Beyond Mean-Field Microkinetics: Toward Accurate and Efficient

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Beyond the mean-field microkinetics: Towards the accurate and efficient theoretical modeling in heterogeneous catalysis Zheng Chen, He Wang, Neil Qiang Su, Sai Duan, Tonghao shen, and Xin Xu ACS Catal., Just Accepted Manuscript • Publication Date (Web): 11 Apr 2018 Downloaded from http://pubs.acs.org on April 11, 2018

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ACS Catalysis

Beyond the mean-field microkinetics: Towards the accurate and efficient theoretical modeling in heterogeneous catalysis Zheng Chen, He Wang, Neil Qiang Su, Sai Duan, Tonghao Shen, Xin Xu*

Collaborative Innovation Center of Chemistry for Energy Materials, Shanghai, Key Laboratory of Molecular Catalysis and Innovative Materials, MOE Key Laboratory of Computational Physical Sciences, Department of Chemistry, Fudan University, Shanghai 200433, China

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Abstract Kinetics as the link between the atomic scale properties and the macroscopic functionalities is indispensable in describing surface chemical reactions and computation-based rational design of catalysts. Kinetic Monte Carlo (KMC) on the explicit lattice can resolve events taking place on the catalytic surfaces at the atomic level. It can explicitly account for the spatial correlations due to lateral interactions among adsorbates, which have been proved to significantly affect the surface chemical reactions. However, the disparity in time scales of various processes (e.g. adsorption/desorption, diffusion and reaction) usually makes the brute force KMC simulations impractical. Here, we proposed a method, namely XPK, to extend the phenomenological kinetics (PK) for the accurate and efficient microkinetic modelling of heterogeneous catalysis. XPK is achieved through a hybrid between the diffusion-only KMC on the explicit lattice to evaluate the reaction propensities and later an implicit lattice KMC in the PK form to evolve the coverages and calculate the final rates. XPK is tested against the explicit lattice KMC using model systems and is applied to describe the volcano curve of ammonia decomposition on the close-packed surfaces of transition metals with lateral interactions among adsorbates being introduced. The results demonstrate the accuracy of XPK, show the significant influences of the lateral interactions on both the shape of volcano curve and the position of volcano top, and highlight the usefulness of XPK in describing complex catalytic

kinetics

of

practical

interest

and

the

predictive

capability

in

computation-based rational design of catalysts.

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ACS Catalysis

KEYWORDS Kinetics, lateral interactions, KMC, time-scale separation, volcano curve, ammonia decomposition

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Introduction There is a continuous demand for industry to develop more active, more selective and less expensive catalysts, as this holds the key to eventually solve the energy problems and the environmental problems for a sustainable development of our society. Thanks to the advance of the experimental techniques for surface characterizations, as well as the progress of the theoretical methods for density functional theory (DFT) calculations, we are now closer than before to the target of the rational catalyst design. There have been a number of successful examples1-9, where the complete kinetics of a catalytic reaction has been evaluated by suitable kinetic methods on the basis of the first principles’ DFT calculations and the agreement between theory and experiment is encouragingly good. Kinetics is indispensable as the link between the atomic scale properties and the macroscopic functionalities in describing surface chemical reactions and in computation-based rational design of catalysts. Thus, effort has been continuously devoted to the development of accurate yet efficient kinetic methods10-14. For the sake of their simplicity, phenomenological kinetics (PK) methods based on mean-field approximations, such as Langmuir–Hinshelwood-type models15, Sabatier analysis16, mean-field micro-kinetic models17, are the most widely used. However, the adsorbate–adsorbate interactions can markedly influence the thermodynamic and kinetic property of the elementary reactions4,

10

, the order of adsorbates18-19, the

favored mechanism20, the catalytic selectivity21, as well as the shape of “volcano curve” for catalyst design22. Hence, more sophisticated statistical mechanical

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treatments, in particular, the explicit lattice kinetic Monte Carlo (KMC)3, 11, 14, 23-24 simulation is necessary where the catalyst sites are mapped onto a lattice. The explicit lattice KMC model can resolve events taking place on a catalytic surface at the atomic scale, explicitly accounting for spatial correlations and the neighboring effects due to lateral interactions among adsorbates10, 12, 14. However, in many cases, there is a large time scale separation in a sequence of elementary steps on the catalyst surfaces. For instance, the diffusion barriers of small atoms and molecules (e.g. H, C, N, O, CO, NO, etc.) on metal surfaces are only 0.12 times the respective adsorbate binding energies25, suggesting that diffusions on the surfaces are usually much faster than chemical reactions on the surfaces. Aside from surface diffusions, there is, in general, a complicated network of surface chemical reactions, some of which (e.g. adsorption/desorption or hydrogenation/dehydrogenation) are fast, whereas some others are slow. As a result, the brute force KMC simulation becomes impractical, because the standard algorithm samples almost always the uninteresting fast events, while the sampling of the slow rate determining step is insufficient for a correct statistical analysis at a feasible period of simulation time. There are several recent reviews discussing the current status and frontiers for KMC simulations of the complex catalytic reactions10, 12, 14, 26-27. Meanwhile, time scale separation still imposes one of the outstanding computational challenges. The commonly used strategy is either to partition the fast events and the slow events, and treat them in multiscale modeling frameworks26, 28-31, or to invoke some single time scale acceleration techniques such as the rate constant rescalings32-36. As has been

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commented,36 even though significant theoretical advances have been made recently, few easily usable algorithms are available for practical systems. It is widely accepted that the adlayer is in equilibrium at a given coverage and temperature when all reactants are in the high diffusion limit37-38. (Unless otherwise stated, all systems discussed in the present work are in the fast diffusion limit). Along this line, there are several useful methods being proposed, e.g. the quasi-chemical approximation (QCA) method37-38, and the coarse time-stepper method28,

39-40

.

