMD) simulations at different temperatures. Interestingly, we found that, for reactions protected by strong steric hindrance effects, the MEP was still followed effectively even at a temperature close to the Cu melting point. In contrast, without such protection, the flexibility of the surface Cu atoms could lead to a significant reduction of the free-energy barrier at a high temperature. Accordingly, some earlier conclusions made about graphene growth mechanisms based on MEP calculations should be revisited. The physical insights provided by this study could deepen our understanding of high-temperature surface reactions.

    Introduction

    Characterizing the elementary chemical reactions precisely is a prerequisite for the understanding of many complex processes such as heterogeneous catalysis and nanostructure growth. According to the transition-state theory,1 a reaction rate is dependent exponentially on the height of the barrier, defined as the free-energy difference between the transition state and the reactant state (ΔG = Gs‡ − Gs). As a common practice, this free-energy barrier is estimated by identifying the minimum energy path (MEP) on the potential energy surface (PES). The MEP starts from the lowest-energy structure of the reactant state and passes through a saddle point on the PES, which is the lowest-energy structure of the transition state. Usually, the activation energy, which is the energy difference between these two structures, makes a dominant contribution to the free-energy barrier. Subsequently, other temperature-dependent contributions can be estimated with a harmonic-oscillator or ideal-gas approximation.2,3 Such an MEP-based protocol is computationally very effective, and thus, routinely used in reaction studies even when the reaction temperature is high. However, the MEP model is questionable at high temperatures since structures far away from the MEP also become essential.

    An accurate description of high-temperature reactions is of considerable importance in methane (CH4) and hydrogen (H2)-fed graphene chemical vapor deposition (CVD) growth on Cu surfaces,4–6 since the typical growth temperature of 1000 °C approaches the melting point of Cu (1084.62 °C). Due to the complexity of the growth process, most previous theoretical studies on graphene growth continue to use the MEP model to describe elementary chemical reactions.7–13 MEP calculations predict that CH4 decomposition on Cu surface, especially for the last dehydrogenation step (CH →C + H), is thermodynamically and kinetically very difficult.14 Accordingly, CH4 decomposition is considered the rate-determining step of the graphene growth, and CH is identified as the dominant feeding species in the process when the partial pressure of H2 is high.15 These important conclusions become questionable if the MEP-based thermodynamics and kinetics outcomes do not guarantee reliability. Therefore, it is highly desirable to go beyond the MEP model16 and carry out a systematic study of methane decomposition on Cu surfaces.

    Herein, we investigated CH4 decomposition on Cu(111) surface and compared the results from the MEP calculations with those obtained from ab initio molecular dynamics (AIMD) simulations at different temperatures. Since AIMD explicitly samples all relevant configurations, it could be applied to both low- and high-temperature reactions. For CH4 dehydrogenation, the AIMD simulations indicated that gas-phase reactions were involved. Therefore, it could not be well described by an MEP model without a gas-phase contribution. Accordingly, we focused more on the three surface reactions involving CHI (I = 1–3) dehydrogenations. At 300 K, the AIMD free-energy barriers for these three reactions could be well reproduced by adding temperature-dependent corrections to the MEP energy barriers, which confirmed that MEP was a reasonable model at this temperature. Considering that a melting Cu surface occurs at 1300 K, one might expect that MEP would no longer be a good approximation at this temperature. Surprisingly, the MEP-based barriers were still very close to the AIMD results for CH2 and CH3 dehydrogenation reactions. A coordination number (CN) analysis suggested that, in these two cases, the overall surface structures were distinct. However, the local chemical environments around the reaction centers were similar to the low-temperature occurrences due to steric hindrance effects from the H atoms. In contrast, the CH dehydrogenation reaction was not well protected by such a steric hindrance effect, as confirmed by the AIMD simulations, which showed that the C atom could be wrapped by up to six Cu atoms at 1300 K, thereby, giving a chemical environment significantly different from that in MEP structures. As a result, the AIMD free-energy barrier was much lower than that predicted by MEP for CH decomposition. Kinetic Monte Carlo (KMC) simulations indicated that such a difference could change the mechanism of graphene growth, especially for those under high H2 partial pressures. These results provided useful insights into the understanding of high-temperature surface reactions.

