**2. Diffusivity of guest molecules confined to the pores from MD simulations**

In order to understand the diffusion of hydrocarbon and other molecules within the confined spaces of the zeolite, it is essential to carry out an investigation into diffusivity of guest molecules within the confined space provided by a zeolite or other porous solids. Such a study is essential for an understanding of the process of separation of hydrocarbon as well as other mixtures.

One of the early studies investigating the diffusivity of guest molecules in zeolites was the diffusion of xenon in zeolite Y and argon in NaCaA [27]. The diffusivity is likely to be strongly influenced by the bottleneck for diffusion. In the case of xenon the bottleneck is the 12-ring window, which has a diameter of around 7.8 Å. In the case of argon the bottleneck is an 8-ring window, which has a diameter of around 4.5 Å. The ratio of the bottleneck to the molecular/atomic diameter for xenon-NaY system is 7.0/4.1 = 1.70 while in the case of argon-NaCaA it is 4.0/ 3.405 = 1.17. Although the diameters of the windows are approximate, it is clear that the window diameter is significantly larger than the diameter of xenon while this is not true for argon where the diameter of the window is only slightly larger than the argon diameter. From these it is evident that xenon in NaY should have a higher diffusvity than argon in NaCaA.

From MD, it was found that xenon in NaY has a diffusivity of 0*:*<sup>19</sup> <sup>10</sup><sup>8</sup> <sup>m</sup><sup>2</sup>*=*<sup>s</sup> while argon in NaCaA has a diffusivity of 0*:*<sup>9</sup> <sup>10</sup><sup>8</sup> m2*=*s. As the diffusivity of

that many new and novel aspects of zeolites are still being discovered and zeolites

Zeolites are porous aluminosilicates capable of accommodating molecules within the pores. They are well known for their catalytic, ion-exchange, and separation properties. They are widely used in petrochemical industries for processing hydrocarbons. Hydrocarbon cracking, transformation, isomerization, etc. are achieved with the help of zeolites [14, 15]. Zeolites are also used in separating hydrocarbon molecules of various sizes [16]. Larger hydrocarbons such as C15 and with still higher number of carbon atoms diffuse slowly through the pores and hence reach the bottom of a zeolite column last. Small molecules such as C1-C5 diffuse fast and exit from the column first [17, 18]. Other molecules of intermediate size have values of diffusivity in between those of C15 and C1-C5 and exit at intermediate times. Thus the various fractions from crude can be separated. This separation is much

Another application of zeolites is its use for ion-exchange and water softening

Apart from the use of zeolite for separation based on the size of the molecule, it is also of considerable use for separations based on the shape of the molecules. This property of zeolites is often referred to as shape selectivity [21]. The pore dimensions of zeolites, which are not always of regular shape, make this possible. An oft quoted example is the separation of xylenes (p-, m-, and o-xylenes) using silicate or ZSM-5 [22, 23]. At sufficiently high temperatures, o-xylene will convert to p-xylene

Zeolites and other host materials also exhibit interesting properties. They exhibit

In order to understand the diffusion of hydrocarbon and other molecules within the confined spaces of the zeolite, it is essential to carry out an investigation into diffusivity of guest molecules within the confined space provided by a zeolite or other porous solids. Such a study is essential for an understanding of the process of

One of the early studies investigating the diffusivity of guest molecules in zeolites was the diffusion of xenon in zeolite Y and argon in NaCaA [27]. The diffusivity is likely to be strongly influenced by the bottleneck for diffusion. In the case of xenon the bottleneck is the 12-ring window, which has a diameter of around 7.8 Å. In the case of argon the bottleneck is an 8-ring window, which has a diameter of around 4.5 Å. The ratio of the bottleneck to the molecular/atomic diameter for xenon-NaY system is 7.0/4.1 = 1.70 while in the case of argon-NaCaA it is 4.0/ 3.405 = 1.17. Although the diameters of the windows are approximate, it is clear that the window diameter is significantly larger than the diameter of xenon while this is not true for argon where the diameter of the window is only slightly larger than the argon diameter. From these it is evident that xenon in NaY should have a higher