However, these methods may fail or are still inefficient when there are strong lateral interactions among adsorbates and when there are many quasi-equilibrium elementary reactions in the reaction network. Grand Canonical Monte Carlo (GCMC) simulations have also been introduced to describe accurately the kinetics at the quasi-equilibrated adsorbate limit7. A recent method that uses the so-called cluster mean field approach has been developed, which is promising in treating the spatial correlations at a progressively higher level of approximations41. Nevertheless, these algorithms often require to specify a priori information about the rate determining step and/or identify in advance the quasi-equilibrated reactions, which limit their applications to general systems with complicated reaction networks. Raimondeau and Vlachos proposed the “high but not too high diffusion rate” method29. The diffusion rate is increased progressively until a plateau in the production rates of interest is reached. Despite of its straightforwardness, the diffusion rates may need to be modified each time to reach the plateau when the surface

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coverage changes, which is a tedious job. Moreover, the plateau is normally reached when diffusion rates are three to four orders of magnitude higher than reaction rates. In such a condition, the computational resources are still not efficiently used31. Chatterjee and Voter developed a temporal acceleration scheme, called Accelerated Superbasin Kinetic Monte Carlo (AS-KMC), by automatically modifying the reaction rates of fast processes without the need for the user to specify these processes in advance33. The applicability of this method has been greatly extended by a recent development of Dybeck et al.35, where the acceleration is accomplished by applying the scalings to all of the events in a given reaction channel rather than to the individual event as done in the original AS-KMC. While the latter algorithm was found to overall perform quite well, some challenging cases were identified which was shown to lead to a breakdown of the acceleration algorithms.42 Mastny, Haseltine and Rawlings proposed a stochastic method to simulate the catalytic surface reactions in the fast diffusion limit31. A reduced master equation was derived for the probability distribution of the coverages, rather than that of the spatial configurations. While this method shows several distinct advantages, it has the deficiency, as pointed out by the authors themselves31, that the dimension of the master equation increases rapidly with lattice size and species number, limiting its application to catalytic surface kinetics of realistic systems. In the present work, we proposed a method, namely XPK, to extend the phenomenological kinetics (PK) for the accurate yet efficient microkinetic modelling of heterogeneous catalysis. XPK adopts the partitioning strategy28, 31, 40 to separate

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fast diffusion events and slow reaction events. It is achieved through a hybrid approach between the diffusion-only KMC on the explicit lattice to evaluate the reaction propensities and a subsequent implicit lattice KMC in the PK form to evolve the coverages and to calculate the final rates. XPK is tested against the explicit lattice KMC using model systems, which demonstrates that XPK provides the same results as the explicit lattice KMC simulation while it is much more efficient by construction. The performance of XPK has been further examined in realistic systems using ammonia decomposition as an archetypical example43-44. The volcano curve has been described on the close-packed surfaces of transition metals with lateral interactions among adsorbates being introduced. The results demonstrate the accuracy of XPK, show the significant influences of the lateral interactions on both the shape of volcano curve and the position of volcano top, and highlight the usefulness of XPK in describing complex catalytic kinetics of practical interest and the predictive capability in computation-based rational design of catalysts.

2. Models and method 2.1 The model system To illustrate our approach, a model system with two different settings of potential energy surface (PES) is used here. This corresponds to the reaction of A + H2 → AH2, a simplified model for hydrogenation reaction commonly observed in heterogeneous catalysis. It consists of the following chemical processes, involving three types of adsorbed species, A*, H* and AH*,

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ACS Catalysis

A(g) + * ↔ A*

∆E1 ,

H 2 (g) + 2* ↔ 2H*

∆E2 ,

A* + H* ↔ AH* + *

Ea3 Ea−3 ,

AH* + H* → AH 2 (g) + 2*

Ea4 ,

where ∆E j represents the reaction energy, which is equal to the difference of the forward

reaction

barrier

Ea j

and

the

reverse

reaction

barrier

Ea− j ,

i.e., ∆E j = Ea j − Ea− j . The corresponding values for ∆E j and Ea j Ea− j in PES I and PES II are listed in Table 1. Note that, in this model, there are seven types of chemical reactions and three types of diffusions. The list of microscopic events in the system is expanded when lateral interactions are included, because, for example, the same chemical reaction may become different microscopic chemical events with different rate coefficients due to the lattice configuration changes. While no lateral interactions are considered in PES I, three types of lateral interactions are considered in PES II, where the energetic parameters are 0.06 eV for (A*, A*), 0.015 eV for (AH*, AH*), and 0.03 eV (A*, AH*). Here we only consider the first nearest-neighbors (1NN) effects and the positive sign refers to a repulsive interaction. 2.2 Rate coefficient calculations Transition state theory (TST)45-48 is often utilized to calculate the rate coefficient for each elementary step with the reaction barrier Eai

ki =

 Ea k BT Q ≠ exp  − i h QR  k BT

 . 

(1)

Here h is the Planck’s constant, kB is the Boltzmann’s constant, and T is the temperature. Q ≠ refers to the partition function of the transition state where the reaction coordinate has been separated out, while QR refers to the partition functions

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of the reactants. For a surface reaction, it is often assumed that the contributions of Q ≠ and QR are similar42. Then, Eq. (1) is simplified to the form as

ki = The

rate

coefficient

for

 Ea k BT exp  − i h  k BT

adsorption,

kads,

 .  can

(2) be

obtained

from

the

adsorption-desorption equilibrium

kads , A =

 µ  k BT exp  − gas , A  , h  k BT 

(3)

where µ gas , A is the chemical potential for species A in the gas phase. Here in this work, we set the chemical potential for species A in the gas phase at 1 bar and 600 K as ‒1.00 eV. For H2, its chemical potential at 1 bar and 600 K is calculated as ‒0.76 eV assuming it to be an ideal gas. For a realistic modeling of surface reactions, lateral interactions among the adsorbates have to be taken into account, leading to the desorption energy of A* that depends on its detailed environment. The effects of lateral interactions upon H* are always small, and therefore are ignored. The detailed desorption energy of each A* is computed by the desorption energy in the zero coverage limit, corrected by

∑

B

ε BA M BA , where ε BA is the lateral interaction energy per species B to A and M BA

is the number of species B neighboring the A species of interest. Lateral interactions affect not only the reactants ( ∑ B ε BR M BR ) and the products ( ∑ B ε BP M BP ), but also the transition state ( ∑ B ε BTS M BTS ).Here R and P represent an assembly of reactants and products, respectively. For instance, if a double site reaction involves (A + C), the energy level of the reactants will be affected by

∑

B

ε BA-C M BA-C

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ACS Catalysis

due to the lateral interactions of the neighboring spectators B to the A-C pair. The energy

barrier

for

each

elementary

step

is

estimated

based

on

the

Brønsted-Evans-Polanyi (BEP) type relationship32, 49-52, which correlates the change of activation barrier height with that of the reaction energy, E ′ai = Eai +α

(∑

B

ε BP M BP − ∑ B ε BR M BR ) ,

E ′a− i = Ea− i + (1 − α )

(∑

B

ε BR M BR − ∑ i ε BP M BP ) .