    Computational details

    All the first-principles calculations were performed using the Vienna ab initio simulation package (VASP)20 with the Perdew–Burke–Ernzerhof exchange-correlation functional.21 Projector augmented wave method22 was used with a plane-wave energy cutoff of 500 eV. The Cu(111) surface was modeled by a four-layer 4 × 4 supercell with a 15 Å vacuum layer. The Brillouin zone was sampled with a 3 × 3 × 1 k-grid. Transition states were located using the climbing image nudged elastic band (NEB) method.23

    Due to the high-energy barrier, a C–H bond breaking event is very difficult to observe in a brute-force AIMD simulation. Enhanced sampling was thus performed using the metadynamics24 technique by adding history-dependent bias potentials in a predefined collective coordinate space. Potential of mean force (PMF) was obtained using the PLUMED package.25 The Cu coordination number (CN) of atom i was defined as a continuous function of the distance to neighboring Cu atom j in the following equation:

    CN = ∑ j 1 − ( r i j r 0 ) 18 1 − ( r i j r 0 ) 36

    The switching parameter r0 was chosen to be 2.4 and 2.0 Å for carbon (C) and hydrogen (H), respectively. Test calculations indicated that the results obtained in this study were not very sensitive to these parameters.

    Graphene growth on the Cu surface at 1300 K was simulated using the standard rejection-free KMC approach.26,27 A 2000 × 2000 lattice was used to represent the Cu(111) surface. To speed up the KMC simulations, a mean field approximation was applied to H adatoms lying on the Cu surface, since they displayed high concentration and rapid diffusion rate. Only the numbers of H adatoms were recorded to adjust the hydrogenation and dehydrogenation rates accordingly. More details about the KMC simulations are described in our previous studies.15

    Results and Discussion

    Dehydrogenation at 300 K

    MEPs of CH4, CH3, CH2, and CH dehydrogenation on Cu (111) were identified via NEB calculations, which gave activation energies of 1.63, 1.55, 1.03, and 1.88 eV, respectively, agreeing well with previous results.15,17–19 These data suggested that the CH decomposition was the most challenging step in the sequential CH4 dehydrogenation reactions. The AIMD metadynamics simulations were performed at 300 K using a specific C–H distance as the collective coordinate to obtain the PMF ( Supporting Information Figure S1). In AIMD trajectories, the overall structure of the Cu(111) surface was well maintained, and Cu atoms oscillated around their equilibrium positions. As expected, the free-energy barriers obtained for CHi (i = 1–4) dehydrogenation were different from the MEP activation energies (Figure 1a), since no temperature effect was included in the latter case.

    Figure 1

    Figure 1 | Activation energy (black) and free-energy barrier (yellow, green, red, and blue) in CH4, CH3, CH2, and CH dehydrogenation at (a) 300 K and (b) 1300 K. (c) Distribution of the transition-state Cu coordination number of H in CH4 dehydrogenation AIMD trajectories at different temperatures.

    To directly compare the MEP and AIMD results, some corrections needed to be applied on top of the MEP activation energy to obtain an estimation of the free-energy barrier at 300 K. We first discuss the dehydrogenation of CH3, CH2, and CH. Vibrational contributions were added within a harmonic approximation to obtain an MEP-based free-energy barrier estimation (MEP-300). Since nuclear quantum effects were not included in our AIMD simulations, we calculated the vibrational free energy in MEP-300 using a classical harmonic oscillator model, even though the free-energy barriers obtained from classical and quantum harmonic oscillator models were similar, typically within a 0.1 eV difference ( Supporting Information Figure S19). For CH3 and CH2 dehydrogenation, the MEP-300 estimation of the free-energy barrier agreed well with the AIMD results (Figure 1a). However, there was still a small discrepancy in the case of the CH dehydrogenation.

    Typically, an MEP starts from the most stable structure of the reactant, which means an fcc hollow-site adsorption for CH2 and CH dehydrogenations and an hcp hollow-site adsorption for CH3 dehydrogenation. However, in AIMD simulation, more than one adsorption site could be observed due to the relatively low-diffusion barriers.28 Therefore, in addition to the MEP associated with the most stable reactant structure, MEPs starting from other relevant structures should also be considered.29 By simply averaging all relevant MEP barriers ( Supporting Information Figure S15) using the Boltzmann weight factor of their initial-state structures, we obtained an effective free-energy barrier named as MEPs-300. MEPs-300 results for CH3, CH2, and CH dehydrogenations were all consistent with their corresponding AIMD results, and the maximum free-energy barrier difference was smaller than 0.11 eV. Therefore, the MEP model could be utilized to describe surface reactions correctly when the temperature is not very high, and for this reason, it favored its routine use in current kinetics studies.

