Exact results for the reactivity of a single-file system

We derive analytical expressions for the reactivity of a Single-File System with fast diffusion and adsorption and desorption at one end. If the conversion reaction is fast, then the reactivity depends only very weakly on the system size, and the conversion is about 100%. If the reaction is slow, then the reactivity becomes proportional to the system size, the loading, and the reaction rate constant. If the system size increases the reactivity goes to the geometric mean of the reaction rate constant and the rate of adsorption and desorption. For large systems the number of nonconverted particles decreases exponentially with distance from the adsorption/desorption end.

We derive analytical expressions for the reactivity of a Single-File System with fast diffusion and adsorption and desorption at one end. If the conversion reaction is fast, then the reactivity depends only very weakly on the system size, and the conversion is about 100%. If the reaction is slow, then the reactivity becomes proportional to the system size, the loading, and the reaction rate constant. If the system size increases the reactivity goes to the geometric mean of the reaction rate constant and the rate of adsorption and desorption. For large systems the number of nonconverted particles decreases exponentially with distance from the adsorption/desorption end.  [1] They have one-dimensional pores through which molecules can diffuse, but with cross sections that are too small to allow passing. As for many other 1D systems, this leads to very interesting kinetic effects. [2] Research on SFS zeolites has focussed on the diffusion, because the mean-square displacement in a SFS is proportional to the square root of time, and not proportional to time. [3,4,5] Molecular Dynamics, [3,6,7] Dynamic Monte Carlo (DMC), [8,9,10] and reaction-diffusion equations [11] have mainly been used. Few studies have included reactions. [10,12,13] We have argued that this is unfortunate, because if diffusion is fast the temporal dependence of the mean-square displacement is not very relevant, but there are still effects from the non-passing of the particles in the pores.
In a previous paper we have investigated steady-state properties of a SFS for different assumptions of the reactive site distribution using DMC as well as analytical techniques. [10] In the current paper we derive exact results for the case that the entire pore is reactive. These results give detailed insight in the relationships of the various system parameters and the reactivity.
Our model consists of S sites forming a onedimensional finite chain. Each site is numbered consecutively from 1 for the site on one end, to S for the site on the other end. Each site is vacant or is occupied by a particle. We have two types of particles: A and B. An A can be converted into a B on any site. Adsorption and desorption of particles can only occur on site 1. Only A's are adsorbed, but both A's and B's desorb. Both types of particle diffuse by making random hops to neighboring sites if vacant.
The evolution of the system is described by a master equation [14,15] where α and β refer to the configuration of the adlayer (a particular distribution of particle over the sites), the P 's are the probabilities of the configurations, t is real time, and the W 's are constants that give the rates with which reactions change the occupations of the sites. W αβ corresponds to the reaction that changes β into α. The rate constants in our model are W ads for adsorption of an A onto site 1 if vacant, W des for desorption of a particle from site 1, W rx for the conversion of an A into a B on any site, and W diff for a hop of a particle to a vacant neighboring site. If the diffusion is infinitely fast, we can derive a simpler master equation for the number of particles in the system.
where P N is the probability that there are N particles in the system. This is a master equation of a one-step Markov process. [15] We are interested in the probability distribution f MK (t) that if at time t = 0 the number of particles is K, this number becomes M for the first time at time t with M < K. If there are N particles in the system and at time t = 0 a particle adsorbs, then f N,N +1 is the probability distribution for the time that this particle desorbs. This is based on the property of a SFS that particles can not pass each other. So f N,N +1 is the probability distribution for the residence time of a particle that adsorbs in a system with N particles.
Let P N K be the solution of Eq. (2) with N the number of particles and the initial condition P N K (0) = δ N K . Let Q N K also be a solution with Q N K (0) = δ N K , but now for the master equation with an adsorbing boundary at M : i.e., we remove the term in Eq. (2) that corresponds to an adsorption then there are M particles in the system. With N ≥ M we have This equation is called the renewal equation. [15] If we take N = M in the renewal equation, then we have Q MK (t) = 0 by definition. So we get the integral equation for f MK . A particle that adsorbs, stays in the system for a period t, and then desorbs, has a probability exp(−W rx t) that it is not converted during the time it was in the system. Because the probability distribution for the time that it stayed in the system is given by f N,N +1 (t), the probability that the particle desorbs without being converted is The reason for using this Laplace transform is that is that it is related to the Laplace transform of the solution of the master equation through the renewal equation. Laplace transforming Eq. (4) yieldŝ If we write the master equation (2) in matrix-vector notation asṖ = WP , and Laplace transform it we get With the initial condition for P MK this yieldŝ so thatf We define the reactivity B prod as the number of particles that is being converted per unit time. To get B prod we have to multiply the probability that a particle is converted by the rate of adsorption and make a weighted average over the number of particles in the system. The probability that a particle is converted equals 1−f N,N +1 (W rx ), and the rate of adsorption equals W ads (1 − N/S). At steady state the number of particles in the system is given by [10] Combining this results in  Figure 1 shows this reactivity for two finite system sizes and three loadings. The loading θ is defined as the probability that a site is occupied at steady state. It's equal to [10] The reactivity is compared to the (average) rate of adsorption and desorption λ, which at steady state is given by If the rates of adsorption and desorption are small compared to the reaction rate constant W rx , then B prod → λ. This means that almost every particle that enters the system is converted. If the rates of adsorption and desorption are much larger, then for λ/W rx → ∞ and S not too large. Figure 1 shows the θ dependence for small S and the S dependence for small θ. It does not show this S and θ dependence for S and θ both large. λ needs to be much larger than the values in the figure for that. For S = 100 we see that the reactivity may be proportional to θ for θ ≤ 0.5, but not around θ = 0.9. For the latter value the reactivity is even smaller than for θ = 0.5. In practice only W ads can usually be varied independently, either through the pressure or the concentration of A's outside the system. The limit W ads → ∞ corresponds to θ = 1 and λ = W des . So we see that there generally is an upper bound on λ.
Comparing S = 5 and S = 100 in Fig. 1 for some finite λ shows that B prod varies less with θ for larger S. We can show that in the limit S → ∞ the reactivity depends only on λ and W rx . If the system becomes infinitely large, then the fluctuations become small with respect to the number of sites. If the system is then at steady state we can write the master equation as Formally we can let the number of particles N run from −∞ to +∞. The matrix s − W can easily be diagonalized. The eigenvalues are s + 2λ(1 − cos k) with −π < k ≤ π, and the corresponding eigenvector has components exp [ikN ]. We can use this to calculate elements of the inverse of the matrix s − W. with This then yields for the probability that a particle adsorbs and desorbs at a later time without being converted the expressionf (Here and in the rest of the paper we set s = W rx in the definition of y.) For the reactivity B prod we have We see that the reactivity no longer depends on the loading θ if the system becomes large, but only on the reaction rate and on λ. For small rates of adsorption and desorption we again find B prod → λ. If the rates of adsorption and desorption are large, then for λ/W rx → ∞. The approach to the limit S → ∞ becomes very slow when λ is large as can be seen in Fig. 1.
The procedure for the first-passage problem above can also be used to derive the A and B profiles: i.e., the distribution of the A's and B's in the chain. We will first deal with the question what the probability is that the nth particle, counting from site 1, is an A. Then we will answer the question that site m is occupied by an A. The answers to the same questions about B's follow trivially from the ones for the A's.
We are only interested in the steady state. In this case we have detailed balance: i.e., the number of transitions per unit time from N to N + 1 particles is equal to those of N + 1 to N . This means that for each sequence in the number of particles where after a time lapse ∆t i the number of particles changes from N i to N i+1 , there is another sequence The second sequence is the time reversed of the first one. Moreover, both sequences are equally likely because of the detailed balance. Consequently, the probability distribution that the number of particles in the system is K at time t = 0 and M with M < K at time −t for the last time equals f MK (t). This means that if there are N particles in the system the probability distribution that particle n is in the system for a time t equals f N −n,N (t). The probability that that particle has not been converted is thenf N −n,N (W rx ) following the reasoning after Eq. (5). This probability can be calculated using Eq. (9). For the profile we need the probability that site m is occupied by particle n. This probability is given by The probability A m that site m is occupied by a particle that has not been converted is then given by where the first summation averages over the number of particles in the system and P for the probability that particle n has not been converted. We see that this probability decreases exponentially.
The probability P occ (n, N ; m, S) becomes P occ (n, m; θ) with θ = N/S for S → ∞. This limit of the combinatorial factors yields The probabilities An that a site is occupied by a non-converted particle as a function of site index n for rate of adsorption and desorption λ = 16, loading θ = 0.2, and various system sizes S.

