| dc.description.abstract | The present chapter has attempted to study the temperature dependence of the levitation effect. It has been shown that this effect is most pronounced at lower temperatures and decreases in intensity at higher temperatures. Energies of activation calculated from Arrhenius plots indicate that this quantity is a minimum for the sorbate size corresponding to the anomalous peak in the D vs. 1/?² plot. Arrhenius plots corresponding to different sorbate sizes intersect, suggesting that temperature could be used as a tunable parameter in separations based on the difference in relative diffusivities of the components of a mixture.
Thus, it appears that the levitation effect persists even at higher sorbate concentrations. This implies that the sorbate-zeolite interactions continue to play a predominant role at these high sorbate concentrations where one would normally expect the sorbate-sorbate interactions to predominate. This result is of considerable significance since many chemical and biological processes occur at high sorbate concentrations. The present result suggests that one should take into account the possibility of existence of the levitation effect while trying to understand diffusion processes in these systems. Industrially, many catalytic reactions that involve the diffusion of products through the pores of zeolites and other porous solids as well as molecular sieve processes for separation of mixtures occur at high sorbate concentrations. The present results indicate that the levitation effect can exist in these systems as well. Further, it should be possible to make use of the levitation effect for separation of mixtures. We will discuss separations based on the levitation effect in Chapter 5.
Also, logarithmic plots of the evolution of the mean squared displacement (msd) with time indicate a distinct transition from the ballistic to the diffusive regime. An analysis of the transition times indicates that the potential energy surface “seen” by a larger sorbate (anomalous regime) might be “flatter”.
To the best of our knowledge, most separations reported in the literature are based on separation of components of a mixture, all of which belong to the linear regime. The present results indicate that it is possible to separate a component that belongs to the anomalous regime very selectively. This separation has been achieved at around 160 K, indicating that separations based on the levitation effect can be effected with a reasonable degree of selectivity especially at fairly low temperatures. It can be seen from Figure 5.4 that at the end of the simulation (after a run of 51 ns), the composition of the mixture that is present to the right of the semi-permeable wall is 17.24% of the 3.48 Å sorbate, 10.34% of the 4.96 Å sorbate and 72.42% of the 6.0 Å sorbate. The composition of this mixture after 30 ns of simulation consists of 16.0%, 8.0% and 76.0% of sorbates of size 3.48, 4.96 and 6.0 Å, respectively. This suggests that 30 ns may be sufficient to achieve the degree of selectivity that is reported in this study.
However, we would like to point out that this study has been carried out at fairly low sorbate concentrations. Recent studies have shown that the levitation effect persists at higher loadings also. It is, therefore, expected that such separations may also be achieved at higher concentrations. Also, we have carried out the simulations reported here using equilibrium molecular dynamics techniques. Since actual separation processes involve concentration gradients, it may be more pertinent to model these by the use of non-equilibrium molecular dynamics techniques.
Principal Conclusions of This Study:
It is suggested that the values of t? and t? should be chosen so as to avoid both the initial and the long-time regions. If the initial part of msd is considered in the estimation of D, then the resulting value will normally be higher than the true long-time D. Unfortunately, in the literature D is obtained from msd by fitting to a range in which t? is usually not very large. Therefore, the values reported in such studies are usually higher than the correct value. Neglect of the initial part ensures that the ballistic part of the curve is avoided. At large values of t, the statistics is poor. The final choice of t? will depend on the length T of the run.
Even if t? and t? are selected carefully, shorter the run length T, greater will be the overestimation of D over and above the correct value. Thus, the value of D from shorter blocks seems to result in an overestimation of the value of D.
Values of D from msd and vacf converge to the same value within the error in the calculation for sufficiently long runs. It is not clear beyond what value of T such convergence is assured.
For a guest-zeolite system with 10 particles, a run of length 1 ns is associated with an error of about 50%. The error reduces to 15–40% when the run length is increased by a factor of nine, i.e., for a 9 ns run.
There does not seem to be any effect of the pbc on the behavior of msd even when the system is about 50 Å in length. It is not clear whether this will still be true even when the simulated phase is a solid or when the edge length is shorter.
A steep or widely varying potential energy surface seems to give rise to a larger error in D. Thus, error in D is system dependent. Error in D for sorbate-zeolite system is about 2–4 times larger than the error for pure fluid.
The calculated statistical inefficiency (s) is around 150 ps, suggesting that block averages of the total potential energy, obtained from blocks longer than 150 ps, are uncorrelated.
For the runs reported here, there seems to be no effect of the starting configuration on the calculated values of D.
It is clear from the present study that the levitation effect reported earlier from this laboratory is not an artifact.
The present calculation suggests that accurate diffusion coefficients can be obtained if the runs are sufficiently long. The results indicate that the runs should be about 10 ns long or t = 9663.1* (for a sorbate of size 2.21 Å) where t* is the reduced time unit for a Lennard-Jones system. The error for such a run could be anywhere between 15 and 40% for a system consisting of ten particles confined to the cavities of a zeolite lattice. For a pure fluid, the errors are likely to be smaller. It is expected that the error for a pure fluid with 108 particles will be 5% for a run of 1 ns long while the error will be about 1% if the run is 10 ns long. So, for pure Lennard-Jones fluids it is seen that runs of 1 ns long can give a reasonable estimate of D.
The implications these results have on the values of D reported is now considered. In the literature, most of the simulations are usually only a few picoseconds long, although in recent times, with the availability of faster CPUs, somewhat longer runs of several hundred picoseconds with particles whose mass is close to that of argon and run lengths of 1 or 2 nanoseconds with xenon-like mass have also been reported. The present results indicate that the error for these could be anywhere between 50% to several hundred percent of the value of D reported. It is, therefore, necessary that the values of D reported and often compared with experimentally measured values of D should be interpreted with caution. The present results indicate that the errors in D can depend on the nature of the system, i.e., the underlying potential energy surface and consequently, the error can be different for different zeolites or guest-host systems. Further, the errors are usually larger than those for the system without the host. | |