(4) (5)

Here E ′ai and E ′a− i are the forward and the reverse barriers, respectively, for reaction i after applying the lateral interaction corrections. The slope α in the BEP relationship lies in between 0 and 1, which is chosen as 0.5 for the model system on PES II in the present work. 2.3 Kinetic Monte Carlo method The execution of an elementary event leads to a change in the system state or spatial configuration of the lattice, which can be described by the so-called master equation (see Eq. 6 in the next section). Here KMC simulations23, 27, 53-56 are employed to generate an ensemble of trajectories of the underlying Markovian processes, where each trajectory propagates the system correctly in the sense that the average over the entire ensemble of the trajectories yields the probability density of the master equation23, 53-56. If a stationary system state, i.e. the steady-state in the context of catalytic kinetics, can be reached, the ensemble average can then be replaced by a time average over a trajectory. Therefore, the actual objective for a KMC computer algorithm is to generate such KMC trajectories. Differences between KMC solvers arise in the ways how the event sequence is

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chosen and how the elapsed system time is determined. The Variable Step Size Method (VSSM) 26, 53, 57, employed in the present work, is probably the most widely used algorithm for KMC simulations. At every KMC step, VSSM updates the sequence of available events {p} and calculates the total rate Rtot = ∑ p R p . The evolution of real time is updated to t = t − ln(r1) / Rtot , where r1 is a random number uniformly distributed on (0,1]. Another random number r2 ∈ [0,1] is generated to select which event will occur with a probability proportional to its rate.

3. The extended phenomenological kinetics (XPK) method 3.1 The master equations of lattice configuration x and species numbers n The detailed information for the kinetics on a catalytic surface can be obtained, in principle, by solving the full state master equation that reflects the spatial lattice configuration change due to the execution of an elementary event as

dP ( x ) = ∑ Wx ← y P ( y ) − ∑ W y ← x P ( x ) . dt y y

(6)

Here the catalytic surface is mapped onto a lattice, while P(x) is the probability that the system is in the lattice configuration x at a given time and Wy←x is the rate of going from lattice configuration x to y. This full state master equation can be derived from first principles23, 53-56 and hence forms a solid basis for all subsequent work. However, the exact master equation can rarely be solved directly, as the number of configurations in this spatial space is huge even for the simplest catalytic system, while the KMC simulations provide a way to construct its numerical realization3, 11, 14, 23-24

. Nevertheless, such a brutal force KMC on the explicit lattice is often impractical 12


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ACS Catalysis

that suffers from the notorious problem of time scale separation10, 12, 30. As chemical productivity is often the property of the main concern, one prevailing way is to examine the evolution of P(n) instead of that of P(x)30-31, by looking at the chemical space instead of the spatial space dP ( n ) = ∑ Wn ← m P ( m ) − ∑ Wm ← n P ( n ) . dt m m

(7)

Here n is a vector representing a chemical state that contains the number of each chemical species on the lattice, while Wm←n is the rate of going from chemical state n to m. As P ( n ) = ∑ P ( x )δ ( n − nx ) , where δ ( n − nx ) equals 1 when nx, a vector in the x

lattice configuration x, contains the same species numbers as in the chemical state n and zero otherwise, the dimensionality is reduced from that of P(x) to that of P(n). Naturally, Wy←x can be decomposed into two parts from either diffusion or reaction events, i.e., W y ← x = D y ← x + K y ← x , while only reaction events can change the species numbers: ∆nx , y = nx − n y . Assuming that no reaction events occur to change n during the fast time scale when diffusion events occur, a working equation for P(n) evolution can be derived31

dP ( n) = ∑ K x ← y P ( y | n − ∆n x , y ) P ( n − ∆n x , y ) − ∑ K y ← x P ( x | n ) P ( n ) , dt x, y x, y

(8)

where P( x | n) represents the conditional probability of being in configuration x at a given chemical state n, which is governed by the master equation evolving with diffusion-only events31 dP ( x | n) = ∑ Dx ← y P ( y | n ) − ∑ D y ← x P ( x | n ) . dt y y

(9)

Under such an assumption, it is expected that the x states reach equilibrium due to

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diffusion before reaction events occur to change n. A detailed analysis based on singular perturbation was presented by Stamatakis and Vlachos for systems with no lateral interactions,58 which showed that the error of the approximation drops linearly with the inverse of the diffusion rate. By comparing Eqs. 7 and 8, the total rate or the reaction propensity, i.e., R (n) = ∑ Wm←n , to escape from a given n to any possible m can be estimated m

through the diffusion-only KMC simulations based on Eq. 9

R ( n) = ∑ K y ← x P ( x | n) .

(10)

x, y

Once the information on R(n) is known for all n, Eq. 8 can then be solved readily. Unfortunately, the dimensionality of P(n) grows rapidly as the number of species and the lattice size increase, the number of diffusion-only KMC simulations that must be run becomes a major limiting factor in solving the master equation for P(n)31.

