We now discuss the results of our KMC simulations of benzene mobility in Na-Y (Si:Al>=2.0). After discussing the calculated OCFs and mean square displacements, we compare the information they provide regarding benzene mobility and zeolite structure.
Benzene in Na-Y (Si:Al=2.0). Since BOR in Na-Y arises from a variety of fundamental hopping processes, KMC simulations are useful for determining which hopping process, if any, controls BOR rates. Figure 11 shows the KMC calculated OCF for benzene in Na-Y (Si:Al=2.0) at T = 300 K comparing two computational approaches. The ``direct" calculation involves using KMC to evaulate the ensemble average. The ``two-step" approach involves first using KMC to calculate poff(t), the probability at time t that benzene gives off-diagonal intensity, neglecting spin diffusion, in two-dimensional exchange NMR; followed by substituting poff(t) into c(t) = 1 - 4poff(t)/3. The two methods give essentially exact agreement. The OCF in Fig. 11 exhibits exponential decay, which arises from pseudo-isotropic motion giving full orientational randomization.
In order to extract quantitative BOR rates, we show ln|C(t)| in Fig. 12. The difficulty in using Monte Carlo to give proper sign cancellation for long times is evident in Fig. 12. Analyzing the short time slopes in Fig. 12 indicates that kBOR = 1.3/dt(SII), where kBOR is the BOR rate and dt(SII) is the mean residence time at the SII site. The factor of 1.3 can be understood as follows. For short times the OCF satisfies C(t) ~= 1 - 4khopt where khop = k(SII) = 1/dt
Benzene in Na-Y (Si:Al=3.0). In this section we demonstrate that studying BOR in Na-Y with half Na(II) occupancy can unambiguously disentangle rates of intracage and intercage motion. Half Na(II) occupancy corresponds in Fig. 7 to Si:Al=3.0, although this point is not crucial as previously discussed. It is crucial, however, to determine the effect of different Na(II) occupancy patterns on benzene mobility. Figure 13 shows ln|C(t)| for benzene in Na-Y (Si:Al=3.0) at T = 300 K, comparing several Na(II) occupancy patterns. The cage disorder system has two Na(II) ions per supercage, while the cell disorder systems have the patterns (1, 0, 5, 2, 0)cell1 and (0, 3, 3, 1, 1)cell2, where (n0, n1, n2, n3,n4) signifies n0 supercages with zero Na(II) ions, n1 supercages with one Na(II) ion, etc. The results in Fig. 13 demonstrate remarkable insensitivity to particular occupancy patterns, suggesting that such an OCF should be measurable with, e.g., two-dimensional exchange NMR on benzene in Ca-Y with half Ca(II) occupancy. Below we interpret the nature of BOR in Na-Y (Si:Al=3.0) and suggest an explanantion for the predicted insensitivity to Na(II) disorder.
The OCF in Fig. 13 exhibits biexponential decay resulting from incomplete Na(II) occupancy. Considering first cage disorder, each supercage contains two \siisp binding sites. Benzene hopping between these two sites gives incomplete decay of the OCF up to ca. 0.12 microseconds because the tetrahedral symmetry that was present with full Na(II) occupancy is broken with half Na(II) occupancy. In order to execute full orientational randomization with half Na(II) occupancy, benzene must visit enough supercages to sample all four tetrahedral orientations. The long time BOR rate is thus controlled by the rate of cage-to-cage motion.
Benzene in Na-Y (Si:Al=3.8). In the preceding section we found that cage and cell disorder give very similar OCFs for benzene in Na-Y (Si:Al=3.0). While this insensitivity to particular Na+ distributions lends credibility to the predicted OCF, it makes detecting different occupancy patterns very difficult. This is important because measuring the distribution of Na+ ions is closely related to measuring Al distributions in disordered zeolites, which remains challenging to modern characterization methods. In this section we demonstrate that studying BOR in Na-Y with one quarter Na(II) occupancy can clearly distinguish between cage and cell disorder.
