Molecular Heterogeneous Catalysis, Wiley (2006), 352729662X

456 Appendices

Each state in the extended system is a unique state in the real system. The real velocity

This modified system of equations can then be integrated in order to follow the timedependent changes in the positions of the particles (molecules) and the state of the system.

Molecular dynamics can then be used to simulate the molecular trajectories for molecules in zeolites, the di usivities for sorbates in zeolites, temporal changes in the pore structure due to changes in system variables (T , P ) or sorption of molecules[59,62]. As was described earlier, the integration time steps (10−15 sec) limit the time scales that can actually be simulated to nanosecond behavior and thus preclude the simulation of longer time processes.

C. SIMULATING KINETICS

The methods discussed so far provide the ability to simulate the surface structure, microand mesoporous structure, physisorption and chemisorption at surface sites, lateral interactions between surface intermediates, activation barriers and overall reaction energies for elementary surface reactions, di usion on surfaces and within porous media, and a host of other elementary or equilibrated processes important toward understanding catalysis. Simulating catalysis, however, requires following the dynamics of the entire surface adlayer and its structure as a function to changes in process conditions, including temperature, pressure and conversion. As was discussed in Chapter 2, catalysis is driven by kinetics. The ability to follow the kinetics for catalytic processes requires the ability to simulate the myriad of physicochemical elementary processes including adsorption, surface reaction, di usion and desorption that occur simultaneously and make up the full catalytic cycle.

The kinetics for catalytic systems can be modeled by one of two general methods. The first is based on continuum concentrations and uses deterministic kinetics whereas the second approach follows the temporal fate of individual molecules over the surface via stochastic kinetics. Both approaches have known advantages and disadvantages, as will be discussed. B These methods provide the constructs for simulating the elementary kinetics. However, in order to do so, they require an accurate and comprehensive initial kinetic database that contains parameters for the full spectra of elementary surface processes that make up the catalytic cycle. The ultimate goal for both approaches would be to call upon quantum mechanics calculations in situ in order to establish the potential energy surface as the simulation proceeds. This, however, is still well beyond our computational capabilities.

Currently, the most straightforward way to bridge electronic structure and surface kinetics requires a decoupling of the time scales that govern electronic transfer processes that control elementary surface reaction steps from the overall catalytic cycle which proceeds at much longer times. Ab initio calculations are used first to calculate the kinetics, energetics and potential mechanisms necessary for an external database. The database could then be called “in situ” within the simulation algorithm.

Computational Methods 457

1. Deterministic Kinetic Modeling

The kinetics for most catalytic systems are described using determinstic models rather than stochastic simulations. Deterministic models are straightforward to develop and to program. The temporal concentrations for all species/intermediates in the network are tracked by solving the full set of di erential equations that describe the rate of formation and disappearance for each component in the system. The rate is defined in terms of concentrations, partial pressures or surface coverages. As such, the deterministic models average the intrinsic kinetics over the atomic structure in order to define the concentrations of reaction intermediates and are therefore considered an early averaging method. This approach ignores the features of the local structure and composition near the active site and their influence on the kinetics. By averaging out over the surface structure, the problem becomes one of solving N -di erential equations. These equations can subsequently be used to solve for the reaction rate which can be plugged into various di erent reactor models which would follow not only the changes in the rate but also the spatiotemporal changes throughout the reactor[80] .

Deterministic kinetic modeling approaches are mean-field approaches, whereby the molecules experience only an averaged interaction[81] of the others. These models are reasonable if the lateral interactions between reactant molecules, reagents or products are absent or if di usional e ects maintain a state of ideal mixing. In the latter case, the kinetic parameters will also be concentration dependent.

The input to most of the microkinetic modeling studies has been experimentally derived rate constants. This is for two reasons. The first has been the lack of available first principles kinetic data and the di culty in simulating them. The second is due to the fact that the accuracy is not within 1 kcal/mol. Despite this drawback, there have been some very interesting studies performed even in the absence of very accurate kinetics.

2. Stochastic Methods

Stochastic methods simulate the dynamic changes that occur in the structure of the adlayer of catalytic surface and thus model the elementary surface kinetics [82−100]. The temporal changes of a system can be followed by solving the stochastic master equation which simulates the dynamic changes in the system as it moves from one state (i) to another state (j). The master equation, which can written as

is nothing more than a balance on the kinetic “forces that drive a system from one state to another as a function of time. Pi is the probability that the system is in state i at time t and Pj is the probability that the system is in state j. Wij and Wji are the transition probabilities which denote the probable rate of transition from state i to state j or j to i, respectively, for the system. The self transition probabilities, Wii and Wjj , are equal to zero. For catalytic systems, the changes in system state can be any elementary surface process that changes the nature of the adlayer including surface di usion, surface reaction, desorption, adsorption and surface reconstruction. In order for the master equation to hold, the system must obey a detailed balance, that is

458 Appendices

The requirements for a detailed balance, however, do not control the kinetics.