3.2 The extended phenomenological kinetics (XPK) method It is instructive to decompose the total rate R(n) in terms of chemical reactions

R ( n ) = ∑∑∑ ki| j (α )P (α | n ) α

j

i

=∑∑ R j (α )P (α | n ) α

(11)

j

= ∑ Rj ( n). j

Here α sums over all spatial states at the given n, i sums over all microscopic reaction events on the lattice that follow the same chemical reaction j, and j sums over all possible types of chemical reactions to escape from state n. Without loss of generality, it can be assumed that there exist a single site (SS) reaction such as A → B and a double site (DS) reaction such as A + C → D + E on the surface. According to equation (11), the corresponding R j ( n ) are: NA

RSS (n) = ∑∑ ki|SS (α )P(α | n) , α

(12)

i

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ACS Catalysis

RDS ( n) = ∑ α

N A ,C (α )

∑

ki| DS (α )P (α | n) ,

(13)

i

where N A is the total number of surface species A, which is the same for all lattice configurations at a given n, and N A,C (α ) is the total number of the neighboring (A,C) pair on configuration α at a given n. For systems without lateral interactions, the rate coefficients are the same for all the microscopic reaction events following the same chemical reaction, i.e. ki|SS = kSS and ki|DS = kDS, while the surface species are distributed randomly. Thus, Eqs. 12 and 13 can be rewritten as

RSS ( n ) = kSS N A ,

RDS ( n ) = kDS

N A NC N s Nba , N s ( N s − 1)

(14) (15)

where N ba is the coordination number of the lattice and we assume that A and C refer to different species. Noting that θ A = N A N s for a lattice of Ns sites, the R j ( n ) terms in Eqs. 14 and 15 can now be corresponding to the well-known PK form in terms of coverage θ . At the large lattice limit, the PK form for the rate equations as the ordinary differential equations is hence obtained. For systems with lateral interactions, the rate coefficients are no longer the same, due to the varying environment, for the microscopic reaction events that follow the same chemical reaction. Here we assume that Eqs. 12 and 13 can still be rewritten in the PK form as

RSS ( n ) = kSSapp N A , app RDS ( n ) = k DS

N A NC N s N ba , N s ( N s − 1)

(16) (17)

In fact, Eqs. 12 and 13 provide a recipe that the apparent rate coefficients can be estimated. Assuming that the diffusion-only KMC simulation has been run for a time interval ∆t at equilibrium, where a set of configurations {α} has been produced. When ∆t is large enough, the probability distribution of P (α | n) is well approximated, as

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{α} is well approaching to the complete configuration set to cover all possible α at the given n. Then the apparent rate coefficients can be calculated as: k

app k DS

Here ∆t (α )

app SS

 ∆t (α ) N A (α )  1 = ∑  ki|SS (α )  , ∑ ∆ t α  N A (α ) i 

  1 ∆t (α ) = ∑ ∆t α  N A (α ) N C (α ) N N a  N ( N − 1) s b s s 

N A ,C (α )

∑ i

(18)

  ki| DS (α )  .   

(19)

is the time that system stays at configuration α such that

∆t = ∑ ∆t (α ) . Hence, the equalities of Eqs. 16 and 17 hold, as the lattice α

app information is encoded via Eqs. 18 and 19 into the apparent rate coefficients k SS

app and k DS . We name the method based on Eqs.16-19 the XPK method, which is

reduced to the standard PK method as described by Eqs. 14 and 15 when no lateral interactions are taken into account for a randomly distributed adlayer.

3.3 The efficiency of the XPK method For a real catalytic system, it is impractical to construct R j ( n) for all n or θ beforehand31. For a SS reaction related to reactant A, we may rewrite Eq. 16 as the total rate per site in terms of θ

R%SS (θ ) = kSSapp (θ )θ A .

(20)

Clearly, the apparent rate coefficient and the total rate depend not only on the surface coverage of reactant A alone, but also on other spectator species as exemplified here as B in below. The effect due to the variation of surface coverages is to be taken into account with the first order Taylor series expansion,

∂ ln kSSapp  ∂ ln kSSapp ∆θ B  app  app  % RSS (θ + ∆θ ) =kSS 1 +  ∆θ A + kSS 1 + θ A . ∂ ln θ A  ∂ ln θ B θ B   

(21)

If the coverage for reactant A (θA) is small, it can be reasoned that one has

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ACS Catalysis

1 >>

app ∂ ln kSS app , which suggests that the influence of ∆θA on kSS is negligible such ∂ ln θ A

app that change on R% SS can be well approximated by kSS (θ ) ∆θ A . If the coverage for

spectator B (θB) is small, it can also be reasoned that one has 1 >>

app ∂ ln k SS ∆θ B . ∂ ln θ B θ B

Hence, a small θB has negligible effect on R% SS . Besides, for surface species, like H*, with which the lateral interactions are small, their effects on R% SS are also small and are ignorable. Similar discussion applied to coverage effects on R% DS involving a DS reaction between species A and C with B as an example of the spectators, app app   ∂ ln k DS ∂ ln k DS app  app  R% DS (θ + ∆θ ) =kDS  θ A ∆θC 1 +  ∆θ AθC + k DS 1 + ∂ ln θ A  ∂ ln θC    app ∂ ln kDS ∆θ B  app  1 + kDS +  θ AθC . ∂ ln θ B θ B  

(22)

With these properties, R%j (θ ) can be constructed readily and efficiently. Instead of calculating all R%j (θ ) beforehand, which is formidable for complex catalytic

systems, the XPK method evaluates R%j (θ + ∆θ ) “on-the-fly” from the Taylor expansion of { k app (θ + ∆θ ) }. In practice, the first order central difference of k app j j is often a good approximation, i.e. Eqs, 21 and 22, to the Taylor expansion, while only the partial derivatives with respect to the surface species with a large surface coverage (LSC) are necessary, while those with a small surface coverage (SSC) is ignorable. The steady state is reached when { k app (θ ) } and { θ } are consistent j within a threshold of ∆θ = 0.03. A simplified flow chart of XPK is given in Figure 1. A brief description for the explicit/implicit lattice KMC simulations can be found in S1.1 as the supporting 17


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information (SI). Here, KMC based on Eq. 6 or 7 site by site is named as the explicit lattice KMC, while KMC based on Eq. 7 via Eqs. 14-17 is named as the implicit lattice KMC, where evolution is in the chemical space while lattice information is implicitly encoded in the apparent rate coefficients.

