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Diffusion of propene in DDR crystals studied by interference microscopy
Chemical Engineering Science 138 (2015) 110–117
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Diffusion of propene in DDR crystals studied by interference microscopy A. Lauerer a, T. Binder a, J. Haase a, J. Kärger a,n, D.M. Ruthven b a b
Faculty of Physics and Earth Sciences, University of Leipzig, Linnéstraße 5, 04103 Leipzig, Germany Department of Chemical and Biological Engineering, University of Maine, Orono, ME, USA
H I G H L I G H T S
G R A P H I C A L
A B S T R A C T
Diffusion of propene in zeolite DDR is studied by microimaging with interference microscopy (IFM). Measurements were performed at three different temperatures (296, 323 and 353 K). Several methods for deducing diffusion coefficients from experimental data are discussed. A new approach of analyzing the rate of uptake and release is presented. The diffusivity is found to be strongly concentration dependent.
art ic l e i nf o
a b s t r a c t
Article history: Received 4 June 2015 Received in revised form 24 July 2015 Accepted 26 July 2015 Available online 1 August 2015
This paper presents the results of a detailed experimental study of the diffusion of propene in large crystals of DDR zeolite (the pure silica analog of ZSM-58) carried out by interference microscopy (IFM). Diffusion is very slow as the propene molecules are only just small enough to pass through the distorted 8-ring windows of the DDR structure. This makes it possible to measure many concentration profiles during the available time and thus to study the diffusional behavior at a level of detail that would be impossible in faster systems. In an ideal DDR crystal diffusion should occur only in the radial direction. In the actual system radial diffusion was dominant but there is also a small but significant axial component which complicates the interpretation of the experimental data. The diffusivity increases strongly with loading but the apparent magnitude of this effect may be increased by the small contribution from axial diffusion. The activation energy for radial diffusion is about 34 kJ/mole. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Diffusion Mass transport Sorption Microimaging Interference microscopy DDR
1. Introduction Over the past decade interference microscopy (IFM) has been developed as a powerful experimental technique for studying the diffusion of guest molecules in nanoporous solids (Kärger et al.,
n
Corresponding author. E-mail addresses: [email protected], [email protected] (J. Kärger), [email protected] (D.M. Ruthven). http://dx.doi.org/10.1016/j.ces.2015.07.029 0009-2509/& 2015 Elsevier Ltd. All rights reserved.
2014, 2012; Lehmann et al., 2003; Heinke et al., 2007; Gueudré et al., 2012; Hibbe et al., 2012; Binder et al., 2013, 2012; Hibbe et al., 2011). In contrast to traditional experimental techniques, in which the intra-crystalline behavior is inferred from extracrystalline observations, IFM provides a means of following the transient intra-crystalline concentration profiles in real time. This makes it possible to measure diffusional fluxes and concentration gradients and hence to deduce the coefficients of transport diffusion directly from Fick's equations, thus eliminating the need to invoke any particular model. For brevity, in the following the
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coefficients of transport diffusion shall be generally referred to as diffusivities. Deviations from ideal Fickian behavior as a result of transport barriers at the crystal surface or within the interior can also be observed directly (Kärger et al., 2012; Hibbe et al., 2011). Since the technique depends on measuring changes in the refractive index resulting from the presence of the guest molecules it is, in general, limited to the study of single-component systems. However, it has recently been shown that, in certain situations, the technique may be extended to study diffusion in a binary adsorbed phase (Binder, 2013; Lauerer et al., 2015). Previous studies have focused mainly on relatively “fast” systems involving the diffusion of small molecules (Binder et al., 2013). However, given sufficient patience, the IFM technique is also well suited to the study of “slow” systems in which the profiles evolve over hours or days rather than seconds or minutes. Such slow systems are rigorously isothermal and allow the profiles to be measured with greater accuracy. This sometimes reveals additional features that would not be obvious in faster systems. The recent addition of a thermostatic control to the microscope stage, allowing measurements to be carried out over a range of temperature, provides an additional incentive for such studies. We report here the results of our first such study in which we have examined the transport diffusion of propylene in large crystals of DDR over the temperature range 23–80 1C. As well as providing a convenient example of a system with strong steric hindrance this system is also potentially of practical importance since the pores of DDR are of the required size to provide a very clean size selective molecular sieve separation of propene from propane (Zhu et al., 2000; Gascon et al., 2008).
2. Experimental section Microimaging by interference microscopy is a powerful method to record the intra-crystalline distribution of guest molecules in large, transparent crystals (10–500 mm) both spatially- and timeresolved. Interference microscopy measures the change in the phase shift Δ Δφ between a light beam passing through the sample and a reference beam passing through the surrounding gas phase. A change in the phase shift is induced by adsorption or desorption of guest molecules (Δc) within the pore system of the nanoporous sample and thus by the change of the refractive index of the material (Δn). In first order approximation the change in the phase shift and the change of the guest molecule concentration are proportional to each other according to Z L Z L Δ ΔφðtÞ p Δnðx; y; z; t Þdz p Δcðx; y; z; t Þdz; ð1Þ 0
0
111
with L denoting the thickness of the crystals in the direction perpendicular to the plane of observation. The experimental setup consists of an interference microscope of the type “Jenapol” (Carl Zeiss AG) with a CCD camera (SenSys KAF 0400, Photometrics) mounted on top, a static vacuum system and a personal computer, by which the camera as well as the phase shifter are controlled. The microscope tube contains a Mach–Zehnder type interferometer combined with a phase shifter and a shearing mechanism. Operating with mono-chromatic light at a wavelength of 589 nm, the interferometer creates the interference pattern by superposition of the respective beams. The method of microimaging by IFM provides a spatial resolution of the guest molecule distribution down to the pixel size of the CCD camera (0.45 mm 0.45 mm). The time resolution of IFM for recording a single concentration map is about 10 s. The vacuum system consists of a vacuum pump, a reservoir tank (for storage of guest molecules); a buffer vessel (for compensation of pressure fluctuations during adsorption), pressure valves and pressure sensors for adjusting the desired guest molecule pressure. Furthermore a connection from the vacuum system to an optical cell is provided. For measurements, the glass cell, containing only a few dozen zeolite crystals, is placed on the sample stage of the microscope. For temperature-controlled measurements, the vacuum system as a whole is enclosed by a thermostatic box. A separate heating system for the optical cell, provided by a heating wire wrapped around it, ensures that the whole system is at the same temperature. The temperature can be varied from room temperature (T ¼23 1C) up to T ¼80 1C. A more detailed description of the IFM technique may be found in Kärger et al. (2014, 2012). The DDR zeolites under study were activated under vacuum at T¼ 200 1C (heating rate: 1 K/min) for several hours. After activation the crystals were cooled down to the desired temperature of the individual experiment (23 1C, 50 1C, and 80 1C). Uptake or release experiments are initiated by step-changes in the pressure of the gas phase surrounding the crystals (at time zero). For desorption measurements the sample was pre-equilibrated with propene, under constant conditions, for several days to ensure full equilibration. Concentration profiles are extracted from the concentration maps recorded by IFM, by taking traverses across the diameter of the crystal. The profiles for each section through the crystal center are radially symmetric (in the ideal case) and essentially identical due to the cylindrical symmetry of the DDR crystals. `The present study was carried out using crystals from the DDR sample used in previous studies of the diffusion of small molecules (Binder et al., 2013, 2012; Binder, 2013; Vidoni and Ruthven, 2012a, 2012b) (DDR II). SEM and IFM photomicrographs are shown in Fig. 1.
Fig. 1. SEM (a) and IFM (b) photomicrograph of a DDR II zeolite. The three DDR II crystallites used in this study had a uniform diameter of 33.3 μm and a thickness of 20 μm approximately. The Si/Al ratio is about 1600 so this material is essentially the pure silica analog of zeolite ZSM-58.