    In the case of the CH4 dehydrogenation, the reactant molecule was mainly in the gas phase during the AIMD simulation ( Supporting Information Figure S10). Therefore, in principle, it was not possible to compare the AIMD result with the MEP model since gas-phase states were not incorporated into the latter analysis. Nevertheless, for completeness, we still made an MEP-300 free-energy barrier correction for the activation energy of MEP, except that the translational and rotational degrees of freedom of the reactant-state of CH4 were treated with ideal-gas models,2,3 since the diffusion and rotation barriers of CH4 on Cu(111) were extremely small. The resulting MEP-based estimation of the CH4 dehydrogenation barrier was 0.29 eV lower than the AIMD result. Such a discrepancy emerged mainly from the gas-phase dehydrogenation contribution in the AIMD results. As shown in Figure 1c, there is a notable probability that in the transition state, CH4 was still in the gas phase (where the Cu CN of H approaches to zero). By considering that in the gas phase, the dehydrogenation of CH4 had an energy barrier as high as 4.12 eV ( Supporting Information Figure S13), even if the possibility of gas-phase dehydrogenation was as low as 7.6% ( Supporting Information Figure S12), a significant increase of the free-energy barrier could still be achieved. Notice that the possibility of gas-phase dehydrogenation was expected to have been overestimated in our AIMD simulation30 due to an underestimation of the van der Waals interaction in the density functional theory (DFT) adopted here.

    Dehydrogenation at 1300 K

    When we increased the temperature to a value (1300 K) close to what was adopted in graphene growth, the ordered Cu(111) structure at 300 K was destroyed totally, and the structures observed in AIMD trajectories were generally very different from those in MEPs (Figure 2). As would be discussed in more detail later, in the case of the CH dehydrogenation, fluctuation of the Cu CN was significantly larger than other dehydrogenation steps. To speed up the convergence, we used the number of bridging Cu atoms between C and H, as an additional collective coordinate in the metadynamics simulation of the CH dehydrogenation. Then, the resulting two-dimensional free-energy surface was projected back to the C–H distance dimension. Figure 1b shows all the free-energy barriers obtained from AIMD simulations at 1300 K.

    Figure 2

    Figure 2 | A typical reactant-state snapshot of the AIMD trajectory for (a) CH4, (b) CH3, (c) CH2, and (d) CH dehydrogenation at 1300 K.

    Since AIMD structures at 1300 K were very different from the ordered Cu(111)-based MEP structures, we thought it would be interesting to examine marked plausible differences between the AIMD free-energy barrier and the free-energy barrier estimated from the MEP results. Unexpectedly, the MEP-1300 barriers were still in good agreement with the AIMD results for CH3 and CH2 dehydrogenations (Figure 1b), even though the structures were very different. We realized that this could be understood by checking the local chemical environment around the reaction center. Although the Cu surface was highly disordered at 1300 K, the local structure of the adsorbed CH2 or CH3 remained largely unchanged, compared with the case of the ordered Cu(111). For example, as shown in Figure 2c, the CH2 in the snapshot had a structure similar to that of the hollow-site adsorption on a smooth Cu(111) surface. However, the reverse was true regarding CH dehydrogenation. It turned out that CH could be trapped easily in a cavity created by the fluctuation of Cu atoms on the surface, which made its local chemical environment significantly different from that on a flat crystalline surface. We attributed the difference between the CH3/CH2 and CH events to have resulted mainly from steric hindrance effect exhibited by CH3/CH2, but not CH: Although Cu atoms on the surface were quite flexible, due to the existence of multiple H atoms, there was only a very limited space for Cu atoms to approach the C atom in CH3 or CH2. In contrast, CH was least protected by the steric hindrance effect, which made it possible to be trapped readily in the cavity formed by the Cu atoms. Since CH could coordinate with more Cu atoms on the high-temperature melting surface, the dehydrogenation free-energy barrier was significantly lowered, compared with the low-temperature case. It also made the free-energy barriers for the CH dehydrogenation predicted by MEP and AIMD very different at 1300 K, since large surface distortion was not included in the MEP model.

    We investigated the local chemical environment more systematically by performing a Cu coordination number analysis for each AIMD trajectory (Figure 3). Notably, some widely used concepts referring to a specific surface structure such as active site were more difficult to apply to the dynamic surface system. Cu CN depended on both the number of neighboring Cu atoms and the distances to them. For instance, when CHi (i = 1–3) statically adsorbed at a hollow site on Cu(111), the former was fixed at three and the latter plays the determining role. Due to the difference of Cu–C distances, the Cu CN of C varied from 2.31 for CH3 to 2.86 for CH2, and it reached up to 2.96 for CH. Nonetheless, on a disordered surface at 1300 K, the CN of C was determined mainly by the number of neighboring Cu atoms. In fact, its mean value (∼ 1.25 for CH3, 2.41 for CH2, and 3.47 for CH) reflected approximately, the number of dangling bonds in CHi (i = 1–3). Regarding the CH4 molecule, since it was adsorbed weakly on the substrate, the CN was almost zero at all the temperatures studied.