The master equation, however, can only be solved analytically for very simple systems such as the gas-phase reaction A→B. The analysis of these systems typically requires numerical simulation of a lattice-based kinetic Monte Carlo model. The lattice gas model

can then be used to formulate the respective transition probabilities in order to solve the master equation[81]. The groups of both Zhdanov[97−99,102−108] and Kreuzer[109−113] have

been instrumental in demonstrating the application of lattice gas models to solve adsorption and desorption processed from surfaces. Once a lattice model has been formulated there are three types of solution:

1.the cluster approximation,

2.the transfer matrix technique, 3.) Monte Carlo simulation.

Zhadanov’s group[97−99,102−108], as well as others, have shown that the cluster approximation is useful for understanding surface structure and surface kinetics. It is limited,

however, in its ability to describe the formation of ordered surface structures. Kreuzer’s group[109−113] has demonstrated the utility of the transfer matrix approach[109−110]. Mathematically, the equations become cumbersome for solution, but this technique provides an important insight into the surface physics that control the kinetics. The final approach involves the numerical simulation of the master equation by Monte Carlo algorithms. MC simulation opens up the possibilities of simulating much more complicated surfacecatalyzed systems, as will be discussed.

There are two basic methods that have been used to simulated kinetics via Monte Carlo approaches[114−116]. The first is termed the fixed-time approach, in which every site on

the surface has a set of probabilities associated with the di erent kinetic events that can

occur at these sites. This could include di usion, reaction, adsorption, and desorption processes. The state of the system then moves in fixed incremental steps of time and

subsequently surveys all of the physicochemical steps to determine which of them can take place within the given (short) time step. This is accomplished by sampling every site and determining whether it changes due to the occurrence of a kinetic process. This is determined by drawing a random number for each potential step and comparing it with

where krs is the rate constant for reaction (r) associated with the specific environment

(s)[114].

The fixed-time approach has proven e ective in modeling well-defined reaction systems which have a sequence of known steps. The benefits of this approach are that the user is able to specify the time step at which the simulation proceeds. This helps to overcome some of the di culties associated with disparate time scales. Very fast processes can be treated as pseudo-equilibrated, thus enabling the simulation to move to the time scales of interest.

One of the drawbacks associated with the fixed-time approach, however, is that at any give point in time one needs to have all of the possible future pathways worked out in order to calculate the probability that within that particular time step a sequence of events occurs. This becomes challenging and expensive computationally as the network of surface processes is rather complex.

Computational Methods 459

A second drawback of the fixed-time approach is that it is mathematically not exact. The accuracy of the simulation is governed by the choice of the time step used. Only in the limit of an infinitesimal time step does the method becomes mathematically exact. For simulations performed at very small time steps, accuracy is not an issue. Although

the fixed time algorithm has proven to be fairly e ective for certain gas-phase reaction systems[114,117,118], nearly all of the published studies on surfaces use what is known as the variable time-step approach[82−122].

In the variable time-step approach, the system moves in event space, thus simulating

the elementary kinetic processes event-by-event whereby the time is updated in variable time increments[115,116,122]. At any instant in time, ti, the rates for all possible events are

By rearranging, we can to solve for the time at which the next event in the system will occur:

i ri

where RN is a random number between 0 and 1 which is chosen by the computer as it defines the random probability of the time for the next event.

The time step chosen is variable and changes throughout the simulation. It can be infinitely small or infinitely large and depends both on the random probability and the overall calculated rate. The variable-time method is mathematically an exact approach and there are no concerns about accuracy due to time step size. Systems which contain fast events have very small time steps and are thus dominated by the time scale scales of the fastest processes. Faster rates of reaction lead to higher probabilities that these steps are chosen. This leads to problems for systems with disparate rates since the simulation will spend nearly all of its time simulating the faster rates without ever simulating the slower processes. This is especially a problem for systems where di usion is fast and reaction is slow and systems which contain fast processes which are nearly equilibrated together with slow processes.

The simulation of surfaces typically requires defining an appropriate lattice. These can either be a simple lattice model or o -lattice simulations which attempt to treat sites more explicitly. The simulation proceeds in essentially the same manner as described with the one exception that we explicitly follow the surface of the lattice. At any given instant in time the entire surface is surveyed in order to construct a detailed list of all possible surface events that can occur, including adsorption, desorption, surface reaction, and di usion. Each possible event is assigned a rate (or rate constant) based on the nature of the event and the explicit molecular environment around each species. The rates (rate constants) for each of these possible events are added together to determine the cumulative probability for that particular event. The computer draws a random number which is then used in Eq. (C5) [a modified version of Eq. (C3)] in order to establish the time step of the next event.

One of the most important features of the Monte Carlo approach is its ability to monitor explicitly of atoms on the surface and within the adlayer. This allows for the direct accounting of specific surface sites and the local reaction environment at these

460 Appendices

sites. The specific arrangement and orientation of surface ad-species can significantly influence the surface kinetics. These interactions are at the heart of coverage e ects. The ability to model lateral interactions between ad-species on the surface and their influence on catalytic performance was discussed in some detail in Chapter 3 and is therefore not repeated again here.

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