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The pore system of DDR is two-dimensional consisting of large cavities connected to adjacent cages on three sides via eight-membered rings. A good description of the structure of zeolite DDR is provided by Zhu et al. (2000). It is evident from Fig. 1(b) that the beveled edges of the crystal make it impossible to study the outer region of the crystal by IFM. This disadvantage is, however, well outweighed by the advantages of this material. The pores of DDR form a two-dimensional network in the plane perpendicular to the hexad axis of the crystal. In an ideal crystal there should be no transport in the direction of the crystal axis (the z-direction) although, as a result of structural defects, in real crystals there is often some transport in that direction. This combination of pore structure and crystal morphology turns out to be particularly convenient for IFM studies since the transient concentration profiles result from essentially pure radial diffusion, with minimal interference from axial transport. The hexagonal crystals can be represented, to a good approximation, as having cylindrical symmetry, thus simplifying the data analysis.
Desorption :
t o0; cðr Þ ¼ c0 ; t 4 0; cðR; t Þ ¼ 0;
∂c ¼ 0; ∂r r ¼ 0 ð3bÞ
The formal solution for a constant diffusivity system has been given by Crank (1975) Adsorption :
1 c 2X J ðr αn Þ 2 ¼ 1 e αn Dt : 0 co Rn¼1 αn J 1 ðRαn Þ
ð4aÞ
Desorption :
1 c 2X J ðr αn Þ 2 ¼ e αn Dt : 0 co R n ¼ 1 αn J 1 ðRαn Þ
ð4bÞ
where αn denotes the roots of the auxiliary equation J 0 ðRαn Þ ¼ 0. The corresponding expression for the uptake curve is 1 X mt 4 2 ¼ 1 :e αn Dt 2 m1 ðR α Þ n n¼1
ð5Þ
These expressions give the uptake curve and the radial concentration profiles at successive times and thus provide a useful model to which the experimental profiles and uptake/release curves can be compared.
3. Results and discussion With appropriate calibration, the refractive index data derived from the IFM measurements yield a series of normalized concentration profiles across the diameter of the crystal at different times, showing the progress towards the final uniform equilibrium state. Representative examples of such profiles are shown in Fig. 2. They have the qualitative form expected for a system in which the sorption rate is controlled by radial diffusion. 3.1. Radial diffusion model For a cylindrical adsorbent particle with no transport in the axial direction, subjected to a step change in sorbate concentration at the external surface at time zero, the relevant form of the diffusion equation is ∂c 1 ∂ ∂c ¼ rD ð2Þ ∂t r ∂r ∂r with the initial and boundary conditions Adsorption: t o 0; o 0; cðr Þ ¼ 0; t 4 0; cðR; t Þ ¼ c0 ;
∂c ¼ 0; ∂r r ¼ 0 ð3aÞ
3.2. Transient Profiles Fig. 2 shows, as an example, the transient concentration profiles during uptake (a) and release (b) of propene in DDR II induced by pressure steps between 0 and 10 mbar at 296 K. A survey over all measured profiles may be found in Section 3.1 of the SI. They all show the expected random fluctuations, but their general form is as expected for a system dominated by radial diffusion. It is immediately obvious that adsorption is substantially faster than desorption. To match the profiles to the constant diffusivity model it is therefore necessary to assume different diffusivities for adsorption and desorption. As a first attempt at quantitative modeling, the complete set of experimental profiles for a given temperature (296 K) was matched to theoretical profiles calculated from Eqs. (4a) and (4b) with constant diffusivities (Dads ¼4.4 10 16 m2 s 1 and Ddes ¼2.4 10 16 m2 s 1), the values estimated by optimizing the fit of the radial profiles to the constant diffusivity model. The constant diffusivity model evidently provides a fairly good representation of the experimental profiles. However, the fit is certainly not perfect and close inspection reveals that the theoretical profiles tend to lag behind the experimental profiles at short times and to lead the experimental profiles at longer times. This tendency is more pronounced at
Fig. 2. Normalized intra-crystalline radial concentration profiles for the adsorption (a) and desorption (b) of propene in a DDR II crystallite induced by a pressure step from 0 to 10 mbar and 10 to 0 mbar at 296 K. The delineated dashed black line in (a) represents the uniform equilibrium distribution of propene molecules within the crystal. The best fitting (constant diffusivity) theoretical profiles calculated from Eqs. (4a) and (4b) with Dads ¼4.4 10 16 m2 s 1 and Ddes ¼2.4 10 16 m2 s 1 are also shown (smooth solid lines).
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50 1C and 80 1C than at 23 1C. More importantly, for a constant diffusivity system, the diffusivity should be the same for adsorption and desorption over the same pressure step. The higher apparent diffusivity for adsorption suggests that the diffusivity is in fact concentration dependent, increasing with loading (Chmelik et al., 2009; Garg and Ruthven, 1972). At short times the measured profiles are flat, as expected, since the measurements extend only over the central region of the crystal. However, in an ideal crystal with no diffusion in the axial direction there should be no desorption (or adsorption) from the central region until the radial profile has reached that region (about 19,000 s in this example). The experimental data suggest that, during that period, a small but measurable amount of propene adsorbs or desorbs. Since the profile remains flat, the obvious inference is that this desorption occurs by diffusion in the axial direction as a result of structural defects in the crystal.
3.3. Uptake/release curves Integration of the concentration profiles across the radius yields the quantity of sorbate adsorbed or desorbed up to that time, and hence the fractional approach to equilibrium (mt =m1 ) and the “uptake curve” (mt =m1 vs t). For an ideal constant diffusivity cylindrical system in which transport occurs only by radial diffusion, the uptake curve at short times is given by rffiffiffiffiffiffi mt 4 Dt ¼ ð6Þ m1 R π pffiffi It is therefore common practice to plot the uptake curves vs t since, in that representation, the curves should be linear in the initial region. Measurement of the initial slope of the “root-t-plot” thus provides a straightforward and widely used way to estimate the diffusivity. Representative examples of such plots are shown in Fig. 3. Consistent with the evidence from the experimental profiles it is clear that, for the same pressure step, adsorption is faster than desorption. Differences between the curves of relative uptake and release over one and the same pressure step indicate a concentration dependence of the diffusivity over the interval of pressure and, hence, of loading covered in the experiment (Kärger et al., 2012; Binder et al., 2012). Uptake rates exceeding the rates of release are known to indicate, in such cases, diffusivities increasing with increasing loading. Also in such cases however, the diffusivity derived from the desorption curve provides a good
Fig. 3. Uptake/release curves for adsorption and desorption of propene in DDR II at 296 K (pressure step 0–10 or 10–0 mbar).
