| dc.description.abstract | In summary, the present work reports results for diatomic species AB in zeolite NaY. The results show that when A = B, there is a maximum in the self-diffusivity of the diatomic molecule at large bond lengths c_lab. This suggests that the Levitation Effect exists for diatomic species AA. Results when A ? B with small asymmetry in the interaction between A and B with the atoms of the zeolite show a weak maximum in the self-diffusivity. When the interaction strength between A and host atoms is made very different from that of B with host atoms, it is observed that such a maximum in D disappears completely. This suggests the absence of the Levitation Effect for such a system.
A few remarks on the symmetry necessary for the diffusivity maximum or Levitation Effect are worth noting. The inversion symmetry which is essential is not the crystallographically defined symmetry based on structure. The necessary symmetry required to ensure that the Levitation Effect is seen is interaction inversion symmetry. Interaction inversion symmetry requires that the force on the diffusant from a given direction is equal and opposite to the force from the diagonally opposite direction. This is a less stringent requirement than crystallographic inversion symmetry. The latter, however, ensures the existence of interaction inversion symmetry. Situations where there is no crystallographic inversion symmetry but there is interaction inversion symmetry are when the forces arising from atoms at different distances add up along a given direction to yield, say, F_p. Now, although the atomic arrangement along the diagonally opposite direction is completely different, it may still add up to F_p=-F(see Figure 2.12). Such an equality should be seen along all directions.
The results presented here suggest that the Stokes–Einstein relationship breaks down for relatively small solute diameters. The SE regime is valid until the ratio of solute to solvent reaches about 0.5. Further decrease in the ratio is followed by a maximum in D arising from the Levitation Effect. The exact value of kat which the transition from the SE to the LE regime occurs may depend on the nature of interaction between the solute and solvent particles. It is also noted that the transition occurs only if dispersion interactions exist; electrostatic interactions were not investigated in this study. Further decrease in kleads to a transition from the LE to the SS regime, where D?.
It is clear that the important quantity determining the range over which each regime is valid is not the actual solute size but the ratio of the solute to the solvent. More precisely, even this ratio is not entirely correct—it is the ratio of the solute size to the neck diameter of the voids present within the solvent [142,157]. However, since the neck diameters are proportional to the size of the particles (for face-centered close-packed solids, neck radius is 0.155 times the radius of the solvent) and since neck radii are not easily estimated, the solvent size can be used as an approximation.
We demonstrate that solutes in this range, called the Levitating regime, are associated with different (lower) activation energies, different behavior of F_s (k,t), and also A(?)/2Dk^2. A microscopic picture consistent with these properties is proposed, which needs verification by direct calculation of additional quantities or properties.
In summary, the SE regime exists for k>?_uu/?_vv=0.48. The LE regime is observed for the range 0.17<k<0.48. The SS regime occurs for very small solute sizes: k<0.17. The LE regime exists over the size ratio (0.17, 0.48). In a close-packed solid, the neck diameter is 0.155R, where R is the radius of the packing spheres; in a liquid, this is likely larger. The maximum in self-diffusivity is seen when the LE condition—that is, mutual cancellation of forces [142,204] occurs—is met (i.e., the solute diameter is comparable to the neck diameter). The upper limit of the LE regime corresponds to the maximum solute size that can diffuse without encountering too many collisions with the solvent, which in turn depends on the structure of voids and necks in the liquid. This points to the need for a systematic study of void structures in dense liquids and solids.
Some of the enhanced diffusivities reported in the literature but not yet explained can be accounted for by this study. Recognizing that three distinct regimes exist in a liquid interacting through van der Waals interactions is important and may help resolve many anomalies observed in diverse systems in nature.
The overall picture from these results is that variation in diffusivity or conductivity with composition occurs due to changes in the doorway or neck radius. Ions or diffusants jump from one void to another, and the crucial structural feature controlling this motion is the radius of the bottleneck that interconnects two voids relative to the size of the diffusant. Although this is the local feature determining diffusivity, for ions, long-range interactions are important. Polarization or electrostatic interactions can alter the effective size of the bottleneck. This provides a unified picture linking ion size, doorway radius, conductivity, structure of the amorphous solid, and activation energy.
We have shown that the maximum in self-diffusivity of the impurity atom as a function of its size exists in amorphous solids, demonstrating the all-pervasive nature of the diffusivity maximum. This has implications for the correlation between ionic conductivity and activation energy in a variety of oxide and chalcogenide glasses. It provides a microscopic picture, explaining observed changes in conductivity with composition in terms of atomic-level changes. Finally, it provides an explanation for the observed correlation between conductivity and activation energy. Further studies are required to understand the effect of temperature and pressure on conductivity and determine if these can be accounted for.
