Enhanced Ab Initio Molecular Dynamics Simulation of the

Article pubs.acs.org/JPCC

Enhanced Ab Initio Molecular Dynamics Simulation of the Temperature-Dependent Thermodynamics for the Diffusion of Carbon Monoxide on Ru(0001) Surface Zhe-Ning Chen,*,† Lin Shen,† Mingjun Yang,† Gang Fu,‡ and Hao Hu*,†,§ †

Department of Chemistry, The University of Hong Kong, Pokfulam Road, Hong Kong S. A. R., China State Key Laboratory of Physical Chemistry of Solid Surfaces, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen 361005, China § The University of Hong Kong-Shenzhen Institute of Research and Innovation, Kejizhong 2nd Road, Shenzhen, China ‡

ABSTRACT: CO diffusion on the metal surface is an elementary process in many heterogeneous catalytic reactions. The thermodynamics and the molecular mechanism of the diffusion process are key factors contributing to the kinetics of the catalysis. Theoretical study based on computer simulations can complement experimental studies to provide much needed thermodynamic and mechanistic information. Here, we report direct ab initio molecular dynamics (MD) simulation to investigate the temperature-dependent thermodynamics of CO diffusion on the Ru(0001) surface combined with an efficient sampling method based on integrated tempering to speed up phase space sampling. We show that reliable and smooth twodimensional potential of mean force surfaces of CO diffusion can be obtained at different temperatures. As expected, with increasing temperature, the distribution of CO adsorbate at different surface sites becomes more and more uniform, while the height of the free energy barrier to CO diffusion decreases. The simulation results were used to elucidate the physics of the temperature-dependent in-plane diffusion. The good agreement between the results of current simulations and previous theoretical studies demonstrates the effectiveness and reliability of free energy simulation with the enhanced ab initio molecular dynamics in heterogeneous surface processes. For complex situations where a simple harmonic model becomes inappropriate and convergent phase space sampling is required, direct MD simulation with enhanced sampling methods can make important contributions to our understanding of the thermodynamics of the surface catalytic processes.

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INTRODUCTION

one key necessary to understand the molecular mechanism of surface catalysis. Despite an enormous amount of experimental effort,8−17 much molecular detail of the diffusion processes is still poorly understood mainly due to the technical difficulties of obtaining direct molecular information from experiment. An alternative is to apply theoretical and computational tools to reveal atomistic details of the relevant chemistry in surface catalysis, which may then be used to provide guidance for experimental research.21−25 Many theoretical studies have been carried out to investigate the interactions between gaseous molecules and surfaces. Most of these studies used the density functional theory (DFT), as this reduced the computational cost. Even then the demand for computational resources is still so high that typically only a small number of static structural models of important stationary states at zero temperature were used in the calculations. The reaction thermodynamics, necessary for correct depicting of the reaction mechanism, were recovered using the statistic

Adsorption of gaseous molecules on a metal surface is an elementary step and often at the initiation of many heterogeneous catalytic processes, such as the Fischer−Tropsch reaction.1−7 The subsequent diffusion of the adsorbate molecule on the metal surface, quite likely following a typical jumping-among-minima behavior, forms the elementary motion for surface catalysis.8−17 Diffusive motion is the primary means for accomplishing two crucial events required for catalysis: the encounter of different reaction partners to form the reactant complex and the arrival at an active site which provides strong affinity for the transition state of the reaction. This feature is characteristic for most surface catalytic processes and is very different from the case of enzyme catalysis where the two tasks are often achieved by the conformational dynamics of the enzyme molecule.18−20 Given the fact that there exist many active sites on the surface, presumably with different binding affinity to the transition state, and thus with different catalytic efficiency, a proper description of the diffusion between these sites is clearly important for a correct understanding of the catalytic process. Study of the thermodynamics and kinetics of the diffusion of gaseous adsorbate on metal surfaces becomes © 2015 American Chemical Society

Received: June 15, 2015 Revised: September 24, 2015 Published: October 29, 2015 26422