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estimate of the limiting diffusivity at zero loading (D0 ) (Chmelik et al., 2009; Garg and Ruthven 1972). In the present system, as a result of the beveled faces, the outer region of the crystal is invisible to IFM (see Fig. 1), resulting in a significant time delay during which the gain or loss of sorbate is not recorded (as may be seen from Fig. 3). To integrate the radial profiles in order to calculate the uptake curves, the profile near the crystal surface must be estimated by interpolation. It is therefore not possible to determine accurately the initial part of the uptake/ release curves. In previous studies (Kärger et al., 2012) we had assumed that the maximum slope of the root-t-plot could be used as an approximation for the initial slope. This is a good approximation for many systems but it turns out to be a questionable assumption for DDR since, except for very large crystals, the gain or loss of sorbate during the initial blind period is significant. This led us to seek an alternative approach to the analysis of uptake/ release kinetics. Although, as a result of the “blind” period we do not know the initial portion of the uptake/release curve accurately, we do have accurate data for the rate of sorption as a function of time. It is therefore logical to extract the diffusivity from the slope of the uptake curve rather than from the uptake curve itself. It turns out that this is reasonably straightforward. The exact expression for the slope of the uptake curve, for a radial diffusion controlled system, is rather complicated. However, we can obtain an excellent approximation by using Crank's short time approximation for the initial portion and the asymptotic long time expression for the final portion (Crank, 1975). rffiffiffiffiffiffiffiffiffi τ mt τ3=2 Short Times : ¼4 ð7Þ τ π m1 3√ π where τ ¼ Dt=R2 . The slope (in the short time region) is therefore given by d mt 2 1 τ ð8Þ 1 √ ¼ dτ m1 2 π √ ðπτÞ The long time slope is given by differentiation of Eq. (5) (retaining the first three terms) d mt Long Times : ð9Þ ¼ 4ðe 5:78τ þe 30:5τ þ e 74:89τ Þ d τ m1 By using Eq. (8) for the short time region and Eq. (9) for the long time region we obtain the variation of the slope of the dimensionless uptake curve with dimensionless time as shown in Fig. 4. The curves actually coincide in the intermediate time region so there is in fact no interpolation. Therefore the short and the long time curves can be easily combined to an “overall solution” which is presented in Fig. 4(a). In order to estimate the limiting value for the diffusional time 2 constant (D0 =R ) we must match the experimental plot of slope (d=dt mt =m1 ) vs time (t) to the theoretical curve relating the dimensionless slope (d=dτ mt =m1 ) to the dimensionless time 2 (τ ¼ Dt=R ). In the procedure eventually adopted we made an initial estimate of D=R2 from the average value of the ratio of the slopes of the (real time) experimental curve [d=dt mt =m1 Þ vs mt =m1 ] and the dimensionless theoretical curve [d=dτ mt =m1 vs mt =m1 , Fig. 4(b)] at several values of mt =m1 We then adjusted the value of D=R2 to match the experimental and theoretical curves (slope vs t and dimensionless slope vs τ) in the time domain (Fig. 4 (a)). The resulting fits, shown in Fig. 5, are evidently quite satisfactory. If the uptake/release is controlled by radial diffusion with a constant diffusivity, the fraction of sorbate adsorbed or desorbed during the blind period should be independent of temperature. Therefore, if the uptake/release curves are plotted against τ (or pffiffiffi τ ) the curves for adsorption and desorption at all temperatures
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Fig. 4. Theoretical curves showing variation of the dimensionless slope of the uptake/release curve (a) with dimensionless time and (b) with fractional approach to equilibrium.
Fig. 5. Experimental slope vs time for adsorption and desorption of propene in DDR at 296, 323 and 353 K showing fit of the experimental data to the dimensionless theoretical response curve for radial diffusion (with constant diffusivity) in an infinite cylinder. See Table 1(c) for D=R2 values.
should coincide. Such a plot is shown in Fig. 6. It is clear that the experimental data do indeed conform to this expectation with a margin of uncertainty not much greater than the experimental scatter. Also shown in Fig. 6 is the theoretical curve for an ideal cylindrical radial diffusion system (with no blind period) calculated from Eq. (5) (with τ ¼ Dt=R2 ). As expected, the experimental curves all lie below the theoretical curve for an ideal system. The difference between this theoretical curve and the experimental data corresponds to the fractional gain or loss of sorbate arising from errors in the assumed radial profiles in the outer region of the crystal. This difference is significant at short times but decreases at longer times as equilibrium is approached. It is clear that the maximum slope of the root-t-plot is generally lower than the initial slope of the theoretical curve. Therefore diffusivities derived from the root-t-plot will be underestimated. Furthermore, the shape of the curve is altered by the “blind period” so it is not possible to match the experimental curves exactly to the theoretical curve. The diffusivities estimated by several methods, as well as some other relevant parameters are shown in Table 1. We consider the values of D0 derived from the rate analysis (c) to be the best
Fig. 6. Experimental desorption curves showing mt =m1 as a function of dimensionless time (τ). The theoretical curve for radial diffusion in an ideal cylinder with no axial diffusion (calculated from Eq. 5) is shown for comparison.
Table 1 Diffusivities for propene in DDR from experimental data (10 16 m2 s 1). Experiment [T(K), Ads/ Des]
296 296 323 353
Ads Des Des Des
Uptake curves
Radial profiles
Deff (a)
D0 (b)
2.48 1.02 4.34 14.2
2.58 1.65 1.084 1.65 5.0 5.24 13.5 16.3
D0 (c)
λ (d) r (e) Deff (f) 0.8 0.8 0.88 0.9
5 4.4 5 2.4 8.33 10.2 10 28.8
Axial D (g)
— 0.1 0.45 1.5
Crystal radius, R¼ 16.65 μm (a) Constant D fit of uptake curves (Eq. (5))-Def f . (b) Maximum slope of root-t-plot (Eq. (6))-D0 . (c) Rate analysis (Eq. (8) and (9), Fig. 5)-D0 . (d) λ values (Eq. (10)), estimated from experimental isotherms. (e) r ¼ Dmax . D0
(f) Optimization fit of experimental radial profiles to constant D model. (g) Axial diffusivities are very approximate estimates from short time radial profiles in central region of the crystal. Predictions of radial profiles in Figs. 9 and 10 are based on D0 values from (c) with λ values (d) from the isotherms.
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estimates. The values of D0 estimated from the maximum slopes of the “root-t”-plots and from fitting the entire experimental uptake curve are somewhat smaller. We attribute this difference to the loss (or gain) of sorbate during the blind period, as already discussed. The diffusivities derived from optimization of the fit of the radial profiles to the constant diffusivity model are substantially larger, reflecting the effect of the concentration dependence of the actual differential diffusivities. The excellent fits of the radial profiles that are obtained with the limiting diffusivities from the rate analysis together with the corresponding values of λ given in Table 1 (a measure of the concentration dependence defined by Eq. (10), which is introduced later in this paper) provide strong evidence that the variable diffusivity model and the limiting diffusivity values are both essentially correct. 3.4. Axial diffusion From the rate at which the concentration profile advances, in the central region of the crystal at short times, it is possible to make an approximate estimate of the axial diffusivity (Dz ) since in this region any desorption that occurs can pbe ffiffi attributed entirely to axial diffusion. The plots of mt =m1 vs t from which the axial diffusivities were estimated are shown in Fig. 7. However, there is considerable scatter in the data and the amount of sorbate desorbed in this regime is less than 5% of the total so the accuracy with which the axial diffusivity can be estimated is low. The axial diffusivity values obtained in this way are included in Table 1 but it should be understood that these are only rough estimates. Nevertheless it is clear that the axial diffusivities are substantially smaller than the radial diffusivities so, in analyzing the experimental data the assumption of pure radial diffusion should be a reasonable approximation. Fig. 8 shows the temperature dependence of the diffusivities in Arrhenius form. The activation energy for radial diffusion thus determined is about 34 kJ/mol. The large uncertainty in the axial diffusivities precludes a valid estimate of the activation energy.
