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Equation 1 is strictly valid only if there are no diffusion fluxes in the observation direction. This should be true for ZSM since the pore system is two-dimensional with no channels in the direction of the crystal axis. The structure is shown in Fig. However, such a separate determination may become possible if the difference in the equilibration rates of the two components is sufficiently large.

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Cages and windows have diameters of 0. In both b and c , the c axis is aligned perpendicular to the plane of the paper a — b plane. Mass transport occurs exclusively in this plane as the window size in the c direction 0. In one limiting case, realized in the present study for the uptake of CO 2 —ethane by zeolite ZSM, the diffusion rate of one component CO 2 is fast enough to ensure essentially instantaneous equilibration of this component over the sample, with the local concentration therefore determined entirely by the partial pressure of this component in the surrounding atmosphere and the local concentration of the second component in this case, ethane.

Under such conditions, equation 1 may be rewritten as. Figure 2 shows a summary of the transient concentration profiles determined in this way for ethane Fig. In the Methods section see also Supplementary Figs 1—3 , we describe in further detail how these profiles have been determined from the primary IFM data by incorporating the information from the two-component adsorption isotherm as provided by the ideal adsorbed solution theory IAST For the system under study, IAST has been found to provide an accurate prediction of the binary equilibrium over the relevant pressure range We will return to a more detailed discussion of the evolution of the transient concentration profiles at the end of the next subsection where the initial period of uphill diffusion and overshooting will be shown to be followed by final equilibration.

With the ethane front penetrating the system, CO 2 is partially desorbed as shown in b. Finally both components reach uniform equilibrium concentrations determined by their partial pressures. A second possibility that allows the evolution of the concentration of a particular component to be followed by IFM during multi-component adsorption is the other limiting case in which the uptake of one component is slow enough to be followed by IFM but still sufficiently fast that the concentration profiles of the other component may be assumed to remain essentially invariant during uptake of the fast component.

For this type of experiment we have, once again, used ethane, now as the fast component, in a mixture with propene as the slow component see Methods and Supplementary Figs 4 and 5.


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With equilibration times of several days, the propene concentration may in fact be considered to remain essentially constant during the uptake of ethane. As an example, Fig. The measured profile has been approximated by a smoothed curve. The onset of ethane uptake as revealed by the concentration profiles shown in Fig. Although guest concentrations at the boundary itself are inaccessible to direct observation as a consequence of the beveled side faces of the crystals see Fig. Remarkably, however, the ethane concentration within the crystal continues to increase further, with concentrations considerably exceeding the concentrations near the crystal surface: the diffusive flux of the ethane molecules is thus seen to continue to be directed towards the crystal interior, even though this is now the direction of increasing ethane concentration.

To understand this observation, we have to recognize that diffusive fluxes are driven by the gradients of the chemical potentials.

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Uphill diffusion, that is, diffusive fluxes in this case of ethane in the direction of increasing concentration, may thus be easily explained by the presence of a second component propene with a concentration profile that decreases sufficiently rapidly in the direction of the ethane flux. Provided that there is no significant barrier at the crystal surface, the ethane concentration close to the surface will assume its equilibrium value as determined by the gas phase composition. With concentrations in the crystal interior exceeding the boundary concentration, the overall concentration of ethane is seen to exceed the equilibrium value as determined by the partial pressures of the constituents of the surrounding gas phase.

With the potential of microimaging, the exploration of overshooting phenomena may now be based on the in situ observation of the evolution of the guest profiles. Within the timescale of ethane uptake as considered in Fig.

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As a prerequisite for this stationary behaviour, the ethane flux must vanish during this period. With equation 3 , the ethane fluxes are seen to vanish when, throughout the crystal, the gradients of the ethane concentration have attained a certain, well-defined maximum value, determined by the gradient of the propene concentration at this same position and the two relevant elements of the diffusion matrix which predict the ethane flux from the two concentration gradients.

To continue the observation of overshooting, one would have to switch to a much longer timescale large enough to cover the uptake or release of propene.

Within this timescale, the concentration of ethane molecules would assume, essentially instantaneously, their equilibrium values determined by the local propene concentrations and the ethane pressure in the surrounding atmosphere. Overshooting terminates with the formation of a homogeneous propene distribution at which point the ethane molecules are also distributed homogeneously throughout the crystal, with both concentrations determined by their partial pressures in the surrounding atmosphere.