DOI: 10.1021/acs.jpcc.5b05722 J. Phys. Chem. C 2015, 119, 26422−26428

Article

The Journal of Physical Chemistry C

where U(R) is the potential energy as a function of the coordinates R of the molecule and nk is a weighting factor for temperature Tk with

mechanics model of simple harmonic vibration. This approach is computationally economical and in spite of its approximations has successfully provided qualitative and even semiquantitative information for a number of reactions. Nevertheless, it is known that a method using only a small number of selected static structures suffers severely from two fundamental issues in the study of the surface catalytic processes. The first one is the accuracy of the harmonic vibration model, that is, whether the contribution of anharmonic effects is negligible.26,27 The second one is the poor sampling of active sites because of a limited number of conformations. In contrast, molecular dynamics (MD) simulation is a well-tested approach to obtain the thermodynamics of the target system by going beyond static structures and sampling phase space more broadly, although at a significantly higher computational cost.28−32 The adsorption and subsequent activation of CO molecules on metal surfaces have been of special interest because of the importance of the process in heterogeneous catalysis.33−40 As a prototype system for molecular chemisorption,41 the diffusion of CO molecule on metal surfaces in fact involves breaking and formation of metal−carbon bonds, very much like a typical chemical reaction process. This fact clearly implies the necessity of employing ab initio electronic structure methods, instead of the more economical classical force fields, in the simulations, to provide a reasonably accurate description of the complicated electronic/chemical interactions in the adsorption and onsurface diffusion process. The use of ab initio methods does very much increase computational cost. In fact, to the best of our knowledge, ab initio MD simulations of a reasonable length to provide meaningful thermodynamics have not been reported for CO diffusion on metal surfaces, presumably for that reason. When CO is adsorbed on a Ru surface, the potential barriers hindering the diffusion of CO between different binding sites originate from the exchange of bonding interactions between the carbon and metal atoms. Speeding up the barrier crossing is vital for improving the efficiency of MD sampling. To this end, enhanced sampling methods must be employed, which will be especially advantageous considering the high cost of quantum mechanical calculations in general. We report here a computational study of the thermodynamics of CO diffusion on Ru(0001) surface using ab initio MD simulations. We show that the integrated tempering sampling (ITS) method significantly improves the sampling efficiency and generates reliable smooth two-dimensional potential-of-mean-force (PMF) surfaces of CO diffusion at 300, 400, and 500 K in nanosecond MD simulations. Our work suggests that ab initio MD simulation, in conjunction with appropriate enhanced sampling methods, is a promising tool for the study of thermodynamics and molecular mechanism of heterogeneous surface catalysis.

βk = 1/kBTk

The configurational partition function of the system at a given temperature is Zk =

(3)

(4)

which ideally will produce a nearly uniform distribution in the temperature range T1 to TN. An effective potential energy can then be defined for the production simulation temperature T0 following N ̃

e−β0U (R) = P(U (R)) =

∑ nk e−β (U(R)) k

k=1

(5)

or N

−1 Ũ (R) = ln ∑ nk e−βk(U (R)) β0 k=1

(6)

According to this equation, one practical advantage of the ITS method is that the value of Ũ (R) and U(R) can be uniquely mapped once the coefficients {n k } are decided. The corresponding force for propagating coordinates in MD simulations with the effective potential energy can be readily computed. As a result, the sampling in the ITS simulation can be interpreted as an averaged sampling from multiple MD simulations, each at a temperature Tk and with a weighting factor of pk =

∫ nk e−βkU(R) dR ∫e

−β0Ũ (R)

dR

=

nk Zk N ∑ j = 1 njZj

(7)

The sampling enhancement of ITS can be attributed to two factors. First, the MD simulation under the effective ITS potential Ũ (R) at β0 includes contributions of samples obtained with the original potential U(R) at higher temperatures Tk(Tk > T0) as seen from eq 6. This feature ensures that the ITS simulation can overcome barriers more efficiently than is possible in an ordinary MD simulation at T0. Second, the lowand high-energy regions of the effective potential Ũ (R) match with those of the original potential U(R) since Ũ (R) is a monotonic function of U(R) from eq 6. This guarantees that the statistically favored low-energy regions of U(R) still possess larger statistical weights in the samples of Ũ (R) in ITS simulations. This feature helps to maintain a good balance of sampling between low- and high-energy regions and thus can provide accurate unbiased canonical distribution P(R) through reweighting from Ũ (R) to U(R). Therefore, due to these two factors, the ITS simulation is equivalent to calculating a trajectory on an attenuated energy surface, that is, a surface having lower barriers than the original potential but agreeing with the original potential in low-energy states. Obviously, this can result in much more frequent barrier transitions.