Fig. 8. Arrhenius plot showing temperature dependence of the diffusivities D0 (determined from analysis of the uptake rate data), and Dz (the axial diffusivities, estimated from Fig. 7). The errors for the axial diffusivities are rough estimations based on analyzing the extreme values for Dz .
present system, under the experimental conditions, the loadings are substantial. Therefore, to represent the behavior properly it is clearly necessary to include a concentration dependent diffusivity in the mathematical model. For systems of this kind in which (because of strong steric hindrance) the inter-cage jumps are rare events, the concentration dependence is often observed to arise mainly from the thermodynamic factor, with the intrinsic mobility remaining approximately constant (Kärger et al., 2012). This is because, for such systems, the effect of the thermodynamic factor is much greater than any effect on the energy of the transition state arising from higher occupancy of the cages. For a Langmuirian system the concentration dependence of the diffusivity is then given by
3.5. Concentration dependent diffusivity D ¼ D0 Both the experimental profiles and the uptake curves suggest that the diffusivity in this system is concentration dependent. This is not surprising since, in an adsorbed phase, the diffusivity is expected to be constant only at low concentrations within Henry's law region. For the
Fig. 7. Uptake and release curves estimated from the advance of the concentration profiles in the central region of the crystal.
d lnp D0 ¼ d lnc 1 λC
ð10Þ
where D0 is the limiting diffusivity (related to the intrinsic mobility), λ ¼ c0 =cs and C ¼ c=c0 . Eq. (10) implies that the ratio Dmax =D0 is given by 1/(1–λ). To investigate whether the deviations from the constant diffusivity model can be accounted for by a concentration dependent diffusivity of this form we used a modified version of the computer program in which we fixed the values of both D0 and λ. D0 was set at the value derived from the uptake rate analysis (Table 1) and λ was set initially at the values estimated from the equilibrium isotherms taken from the literature (Gascon et al., 2008) (see Supplementary Information, Section 2). The predicted radial profiles at the relevant times were then calculated (without any readjustment of the parameters) and compared with the experimental profiles. The predicted profiles gave a good representation of the experimental profiles but it was found that the fit (especially of the center-line profiles) could be improved by using the slightly larger λ values shown in Table 1. The calculations are bulky and time consuming so no detailed optimization was attempted. For all three temperatures and for both adsorption and desorption the profiles calculated in this way provide an excellent representation of the experimental profiles. In particular the adsorption and desorption profiles at 23 1C have characteristically different shapes but are well matched using the same values of D0 and λ. Representative examples showing the comparison of the experimental and theoretical profiles are shown in Figs. 9 and 10. The tendency of the constant diffusivity theoretical profiles to lag the experimental
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Fig. 9. Three-dimensional plots showing the optimized fit of the adsorption (a) and desorption (b) profiles at 296 K (pressure step 0–10 or 10–0 mbar) to the variable diffusivity model. D0 ¼ 1.65 10 16 m2 s 1, λ¼ 0.80.
the experimental profile runs very slightly ahead of the theoretical profile (variable diffusivity model) at short times.
4. Conclusions
Fig. 10. Two-dimensional concentration profiles for desorption of propene in DDR at 353 K for a pressure step from 210 to 0 mbar. D0 ¼ 1.63 10 15 m2 s 1 and λ ¼0.90 have been used as the parameters for the variable diffusivity model.
profiles at short times and to lead them at longer times is eliminated by the inclusion of a concentration dependent diffusivity.
3.6. Center-line loading vs time Concentration dependent diffusivities are known to affect the shape of concentration profiles, notably also towards the crystal center. In order to investigate this further we have examined in detail the behavior at the crystal axis (the center-line). For all temperatures investigated Fig. 11 shows a comparison of the time dependence of the experimental loading (of desorption) at the central axis (estimated by averaging the profiles over the central region) and the theoretical time dependence predicted from the constant diffusivity and variable diffusivity models (with the parameters shown in Table 1). The experimental center-line concentration profiles (plotted vs dimensionless time) coincide closely with the theoretical profiles calculated according to the variable diffusivity model. The constant diffusivity profile (from Crank's solution (Crank, 1975)) has a different shape and can never be properly matched to the experimental profiles whatever value is used for D/R2. In addition to this behavior we do clearly note (see the “zoom-in” on the right side of Fig. 11) for the very initial of molecular uptake the effect of axial diffusion as considered in detail already above. As a consequence,
The diffusion of propene in large DDR crystals has been studied in detail by interference microscopy. The critical diameter of the propene molecule is close to the dimensions of the windows connecting the larger cavities within the radial-symmetric pore system of zeolite DDR. Diffusion of propene is therefore very slow (with a substantial energy of activation) and can be followed over a long time span during ad- and desorption. Measurements were performed at three different temperatures (23, 50 and 80 1C) with the respective pressure step being adjusted so that the equilibrium loading was approximately constant. It is generally possible to obtain a reliable estimate of the limiting diffusivity (D0 ) from the initial slope of the desorption curve. However, this was not possible for the present system because of the loss of signal at the boundary of the crystal (due to the beveled side faces). An alternative approach was therefore developed based on analysis of the rates of uptake or release rather than of the uptake curve itself. The measured concentration profiles show that diffusion occurs predominantly by radial diffusion with an activation energy of approximately 34 kJ/mol. The diffusivity increases strongly with loading but, within the accuracy of the measurements, this effect appears to be accounted for by the thermodynamic factor and the corrected diffusivity (D0 ) appears to be essentially constant (at a given temperature). This pattern of behavior is typical of a system in which the molecule in the transition state (in the window between adjacent cages) is severely constrained. The simple Langmuirian model (Eq. 10) with constant D0 (estimated from the desorption rate measurements) provides an excellent representation of the transient concentration profiles at all three temperatures. The center-line profiles at short times suggest a small contribution from axial diffusion, presumably arising from structural defects in the actual crystals. However, the data suggest that the axial diffusivity is at least an order of magnitude smaller than the radial diffusivity and so, although its effect is detectable, it is too small to measure with any accuracy and any modification of the overall profiles is minimal. Although the motivation for this study was mainly academic it also has practical relevance since the pore size of DDR is such that it
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Fig. 11. Desorption curves for the central axis of the crystal showing the comparison between the experimental curves and the theoretical curves predicted from both the constant diffusivity and variable diffusivity models using the parameters given in Table 1 (left side). On the right side a “zoom-in” view of the data points within the gray oval is shown. In this short-time regime the experimental data points are slightly higher than the simulated ones, thus indicating the influence of diffusion in axial direction.
provides a very clean molecular sieve separation between propene and propane.
Notation c co cs C D Do Dz Jα(x) L mt m1 p r R t
λ τ
sorbate concentration in crystal/[mol/kg] initial or final (equilibrium) value of c/[mol/kg] saturation capacity (Langmuir model)/[mol/kg] dimensionless concentration c/co diffusivity/[m2/s] limiting (or thermodynamically corrected) diffusivity (Eq. (6))/[m2/s] axial diffusivity/[m2/s] Bessel functions of the first kind of order α crystal thickness/[mm] mass of sorbate adsorbed or desorbed at time t/[mol/kg] mass of sorbate adsorbed or desorbed as t-1/[mol/kg] equilibrium (partial) pressure of sorbate/[bar] radial coordinate/[m] crystal radius/[m] time/[s] co/cs Dot/R2(dimensionless time variable)
Acknowledgment The research presented in this study has been funded by the German Science Foundation. Furthermore, the authors express their gratitude for the continuous support by the Alexander von Humboldt Foundation.
Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.ces.2015.07.029.