4. Results and discussion 4.1 Comparing the explicit/implicit lattice KMC simulations for a system without lateral interactions To compare the simulation results of the full state master equation (simulated by the explicit lattice KMC) and the PK form (simulated by the implicit lattice KMC), model reaction system with PES I without lateral interactions is applied. This is actually a special case of XPK, as the rate coefficients remain constants during the reactions, removing the need for the diffusion-only KMC simulation as the first step in XPK. The rate equations in the PK form on PES I can be written as:

dθ A* = k1θ* − k−1θ A* − N ba k3θ A*θ H* + N ba k−3θ AH*θ* , dt dθ H* 2 = Nba k2θ*2 − N ba k−2θ H* − N ba k3θ A*θ H* + N ba k− 3θ AH*θ* − N ba k4θ AH*θ H* , dt dθ AH* = N ba k3θ A*θ H* − N ba k− 3θ AH*θ* − Nba k4θ AH*θ H* , dt

(23) (24) (25)

where the rate coefficients, k1, k2, k3 and k4 are associated with the adsorption of A, adsorption of H2, hydrogenation of A*, and hydrogenation of AH*, while k-1, k-2 and

k-3 are associated with the reverse reactions, respectively. We assume that the reactions take place on a hexagonal lattice. Hence, Nba is chosen as 6. The results of

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both the explicit and implicit lattice KMC simulations at 600 K on PES I for a set of gas-phase conditions are shown in Figure 2 and Table 2. Figure 2 shows the simulated TOF (turnover frequency) to form AH2. The solid lines are labeled as the A-lines, which correspond to the TOF changes with the input pressure of A from 0.20 to 1.00 MPa with that of H2 being fixed at 0.80 MPa. The dashed lines are labeled as the H2-lines, which correspond to the TOF changes with the input pressure of H2 from 0.20 to 1.00 MPa with that of A being fixed at 0.40 MPa. The coverages of A*, H* and AH* in the respective steady states under different A and H2 pressures are summarized in Table 2. From Figure 2 and Table 2, it is clear that the simulation results from the explicit lattice KMC and the implicit lattice KMC are virtually identical, providing numerical demonstration of the feasibility to reduce the explicit lattice KMC to implicit lattice KMC in the system without lateral interactions on a randomly distributed adlayer5, 58. As the implicit lattice KMC avoids to sample the configuration space directly, it is much more efficient, by construction, as compared to the explicit lattice KMC (See S1.2 in the SI for a more detailed discussion). In this system, the reaction coefficient k-3 for AH* + * is over 100 times larger than k4 for AH* + H*, which suggests that the quasi-equilibrium approximation can be used. Then the overall formation rate of AH2(g) can be described by

rAH 2 =

 E eff  θ 2 k BT exp  a  H* θ A* N s Nba , h  k BT  θ *

(26)

where Eaeff = ( Ea 3 − Ea −3 ) + Ea 4 for the compound reaction of (A* + 2H* AH2(g) + 2*). The blue circles in Figure 2 denote the implicit lattice KMC simulation results 19


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with the formation rate as expressed in Eq. 26. Clearly, the TOF results obtained with the compound reaction are the same as those predicted step by step as in Eqs. 23-25. Hence, the compound reaction rate based on the quasi-equilibrium assumption can be introduced conveniently to treat the time scale separation between the fast reaction events and the slow rate determine step in an implicit lattice KMC simulation, where the explicit lattice KMC simulation suffers in such a situation.

4.2 Applying XPK to the model system with lateral interactions 4.2.1 The performance of XPK XPK is now employed to investigate the hydrogenation kinetics on PES II with lateral interactions. An example for evaluating the apparent rate coefficients { k app j } using the diffusion-only KMC simulations on the explicit lattice is shown in Figure S1 in the SI. When the surface reaches the equilibrium, the { k app } values are j calculated according to Eqs. 18 and 19. Then the evolutions of coverages are simulated by the implicit lattice KMC. The results are summarized in Figure 3 and Table 3. As shown in Figure 3, the simulated results for the TOF changes upon pressure changes from XPK and the explicit lattice KMC are nearly identical even when lateral interactions have been introduced. And so does the coverage of each species as presented in Table 3. It is very convenient to use the PK form. However, as shall be noted from Table S1 in the SI, the number for each pair of species from XPK shows obvious departure from the corresponding pair number evaluated from the explicit lattice KMC. Due to the existence of the lateral interactions, the pair distribution has to be different from

20


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that evaluated by the randomly distributed form of θ Aθ C ( N s N s − 1) N s N ba as in XPK. Nevertheless, the key issue is to have a correct description of the rate for a DS app reaction. Hence, a pair correction must have been absorbed into { k DS }, which

highlights the particular importance of the diffusion-only explicit lattice KMC simulations as the first step in XPK, as well as the consistency between { k app j } and { θ i }. This guarantees the accuracy of XPK for some important kinetic properties of interest such as TOF and { θ i }. As discussed in section 3.3, during the evolution of {θi} by an implicit lattice KMC, it is assumed that the changes of { k app j } caused by the variation of a SSC species can be ignored. This approximation reduces the computational complexity of catalytic kinetics on surfaces to a great extent. To verify the approximation numerically, { k app j } extracted from the diffusion-only explicit lattice KMC under several specific coverages of A*, H* and AH* are presented in Table 4 and the corresponding values of

∂ ln k app j ∂ ln θi

are calculated. On PES II with lateral interactions,

AH* is referred to as the LSC species, while A* the SSC species. As shown in Table 4, within the range of θAH* from 0.546 to 0.646, the

∂ ln k app j ∂ ln θAH*

values are in the range

from 0.17 to 2.35. These effects on { k app j } are appreciable, which cannot be simply ignored. On the other hand, within the range of θA* from 0.001 to 0.010, the

∂ ln k app j ∂ ln θA*

values are in the range from 0.00 to 0.03. These effects on { k app j } are

generally less significant and are often ignorable, although the lateral repulsions experienced by A* are actually higher than those by AH*. These results support the 21


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discussion in section 3.3, suggesting that XPK is an efficient method for complex kinetics on surfaces, as even for a very complicated network, there shall be a few LSC species and many SSC species, while only LSC’s correlations with { k app j } have to be considered by using the diffusion-only explicit lattice KMC simulations. A direct efficiency comparison between XPK and the explicit lattice KMC is provided in S1.2 as the SI. On top of PES II, it is shown that the time cost of XPK is about two orders of magnitude lower than that of the explicit KMC.