N k

k=1

k

n1Z1 = n2Z 2 = ··· = nN ZN

INTEGRATED TEMPERING SAMPLING METHOD The ITS method was originally developed by Gao and coworkers.42−46 We briefly review the method here. In this method, a generalized non-Boltzmann ensemble is constructed by summing canonical distributions over a set of temperatures. This leads to a generalized distribution with a probability according to

∑ nk e−β U(R)

∫ e−β U(R) dR

The values of nk in eq 1 can be determined by applying the condition of

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P(U (R)) =

(2)

(1) 26423


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COMPUTATIONAL DETAILS The Ru surface was modeled by a periodic slab consisting of five layers in the (0001) direction, with a 2 × 2 supercell in the two lateral directions. The top two layers of metal were completely relaxed and then kept fixed at the minimized positions for the description of CO adsorption on the top layer. As shown in Figure 1, there are four different CO binding sites

and corresponding normal modes. A geometrical displacement of 0.01 Å was used for all vibrational calculations. The ab initio MD simulations were carried out in the canonical ensemble with the Nosé−Hoover thermostat60,61 using a time step of 1 fs. The ITS method was employed to improve the sampling efficiency.42−46,62 In the ITS method, the series of temperatures Tk were selected according to ⎛ T ⎞k − 1/ N − 1 Tk = T1⎜ N ⎟ ⎝ T1 ⎠

(8)

Here T1 and TN are the lowest and highest value in the temperature range in the ITS scheme. For the ITS-MD simulations at 300, 400, and 500 K, T1 was set to 200, 300, and 400 K, TN was set to 1000 K, and N was set to 160, 140, and 120, respectively. The values of corresponding nk should be determined by applying the condition specified in eq 4. Since the partition function is unknown a priori, the values of nk generally have to be obtained through a series of short trial MD simulations at different Tk or an iterative procedure discussed in the original paper of ITS.42−46 Both methods become too expensive to be affordable in the ab initio MD simulations. In the current work the nk values were estimated in a different but simpler approach. That is, one assumes the minimized potential energy surface as a two-dimensional (2-D) function of the in-plane coordinates of the carbon atom could be used to represent the adsorption interaction of the system; thus, the partition function at each temperature is estimated directly from the 2-D potential energy surface (PES) using numerical integration. With the estimated partition functions, nk was directly computed using eq 4. It was found previously that the accuracy of nk is not critical as it has been shown that different sets of nk can generate the same free energy difference with similar simulation efficiency.62

Figure 1. Model of free Ru(0001) surface with a 2 × 2 supercell in the lateral directions. The four different sites, 1 top site, 1 twofold bridge site, and 2 threefold hollow sites (hcp and fcc), are marked.

on the Ru(0001) surface: 1 top site, 1 twofold bridge site, and 2 threefold hollow sites (hcp and fcc). Previous studies have shown that the top and hcp sites are the two thermodynamically favored CO binding sites.47−52 All the electronic structure calculations were performed using the plane-wave periodic density functional theory implemented in the Vienna ab initio simulation package (VASP).53−55 The projector-augmented-wave (PAW) method developed by Blöchl56 to describe the electron−ion interactions using plane-wave basis sets was employed. The generalized gradient approximation (GGA) with the Perdew−Burke−Ernzerh (PBE) exchange-correlation functional was used.57 The kinetic energy cutoff was set to 400 eV. The convergence criteria for the energy calculations were set to a self-consistent field (SCF) tolerance of 1.0 × 10−5 eV. Integration over the Brillouin zone was performed by using the Monkhorst−Pack scheme58 with 3 × 3 × 1 k-points, together with a Gaussian smearing broadening of 0.1 eV. The static structure calculations were carried out with a maximum Hellmann−Feynman force tolerance of 0.01 eV/Å. The climbing image nudged-elastic band (CI-NEB) method59 was utilized to locate the transition-state structure. The numerical calculation of the second derivatives of the harmonic potential energy surface provided the vibrational frequencies