With uptake and release times of several days Fig. To continue the discussion of the process of overshooting, we return to Fig. Its concentration at all points within the crystal may therefore be assumed to reach its equilibrium value, as determined by the external CO 2 pressure and the ethane concentration at that position.

Owing to the very high CO 2 diffusivities, this equilibration occurs essentially instantaneously. In the previous section, we exploited this fact to allow the determination of the concentration profiles for both components from the primary IFM data. Already in the very first CO 2 concentration profiles, we recognize the same feature as was seen in Fig.

Following the example given by Fig. We note that CO 2 diffuses so rapidly that the uphill diffusion process has already been accomplished before we were able to record the first profiles. The propagation of the ethane diffusion front is accompanied by a continuous decrease of its slope.

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Already with the very first profile shown in Fig. The progressive decay in the CO 2 concentration indicates that, right from the beginning of our measurements, the CO 2 fluxes are directed towards the crystal surface, that is, in the downhill direction for CO 2. In our mixture uptake experiments with ethane and CO 2 , we are thus able to exactly follow this part of the history of overshooting that was inaccessible with mixtures of propene and ethane, that is, the approach to final equilibrium which, after overshooting, is controlled by the equilibration of the slower component.

Over the last few decades, benefitting from gains in the accuracy of experimental diffusion measurements, the prediction of diffusivities in zeolitic host—guest systems by molecular dynamics simulations 44 , 45 , 46 , 47 , 48 has made impressive progress. Such simulations, involving solutions of the Newtonian equation under the force field arising from molecule—wall and molecule—molecule interactions, are often performed close to the limit of the available computational facilities. Following the framework provided by classical transition state theory 49 , 50 , 51 , such predictions require only the information provided by the binary adsorption equilibrium isotherm for the host—guest system.

First-order predictions of this type are especially useful for binary and multi-component systems, for which rigorous molecular dynamics simulations are complicated by the presence of more than one diffusing species. The approach provided by transition state theory may be rationalized by exploiting the condition of dynamic equilibrium between the populations in the zeolite cavities and in the windows.

As a consequence, the mean transit time through a window is unaffected by the guest concentration and the relative window population increases in proportion to the pressure of the considered species in the surrounding atmosphere. The mean molecular lifetime within the individual cavities is thus immediately seen to be proportional to the ratio between the actual concentration and the partial pressure of the species under consideration. In this approach, a key quantity controlling the rate of molecular propagation is thus directly related to the equilibrium adsorption properties of the system.

In the Methods section, we show that this correlation may be represented by the relations. Figure 4 shows the transient concentration profiles of ethane resulting from equations 4 and 5 as the solutions of the diffusion equation under the initial and boundary conditions considered in our experiments see Methods. For comparison, the corresponding experimental data are also indicated.

The calculations using equations 4 and 5 are seen to reproduce all the main features as experimentally observed. In particular, the overshooting behaviour shown in Fig. We also note that the most severe differences between the calculations and the experimental findings are observed towards the central axis of the crystals. This, however, is the behaviour to be expected based on the previous single-component uptake and release experiments with this type of crystal. Ideal ZSM crystals allow diffusion only in the radial direction see Fig.

However, uptake experiments with light hydrocarbons showed that, in real crystals, there must be a small flux in the axial direction since the concentration at the centre starts to increase before the diffusion fronts coming from the side faces have reached the centre 52 , This effect is not considered in our model, which is based on the assumption that the crystals have the ideal structure. Dashed lines show the experimental data of Fig. The application of IFM, which allows the measurement of transient concentration profiles in nanoporous crystals with submicron spatial resolution, has hitherto been restricted to single-component systems since, in general, the profiles for the individual components of a mixed adsorbed phase cannot be resolved.

However, by considering the adsorption of mixtures of two components with very different intracrystalline diffusivities CO 2 and ethane, and ethane and propene, both in zeolite DDR , we succeeded in overcoming this limitation, thus making it possible to record the individual transient concentration profiles in a binary system with unprecedented spatial resolution.