4.2.2 The effects of lateral interactions To understand what role lateral interactions play in the heterogeneous catalysis, direct comparisons are shown in Figure 4 and Table 5 for the simulation results on PES II with and without lateral interactions. Figure 4 shows how the TOFs of AH2 vary with the pressure changes at 600 K. The A-lines (solid) are for the input pressure of A from 0.20 to 1.00 MPa with that of H2 being fixed at 0.80 MPa, while the H2-lines (dashed) are for the input pressure of H2 from 0.20 to 1.00 MPa with that of A being fixed at 0.40 MPa. The top two lines (red) are with lateral interactions, while the bottom two lines (blue) are with no lateral interactions. As is clear in Figure 4, the pressure dependences of the TOFs are significantly different whether the lateral interactions are considered or not. Even the trends of TOF dependences on the pressure changes of A are obviously different. Table 5 shows how coverages vary with the pressure changes at 600 K. Without lateral interactions, the coverages of species AH* in the steady states are much higher when comparing to the corresponding values with lateral interactions. The absence of

22


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lateral interactions make AH* artificially too stable, which occupy some extra vacant sites and even replace some H*. Lateral interactions prevent the surfaces from being an easy victim to AH*, making the surface not poisoned by AH* with increasing pressure of either A or H2. The availability of vacant sites is important to maintain the catalytic activity for a catalyst59-61. In fact, lateral interactions ensure the VIII transition metal surfaces hard to be poisoned by CO*, even though their adsorption energies for CO (‒1.19 ~ ‒1.64 eV)62 are rather large at the zero coverage. It is often accepted that lateral interactions change the coverages that account for the major differences of kinetics with and without considerations of lateral interactions. The middle two lines (olive) in Figure 4 correspond to the implicit lattice KMC results with no lateral interactions, but with the coverages of adsorbates fixed at the same values from XPK with lateral interactions. As is clear from Figure 4, TOF dependences on pressure changes as represented by the middle lines are still quite different from the top lines. These suggest that coverage change alone is unable to explain the full effect of lateral interactions. With lateral interactions, rate coefficients shall not be taken as rate constants, which shall depend on the local environment where a reaction takes place. This observation highlights the usefulness of XPK where { θ i } and { k app j } are coupled to reach the steady state iteratively, which goes beyond the mean-field microkinetics for an accurate description of kinetics of the real catalytic systems. The effect of lateral interactions may approximately be incorporated into the mean-field microkinetics by means of coverage dependency21. A comparison between

23


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the mean-field microkinetic modeling and XPK with respect to coverage dependencies is provided in S1.3 as the SI. The results once again highlight the importance to properly simulate the local reaction environment.

4.3 Applying XPK to describe the volcano-curve of ammonia decomposition The volcano curve is one of the most fundamental concepts in heterogeneous catalysis and is significantly important to search for the optimal catalysts6,

63

.

Particularly, it has been pointed out that lateral interactions among adsorbates can affect the shape of the volcano curve significantly22. Herein, on the basis of scaling relations64-65, XPK is employed to describe the volcano-curve of ammonia decomposition over the close-packed transition metal surfaces. The simulations are performed under 0.10 MPa NH3 at 850 K. Ammonia decomposition proceeds by means of a sequence of dehydrogenation processes, followed by recombination of N and H to form N2 and H2, respectively. The scaling relationships between N atom adsorption energies and the adsorption energies of the other surface species (including transition states) are applied66,67. Based on these factors, a set of approximate lateral interactions among surface species has been developed. The details about scaling relationships are presented in S1.4 and values for lateral interactions are summarized in Table S2 in the SI. If lateral interactions were omitted, we see the typical volcano-curves as shown in Figure 5(a), where the activity, represented as Log(TOF), decreases to the right of the volcano-top when N atom binding is too weak to activate and dissociate N2, while the activity also decreases to the left of the volcano-top when N atom binding is too

24


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strong. The volcano-top with the highest activity is located around –5 eV for the metal having an appropriate strength for N atom binding. Figure 5(b) shows the corresponding changes of the coverage θ N * . While θ N * decreases sharply to the right of –5 eV as a result of decreasing N atom binding; θ N * increases sharply to the left of –5 eV as a result of increasing N atom binding. Clearly, in the latter case, the surface is blocked by N* with little vacant site. There are actually two such volcano-curves shown in Figure 5(a): one from explicit lattice KMC and the other from implicit lattice KMC that are on top of each other. Once again, we see that explicit and implicit lattice KMC lead to the same simulation results on TOFs, and also on θ N * as shown in Figure 5(b), without considering lateral interactions, while implicit lattice KMC is much more efficient by construction. Figure 5 also displays the simulation results from XPK when lateral interactions are considered. Lateral interactions do not influence the shape of the volcano-curve on the right-hand side, where the N atom binding is weak and the N* coverage is low. On the contrary, the shape of volcano-curve on the left-hand side is significantly changed when lateral interactions are taken into account. Without lateral interactions being considered, the activities for catalysts on the left-hand side are significantly underestimated, where the catalysts are poisoned quickly. With lateral interactions being considered, the activities for catalysts on the left-hand side decrease slowly, as lateral interactions help to resist the strong N atom binding to free some vacant sites and restore the activities. Frey et al.19 also reported a similar effect in catalytic NO oxidation, where the presence of lateral interactions “softens” the shape of the

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volcano. Notably, with lateral interactions considered, there is a significant move of the volcano-top, as shown in Figure 5(a), from the original position around –5.0 eV of N* binding energy to –6.0 eV. The position of the volcano-top is crucial for the design of new catalysts when reactivity descriptors have been identified and kinetic models are used to identify catalysts with the desired properties. The picture that the volcano-curve on the left-hand side is flatter than that on the right-hand side is in agreement with the experiment results such as CO methanation16 or ammonia synthesis68. Similar observation has also been reported in CO oxidation study with a mean field micro-kinetic model to account for lateral interactions22. The fact that the metals lie on the left-hand side still show comparable activity to the volcano-top is important to the rational catalyst design because other advantages in terms of price or stability of the materials can be balanced22.