From MD, it was found that xenon in NaY has a diffusivity of 0*:*<sup>19</sup> <sup>10</sup><sup>8</sup> <sup>m</sup><sup>2</sup>*=*<sup>s</sup> while argon in NaCaA has a diffusivity of 0*:*<sup>9</sup> <sup>10</sup><sup>8</sup> m2*=*s. As the diffusivity of

window effect, single file diffusion, and levitation effect [24–26]. Here we will focus on the levitation effect, which refers to the dependence of diffusivity on the

**2. Diffusivity of guest molecules confined to the pores from MD**

separation of hydrocarbon as well as other mixtures.

[19, 20]. Ions such as Ca2+ can be exchanged with Na+ already present in the zeolites, thus removing these ions from water. Zeolites are therefore used in deter-

continues to be an exciting field of research with a bounty of surprises.

more energy efficient as compared to separation by distillation.

gents for washing clothes.

*Zeolites - New Challenges*

and ZSM-5 also acts as a catalyst.

guest or diffusant diameter.

diffusvity than argon in NaCaA.

**54**

**simulations**

argon in NaCaA is higher than xenon in NaY, further investigations were carried out to find the reasons for this. To start with the energy barrier at the bottleneck was computed. This is shown in **Figure 1** [27]. From the figure it is seen that the energy barrier for xenon at the window in Y is positive while the barrier at the window is negative for argon in A zeolite. This explains why the diffusivity of argon in A zeolite is higher than xenon in Y zeolite. The trends seen in the observed barrier appears to be due to the strong interaction of argon with the oxygens of the 8-ring window. As argon is about the same diameter as the window, its strength of interaction with the oxygens is optimum being close to ϵ, which occurs at a distance at which the Lennard-Jones curve is minimum in energy. This is not the case for xenon in zeolite Y where xenon can be close to only some of the oxygens of the 12-ring window. This is the first indication that nongeometrical factors can influence the diffusivity. This study shows that sorbate-zeolite interaction plays an important role.

This study suggests that an understanding of diffusivity as a function of the diameter of the guest species might show something interesting. Such a study was carried out and the results were indeed found to be interesting [28]. A molecular dynamics study of monatomic guest molecules confined to zeolite NaY and NaCaA were carried out in which the diameter of the guest molecule was varied. The diffusivities of the guest species was computed from the time evolution of the mean square displacements. A plot of diffusivity as a function of the reciprocal of square of the guest diameter is shown in **Figure 2** for guests in both zeolites Y and A [28]. It is seen that the diffusivities decrease linearly with increase in the reciprocal of the square of the diameter of the guest molecule for small diameters. This is referred to as the linear regime (LR). As the diameter increases, it is seen that the diffusivity suddenly increases and later decreases sharply exhibiting a maximum in diffusivity. This is referred to as the anomalous regime (AR). This increase followed by a decrease in diffusivity was surprising and needed further investigations.

As can be seen the location of the guest diameter at which the maximum occurs is different in both zeolite Y and A (see **Figure 3** [28]). In order to understand the reasons for the maximum in diffusivity we have tried to search in literature any report that refers to such an observation. Derouane and coworkers have reported a finding arrived at through a theoretical analysis. They showed that the nesting

#### **Figure 1.**

*Potential energy landscape of (a) xenon in zeolite NaY at 190 K and (b) argon in zeolite NaCaA at 140 K. the energy landscapes are computed from molecular dynamics simulations.*

#### **Figure 2.**

*Diffusion coefficient of guest particles is plotted as a function of 1/σ<sup>2</sup> , where σ is the guest diameter. D for guest confined to (a) zeolite NaCaA at 140 K and to (b) zeolite NaY at 190 K temperature, where 1/σ<sup>2</sup> is the inverse square of the vdw radius of guest particles.*