A second set of experiments for the mixture ethane—CO 2 , in which ethane is now the slow component, allowed the attenuation and final disappearance of the overshoot to be studied in detail. In addition to the advantages for IFM studies arising from the symmetry of the pore system, the windows of zeolite DDR are small enough to allow a straightforward first-order prediction of the relevant diffusivities as a function of the guest composition from transition state theory.

The predicted transient concentration profiles show good agreement with the experimental data, thus confirming that the diffusional behaviour of the binary system can be adequately predicted from the binary equilibrium isotherm and the single-component diffusivities at zero loading, without recourse to detailed molecular simulations. In addition to the relevance of these results for fundamental research, the systems studied are also of particular technological interest due to the potential application of DDR as a kinetically selective adsorbent for removal of CO 2 from natural gas and for separation of light hydrocarbons.

IFM makes it possible to record transient concentration profiles of guest molecules in nanoporous host systems with a high spatial and temporal resolution. It is based on the principle that the optical density of light passing through a transparent porous crystal depends on the nature and amount of guest molecules present in its pore system.

By using an interferometer of Mach—Zehnder type with shearing mechanism and phase shifter, both light beams are superimposed and an interference pattern is generated. Furthermore, the set-up includes a static vacuum system consisting of vacuum pump, reservoir tank for guest molecules, several pressure gauges and sensors for adjusting the pressure to the required level. A few dozen DDR crystallites under study are placed in an optical cell, which is connected to the vacuum system. Uptake or release experiments are initiated by step changes in the gas phase surrounding the crystals.

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The crystal under study should be transparent and exhibit nearly parallel and plane surfaces at top and bottom. Further details of the IFM technique may be found in literature 14 , For mixtures of guest molecules, the change in the recorded signal contains contributions from both components which cannot be measured separately. However, the concentration profiles of both components can be easily calculated if the mobility of both components differs substantially.

This is in particular true if the pore size is in the range of the critical diameter of the guest molecules—a condition that is usually found in technical applications based on the molecular sieving effect. Then, small differences in the guest size lead to dramatic differences in the diffusivities of both components. This opens the field for detailed investigations of various guest mixtures in different host systems of practical relevance. The DDR zeolites of the batch under study were very uniform in size. For further details of the synthesis, we refer to literature 22 , All measurements presented here were carried out at this temperature.

Supplementary Fig. Equation 2 , correlating the phase shift with the changes in concentration of the individual components, becomes. The factors k ethane CO 2 relating the phase shifts to the concentration changes may be determined by comparing the conventionally measured absolute isotherms with the relative isotherms from the IFM signals for the pure components.

Differences in the heights L of the crystals must be accounted for. With the crystals used in this study, however, this precaution turned out to be unnecessary as the crystal height was very uniform. The almost-perfect agreement between the shape of the macroscopically determined adsorption isotherms left ordinate scale in Supplementary Fig. This is illustrated in Supplementary Fig.


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The required second equation is provided by the assumption of local equilibrium for the fast diffusing component CO 2. Our assumption of local equilibrium of CO 2 throughout the crystal makes this choice of the notation more appropriate for our further discussion in which we consider the concentration of CO 2 as a function of the local ethane concentration and the externally applied pressure of CO 2 :. With the value of the CO 2 pressure known for a given sorption experiment, by equation 8 , the local concentration of CO 2 is seen to be a function of only the local ethane concentration.

Hence, with the functional relationship between the local concentrations of ethane and carbon dioxide, shown to exist by virtue of equation 8 , we have sufficient information to deduce from equation 7 the concentration profiles of both components from the profiles of the overall phase shifts. IAST 35 has proved to be the most universally applicable approach for predicting the equilibrium concentrations in two-component adsorption from the single-component adsorption isotherms for both components.

In detailed investigations employing Grand Canonical Monte Carlo simulations for mixture adsorption in DDR 36 , IAST has been found to serve as a good approach in the range of small and modest gas phase pressures as considered in this study. The relation between the two concentrations is thus seen to yield a simple linear dependency.

By inserting the respective expression for or c ethane provided by equation 9 into equation 7 , the individual intracrystalline concentration for each component is obtained as presented in the Fig. Conversion of the phase shifts into absolute single-component concentration values was based on the adsorption isotherms of Hedin et al.

Determination of the total propene loading did, moreover, necessitate the performance of a whole cycle of ad- and desorption processes as shown in Supplementary Fig. Total propene loading results as the very last step in the series of experiments since reactivation of the crystal under study is, under the given experimental conditions, essentially impossible within a reasonable timescale.