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RESULTS AND DISCUSSION We first show that the ITS method indeed significantly improves the sampling efficiency. A 30 ps trajectory of the carbon atom is plotted to compare the sampling efficiency with and without the ITS method (Figure 2). In normal MD simulation without use of the ITS method, the adsorbate CO remains steadily in a single binding site (Figure 2a). In contrast, the CO molecule quickly moves over a broad region on the surface in a MD simulation of similar length with the ITS method (Figure 2b). The strikingly different performance demonstrates that the ITS method greatly facilitates the barrier crossing that normally limits the diffusion of the CO molecule

Figure 2. Thirty picosecond MD trajectories of the CO carbon atom along fractional coordinates a and b of the Ru(0001) surface at 300 K. (a) Normal MD results; (b) ITS-MD results. 26424


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A quantitative comparison could be made from PMFs to generate the temperature-dependent kinetics of CO diffusion between the top and hcp sites, as shown in Figure 4. Our results

on the surface. This feature is highly effective in the current project as this makes it possible to obtain reliable sampling within limited length of costly ab initio MD simulations. Thermodynamic properties were obtained from a total of 2 ns long ITS-MD simulations at each of three different temperatures, that is, 300, 400, and 500 K. The 2-D free energy surfaces, or PMF surfaces, for CO diffusion are illustrated in Figure 3 with respect to the fractional coordinates

Figure 4. Thermodynamics of the most feasible path for the CO diffusion from top to hcp site on Ru(0001) surface at different simulation temperatures.

show that the binding free energy difference, as well as the height of diffusion barrier, becomes smaller as the temperature increases. The PMF difference for CO binding at the two thermodynamically important sites (top and hcp) changes from 1.6 to 1.4 kcal/mol when the temperature increases from 300 to 500 K. Meanwhile, the free energy barrier for CO diffusion from top to hcp site decreases from 4.2 to 3.7 kcal/mol. Entropy is an important component of the CO binding free energy. The entropic component is determined by the physical nature of the motional modes, which also contributes to the dynamics of diffusion between different surface sites. When a free CO molecule binds to the surface, the five global motional modes (three translational and two rotational modes) convert to five new modes. First, because of the attracting interaction between CO and metal surface, two of the three translational modes of CO, that is, along the x- or y-direction of the surface plane, now behave as a mixture of vibration and obstructed translation. The third translation mode evolves into a mostly vibration-like off-plane motion of the Ru−C bond. For the same reason, the two rotations are no longer motions of free rotor. The change of mode of motions has a profound impact on the entropy of the system in different states. In the method using static structural models, the five new motional modes plus the CO vibration are identified on the PES with the frequencies calculated according to the harmonic vibration model and compiled in Table 1. Our calculated results are in good agreement with the previous theoretical66 and

Figure 3. 2-D PMF and PES of CO diffusion on the Ru(0001) surface along the fractional coordinates a and b of carbon atom on the surface plane. (a) PMF at 300 K; (b) PMF at 400 K; (c) PMF at 500 K; (d) optimized PES. The bottom bar shows the relative free energy scale in kcal/mol.

a and b of the carbon atom on the Ru(0001) surface plane. The 2-D minimal PES is also illustrated in Figure 3. As was expected, the shapes of PMF and PES are very similar to each other. Our calculation results clearly indicate the existence of two distinct thermodynamically stable CO binding sites. The top site is more favored in both MD simulations and PES calculations. This is in agreement with many previous experimental and theoretical studies for low-coverage CO adsorption on the Ru(0001) surface.47−52 The resemblance between the PMFs, PES, and previous results suggests that our current MD simulation approach is reliable. Most reactions take place at finite temperature, and temperature plays a critical role in determining the reaction kinetics and thermodynamics. The 2-D PMF of CO binding grows flatter and flatter with increasing temperature (Figure 3), and the corresponding distribution of CO adsorbate at different surface sites becomes more and more uniform. The relative height of the free energy barrier for CO diffusion also decreases; that is, CO diffusion on the surface becomes easier at high temperatures. Previous experiments reported similar spectroscopic changes due to increases of temperature and of CO coverage; this effect was suggested to be the result of the increased presence of CO at higher coordination sites in both cases.63−65 Our MD simulation results are in line with these reports.

