**2.2 Studies on real molecular systems**

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

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

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

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

*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*

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

of both the species.

*Zeolites - New Challenges*

levitation effect [35].

**Figure 10.**

**Figure 11.**

**60**

*natural logarithmic scale.*

disappearance of the maximum in the zeolite.

shifted to lower values of guest diameter.

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

Simulations of pentane isomers, *n*-pentane, and isopentane in AlPO-5, which has 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 what one expects.

## **2.3 Experimental verification of the diffusivity maximum**

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

*n*-pentane, isopentane, and neopentane. The experimental QENS spectra are given in **Figure 12** [42, 43]. Also shown are the variation in half width at half maximum (HWHM) as a function of *Q<sup>2</sup>* . From the broadening of the spectra as a function of *Q2* , one obtains the diffusivity. The HWHM increases fastest for neopentane, followed by isopentane and last is *n*-pentane. These suggest that neopentane has the highest diffusivity, followed by isopentane and *n*-pentane in that order.

#### **2.4 Hexane isomers in faujasite**

Diffusion of hexane isomers in zeolite NaY was carried out with the help of MD technique. Linear hydrocarbons *n*-hexane (nC6), singly branched isomers 2-methyl pentane (2MP) and 3-methyl pentane (3MP) as well as the doubly branched isomers 2,2-dimethyl butane (22DMB) and 2,3-dimethyl butane (23DMB) were studied [44]. The calculated adsorption energies from MD of different isomers of hexane are listed in **Table 1**.

There is little differene in the adsorption energies of the isomers. Hence, it is difficult to separate the isomers from each other using a method based on adsorption. Instead, a kinetic-based approach might be helpful for separating the mixtures consisting of hexane isomers. In **Figure 13** the mean squared displacements of the isomers are shown for 2.25 ns. The lines are all straight suggesting good statistics [44]. The diffusivities of the isomers at these temperatures are listed in **Table 2**.

The cross-sectional diameter of the isomers can be computed from the geometry of the isomers. These are listed in **Table 3** along with the *γ* values for the isomers [44]. As can be seen except for *n*-hexane other isomers are all in the anomalous regime.

#### **Figure 12.**

*(a) Spectra of QENS obtained for neopentane at 300 K for different values of the wave vector transfer Q and (b) HWHM vs.* Q*<sup>2</sup> corresponding to the translational motion of the pentane isomers in zeolite NaY at 300 K, neopentane (triangles), isopentane (squares), and n-pentane (circles).*

From the Arrhenius plots of diffusivity (see **Figure 14**), we have obtained the activation energies [44]. These are listed in **Table 4** [44]. It is seen that the activation energies of *n*-hexane is the highest at 11.2 kJ/mol. 2MP has an activation energy

*Molecular diameter perpendicular to the long axis (σ*⊥*) and the levitation parameter (γ) values of all hexane*

**Isomer 250 K 300 K 330 K 350 K 400 K** *n*-Hexane 1.43(0.24) 3.59(0.35) 5.07(0.40) 7.27(0.61) 10.47(0.50) 2-Methylpentane 1.79(0.09) 3.17(0.21) 4.58(0.50) 6.44(0.55) 9.78(0.57) 3-Methylpentane 2.04(0.12) 3.92(0.24) 5.49(0.42) 6.23(0.44) 10.44(0.66) 2,3-Dimethylbutane 2.57(0.19) 4.68(0.32) 6.63(0.30) 7.26(0.61) 10.60(0.8) 2,2-Dimethylbutane 2.91(0.38) 5.67(0.29) 7.39(0.39) 7.75(0.46) 11.43(0.38)

**Isomer** *σ***<sup>⊥</sup> (Å)** *γ n*-Hexane 6.905 0.69 2-Methylpentane 8.09 0.80 3-Methylpentane 9.55 0.95 2,3-Dimethylbutane 9.96 0.98 2,2-Dimethylbutane 10.31 1.02

**Figure 13.**

**Table 2.**

**Table 3.**

**Figure 14.**

**63**

*Mean square displacement of hexane isomers in zeolite NaY at 250 K.*

*Anomalous Diffusivity in Porous Solids: Levitation Effect*

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

*Diffusion coefficients of hexane isomers in zeolite Y.*

*isomers, when adsorbed in zeolite NaY.*

*Arrhenius plot of various hexane isomers in zeolite NaY.*


#### **Table 1.**

*Adsorption energies E*ads *of hexane isomers in zeolite NaY.*

*Anomalous Diffusivity in Porous Solids: Levitation Effect DOI: http://dx.doi.org/10.5772/intechopen.92685*