References Binder, T., 2013. Mass Transport in Nanoporous Materials: New Insights from Micro-Imaging by Interference Microscopy (Ph.D. thesis). University of Leipzig, Germany. Binder, T., Hibbe, F., Chmelik, C., Kärger, J., Martinez-Joaristi, A., Gascon, J., Kapteijn, F., Ruthven, D.M., 2012. Micro-imaging of transient guest profiles in nanoporous host systems of cylindrical symmetry. J. Chem. Phys. 137, 1647041–164704-8. Binder, T., Chmelik, C., Kärger, J., Martinez-Joaristi, A., Gascon, J., Kapteijn, F., Ruthven, D.M., 2013. A diffusion study of small hydrocarbons in DDR zeolites by micro-imaging. Microporous Mesoporous Mater. 180, 219–228. Chmelik, C., Heinke, L., Kortunov, P., Li, J., Olson, D., Tzoulaki, D., Weitkamp, J., Kärger, J., 2009. Ensemble measurement of diffusion: novel beauty and evidence. ChemPhysChem 10, 2623–2627. Crank, J., 1975. Mathematics of Diffusion Chapter V. 2nd ed. Oxford University Press, London. Garg, D.R., Ruthven, D.M., 1972. The effect of the concentration dependence of diffusivity on zeolitic sorption curves. Chem. Eng. Sci. 27, 417–423. Gascon, J., Blom, W., van Miltenbergh, A., Ferreira, A., Berger, R., Kapteijn, F., 2008. Accelerated synthesis of all-silica DD3R and its performance in the separation of propylene/propane mixtures. Microporous Mesoporous Mater. 115, 585–593. Gueudré, L., Binder, T., Chmelik, C., Hibbe, F., Ruthven, D.M., Kärger, J., 2012. Microimaging by interference microscopy: a case study of orientation-dependent guest diffusion in MFI-type zeolite host crystals. Materials 5, 721–740. Heinke, L., Kortunov, P., Tzoulaki, D., Kärger, J., 2007. The options of interference microscopy to explore the significance of intracrystalline diffusion and surface permeation for overall mass transfer on nanoporous materials. Adsorption 13, 215–223. Hibbe, F., Chmelik, C., Heinke, L., Pramanik, S., Li, J., Ruthven, D.M., Tzoulaki, D., Kärger, J., 2011. The Nature of Surface Barriers on Nanoporous Solids Explored by Microimaging of Transient Guest Distributions. J. Am. Chem. Soc. 133, 2804–2807. Hibbe, F., Caro, J., Chmelik, C., Huang, A., Kirchner, T., Ruthven, D.M., Valiullin, R., Kärger, J., 2012. Monitoring molecular mass transfer in cation-free nanoporous host-crystals of type AlPO-LTA. J. Am. Chem. Soc. 134, 7725–7732. Kärger, J., Ruthven, D.M., Theodorou, D.N., 2012. Diffusion in Nanoporous Materials. Wiley–VCH, Weinheim, pp. 107–108. Kärger, J., et al., 2014. Microimaging of transient guest profiles to monitor mass transfer in nanoporous materials. Nat. Mater. 13, 333–343. Lauerer, A., Binder, T., Chmelik, C., Miersemann, E., Haase, J., Ruthven, D.M., Kärger, J., 2015. Uphill diffusion and overshooting in the adsorption of binary mixtures in nanoporous solids. Nat. Commun. 6, 7697. http://dx.doi.org/10.1038/ ncomms8697. Lehmann, E., Vasenkov, S., Kärger, J., Zadrozna, G., Kornatowski, J., Weiss, Ö., Schüth, F., 2003. Inhomogeneous distribution of water adsorbed under low pressure in CrAPO-5 and SAPO-5: an interference microscopy study. J. Phys. Chem. B 107, 4685–4687. Vidoni, A., Ruthven, D.M., 2012a. Diffusion of C2H6 and C2H4 in DDR zeolite. Ind. Eng. Chem. Res 51, 1383–1390. Vidoni, A., Ruthven, D.M., 2012b. Diffusion of methane in DD3R zeolite. Microporous Mesoporous Mater. 159, 57–65. Zhu, W., Kapteijn, F., Moulijn, J.A., den Exter, M.C., Jansen, J.C., 2000. Shape selectivity in adsorption on the all-silica DD3R. Langmuir 16, 3322–3329.
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Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces
Diffusion of propene in DDR crystals studied by interference microscopy A. Lauerer a, T. Binder a, J. Haase a, J. Kärger a,n, D.M. Ruthven b a b
Faculty of Physics and Earth Sciences, University of Leipzig, Linnéstraße 5, 04103 Leipzig, Germany Department of Chemical and Biological Engineering, University of Maine, Orono, ME, USA
H I G H L I G H T S
G R A P H I C A L
A B S T R A C T
Diffusion of propene in zeolite DDR is studied by microimaging with interference microscopy (IFM). Measurements were performed at three different temperatures (296, 323 and 353 K). Several methods for deducing diffusion coefficients from experimental data are discussed. A new approach of analyzing the rate of uptake and release is presented. The diffusivity is found to be strongly concentration dependent.
art ic l e i nf o
a b s t r a c t
Article history: Received 4 June 2015 Received in revised form 24 July 2015 Accepted 26 July 2015 Available online 1 August 2015
This paper presents the results of a detailed experimental study of the diffusion of propene in large crystals of DDR zeolite (the pure silica analog of ZSM-58) carried out by interference microscopy (IFM). Diffusion is very slow as the propene molecules are only just small enough to pass through the distorted 8-ring windows of the DDR structure. This makes it possible to measure many concentration profiles during the available time and thus to study the diffusional behavior at a level of detail that would be impossible in faster systems. In an ideal DDR crystal diffusion should occur only in the radial direction. In the actual system radial diffusion was dominant but there is also a small but significant axial component which complicates the interpretation of the experimental data. The diffusivity increases strongly with loading but the apparent magnitude of this effect may be increased by the small contribution from axial diffusion. The activation energy for radial diffusion is about 34 kJ/mole. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Diffusion Mass transport Sorption Microimaging Interference microscopy DDR
1. Introduction Over the past decade interference microscopy (IFM) has been developed as a powerful experimental technique for studying the diffusion of guest molecules in nanoporous solids (Kärger et al.,
n
Corresponding author. E-mail addresses: [email protected], [email protected] (J. Kärger), [email protected] (D.M. Ruthven). http://dx.doi.org/10.1016/j.ces.2015.07.029 0009-2509/& 2015 Elsevier Ltd. All rights reserved.
2014, 2012; Lehmann et al., 2003; Heinke et al., 2007; Gueudré et al., 2012; Hibbe et al., 2012; Binder et al., 2013, 2012; Hibbe et al., 2011). In contrast to traditional experimental techniques, in which the intra-crystalline behavior is inferred from extracrystalline observations, IFM provides a means of following the transient intra-crystalline concentration profiles in real time. This makes it possible to measure diffusional fluxes and concentration gradients and hence to deduce the coefficients of transport diffusion directly from Fick's equations, thus eliminating the need to invoke any particular model. For brevity, in the following the
A. Lauerer et al. / Chemical Engineering Science 138 (2015) 110–117
coefficients of transport diffusion shall be generally referred to as diffusivities. Deviations from ideal Fickian behavior as a result of transport barriers at the crystal surface or within the interior can also be observed directly (Kärger et al., 2012; Hibbe et al., 2011). Since the technique depends on measuring changes in the refractive index resulting from the presence of the guest molecules it is, in general, limited to the study of single-component systems. However, it has recently been shown that, in certain situations, the technique may be extended to study diffusion in a binary adsorbed phase (Binder, 2013; Lauerer et al., 2015). Previous studies have focused mainly on relatively “fast” systems involving the diffusion of small molecules (Binder et al., 2013). However, given sufficient patience, the IFM technique is also well suited to the study of “slow” systems in which the profiles evolve over hours or days rather than seconds or minutes. Such slow systems are rigorously isothermal and allow the profiles to be measured with greater accuracy. This sometimes reveals additional features that would not be obvious in faster systems. The recent addition of a thermostatic control to the microscope stage, allowing measurements to be carried out over a range of temperature, provides an additional incentive for such studies. We report here the results of our first such study in which we have examined the transport diffusion of propylene in large crystals of DDR over the temperature range 23–80 1C. As well as providing a convenient example of a system with strong steric hindrance this system is also potentially of practical importance since the pores of DDR are of the required size to provide a very clean size selective molecular sieve separation of propene from propane (Zhu et al., 2000; Gascon et al., 2008).