    Figure 3

    Figure 3 | Average Cu coordination numbers of C and H atoms in (a and b) CH4, (c and d) CH3, (e and f) CH2, and (i and j) CH dehydrogenation reactions as a function of the C–H distance at different temperatures. The initial, transition and final states are marked by circles.

    Generally, the Cu CNs of both C and H increased along the reaction pathways with an elongation of the C–H bond, due to a stronger interaction with the surface at a longer C–H distance. How the Cu CN of C changed with temperature depended on the steric hindrance effect. For species with a strong steric hindrance, such as CH3, only limited space was available for Cu atoms to approach C. To utilize this space effectively, ordering of neighboring Cu atoms was essential. Therefore, a flat surface with stable adsorption will yield more effective coordination compared with the melting surface. As a result, Cu CN of C decreased at elevated temperature (Figure 3c). Meanwhile, the steric hindrance in CH2 was relatively weaker, compared with CH3, and thus, the Cu CNs at 300 and 1300 K were essentially the same.

    For the smallest species, CH, the Cu CN of C at 1300 K was significantly larger than that at 300 K or zero temperature. The CN difference is especially large at the transition state, indicating the existence of distinct local chemical environments. Such an environmental difference also reflected in the Cu CN of H, with a much more significant temperature effect with CH decomposition, compared with the CH2 and CH3 dehydrogenation. This local chemical environment effect is the main reason why the CH dehydrogenation free-energy barrier predicted by AIMD at 1300 K was very different from that estimated from MEP. Indeed it could be observed even when the surface remained to be ordered. For example, when we used the MEP model to study CH decomposition on Cu(111), Cu(100), and Cu(410) (with an increasing coordination number), we found that a larger CN gave a lower energy barrier ( Supporting Information Figure S18), consistent with results from a previous study.31

    The gas-phase reaction was no longer negligible for CH4 dehydrogenation at 1300 K.30 However, in AIMD simulations, there was a reduced possibility of the reactant-state CH4 molecule being in the gas phase, compared with that at 300 K ( Supporting Information Figure S12). Notice that a wall potential was applied on top of the surface to prevent the CH4 from moving far away from the surface. Therefore, the geometry fluctuation of surface Cu atoms would increase the chance of CH4 to be contacted with Cu atoms. Again, for CH4 dehydrogenation, a comparison of AIMD results, including gas-phase reactions (Figure 1c), with the MEP results without a gas-phase contribution, is, in principle, impossible. However, the estimation of the CH4 dehydrogenation free-energy barrier (MEPs-1300) obtained using the same protocol as in the 300 K case, surprisingly, agreed with the AIMD result, which was actually an error cancellation effect. In the reactant state, the ideal-gas model is not expected to describe the rotational and translational degrees of freedom of CH4 fairly well on a fluctuating surface, which caused an overstabilized reactant state. On the other hand, since there is no contribution of gas-phase dehydrogenation, in the MEP model, the transition state was also overstabilized.

    Using the free-energy barriers shown in Figure 1, the rate constants of the dehydrogenation reactions could be estimated employing the transition-state theory.1,29,32 As listed in Table 1, the dehydrogenation rates at 1300 K were always higher than those at 300 K. Typically, the rates predicted from AIMD- and MEP-based free-energy barriers were similar except for CH4 dehydrogenation at 300 K and CH dissociation at 1300 K. The former was mainly due to an effect of gas-phase reaction. With our focus on pure surface reactions, the largest discrepancy arose from the 1300 K CH decomposition, where there were four orders of magnitude difference between the MEP and the AIMD results. Since MEP failed to describe the local chemical environment at high temperatures when the steric hindrance effect was weak, the decomposition rate of CH at 1300 K was underestimated considerably by the MEP model.

    Table 1 | Reaction Rate Constant for CHi (i = 1–4) Dehydrogenation at 300 K (I) and 1300 K (II) Obtained from the MEP Model and PMF in AIMD Simulations. All Values are in s−1
    CH4 CH3 CH2 CH
    I MEP 5.31E−18 1.26E−12 2.11E−5 1.19E−15
    PMF 1.27E−21 2.71E−11 6.99E−5 1.16E−14
    II MEP 2.17E6 2.22E7 5.04E8 1.99E6
    PMF 4.81E6 1.37E8 2.46E9 4.96E10

    MEP, minimum energy path; PMF, potential of mean force; AIMD, ab initio molecular dynamics.