**Figure 13.** *Mean square displacement of hexane isomers in zeolite NaY at 250 K.*


#### **Table 2.**

*n*-pentane, isopentane, and neopentane. The experimental QENS spectra are given in **Figure 12** [42, 43]. Also shown are the variation in half width at half maximum

, one obtains the diffusivity. The HWHM increases fastest for neopentane, followed by isopentane and last is *n*-pentane. These suggest that neopentane has the

Diffusion of hexane isomers in zeolite NaY was carried out with the help of MD technique. Linear hydrocarbons *n*-hexane (nC6), singly branched isomers 2-methyl pentane (2MP) and 3-methyl pentane (3MP) as well as the doubly branched isomers 2,2-dimethyl butane (22DMB) and 2,3-dimethyl butane (23DMB) were studied [44]. The calculated adsorption energies from MD of different isomers of

There is little differene in the adsorption energies of the isomers. Hence, it is difficult to separate the isomers from each other using a method based on adsorption. Instead, a kinetic-based approach might be helpful for separating the mixtures consisting of hexane isomers. In **Figure 13** the mean squared displacements of the isomers are shown for 2.25 ns. The lines are all straight suggesting good statistics [44]. The diffusivities of the isomers at these temperatures are listed in **Table 2**. The cross-sectional diameter of the isomers can be computed from the geometry of the isomers. These are listed in **Table 3** along with the *γ* values for the isomers [44]. As can be seen except for *n*-hexane other isomers are all in the anomalous regime.

*(a) Spectra of QENS obtained for neopentane at 300 K for different values of the wave vector transfer Q and (b) HWHM vs.* Q*<sup>2</sup> corresponding to the translational motion of the pentane isomers in zeolite NaY at 300 K,*

**Isomer** *Eads* **(kJ/mol)** *n*-Hexane 46.7 2-Methylpentane 45.9 3-Methylpentane 46.4 2,3-Dimethylbutane 45.9 2,2-Dimethylbutane 45.8

*neopentane (triangles), isopentane (squares), and n-pentane (circles).*

*Adsorption energies E*ads *of hexane isomers in zeolite NaY.*

highest diffusivity, followed by isopentane and *n*-pentane in that order.

. From the broadening of the spectra as a function of

(HWHM) as a function of *Q<sup>2</sup>*

*Zeolites - New Challenges*

**2.4 Hexane isomers in faujasite**

hexane are listed in **Table 1**.

*Q2*

**Figure 12.**

**Table 1.**

**62**

*Diffusion coefficients of hexane isomers in zeolite Y.*


#### **Table 3.**

*Molecular diameter perpendicular to the long axis (σ*⊥*) and the levitation parameter (γ) values of all hexane isomers, when adsorbed in zeolite NaY.*

From the Arrhenius plots of diffusivity (see **Figure 14**), we have obtained the activation energies [44]. These are listed in **Table 4** [44]. It is seen that the activation energies of *n*-hexane is the highest at 11.2 kJ/mol. 2MP has an activation energy

**Figure 14.** *Arrhenius plot of various hexane isomers in zeolite NaY.*


#### **Table 4.**

*Activation energies E*<sup>a</sup> *of hexane isomers in zeolite NaY.*

of 9.5 kJ/mol followed by 3MP (8.8 kJ/mol). The doubly branched isomers 23DMB (7.8 kJ/mol) and 22DMB (7.4 kJ/mol) have the lowest activation energy. These trends in the activation energies are according to what one expects based on the levitation effect.

**Figure 16.**

**Table 5.**

**Table 6.**

**65**

*Change of order of diffusivity for various hexane isomers in zeolite NaY.*

*Anomalous Diffusivity in Porous Solids: Levitation Effect*

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

**Molecules inverted T***inv* **(K)** 2MP/nC6 300 MP/nC6 347 DMB/nC6 387 DMB/nC6 418 DMB/3MP 528 DMB/2MP 549 MP/2MP 586 DMB/2MP 613 DMB/3MP 627 DMB/23DMB 1172

*Inversion temperatures for diffusivities (T*inv*) of hexane isomers inside zeolite NaY.*

*Different regions according to order of diffusivities of hexane isomers in zeolite NaY.*

**Region Temp range (K) Order of diffusivities**

I T < 300 nC6 < 2MP < 3MP < 23DMB < 22DMB II 300,347 2MP < nC6 < 3MP < 23DMB < 22DMB III 347,387 2MP < 3MP < nC6 < 23DMB < 22DMB IV 387,418 2MP < 3MP < 23DMB < nC6 < 22DMB V 418,528 2MP < 3MP < 23DMB < 22DMB < nC6 VI 528,549 2MP < 23DMB < 3MP < 22DMB < nC6 VII 549,586 23DMB < 2MP < 3MP < 22DMB < nC6 VIII 586,613 23DMB < 3MP < 2MP < 22DMB < nC6 IX 613,627 23DMB < 3MP < 22DMB < 2MP < nC6 X 6,271,172 23DMB < 22DMB < 3MP < 2MP < nC6 XI T > 1172 22DMB < 23DMB < 3MP < 2MP < nC6

The potential energy landscape of the various isomers as they diffuse during the passage through the 12-ring window of zeolite NaY is given in **Figure 15** [44]. It is seen that *n*-hexane alone has a maximum in the potential energy at the 12-ring window. 23DMB has a small maximum but overall it is a negative barrier in the vicinity of the window.