Table 1. Frequencies (cm−1) of Center-of-Mass Translation (Vx, Vy, and Vz) and Internal Vibrational and Rotational Modes of CO at Different Sites of Ru(0001) Surface Obtained within the DFT Framework

26425

site

νx

νy

νz

νrotx

νroty

νvib

top hcp TS gas

74.6 161.4 161.9 i

61.4 150.8 75.5

409.4 323.9 384.4

394.3 213.6 185.3

391.3 189.2 262.9

1978.4 1724.9 1825.9 2123.4


Article

The Journal of Physical Chemistry C experimental results.67,68 The frequencies of the five new motions are significantly lower that of free CO vibration, confirming the validity of using classical MD simulation to sample phase space.69−75 The frequency of the two in-plane modes is lower in the top site, which is consistent with the fact that the top site has less steric interactions between the metal atom and the CO molecule. Thus, the in-plane motion is easier in the top site. Meanwhile, the higher frequency for the Ru−C vibration in the top site suggests a stronger chemical binding interaction for CO in this site. The frequency of the C−O vibration is lower in the hcp site, where the binding interaction is weaker. This is presumably due to stronger back-donation of d electrons from Ru surface to the 2π orbitals of CO in higher coordinated sites.63 The temperature-induced red shift of the CO oscillator observed in experiments thus suggested higher possibility for CO binding in the higher coordinated sites other than the top sites at higher temperature.63−65 The frequency of the newly formed Ru−C vibration is significantly lower than that of a typical chemical bond (Table 1). This raises a concern if the anharmonic effects are important in CO adsorption. As shown in Figure 5, the temperature

Figure 6. Temperature-dependent average bond length for Ru−C and C−O bonds obtained by direct MD simulations and averaging over two static binding sites.

negligible in the current system, such that the Boltzmannweighted result show the same temperature dependence on the MD simulation results. However, although for the Ru−C bond the results from MD simulations closely match those from static calculations, the results for the C−O bond show a large difference. This could result from either an inefficiency of ordinary MD simulation in describing the quantum mechanics of stiff bonds,69−75 as nuclear motions are still treated classically, or an artifact of the thermostat employed in the current work.76 Both issues can be corrected in future simulations. Nevertheless, the similar slopes for both bonds suggest that at least qualitatively the two methods captured the same essence of the temperature dependence of structural variations. Note that the temperature dependence of the bond length calculated in the static approach will be nearly flat without the Boltzmann-weighted averaging because of the small anharmonic effect in the current system. This analysis suggests again the importance of combining the contributions of multiple statistically important states, that is, proper phase sampling. Of course, when the number of statistically important binding sites grows, direct MD simulation will have greater advantage in providing more accurate results than the other approach does. The thermodynamic results obtained in the current simulation work can be used to explore some fundamental issues of CO diffusion on Ru surface. One of the most important questions in the current system regards a correct description of CO diffusion on the surface. That is, which motional mode(s) may one use to describe the on-surface diffusion? Past studies often characterized the diffusion as a free inplane translation.77−80 The PMF results for the CO diffusion

Figure 5. Temperature variation of the free-energy difference (ΔG) and barrier height (ΔG≠) for diffusion between top and hcp sites.