Since, close to the boundary, the crystals are not transparent, the recorded profiles do not extend right to the crystal surface. The propene concentration directly at the crystal boundaries is assumed to correspond to the total propene loading or , since no dominating surface barrier is expected. The maximum ethane concentration for this crystal has been estimated in the following way: in the centre part of the propene profile after presorption see Fig. Thus, the ethane concentration in the centre region of the saturation profile was found to be about.

With these considerations, we arrive at the concentration profiles in absolute concentration values as presented in Fig. For a solution, the equation must be regularized and appropriate initial and boundary conditions need to be defined. For both experiments presented in the manuscript, the problem that needs to be solved for achieving the evolution of the ethane concentration with time, c 1 r , t , is formulated as follows:. A numerical solution for c 1 r , t according to equation 12 can be achieved for two special cases: first, if c 2 is independent of time, it can easily be replaced by a fit curve depending on only the radius r , as it would be the case for propene.

Second, if c 2 can directly be expressed by a function of c 1 r , t. This is the case for CO 2 , which is assumed to be instantaneously equilibrated. Furthermore, eligible models for D 11 c 1 , c 2 and D 12 c 1 , c 2 are needed. These models will be provided by analysis of the data maintained by the approaches made for D 11 and D 12 Supplementary Figs 6 and 7. We exploit the special geometry of the pore structure of ZSM see Fig.

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Diffusion of reactive molecules in solids and melts / Robert H. Doremus. - Version details - Trove

Given the existence of narrow windows between the individual cages, the diffusion of guest molecules in the ZSM pore system may be described by a simple jump model 40 , Thus, one may note. One may immediately arrive at these relations by inserting the expression of the self-diffusivity as derived by transition state theory Equation 4. To determine the partial derivatives appearing in equations 22 and 23 , we have to know the dependence p 1 c 1 , c 2 of the pressure of the relevant diffusant on the concentrations of both components under equilibrium.

In co-adsorption with ethane, CO 2 may be assumed to be instantaneously equilibrated throughout the crystal. Correspondingly, also the equilibrium ethane pressure p 1 at any position corresponding to the local concentrations of the two components is fixed and determined by these concentrations. IAST was applied for either constant c 2 and a fine-meshed list of p 1 values, or for constant c 1 and a fine-meshed list of p 2 values. Subsequently, numerical differentiation of the respective lists was performed to achieve the list of values for and.

The, thus, calculated values of and are found to reveal a linear dependence as shown in Supplementary Fig. The numerical solution was based on the approach. By recording the transient concentration profiles during molecular uptake and release, microimaging also provides all information necessary for the measurement of the intracrystalline diffusive fluxes. Thus, for the system under study, molecular fluxes entering the individual crystal in a time interval t 1 … t 2 are easily seen to be correlated with the evolution of the intracrystalline concentrations by the relation.

For a proper calculation of the integral, the measured data were fitted by either fifth-order polynomials or asymmetric double sigmoidal functions. The resulting fluxes have to be attributed to mean ethane concentrations concentrations between two adjacent profiles. How to cite this article: Lauerer, A. Uphill diffusion and overshooting in the adsorption of binary mixtures in nanoporous solids.

Heitjans P. Cussler, E. Press Shimomura, S. Selective sorption of oxygen and nitric oxide by an electron-donating flexible porous coordination polymer. Buurmans, I. Heterogeneities of individual catalyst particles in space and time as monitored by spectroscopy. Mitchell, S. Visualization of hierarchically structured zeolite bodies from macro to nano length scales. Herm, Z. Separation of hexane isomers in a metal-organic framework with triangular channels. The bias dependence of the hysteretic strain response accessed through first-order reversal curve FORC measurements demonstrates that the process is activated at a certain critical voltage and is linear above this voltage everywhere on the surface.

The relevant parameters such as critical voltage and effective mobility can be extracted for each location and correlated with the microstructure. The evolution of these behaviors with the charging of the amorphous Si anode in a thin-film Li-ion battery is explored.

A broad applicability of this method to other ionically conductive systems is predicted. Read Article at publisher's site. How does Europe PMC derive its citations network? Protein Interactions.

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