One quarter Na(II) occupancy corresponds on average to one Na(II) per supercage, and in Fig. 7 to Si:Al=3.8. Figure 14 shows ln|C(t)| for benzene in Na-Y (Si:Al=3.8) at T = 300 K, showing qualitative sensitivity to Na(II) occupancy patterns. Since cage disorder entails precisely one Na(II) in each cage, benzene must visit adjacent supercages to commence orientational randomization. As such, the OCF exhibits single exponential decay controlled by the SII->W rate coefficient. Indeed, for cage disorder kBOR = 2k(SII->W) as was found for long time BOR in Na-Y (Si:Al=3.0).
The cell disorder system in Fig. 14 has a (3, 2, 3, 0, 0) occupancy pattern, enabling partial intracage BOR because three supercages contain two Na(II) cations. Since full BOR in the cell disorder system requires cage-to-cage motion, biexponential decay arises as was predicted for benzene in Na-Y (Si:Al=3.0). Figure 14 also predicts a slight difference in he diffusion coefficients obtained from cage and cell disorder. The striking conclusion drawn from Fig. 14 is that studying BOR in Na-Y with one quarter Na(II) occupancy can clearly distinguish between qualitatively different Na+ distributions.
Figure 15 shows KMC calculated self diffusion coefficients for benzene in Na-Y for various temperatures and Si:Al ratios, comparing cage and cell disorder patterns designated by large filled symbols and small empty symbols, respectively. The results in Fig. 15 predict that benzene diffusion coefficients increase monotonically with the Na-Y Si:Al ratio, and that the increase is gentle for Si:Al=2.0--4.0 and more steep near Si:Al=5.0. The diffusion coefficients for Si:Al=2.0--4.0 are given by Dself = kcagea2/6 where kcage = 2k(SII->W) and a ~= 11 angstroms is the cage-to-cage length, as explained in our recent theoretical studies of benzene diffusion in Na-Y. Figure 15 also indicates that the calculated diffusion coefficients are virtually insensitive to particular Na(II) distributions. As such, our calculations suggest that diffusion coefficients alone cannot distinguish between cage and cell cation distributions.
Underlying these diffusion coefficients are mean square displacements which may contain more information. Figure 16 shows benzene mean square displacements in Na-Y at T = 300 K for Si:Al=3.0, 3.8 and 4.8. The Si:Al ratios 3.0 and 3.8 use cage disorder giving two and one Na(II) per supercage, respectively, while Si:Al=4.8 gives one Na(II) per unit cell, thus eliminating cation disorder. The only curious aspect in Fig. 16 is the sizable y-intercept for Si:Al=4.8, suggesting rapid diffusion at short times in this system.
To pursue this, we repeated the KMC calculation for Si:Al=4.8 using logarithmic binning to resolve short time dynamics. The log-log mean square displacement plot in Fig. 17 shows KMC data from two long runs (thin lines), and a smoothed version of the longer KMC run (thick line) portraying the converged mean square displacement. In linear regions of such a log-log plot, unit slope indicates normal diffusion with the y-intercept giving log10(6Dself). Figure 17 shows an early diffusive regime where benzene jumps rapidly among supercages containing no Na(II) ions, eventually becoming trapped at the lone SII site. The early diffusive regime is analogous to benzene mobility in siliceous-Y, although the rapid cage-to-cage hopping mechanism in our model is not specifically tailored for that system. The trap duration is the mean residence time in the supercage containing Na(II), and the plateau <R2(t)> value is the y-intercept in Fig. 16. Normal diffusion ensues for t > ttrap giving the mean square displacement in Fig. 16 and the diffusion coefficient in Fig. 15. The results in Fig. 17 indicate that mean square displacements, while providing diffusion coefficients, can contain other interesting information as well.
To determine whether measuring mean square displacements can probe cation distributions in Na-Y, we show in Fig. 18 log-log benzene mean square displacement plots for Si:Al=3.0 and 3.8 comparing cage and cell disorder. These results contain interesting information about rapid and sluggish diffusion as was found in Fig. 17, but also demonstrate qualitative insensitivity to particular Na(II) occupancy patterns. Our results therefore suggest that diffusion measurements cannot be used to infer cation distributions in Na-Y.