### *2.4.1 Kinetic-based separation of hexane isomers*

Arrhenius plots of diffusivities for various isomers cross each other at some temperature. Consider two isomers. They cross at some temperature referred to as the inversion temperature. Above the temperature if one isomer has a higher diffusivity, the same isomer below the inversion temperature will have the lower diffusivity among the two isomers. The Arrhenius plots of various isomers are plotted in **Figure 16** [44]. Various pairs of isomers cross each other at different temperatures. These are listed in **Table 5** [44]. Thus, from the table it is evident that pairs 2MP/*n*hexane have an inversion temperature of 300 K. As *n*-hexane has higher activation energy of the two, at T < 300 K, 2MP will exit from a column first and then *n*hexane. At T > 300 K, *n*-hexane will have higher diffusivity and exit from the column first, followed by 2MP. The order of exit of the various isomers from a zeolite single crystal is seen to be different at different temperatures. There are in all 11 regions, which correspond to different order in which the isomers will exit. The order of diffusivities or the order of exit of the various isomers is listed in **Table 6** [44]. As the temperature is increased from below 300 K upto 1172 K, the order of diffusivities of various isomers is given.

#### **Figure 15.**

*Potential energy landscape for hexane isomers in zeolite NaY at 250 K. these plots are obtained by averaging over all cage-to-cage jumps.*

*Anomalous Diffusivity in Porous Solids: Levitation Effect DOI: http://dx.doi.org/10.5772/intechopen.92685*

**Figure 16.**

of 9.5 kJ/mol followed by 3MP (8.8 kJ/mol). The doubly branched isomers 23DMB (7.8 kJ/mol) and 22DMB (7.4 kJ/mol) have the lowest activation energy. These trends in the activation energies are according to what one expects based on the

**Isomer E***<sup>a</sup>* **(kJ/mol)** *n*-Hexane 11.2 2-Methylpentane 9.5 3-Methylpentane 8.8 2,3-Dimethylbutane 7.8 2,2-Dimethylbutane 7.4

The potential energy landscape of the various isomers as they diffuse during the passage through the 12-ring window of zeolite NaY is given in **Figure 15** [44]. It is seen that *n*-hexane alone has a maximum in the potential energy at the 12-ring window. 23DMB has a small maximum but overall it is a negative barrier in the

Arrhenius plots of diffusivities for various isomers cross each other at some temperature. Consider two isomers. They cross at some temperature referred to as the inversion temperature. Above the temperature if one isomer has a higher diffusivity, the same isomer below the inversion temperature will have the lower diffusivity among the two isomers. The Arrhenius plots of various isomers are plotted in **Figure 16** [44]. Various pairs of isomers cross each other at different temperatures. These are listed in **Table 5** [44]. Thus, from the table it is evident that pairs 2MP/*n*hexane have an inversion temperature of 300 K. As *n*-hexane has higher activation energy of the two, at T < 300 K, 2MP will exit from a column first and then *n*hexane. At T > 300 K, *n*-hexane will have higher diffusivity and exit from the column first, followed by 2MP. The order of exit of the various isomers from a zeolite single crystal is seen to be different at different temperatures. There are in all 11 regions, which correspond to different order in which the isomers will exit. The order of diffusivities or the order of exit of the various isomers is listed in **Table 6** [44]. As the temperature is increased from below 300 K upto 1172 K, the order of

*Potential energy landscape for hexane isomers in zeolite NaY at 250 K. these plots are obtained by averaging*

levitation effect.

*Zeolites - New Challenges*

**Table 4.**

vicinity of the window.

*2.4.1 Kinetic-based separation of hexane isomers*

*Activation energies E*<sup>a</sup> *of hexane isomers in zeolite NaY.*

diffusivities of various isomers is given.

**Figure 15.**

**64**

*over all cage-to-cage jumps.*

*Change of order of diffusivity for various hexane isomers in zeolite NaY.*


#### **Table 5.**

*Inversion temperatures for diffusivities (T*inv*) of hexane isomers inside zeolite NaY.*


#### **Table 6.**

*Different regions according to order of diffusivities of hexane isomers in zeolite NaY.*
