**2.1 Effect of temperature on the levitation effect**

The diffusivity maximum or the levitation effect is a consequence of the existence of dispersion forces. If the attractive part of the guest-zeolite interaction is switched off, then the diffusivity maximum disappears. This is shown in **Figure 8** [28]. Temperature plays an important role in the behavior of the diffusion coefficient as a function of the guest diameter.

At very higher temperatures the diffusivity maximum altogether disappears. This can be seen in **Figure 9** [31]. What determines the temperature at which the diffusivity maximum will vanish? It is the strength of interaction between the guest and the zeolite. At relatively higher temperatures when *kBT* > > *Ugz* the diffusivity maximum vanishes and only a monotonic dependence on guest diameter is seen.

At low temperatures, the diffusivity maximum is very pronounced with the diffusivity of the guest of 6.0 Å (in zeolite Y) showing several orders of magnitude higher value than the diffusivity of the 4.96 Å guest. This is shown in **Figure 9** [31].

#### **Figure 8.**

The activation energy for diffusion can be obtained from an Arrhenius plot of log(D) vs. 1/T, where T is the temperature and D is the diffusivity. Variation of activation energy as a function of the guest diameter has been plotted in **Figure 6** [30]. It is seen that the activation energy is higher for the linear regime than for the anomalous regime guests. It is also seen that activation energy is maximum for the size with minimum diffusivity and minimum for the guest size in the AR with

The observed behavior has been termed the levitation effect (LE) as its origin is in the dispersion forces, which cancel each other leading to reduced forces on the guest molecule with maximum diffusivity. Unlike diffusion of p-xylene and o- and m-xylenes whose diffusivities are controlled by the steric repulsion and therefore only p-xylene manages to diffuse, here the diffusivities are controlled by the

Arrhenius plots for two sizes, namely, 4.96 Å and 6.0 Å are shown in **Figure 7** [28]. The smaller sized guest atom has a higher slope and activation energy than the larger sized guest atom. The activation energies for the smaller and larger sized

dispersion forces, which are always attractive in nature.

guest atoms are respectively 5.89 kJ/mol and 3.26 kJ/mol.

*Variation of activation enegy E*<sup>a</sup> *as a function of vdw radius of guests, σ in zeolite NaY.*

*Arrhenius plot of ln D for two different values of γ = 0.67 and 0.89 in zeolite NaY.*

maximum diffusivity.

*Zeolites - New Challenges*

**Figure 6.**

**Figure 7.**

**58**

*Variation of diffusion coefficient D as a function of 1/σ*<sup>2</sup> *in (a) zeolite NaCaA and (b) zeolite NaY, where σ is the van der Waals diameter of guest atoms. This plot is obtained without taking into account the dispersion force between guest and host atoms during molecular dynamics simulation. This clearly shows that the levitation effect is a resulting phenomenon from dispersion force of attraction.*

#### **Figure 9.**

*Diffusion coefficient D as a function of 1/σ*<sup>2</sup> *in zeolite NaY at different temperatures. It is depicted that the maxima in diffusivity disappear with increase in temperature.*

#### *Zeolites - New Challenges*

At very low temperatures, the diffusivity maximum is seen to be very pronounced [32]. This can be seen from **Figure 10**. The difference between the anomalous regime guest and the linear regime guest is now several orders of magnitude. This can not be easily utilized in the practice because of the low diffusivities of both the species.

**2.2 Studies on real molecular systems**

*DOI: http://dx.doi.org/10.5772/intechopen.92685*

*Anomalous Diffusivity in Porous Solids: Levitation Effect*

what one expects.

**61**

Until now simulations have been on monatomic guest species diffusion in the pores of the zeolites. These guest molecules are not of interest in real laboratory or industry. Simulations were therefore carried out on hydrocarbons molecules within zeolites to see if the observed anomalous diffusion can be observed in these real hydrocarbons. Simulations were carried out on pentane isomers: *n*-pentane,

isopentane, and neopentane. These are anisotropic molecules. Hence, the dimensions of these molecules along different directions are different. For *n*-pentane the direction that is relevant is the dimension of the molecule perpendicular to its long axis. The relevant dimension for the other molecule, which is isopentane, is also the dimension perpendicular to its long axis. For neopentane, which is tetrahedral in shape, the molecular diameter is the relevant dimension. There have been attempts to compute and list the various dimensions of hydrocarbon and other molecules [36]. These can