Conclusion Reliable kinetic model is of great significance in relating the atomic scale properties and the macroscopic functionalities, which provides the foundation for computation-based rational design of catalysts. In the present work, we proposed a method, namely XPK, to go beyond microkinetic modelling for the accurate and efficient simulations of heterogeneous catalysis. XPK is achieved through a hybrid between the diffusion-only KMC on the explicit lattice to evaluate the reaction propensities and later an implicit lattice KMC

26


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in the PK form to evolve the coverages as well as to evaluate the reaction rates. It embodies the atomic details into the apparent rate coefficients { k app }, which depend j on the local environment where the reactions take place and couple with the coverage evolution to reach consistency. XPK is tested against the explicit lattice KMC using model systems where perfect agreement has been reached that validates the XPK method. It is applied to describe the volcano-curve of ammonia decomposition on the close-packed surfaces of transition metals with and without lateral interactions among adsorbates being introduced. The results demonstrate the accuracy of XPK, show the significant influences of the lateral interactions on both the shape of volcano-curve and the position of volcano-top, and highlight the usefulness of XPK in describing complex catalytic

kinetics

of

practical

interest

and

the

predictive

capability

in

computation-based rational design of catalysts.

Acknowledgment We thank Dr. Igor Ying Zhang for his initial work on this project. We thank Prof. Zhonghui Hou and Dr. Huijun Jiang for their insightful discussion. This work was supported by National Natural Science Foundation of China (21688102,

91027044),

and

the

Ministry

of

Science

and

Technology

(2013CB834606).

Reference

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Effects in Solids, Sickafus, K. E.; Kotomin, E. A.; Uberuaga, B. P., Eds. Springer Netherlands: Dordrecht, 2007; pp 1-23. (57) Gillespie, D. T. Exact stochastic simulation of coupled chemical reactions. J.

Phys. Chem. 1977, 81, 2340-2361. (58) Stamatakis, M.; Vlachos, D. G. Equivalence of on-lattice stochastic chemical kinetics with the well-mixed chemical master equation in the limit of fast diffusion.

Comput. Chem. Eng. 2011, 35, 2602-2610. (59) Dumesic, J. A. Analyses of reaction schemes using de donder relations. J. Catal. 1999, 185, 496-505. (60) Logadottir, A.; Rod, T. H.; Nørskov, J. K.; Hammer, B.; Dahl, S.; Jacobsen, C. J. H. The Brønsted–Evans–Polanyi relation and the volcano plot for ammonia synthesis over transition metal catalysts. J. Catal. 2001, 197, 229-231. (61) Cheng, J.; Hu, P. Theory of the kinetics of chemical potentials in heterogeneous

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ACS Catalysis

catalysis. Angew. Chem. Int. Ed. 2011, 50, 7650-7654. (62) Wellendorff, J.; Silbaugh, T. L.; Garcia-Pintos, D.; Nørskov, J. K.; Bligaard, T.; Studt, F.; Campbell, C. T. A benchmark database for adsorption bond energies to transition metal surfaces and comparison to selected DFT functionals. Surf. Sci. 2015, 640, 36-44. (63) Ertl, G.; Knözinger, H.; Weitkamp, J. In Handbook of Heterogeneous Catalysis. Wiley–VCH, Weinheim: 1997. (64) Abild-Pedersen, F.; Greeley, J.; Studt, F.; Rossmeisl, J.; Munter, T. R.; Moses, P. G.; Skúlason, E.; Bligaard, T.; Nørskov, J. K. Scaling properties of adsorption energies for hydrogen-containing molecules on transition-metal surfaces. Phys. Rev. Lett. 2007, 99, 016105. (65) Wang, S.; Petzold, V.; Tripkovic, V.; Kleis, J.; Howalt, J. G.; Skulason, E.; Fernandez, E. M.; Hvolbaek, B.; Jones, G.; Toftelund, A.; Falsig, H.; Bjorketun, M.; Studt, F.; Abild-Pedersen, F.; Rossmeisl, J.; Norskov, J. K.; Bligaard, T. Universal transition state scaling relations for (de)hydrogenation over transition metals. Phys.

Chem. Chem. Phys. 2011, 13, 20760-20765. (66) Đnoğlu, N.; Kitchin, J. R. Simple model explaining and predicting coverage-dependent atomic adsorption energies on transition metal surfaces. Phys.

Rev. B 2010, 82, 045414. (67) Kitchin, J. R. Correlations in coverage-dependent atomic adsorption energies on Pd(111). Phys. Rev. B 2009, 79, 205412. (68) Vojvodic, A.; Medford, A. J.; Studt, F.; Abild-Pedersen, F.; Khan, T. S.; Bligaard,

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T.; Nørskov, J. K. Exploring the limits: A low-pressure, low-temperature Haber–Bosch process. Chem. Phys. Lett. 2014, 598, 108-112.

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ACS Catalysis

Table 1. Settings for reaction energies and reaction energy barriers (in eV) in PES I and PES II. ∆E1

∆E2

Ea3

Ea-3

Ea4

PES I

-1.00

-0.60

1.24

0.94

1.18

PES II

-0.70

-0.60

0.94

1.34

1.34

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Table 2. Coverages of A*, H* and AH* in the respective steady states under different A and H2 pressures, simulated by explicit/implicit lattice KMC. No lateral interactions have been considered, and PES I has been adopted. Pressure (MPa)

Explicit lattice KMC

Implicit lattice KMC

A

H2

θA*

θH*

θAH*

θA*

θH*

θAH*

0.20

0.80

0.550

0.173

0.001

0.549

0.174

0.001

0.40

0.80

0.709

0.112

0.001

0.710

0.111

0.001

0.60

0.80

0.785

0.082

0.001

0.786

0.082

0.002

0.80

0.80

0.829

0.065

0.002

0.829

0.065

0.002

1.00

0.80

0.858

0.054

0.002

0.858

0.054

0.002

0.40

0.20

0.752

0.059

0.001

0.751

0.059

0.001

0.40

0.40

0.734

0.082

0.001

0.733

0.082

0.001

0.40

0.60

0.720

0.098

0.001

0.721

0.098

0.001

0.40

0.80

0.709

0.112

0.001

0.710

0.111

0.001

0.40

1.00

0.700

0.123

0.001

0.700

0.123

0.001

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ACS Catalysis

Table 3. Coverages of A*, H* and AH* in the respective steady states under different A and H2 pressures, simulated by the explicit lattice KMC and XPK. Lateral interactions have been considered, and PES II has been adopted. Pressure (MPa)