#### **Figure 3.**

*Diffusion coefficient, D as a function of vdw radius of guest particles, σ is plotted in (a) zeolite NaCaA at 140 K and in (b) zeolite NaY at 190 K temperature.*

effect can lead to *floating molecules* when the pore diameter is comparable to the molecular diameter. Earlier Kemball found that when guest molecules are sorbed inside host materials such as zeolites or other adsorbents it is seen that some undergo little loss of entropy. In such systems, he suggested, the guest molecules will exhibit high mobility or *superdiffusivity*.

In order to obtain a better understanding, we define a dimensionless parameter

$$\gamma = \frac{2 \times 2^{1/6} \sigma\_{\rm gr}}{\sigma\_w} \tag{1}$$

leading to rather small force on the guest due to the zeolite. This situation is akin to the guest being a free particle even when confined within the zeolite and therefore

*Schematic figure indicating the position of bigger and smaller particles while passing through the zeolite window. It is shown that the bigger particle passes through the symmetry position, the center of the window, whereas the smaller particle is near to the periphery of the window. Therefore, the forces along a given direction is equal and opposite to that exerted on the bigger particle from the diagonally opposite direction, resulting in higher diffusion than for the smaller particle, which is attracted to the periphery, and therefore experiences a net*

*Diffusion coefficient, D is plotted as a function of levitation parameter, γ (see text) in (a) zeolite NaCaA at 140*

When *γ* < <1 the guest passes through the window at the periphery. These points do not possess inversion symmetry and therefore there is no cancelation of forces exerted on the guest by the zeolite. This leads to lower diffusivity of these

The reason for the observed maximum in diffusivity arises from the lowered force on the guest molecule as compared to the smaller guest molecule, which encounters a higher force on itself due to the zeolite. These translate to a less undulating potential energy landscape with shallower minima and maxima for the larger guest molecule for which *γ* is close to unity. In the case of smaller guest molecule the larger force implies a highly undulating potential energy landscape with deep valleys and high mountains. However, in a recent report the lower force on the guest molecule at the window has not been found [29]. More studies are required to understand the origin of the observed diffusivity maximum.

has a high diffusivity.

*attraction. This is shown in both 2D and 3D.*

guest molecules.

**57**

**Figure 4.**

**Figure 5.**

*K and in (b) zeolite NaY at 190 K temperature.*

*Anomalous Diffusivity in Porous Solids: Levitation Effect*

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

where the numerator gives the distance at which the interactions between the guest and the zeolite atoms are optimum, that is, when this gives an interaction energy of ϵ*gz*. The denominator is the window radius. Thus, the dimensionless parameter suggests that when these two are equal (*γ* = 1) and when they are not equal (*γ* < <1). We now plot the diffusivity as a function of *γ*. This is shown in **Figure 4** [28]. We see that the maximum is seen when *γ* is between 0.9 and 1.0 for *both* the zeolites. Thus, the guest-zeolite interaction is optimum at 21*<sup>=</sup>*<sup>6</sup>*σgz* and when this equals *σw*/2, the diffusivity maximum is seen. This situation when *γ* is close to unity is illustrated in **Figure 5** along with the situation when *γ* < <1. Now when *γ* is close to unity the guest molecule is passing through the center of the window. Such a position has inversion symmetry, which leads to mutual cancelation of forces

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

#### **Figure 4.**

*Diffusion coefficient, D is plotted as a function of levitation parameter, γ (see text) in (a) zeolite NaCaA at 140 K and in (b) zeolite NaY at 190 K temperature.*

#### **Figure 5.**

effect can lead to *floating molecules* when the pore diameter is comparable to the molecular diameter. Earlier Kemball found that when guest molecules are sorbed inside host materials such as zeolites or other adsorbents it is seen that some undergo little loss of entropy. In such systems, he suggested, the guest molecules