dependence of the PMF is similar to that of the free energy computed by static calculations with the harmonic vibration model. This result suggests that any anharmonic effect may be negligible in the current system. This is probably because of the relatively simple potential energy surface of the system studied here. Clearly, for more complicated situations such as when there is more than one molecule adsorbate (of the same or different type) on the surface, the potential energy surface becomes increasingly complex, and the anharmonic may well become non-negligible at high temperatures, in which case simple static model calculations will become inaccurate and inefficient and MD simulation will be more accurate. The performance of the MD simulation and the approach using static structural models can also be examined by comparing the structural properties obtained in both calculations. While MD simulation directly provides a statistically averaged result, in the approach of static structural models, the equilibrium structures obtained in two binding sites are averaged by the relative Boltzmann weight of each site to extrapolate a statistically meaningful result. As shown in Figure 6, the temperature-dependent structural variations are similar for the two computational approaches. This agreement again supports the conclusion that the anharmonic effects are 26426


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sampling is much desired to provide accurate thermodynamics and dynamics of the adsorption and diffusions. Our results show that MD simulations with enhanced sampling can be very useful in studying these surface processes over a broad temperature range.

(Figures 3 and 4) clearly question the validity of such a simplified model since there are significant barriers separating different binding sites. Accurate calculation of hindered diffusion is a difficult topic left for future simulation research. But the results obtained here allow us to address a related important question. Since the obstructive effect of the barrier on the in-plane diffusion gets weaker and weaker when temperature rises, the hindered diffusion will converge to free translations at high temperatures. It is then interesting to determine the value of such a transition temperature at which the diffusion barrier becomes insignificant and the gaseous molecules diffuse like in free translation. The transition temperature is estimated here by comparing the mean speed of 2-D translation of a free ideal gas particle and for barrier crossing. When the temperature is so high that the diffusion speed is of the same magnitude as the mean translation speed of an ideal gas molecule, one hardly makes any difference between free translation and obstructed diffusion. The mean speed of a 2-D ideal gas particle is videal =

πRT 2M

■

CONCLUSION In summary, we report direct ab initio MD simulations to investigate the temperature-dependent thermodynamics of CO diffusion on the Ru(0001) surface. Compared to the normal MD simulations, it is shown that ITS-enhanced MD simulations sample phase space much more efficiently. Reliable and smooth two-dimensional potentials of mean force surfaces for CO diffusion at three different temperatures have been obtained to provide direct thermodynamic information for the on-surface diffusion process. To the best of our knowledge, this is the first report of direct MD simulations of temperaturedependent thermodynamics of CO diffusion on the Ru(0001) surface. Our comparison between MD simulation and staticstructures-based method for computing the temperaturedependent free energy difference and the barrier height between stable binding sites provides an important benchmark for future simulation studies. The current results also suggest that only at high temperatures, for example, above 850 K, a model of free translation might be used to describe the onsurface CO diffusion process. Our study clearly demonstrates that direct ab initio MD simulations, in conjunction with efficient enhance sampling methods, become readily available as an effective and reliable approach to study the thermodynamics of heterogeneous catalysis.

(9)

where R is the gas constant and M is the molar mass. On the other hand, the diffusion barrier between two sites can be used to estimate an upper limit of the mean diffusion speed, vad, which is a function of the rate constant of barrier crossing, k, and the geometrical distance between the top and hcp sites, a. Here k=

⎡ Δ≠G ⌀ ⎤ kBT exp⎢ − r m ⎥ h ⎣ RT ⎦

■

(10)

Δ≠r G⌀m

where is the activation free energy for the barrier crossing determined currently.

vad = ka

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (H.H.). *E-mail: [email protected] (Z.-N.C.).

(11)

Notes

The authors declare no competing financial interest.

On the basis of the results computed here, the temperature for the convergence of these two motional modes is estimated to be 850 K (Figure 7). Thus, the free translation model seems inappropriate to describe the on-surface diffusion motions at low temperatures. At temperatures significantly lower than this limit, theoretical simulation with extensive phase space

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ACKNOWLEDGMENTS We are thankful for financial support from the HKU strategy research themes on the topics of “Clean Energy”, the HKU Small Project Funding (201309176150) to Z.-N. C., and the Chinese National Science Foundation. We are grateful to Dr. Jan Hermans for help reading and revising the paper.

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Figure 7. Temperature variation of the mean speed of 2-D free translation and the diffusion between top and hcp sites. 26427


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