Explicit lattice KMC

XPK

A

H2

θA*

θH*

θAH*

θA*

θH*

θAH*

0.20

0.80

0.000

0.193

0.499

0.000

0.202

0.476

0.40

0.80

0.000

0.152

0.606

0.000

0.156

0.596

0.60

0.80

0.000

0.131

0.658

0.000

0.133

0.654

0.80

0.80

0.000

0.119

0.690

0.000

0.113

0.707

1.00

0.80

0.000

0.105

0.726

0.000

0.102

0.736

0.40

0.20

0.001

0.102

0.573

0.001

0.102

0.576

0.40

0.40

0.000

0.122

0.602

0.000

0.124

0.598

0.40

0.60

0.000

0.142

0.598

0.000

0.141

0.599

0.40

0.80

0.000

0.152

0.606

0.000

0.156

0.596

0.40

1.00

0.000

0.165

0.600

0.000

0.166

0.597

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Page 40 of 47

Table 4. Coverage effects on the apparent rate coefficients { k app }1. AH* is referred to as the large j surface coverage (LSC) species, while A* the small surface coverage (SSC) species. Lateral interaction is considered, and the energy surface is related to PES II. app The slope of ln k j vs ln θi

varying θΑΗ*2 varying θΑ*3

k −app 1

k 2app

k −app 2

k3app

k −app 3

k −app 4

2.35 0.02

-0.20 -0.00

-0.17 -0.00

0.64 0.03

-0.51 -0.01

0.43 0.00

1

The apparent rate coefficient k1 of A adsorption is a constant at constant chemical potential µA.

2

θΑΗ∗ varies from 0.546 to 0.646 with θΑ∗ and θΗ∗ being fixed at 0.005 and 0.156, respectively. θΑ∗ varies from 0.001 to 0.010 with θΑΗ∗ and θΗ∗ being fixed at 0.596 and 0.156, respectively.

3

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ACS Catalysis

Table 5. Coverages of A*, H* and AH* in the steady states under different A and H2 pressures, calculated based on PES II with lateral interactions and without lateral interactions. Pressure (MPa)

With lateral interactions

Without lateral interactions

A

H2

θ*

θA*

θH*

θAH*

θ*

θA*

θH*

θAH*

0.20

0.80

0.322

0.000

0.202

0.476

0.153

0.001

0.097

0.749

0.40

0.80

0.248

0.000

0.156

0.596

0.087

0.001

0.055

0.857

0.60

0.80

0.213

0.000

0.133

0.654

0.061

0.001

0.038

0.900

0.80

0.80

0.180

0.000

0.113

0.707

0.047

0.001

0.030

0.922

1.00

0.80

0.162

0.000

0.102

0.736

0.038

0.001

0.024

0.937

0.40

0.20

0.321

0.001

0.102

0.576

0.129

0.002

0.041

0.828

0.40

0.40

0.278

0.000

0.124

0.598

0.106

0.001

0.047

0.846

0.40

0.60

0.260

0.000

0.141

0.599

0.094

0.001

0.051

0.854

0.40

0.80

0.248

0.000

0.156

0.596

0.087

0.001

0.055

0.857

0.40

1.00

0.237

0.000

0.166

0.597

0.082

0.001

0.058

0.859

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Page 42 of 47

Figure 1. A brief flow chart of the XPK method

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ACS Catalysis

Figure 2. TOF of AH2 at 600 K on PES I with no lateral interactions. The A-lines (solid) are for the input pressure of A from 0.20 to 1.00 MPa with that of H2 at 0.80 MPa, where are from the explicit lattice KMC simulations, ● from the implicit lattice KMC simulations, and ● from implicit lattice KMC simulations with the compound rate equation. The H2-lines (dashed) are for the input pressure of H2 from 0.20 to 1.00 MPa with that of A at 0.40 MPa, where are from the explicit lattice KMC simulations, ○ from the implicit lattice KMC simulations, ○ from implicit lattice KMC simulations with the compound rate equation.

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Page 44 of 47

Figure 3. TOF of AH2 at 600 K on PES II with lateral interactions. The A-lines (solid) are for the input pressure of A from 0.20 to 1.00 MPa with that of H2 at 0.80 MPa, where are from the explicit lattice KMC simulations, and ● from the XPK simulations. The H2-lines (dashed) are for the input pressure of H2 from 0.20 to 1.00 MPa with that of A at 0.40 MPa, where are from the explicit lattice KMC simulations, and ○ from the XPK simulations.

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ACS Catalysis

Figure 4. TOF of AH2 at 600 K on PES II. The A-lines (solid) are for the input pressure of A from 0.20 to 1.00 MPa with that of H2 at 0.80 MPa, while the H2-lines (dashed) are for the input pressure of H2 from 0.20 to 1.00 MPa with that of A at 0.40 MPa. Here and are from the implicit lattice KMC simulations without lateral interactions, ●and ○ are from the XPK simulations with lateral interactions, while ▼ and are from the implicit lattice KMC simulations with fixed coverages of the reactants obtained by XPK.

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Page 46 of 47

Figure 5. (a) The volcano curve and (b) coverages of N at different N adsorption energies for the ammonia decomposition model based on scaling relations at 0.10 MPa NH3, and 850 K. (the line

∆/ from explicit/implicit lattice KMC simulations without lateral interactions (LI); ● from XPK simulations with N-N repulsive interactions equal to ‒0.2 eV).

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ACS Catalysis

The XPK method is proposed to extend the phenomenological kinetics (PK) for the accurate and efficient microkinetic modelling of heterogeneous catalysis. 62x51mm (300 x 300 DPI)


Beyond Mean-Field Microkinetics: Toward Accurate and Efficient

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