The explanation for the correlation between conductivity and activation energy is the Levitation Effect. This effect is generic and universal, explaining why a wide variety of glasses exhibit such a correlation. It is evident that different-sized impurity atoms exhibit completely and dramatically different dependence on density. A logical explanation for this behavior has been provided in terms of the Levitation Effect. We noted that large-sized impurity atoms exhibit the expected decrease in Dwith ?. One may ask, what is “large”? In other words, what size can be considered large, and for which size does Ddecrease with ?? Similarly, what sizes may be considered intermediate, and what sizes can be said to be small? How does one determine the boundaries between small and intermediate, and between intermediate and large? This is an important question.
Firstly, we note that the precise range for small, intermediate, and large sizes is determined by the void structure of the host matrix. An impurity atom may be said to be small (a_aa<2r_"neck" ) if the impurity atom is smaller than the neck diameter of the host void structure. However, if the impurity atom radius is much smaller (a_aa?2r_"neck" ) than the neck radii, then it may lie in the linear regime (see Figure 5.4), where an increase in density, and therefore in the Levitation parameter ?, will lead to a decrease in D. This behavior is the same as that seen for large-sized impurity atoms.
An intermediate-sized impurity atom is one whose size is close to the size of the neck radius in the host matrix. Finally, an impurity atom is large if its size exceeds the neck radii.
We also explained the observed behavior of Dwith ?. The Levitation parameter is most strongly influenced by the void and neck distributions. In defining the Levitation parameter ?, only the average neck radius is considered; changes in the distribution (broad, narrow, or skewed) are not included. As a result, variations in ?alone provide only a first-order influence on D. A weaker, albeit important, influence may also need to be considered to explain experimentally observed trends. In solids, the range over which density can be varied is limited, and therefore changes in D(?^*)behavior with diffusant size can be difficult to observe. The degree of disorder, included through factor C, also influences the value of D.
The Levitation Effect is indeed a generic and ubiquitous effect. It has been observed in liquids where the neighboring hosts are as mobile as the impurity or solute. This suggests that the effect does not disappear despite static and dynamic disorder seen in amorphous solids and liquids. Here, the free energy landscape picture discussed in Section 1.6 of Chapter 1 or by Odagaki and coworkers [228] is more appropriate, since the free energy landscapes in a liquid and an amorphous solid are not qualitatively different.
The present study suggests that there is a maximum in self-diffusivity for solutes diffusing within the interstitial spaces provided by a body-centered cubic solid.
The maxima are observed when the solute/solvent size ratio is in the range 0.25–0.33. For the first time, we report the existence of more than one maximum in self-diffusivity as a function of solute size. Two maxima are seen when the solute–solvent interaction strength is large. Two maxima are also observed in face-centered cubic arrangements as well as in liquids; for the latter, two maxima appear when the solute–solvent interaction strength is higher relative to diffusion in solids. The relative heights of the two maxima are determined by the activation energies.
We emphasize that the present study does not address the regime of large solutes, where strain energy becomes significant. In previous studies, it was thought that the distribution of bottleneck diameters alone determined the diffusivity maximum. The present study shows that, in addition to f(r_n), the solute–solvent interaction strength also influences the observed size dependence of self-diffusivity.
The main conclusions of this study are:
At sufficiently low density, the size-dependent diffusivity maximum disappears altogether.
The presence of dynamic disorder at high temperature does not lead to backscattering in the velocity autocorrelation function (VACF) for solutes near the diffusivity maximum.
Significant backscattering is seen in VACF for large-radius (LR) solutes but not for average-radius (AR) solutes in the solid phase.
In the solid phase, the exponent ?in ?u^2 (t)??t^?exhibits a negative value during the ballistic-to-diffusive transition for LR solutes but not for AR solutes.
The solute diameter a_"max" (equal to a_uufor the solute at diffusivity maximum) shifts to smaller sizes at high temperature despite a decrease in density, due to an increase in disorder.
Multiple maxima are seen in the solid phase for high ?_uv.
A significant shift to lower a_mis seen across the solid–liquid transition, corresponding to lower density.
Correspondingly, a shift in a_"max" to smaller values is observed.
The study further suggests that confined molecules will exhibit a maximum not only in self-diffusivity but also in rotational diffusivity. The condition for this maximum is the same as that for translational diffusivity. However, for translational diffusivity, the relevant parameter is the narrowest part of the void dimension, usually the windows or bottlenecks within the zeolite. For rotational diffusivity, the relevant void network is typically the diameter of the cage or sphere in which the molecule is confined.
These results are relevant to real systems. For example, methane and halogen-substituted methanes such as CF?, CCl?, CBr?, and CI? provide examples of guest molecules with varying I_"ax" . Nearly spherical cavities can be found in zeolites Y, A, and ?. When halomethanes are confined within these spherical cavities, commonly known as ?-cages, the rotational motion of the guests is primarily determined by I_"ax" , in addition to other factors such as mass and guest–zeolite interactions | |