Substituting this expression and (25) in (24) yields
Again we find an exponential decrease but now with a characteristic length of ∆ = 1/[ln(y + 1) − ln(y + 1 − 2θ)]. For high loadings (θ → 1) we find ∆ ≈ y/2. For low loading (θ → 0) we get ∆ ≈ (y + 1)/(2θ). So the characteristic length is larger for low loadings and higher rate of adsorption and desorption. Figure 2 shows some typical profiles. The straight line corresponds to the exponential decrease of Eq. (27). The result for finite system sizes can be understood from the fact that smaller systems are less reactive because there are fewer sites at which conversion can take place. As the reactivity must be equal to the number of desorbing converted particles, W des B 1 must be smaller for smaller S, and because B 1 = θ − A 1 the curves in Fig. 2 must start out at larger values of A 1 . The reactivity also equals W rx S n=1 A n . This means that S n=1 A n must be smaller for smaller S. Because A 1 is larger for smaller S, A n must decrease faster. Figure 2 confirms the slow convergence of the system to the limit S → ∞. At S = 100 the values of A n drop down to 2 · 10 −5 . At such low value one would not expect an influence of these sites on the kinetics, but one clearly sees differences between S = 100 and S → ∞ for all sites.
We would finally like to comment on three extensions of our model: adsorption and desorption at both ends, finite diffusion, and other reactions. A system with 2S sites and adsorption and desorption at both ends has a reactivity that is twice the one of a system with S sites and adsorption and desorption at just one end provided that S ≫ ∆. DMC simulation show that if S is near or smaller than ∆, then the system open at both ends is relatively more reactive.
Finite diffusion is expected to be equivalent to infinite diffusion as long as it is much faster than the other reactions. If this is not the case, then it may become a rate determining step. The system then becomes transport limited. As a consequence the reactivity can only go down. We have seen that in our simulations. [10] We expect that our results will change only little when we change the details of the reaction or when interaction between the particles are included. [16] Our results are a consequence of the non-passing of the particles and of the fact that particles that stay longer in the system have a higher probability of being converted. We therefore also do not expect our results to change when we change our model of discrete sites to some continuous model.