2. Experimental section Microimaging by interference microscopy is a powerful method to record the intra-crystalline distribution of guest molecules in large, transparent crystals (10–500 mm) both spatially- and timeresolved. Interference microscopy measures the change in the phase shift Δ Δφ between a light beam passing through the sample and a reference beam passing through the surrounding gas phase. A change in the phase shift is induced by adsorption or desorption of guest molecules (Δc) within the pore system of the nanoporous sample and thus by the change of the refractive index of the material (Δn). In first order approximation the change in the phase shift and the change of the guest molecule concentration are proportional to each other according to Z L Z L Δ ΔφðtÞ p Δnðx; y; z; t Þdz p Δcðx; y; z; t Þdz; ð1Þ 0
0
111
with L denoting the thickness of the crystals in the direction perpendicular to the plane of observation. The experimental setup consists of an interference microscope of the type “Jenapol” (Carl Zeiss AG) with a CCD camera (SenSys KAF 0400, Photometrics) mounted on top, a static vacuum system and a personal computer, by which the camera as well as the phase shifter are controlled. The microscope tube contains a Mach–Zehnder type interferometer combined with a phase shifter and a shearing mechanism. Operating with mono-chromatic light at a wavelength of 589 nm, the interferometer creates the interference pattern by superposition of the respective beams. The method of microimaging by IFM provides a spatial resolution of the guest molecule distribution down to the pixel size of the CCD camera (0.45 mm 0.45 mm). The time resolution of IFM for recording a single concentration map is about 10 s. The vacuum system consists of a vacuum pump, a reservoir tank (for storage of guest molecules); a buffer vessel (for compensation of pressure fluctuations during adsorption), pressure valves and pressure sensors for adjusting the desired guest molecule pressure. Furthermore a connection from the vacuum system to an optical cell is provided. For measurements, the glass cell, containing only a few dozen zeolite crystals, is placed on the sample stage of the microscope. For temperature-controlled measurements, the vacuum system as a whole is enclosed by a thermostatic box. A separate heating system for the optical cell, provided by a heating wire wrapped around it, ensures that the whole system is at the same temperature. The temperature can be varied from room temperature (T ¼23 1C) up to T ¼80 1C. A more detailed description of the IFM technique may be found in Kärger et al. (2014, 2012). The DDR zeolites under study were activated under vacuum at T¼ 200 1C (heating rate: 1 K/min) for several hours. After activation the crystals were cooled down to the desired temperature of the individual experiment (23 1C, 50 1C, and 80 1C). Uptake or release experiments are initiated by step-changes in the pressure of the gas phase surrounding the crystals (at time zero). For desorption measurements the sample was pre-equilibrated with propene, under constant conditions, for several days to ensure full equilibration. Concentration profiles are extracted from the concentration maps recorded by IFM, by taking traverses across the diameter of the crystal. The profiles for each section through the crystal center are radially symmetric (in the ideal case) and essentially identical due to the cylindrical symmetry of the DDR crystals. `The present study was carried out using crystals from the DDR sample used in previous studies of the diffusion of small molecules (Binder et al., 2013, 2012; Binder, 2013; Vidoni and Ruthven, 2012a, 2012b) (DDR II). SEM and IFM photomicrographs are shown in Fig. 1.
Fig. 1. SEM (a) and IFM (b) photomicrograph of a DDR II zeolite. The three DDR II crystallites used in this study had a uniform diameter of 33.3 μm and a thickness of 20 μm approximately. The Si/Al ratio is about 1600 so this material is essentially the pure silica analog of zeolite ZSM-58.
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The pore system of DDR is two-dimensional consisting of large cavities connected to adjacent cages on three sides via eight-membered rings. A good description of the structure of zeolite DDR is provided by Zhu et al. (2000). It is evident from Fig. 1(b) that the beveled edges of the crystal make it impossible to study the outer region of the crystal by IFM. This disadvantage is, however, well outweighed by the advantages of this material. The pores of DDR form a two-dimensional network in the plane perpendicular to the hexad axis of the crystal. In an ideal crystal there should be no transport in the direction of the crystal axis (the z-direction) although, as a result of structural defects, in real crystals there is often some transport in that direction. This combination of pore structure and crystal morphology turns out to be particularly convenient for IFM studies since the transient concentration profiles result from essentially pure radial diffusion, with minimal interference from axial transport. The hexagonal crystals can be represented, to a good approximation, as having cylindrical symmetry, thus simplifying the data analysis.
Desorption :
t o0; cðr Þ ¼ c0 ; t 4 0; cðR; t Þ ¼ 0;
∂c ¼ 0; ∂r r ¼ 0 ð3bÞ
The formal solution for a constant diffusivity system has been given by Crank (1975) Adsorption :
1 c 2X J ðr αn Þ 2 ¼ 1 e αn Dt : 0 co Rn¼1 αn J 1 ðRαn Þ
ð4aÞ
Desorption :
1 c 2X J ðr αn Þ 2 ¼ e αn Dt : 0 co R n ¼ 1 αn J 1 ðRαn Þ
ð4bÞ
where αn denotes the roots of the auxiliary equation J 0 ðRαn Þ ¼ 0. The corresponding expression for the uptake curve is 1 X mt 4 2 ¼ 1 :e αn Dt 2 m1 ðR α Þ n n¼1
ð5Þ
These expressions give the uptake curve and the radial concentration profiles at successive times and thus provide a useful model to which the experimental profiles and uptake/release curves can be compared.
3. Results and discussion With appropriate calibration, the refractive index data derived from the IFM measurements yield a series of normalized concentration profiles across the diameter of the crystal at different times, showing the progress towards the final uniform equilibrium state. Representative examples of such profiles are shown in Fig. 2. They have the qualitative form expected for a system in which the sorption rate is controlled by radial diffusion. 3.1. Radial diffusion model For a cylindrical adsorbent particle with no transport in the axial direction, subjected to a step change in sorbate concentration at the external surface at time zero, the relevant form of the diffusion equation is ∂c 1 ∂ ∂c ¼ rD ð2Þ ∂t r ∂r ∂r with the initial and boundary conditions Adsorption: t o 0; o 0; cðr Þ ¼ 0; t 4 0; cðR; t Þ ¼ c0 ;
∂c ¼ 0; ∂r r ¼ 0 ð3aÞ
3.2. Transient Profiles Fig. 2 shows, as an example, the transient concentration profiles during uptake (a) and release (b) of propene in DDR II induced by pressure steps between 0 and 10 mbar at 296 K. A survey over all measured profiles may be found in Section 3.1 of the SI. They all show the expected random fluctuations, but their general form is as expected for a system dominated by radial diffusion. It is immediately obvious that adsorption is substantially faster than desorption. To match the profiles to the constant diffusivity model it is therefore necessary to assume different diffusivities for adsorption and desorption. As a first attempt at quantitative modeling, the complete set of experimental profiles for a given temperature (296 K) was matched to theoretical profiles calculated from Eqs. (4a) and (4b) with constant diffusivities (Dads ¼4.4 10 16 m2 s 1 and Ddes ¼2.4 10 16 m2 s 1), the values estimated by optimizing the fit of the radial profiles to the constant diffusivity model. The constant diffusivity model evidently provides a fairly good representation of the experimental profiles. However, the fit is certainly not perfect and close inspection reveals that the theoretical profiles tend to lag behind the experimental profiles at short times and to lead the experimental profiles at longer times. This tendency is more pronounced at
Fig. 2. Normalized intra-crystalline radial concentration profiles for the adsorption (a) and desorption (b) of propene in a DDR II crystallite induced by a pressure step from 0 to 10 mbar and 10 to 0 mbar at 296 K. The delineated dashed black line in (a) represents the uniform equilibrium distribution of propene molecules within the crystal. The best fitting (constant diffusivity) theoretical profiles calculated from Eqs. (4a) and (4b) with Dads ¼4.4 10 16 m2 s 1 and Ddes ¼2.4 10 16 m2 s 1 are also shown (smooth solid lines).