Simulations of pentane isomers, *n*-pentane, and isopentane in AlPO-5, which has

The diffusivity of a species changes with its mass as well as other parameters such as size, temperature, etc. In simulations, the diameter of the diffusing species were changed without changing its mass. Ideally, an experimental verification of the levitation effect should do the same, that is change the diameter without changing the mass. But in real laboratory this appears almost impossible. However, Dr. S.G.T. Bhat during one of our discussions mentioned that this indeed is possible. He suggested use of isomers of a hydrocarbon all of which will have the same mass but differ in their cross-sectional diameter [39]. The choice of the experiments was also crucially important. Different techniques of measuring the diffusivity such as uptake, NMR, ZLC, or QENS yield different values of for the diffusivity of the same species. Kärger and coworkers have investigated the reasons for this [40, 41]. They have suggested that this is due to the difference in the sampling time and length scales. As MD sample over picoseconds to nanoseconds, a technique which samples for similar time scale would be ideal. As QENS samples over the same period, we choose to carry experiments with this technique. We chose zeolite NaY with three isomers of pentane, namely, *n*-pentane, isopentane, and neopentane. The diameters of these were calculated from their geometry and Lennard-Jones interaction parameters. Knowing the 12-ring window diameter of faujasite, we computed the levitation parameters *γ* for these isomers, which are 0.71, 0.86, and 0.96 for

be helpful in computing the *γ* values for various guest-zeolite systems.

**2.3 Experimental verification of the diffusivity maximum**

one-dimensional channels, have been reported [37]. These studies show that isopentane has a higher diffusivity as compared to *n*-pentane. Thus, anomalous diffusion is seen even in AlPO-5. The diffusivities obtained are 2*:*<sup>7</sup> <sup>10</sup><sup>8</sup> *<sup>m</sup>*<sup>2</sup>*=<sup>s</sup>* and <sup>3</sup>*:*<sup>33</sup> <sup>10</sup><sup>8</sup> *<sup>m</sup>*<sup>2</sup>*=<sup>s</sup>* for *<sup>n</sup>*-pentane and isopentane respectively at 300 K. The potential parameters employed in this study were the unified potential parameters of Jorgensen [38]. The potential parameters are very similar to the OPLS parameters later proposed by Jorgensen. Masses of the isomers are identical and therefore the difference in the diffusivity arises from the difference in *γ*. For AlPO-5, it is seen that the *γ* values are 0.71 and 0.88 respectively for *n*-pentane and isopentane. The value of *γ*, which separates linear and anomalous regime, is around 0.75. This boundary will vary and depends on the zeolite but as a rule of thumb, a value of 0.75 may be used. The value of 0.71 lies in the linear regime while 0.88 lies in the anomalous regime. Thus, *n*-pentane with lower *γ* has a lower diffusivity, which is

Smit and coworkers have reported a study on carbon nanotubes (CNTs). They investigated diffusion of methane in CNTs of different diameters. They found that the diffusion coefficient is maximum in the CNTs with similar diameter as the methane. They also carried out a simulation at higher temperatures when the height of the diffusivity maximum decreased and eventually disappeared similar to the disappearance of the maximum in the zeolite.

Many of these simulations have been carried out with the zeolite framework fixed. Will the diffusivity maximum persist when the framework is flexible? For this, simulations with flexible framework were carried out and the results are shown in **Figure 11** [33]. As can be seen the diffusivity maximum persists in spite of the framework flexibility. The height of the maximum is marginally lower and slightly shifted to lower values of guest diameter.

Kar and Chakravarty reported instantaneous normal mode analysis of guest of different diameters to understand the levitation effect. They could reproduce the velocity autocorrelation functions of various guest molecules in zeolite NaY [34]. Bhattacharyya and coworkers have carried out a mode coupling analysis of the levitation effect [35].

#### **Figure 10.**

*Diffusion coefficient D as a function of 1/σ*<sup>2</sup> *in zeolite NaY at 10 K. The diffusivity enhances to 17 orders in natural logarithmic scale.*

**Figure 11.** *Plot of diffusion coefficient D as a function of σ in flexible zeolite NaA at 140 K.*