    Effects on graphene growth modeling

    Since the MEP model is used routinely in previous theoretical studies on graphene growth, it was important to check the effect of the underestimation of the CH decomposition rate on graphene growth mechanisms. For this purpose, we performed KMC simulations using updated CH dehydrogenation rate. Other reactions involved in graphene growth were expected to be reasonably well described with the MEP model. For example, attachment/detachment of carbon species at/from graphene island edges is expected to be protected by the steric hindrance effect from the graphene island. Nevertheless, the simple protocol adopted here with only the CH dehydrogenation rate updated is not expected to give reliable graphene growth mechanisms. The main purpose of the KMC simulations presented here was to demonstrate how significant the effect of more accurate kinetic parameters could be.

    KMC simulations were performed at 1300 K under different H2 and CH4 partial pressures using MEP kinetic parameters in case A. In case B, the same parameters were used, except that the CH decomposition rate was updated with the AIMD result. As an example, the results obtained under 10 Torr of both H2 and CH4 partial pressures are shown in Figure 4. In both cases, sequential dehydrogenation steps, involving the conversion of CH4 to CH are observed. However, after the formation of CH, the reaction pathways become very different, reflected in the competition of three possible subsequent pathways, as follows: (1) further dehydrogenation, (2) CH combination reaction to form C2H2, and (3) graphene edge CH attachment.15 In case A, further dehydrogenation to form C monomer was suppressed. However, this reaction dominated in case B. Although CH attachment to graphene edge occurred 419 times in case B, there was also 412 times of CH detachment observed in the KMC trajectory. Therefore, the net CH attachment (seven times) was negligible, compared with the CH dehydrogenation. For the CH combination reaction to form C2H2, we even observed that its reverse reaction occurred two more times. The dominance of the CH decomposition pathway could be explained by the much faster CH decomposition in case B, compared with case A.

    Figure 4

    Figure 4 | (a) Kinetic pathways from KMC simulations in case A using kinetic parameters from MEP energy barriers. Both H2 and CH4 pressures are 10 Torr. Diffusion events and H2 adsorption/desorption is not shown. Gas-phase species are denoted with “g” character. Black numbers are occurring times of reactions and reverse reactions indicated by the green arrows, while white numbers give the net occurring times. (b) KMC kinetic pathways in case B with the CH decomposition rate obtained from PMF. KMC, kinetic Monte Carlo; MEP, minimum energy path; PMF, potential of mean force.

    From the growth mechanism point of view, a fundamental issue was to identify the dominant feeding species of graphene growth. In our previous studies, we utilized MEP-based kinetic parameters to perform KMC simulation, which identified C2 or CH as the dominant feeding species of graphene growth under low and high partial pressures of H2.15,33 We revisited this inference since the CH decomposition was underestimated substantially previously. Our current findings showed that a faster CH decomposition led to a decrease in the steady-state CH concentration. As shown in Figure 4b, even when the H2 partial pressure was already high, CH might still not be the dominant feeding species of graphene growth.

    Notice that results presented here do not mean all previous studies on graphene growth are suspicious. The physical insight we have obtained here regarding the effect of steric hindrance can distinguish between questionable and reliable conclusions. For instance, the hydrogen-saturation-induced stabilization of the graphene edge was determined mainly by rates of attachment/detachment reactions at metal-passivated and hydrogen-saturated graphene edges.15 These reactions were protected by a strong steric hindrance from the graphene island. Therefore, such conclusions, although obtained previously based on the MEP model, are deemed to be reliable.

    Conclusion

    We have performed AIMD simulations to study CH4 dissociation on the Cu surface, focusing on how well the MEP model could be used to describe high-temperature reactions. It turned out that it is still a reasonable approximation even when the Cu surface is already in a melting phase, as long as there is powerful steric hindrance protection. For small-species reactions that are not well protected by steric hindrance such as CH dehydrogenation, the free-energy barrier predicted by AIMD simulation could be lower significantly, than the value predicted by MEP-based models. Therefore, some conclusions drawn previously on graphene growth, based on the overestimated CH decomposition barrier, require revisitation. This study provides useful insights into an understanding of high-temperature reactions.

    Supporting Information

    Supporting Information is available.

    Conflict of Interest

    The authors declare no competing financial interest.

    Acknowledgments

    This work was partially supported by NSFC (21825302), MOST (2016YFA0200604), and by USTC-SCC, Tianjin, and Guangzhou Supercomputer Centers.

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