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50 1C and 80 1C than at 23 1C. More importantly, for a constant diffusivity system, the diffusivity should be the same for adsorption and desorption over the same pressure step. The higher apparent diffusivity for adsorption suggests that the diffusivity is in fact concentration dependent, increasing with loading (Chmelik et al., 2009; Garg and Ruthven, 1972). At short times the measured profiles are flat, as expected, since the measurements extend only over the central region of the crystal. However, in an ideal crystal with no diffusion in the axial direction there should be no desorption (or adsorption) from the central region until the radial profile has reached that region (about 19,000 s in this example). The experimental data suggest that, during that period, a small but measurable amount of propene adsorbs or desorbs. Since the profile remains flat, the obvious inference is that this desorption occurs by diffusion in the axial direction as a result of structural defects in the crystal.
3.3. Uptake/release curves Integration of the concentration profiles across the radius yields the quantity of sorbate adsorbed or desorbed up to that time, and hence the fractional approach to equilibrium (mt =m1 ) and the “uptake curve” (mt =m1 vs t). For an ideal constant diffusivity cylindrical system in which transport occurs only by radial diffusion, the uptake curve at short times is given by rffiffiffiffiffiffi mt 4 Dt ¼ ð6Þ m1 R π pffiffi It is therefore common practice to plot the uptake curves vs t since, in that representation, the curves should be linear in the initial region. Measurement of the initial slope of the “root-t-plot” thus provides a straightforward and widely used way to estimate the diffusivity. Representative examples of such plots are shown in Fig. 3. Consistent with the evidence from the experimental profiles it is clear that, for the same pressure step, adsorption is faster than desorption. Differences between the curves of relative uptake and release over one and the same pressure step indicate a concentration dependence of the diffusivity over the interval of pressure and, hence, of loading covered in the experiment (Kärger et al., 2012; Binder et al., 2012). Uptake rates exceeding the rates of release are known to indicate, in such cases, diffusivities increasing with increasing loading. Also in such cases however, the diffusivity derived from the desorption curve provides a good
Fig. 3. Uptake/release curves for adsorption and desorption of propene in DDR II at 296 K (pressure step 0–10 or 10–0 mbar).
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estimate of the limiting diffusivity at zero loading (D0 ) (Chmelik et al., 2009; Garg and Ruthven 1972). In the present system, as a result of the beveled faces, the outer region of the crystal is invisible to IFM (see Fig. 1), resulting in a significant time delay during which the gain or loss of sorbate is not recorded (as may be seen from Fig. 3). To integrate the radial profiles in order to calculate the uptake curves, the profile near the crystal surface must be estimated by interpolation. It is therefore not possible to determine accurately the initial part of the uptake/ release curves. In previous studies (Kärger et al., 2012) we had assumed that the maximum slope of the root-t-plot could be used as an approximation for the initial slope. This is a good approximation for many systems but it turns out to be a questionable assumption for DDR since, except for very large crystals, the gain or loss of sorbate during the initial blind period is significant. This led us to seek an alternative approach to the analysis of uptake/ release kinetics. Although, as a result of the “blind” period we do not know the initial portion of the uptake/release curve accurately, we do have accurate data for the rate of sorption as a function of time. It is therefore logical to extract the diffusivity from the slope of the uptake curve rather than from the uptake curve itself. It turns out that this is reasonably straightforward. The exact expression for the slope of the uptake curve, for a radial diffusion controlled system, is rather complicated. However, we can obtain an excellent approximation by using Crank's short time approximation for the initial portion and the asymptotic long time expression for the final portion (Crank, 1975). rffiffiffiffiffiffiffiffiffi τ mt τ3=2 Short Times : ¼4 ð7Þ τ π m1 3√ π where τ ¼ Dt=R2 . The slope (in the short time region) is therefore given by d mt 2 1 τ ð8Þ 1 √ ¼ dτ m1 2 π √ ðπτÞ The long time slope is given by differentiation of Eq. (5) (retaining the first three terms) d mt Long Times : ð9Þ ¼ 4ðe 5:78τ þe 30:5τ þ e 74:89τ Þ d τ m1 By using Eq. (8) for the short time region and Eq. (9) for the long time region we obtain the variation of the slope of the dimensionless uptake curve with dimensionless time as shown in Fig. 4. The curves actually coincide in the intermediate time region so there is in fact no interpolation. Therefore the short and the long time curves can be easily combined to an “overall solution” which is presented in Fig. 4(a). In order to estimate the limiting value for the diffusional time 2 constant (D0 =R ) we must match the experimental plot of slope (d=dt mt =m1 ) vs time (t) to the theoretical curve relating the dimensionless slope (d=dτ mt =m1 ) to the dimensionless time 2 (τ ¼ Dt=R ). In the procedure eventually adopted we made an initial estimate of D=R2 from the average value of the ratio of the slopes of the (real time) experimental curve [d=dt mt =m1 Þ vs mt =m1 ] and the dimensionless theoretical curve [d=dτ mt =m1 vs mt =m1 , Fig. 4(b)] at several values of mt =m1 We then adjusted the value of D=R2 to match the experimental and theoretical curves (slope vs t and dimensionless slope vs τ) in the time domain (Fig. 4 (a)). The resulting fits, shown in Fig. 5, are evidently quite satisfactory. If the uptake/release is controlled by radial diffusion with a constant diffusivity, the fraction of sorbate adsorbed or desorbed during the blind period should be independent of temperature. Therefore, if the uptake/release curves are plotted against τ (or pffiffiffi τ ) the curves for adsorption and desorption at all temperatures
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Fig. 4. Theoretical curves showing variation of the dimensionless slope of the uptake/release curve (a) with dimensionless time and (b) with fractional approach to equilibrium.
Fig. 5. Experimental slope vs time for adsorption and desorption of propene in DDR at 296, 323 and 353 K showing fit of the experimental data to the dimensionless theoretical response curve for radial diffusion (with constant diffusivity) in an infinite cylinder. See Table 1(c) for D=R2 values.
should coincide. Such a plot is shown in Fig. 6. It is clear that the experimental data do indeed conform to this expectation with a margin of uncertainty not much greater than the experimental scatter. Also shown in Fig. 6 is the theoretical curve for an ideal cylindrical radial diffusion system (with no blind period) calculated from Eq. (5) (with τ ¼ Dt=R2 ). As expected, the experimental curves all lie below the theoretical curve for an ideal system. The difference between this theoretical curve and the experimental data corresponds to the fractional gain or loss of sorbate arising from errors in the assumed radial profiles in the outer region of the crystal. This difference is significant at short times but decreases at longer times as equilibrium is approached. It is clear that the maximum slope of the root-t-plot is generally lower than the initial slope of the theoretical curve. Therefore diffusivities derived from the root-t-plot will be underestimated. Furthermore, the shape of the curve is altered by the “blind period” so it is not possible to match the experimental curves exactly to the theoretical curve. The diffusivities estimated by several methods, as well as some other relevant parameters are shown in Table 1. We consider the values of D0 derived from the rate analysis (c) to be the best
Fig. 6. Experimental desorption curves showing mt =m1 as a function of dimensionless time (τ). The theoretical curve for radial diffusion in an ideal cylinder with no axial diffusion (calculated from Eq. 5) is shown for comparison.