*Diffusion coefficient, D as a function of vdw radius of guest particles, σ is plotted in (a) zeolite NaCaA at 140 K*

*confined to (a) zeolite NaCaA at 140 K and to (b) zeolite NaY at 190 K temperature, where 1/σ<sup>2</sup> is the inverse*

In order to obtain a better understanding, we define a dimensionless parameter

*<sup>γ</sup>* <sup>¼</sup> <sup>2</sup> � <sup>21</sup>*<sup>=</sup>*<sup>6</sup>*σgz σw*

where the numerator gives the distance at which the interactions between the guest and the zeolite atoms are optimum, that is, when this gives an interaction energy of ϵ*gz*. The denominator is the window radius. Thus, the dimensionless parameter suggests that when these two are equal (*γ* = 1) and when they are not equal (*γ* < <1). We now plot the diffusivity as a function of *γ*. This is shown in **Figure 4** [28]. We see that the maximum is seen when *γ* is between 0.9 and 1.0 for *both* the zeolites. Thus, the guest-zeolite interaction is optimum at 21*<sup>=</sup>*<sup>6</sup>*σgz* and when this equals *σw*/2, the diffusivity maximum is seen. This situation when *γ* is close to unity is illustrated in **Figure 5** along with the situation when *γ* < <1. Now when *γ* is close to unity the guest molecule is passing through the center of the window. Such a position has inversion symmetry, which leads to mutual cancelation of forces

(1)

*, where σ is the guest diameter. D for guest*

will exhibit high mobility or *superdiffusivity*.

*and in (b) zeolite NaY at 190 K temperature.*

*Diffusion coefficient of guest particles is plotted as a function of 1/σ<sup>2</sup>*

*square of the vdw radius of guest particles.*

*Zeolites - New Challenges*

**Figure 2.**

**Figure 3.**

**56**

*Schematic figure indicating the position of bigger and smaller particles while passing through the zeolite window. It is shown that the bigger particle passes through the symmetry position, the center of the window, whereas the smaller particle is near to the periphery of the window. Therefore, the forces along a given direction is equal and opposite to that exerted on the bigger particle from the diagonally opposite direction, resulting in higher diffusion than for the smaller particle, which is attracted to the periphery, and therefore experiences a net attraction. This is shown in both 2D and 3D.*

leading to rather small force on the guest due to the zeolite. This situation is akin to the guest being a free particle even when confined within the zeolite and therefore has a high diffusivity.

When *γ* < <1 the guest passes through the window at the periphery. These points do not possess inversion symmetry and therefore there is no cancelation of forces exerted on the guest by the zeolite. This leads to lower diffusivity of these guest molecules.

The reason for the observed maximum in diffusivity arises from the lowered force on the guest molecule as compared to the smaller guest molecule, which encounters a higher force on itself due to the zeolite. These translate to a less undulating potential energy landscape with shallower minima and maxima for the larger guest molecule for which *γ* is close to unity. In the case of smaller guest molecule the larger force implies a highly undulating potential energy landscape with deep valleys and high mountains. However, in a recent report the lower force on the guest molecule at the window has not been found [29]. More studies are required to understand the origin of the observed diffusivity maximum.

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 maximum diffusivity.

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

*Anomalous Diffusivity in Porous Solids: Levitation Effect*

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

coefficient as a function of the guest diameter.

*is a resulting phenomenon from dispersion force of attraction.*

*maxima in diffusivity disappear with increase in temperature.*

**Figure 8.**

**Figure 9.**

**59**

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**

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].

*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*

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

[28]. Temperature plays an important role in the behavior of the diffusion

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 dispersion forces, which are always attractive in nature.

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

**Figure 6.** *Variation of activation enegy E*<sup>a</sup> *as a function of vdw radius of guests, σ in zeolite NaY.*

**Figure 7.** *Arrhenius plot of ln D for two different values of γ = 0.67 and 0.89 in zeolite NaY.*