Table 1 Diffusivities for propene in DDR from experimental data (10 16 m2 s 1). Experiment [T(K), Ads/ Des]
296 296 323 353
Ads Des Des Des
Uptake curves
Radial profiles
Deff (a)
D0 (b)
2.48 1.02 4.34 14.2
2.58 1.65 1.084 1.65 5.0 5.24 13.5 16.3
D0 (c)
λ (d) r (e) Deff (f) 0.8 0.8 0.88 0.9
5 4.4 5 2.4 8.33 10.2 10 28.8
Axial D (g)
— 0.1 0.45 1.5
Crystal radius, R¼ 16.65 μm (a) Constant D fit of uptake curves (Eq. (5))-Def f . (b) Maximum slope of root-t-plot (Eq. (6))-D0 . (c) Rate analysis (Eq. (8) and (9), Fig. 5)-D0 . (d) λ values (Eq. (10)), estimated from experimental isotherms. (e) r ¼ Dmax . D0
(f) Optimization fit of experimental radial profiles to constant D model. (g) Axial diffusivities are very approximate estimates from short time radial profiles in central region of the crystal. Predictions of radial profiles in Figs. 9 and 10 are based on D0 values from (c) with λ values (d) from the isotherms.
A. Lauerer et al. / Chemical Engineering Science 138 (2015) 110–117
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estimates. The values of D0 estimated from the maximum slopes of the “root-t”-plots and from fitting the entire experimental uptake curve are somewhat smaller. We attribute this difference to the loss (or gain) of sorbate during the blind period, as already discussed. The diffusivities derived from optimization of the fit of the radial profiles to the constant diffusivity model are substantially larger, reflecting the effect of the concentration dependence of the actual differential diffusivities. The excellent fits of the radial profiles that are obtained with the limiting diffusivities from the rate analysis together with the corresponding values of λ given in Table 1 (a measure of the concentration dependence defined by Eq. (10), which is introduced later in this paper) provide strong evidence that the variable diffusivity model and the limiting diffusivity values are both essentially correct. 3.4. Axial diffusion From the rate at which the concentration profile advances, in the central region of the crystal at short times, it is possible to make an approximate estimate of the axial diffusivity (Dz ) since in this region any desorption that occurs can pbe ffiffi attributed entirely to axial diffusion. The plots of mt =m1 vs t from which the axial diffusivities were estimated are shown in Fig. 7. However, there is considerable scatter in the data and the amount of sorbate desorbed in this regime is less than 5% of the total so the accuracy with which the axial diffusivity can be estimated is low. The axial diffusivity values obtained in this way are included in Table 1 but it should be understood that these are only rough estimates. Nevertheless it is clear that the axial diffusivities are substantially smaller than the radial diffusivities so, in analyzing the experimental data the assumption of pure radial diffusion should be a reasonable approximation. Fig. 8 shows the temperature dependence of the diffusivities in Arrhenius form. The activation energy for radial diffusion thus determined is about 34 kJ/mol. The large uncertainty in the axial diffusivities precludes a valid estimate of the activation energy.
Fig. 8. Arrhenius plot showing temperature dependence of the diffusivities D0 (determined from analysis of the uptake rate data), and Dz (the axial diffusivities, estimated from Fig. 7). The errors for the axial diffusivities are rough estimations based on analyzing the extreme values for Dz .
present system, under the experimental conditions, the loadings are substantial. Therefore, to represent the behavior properly it is clearly necessary to include a concentration dependent diffusivity in the mathematical model. For systems of this kind in which (because of strong steric hindrance) the inter-cage jumps are rare events, the concentration dependence is often observed to arise mainly from the thermodynamic factor, with the intrinsic mobility remaining approximately constant (Kärger et al., 2012). This is because, for such systems, the effect of the thermodynamic factor is much greater than any effect on the energy of the transition state arising from higher occupancy of the cages. For a Langmuirian system the concentration dependence of the diffusivity is then given by
3.5. Concentration dependent diffusivity D ¼ D0 Both the experimental profiles and the uptake curves suggest that the diffusivity in this system is concentration dependent. This is not surprising since, in an adsorbed phase, the diffusivity is expected to be constant only at low concentrations within Henry's law region. For the
Fig. 7. Uptake and release curves estimated from the advance of the concentration profiles in the central region of the crystal.
d lnp D0 ¼ d lnc 1 λC
ð10Þ
where D0 is the limiting diffusivity (related to the intrinsic mobility), λ ¼ c0 =cs and C ¼ c=c0 . Eq. (10) implies that the ratio Dmax =D0 is given by 1/(1–λ). To investigate whether the deviations from the constant diffusivity model can be accounted for by a concentration dependent diffusivity of this form we used a modified version of the computer program in which we fixed the values of both D0 and λ. D0 was set at the value derived from the uptake rate analysis (Table 1) and λ was set initially at the values estimated from the equilibrium isotherms taken from the literature (Gascon et al., 2008) (see Supplementary Information, Section 2). The predicted radial profiles at the relevant times were then calculated (without any readjustment of the parameters) and compared with the experimental profiles. The predicted profiles gave a good representation of the experimental profiles but it was found that the fit (especially of the center-line profiles) could be improved by using the slightly larger λ values shown in Table 1. The calculations are bulky and time consuming so no detailed optimization was attempted. For all three temperatures and for both adsorption and desorption the profiles calculated in this way provide an excellent representation of the experimental profiles. In particular the adsorption and desorption profiles at 23 1C have characteristically different shapes but are well matched using the same values of D0 and λ. Representative examples showing the comparison of the experimental and theoretical profiles are shown in Figs. 9 and 10. The tendency of the constant diffusivity theoretical profiles to lag the experimental
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Fig. 9. Three-dimensional plots showing the optimized fit of the adsorption (a) and desorption (b) profiles at 296 K (pressure step 0–10 or 10–0 mbar) to the variable diffusivity model. D0 ¼ 1.65 10 16 m2 s 1, λ¼ 0.80.
the experimental profile runs very slightly ahead of the theoretical profile (variable diffusivity model) at short times.
4. Conclusions
Fig. 10. Two-dimensional concentration profiles for desorption of propene in DDR at 353 K for a pressure step from 210 to 0 mbar. D0 ¼ 1.63 10 15 m2 s 1 and λ ¼0.90 have been used as the parameters for the variable diffusivity model.
profiles at short times and to lead them at longer times is eliminated by the inclusion of a concentration dependent diffusivity.
3.6. Center-line loading vs time Concentration dependent diffusivities are known to affect the shape of concentration profiles, notably also towards the crystal center. In order to investigate this further we have examined in detail the behavior at the crystal axis (the center-line). For all temperatures investigated Fig. 11 shows a comparison of the time dependence of the experimental loading (of desorption) at the central axis (estimated by averaging the profiles over the central region) and the theoretical time dependence predicted from the constant diffusivity and variable diffusivity models (with the parameters shown in Table 1). The experimental center-line concentration profiles (plotted vs dimensionless time) coincide closely with the theoretical profiles calculated according to the variable diffusivity model. The constant diffusivity profile (from Crank's solution (Crank, 1975)) has a different shape and can never be properly matched to the experimental profiles whatever value is used for D/R2. In addition to this behavior we do clearly note (see the “zoom-in” on the right side of Fig. 11) for the very initial of molecular uptake the effect of axial diffusion as considered in detail already above. As a consequence,
The diffusion of propene in large DDR crystals has been studied in detail by interference microscopy. The critical diameter of the propene molecule is close to the dimensions of the windows connecting the larger cavities within the radial-symmetric pore system of zeolite DDR. Diffusion of propene is therefore very slow (with a substantial energy of activation) and can be followed over a long time span during ad- and desorption. Measurements were performed at three different temperatures (23, 50 and 80 1C) with the respective pressure step being adjusted so that the equilibrium loading was approximately constant. It is generally possible to obtain a reliable estimate of the limiting diffusivity (D0 ) from the initial slope of the desorption curve. However, this was not possible for the present system because of the loss of signal at the boundary of the crystal (due to the beveled side faces). An alternative approach was therefore developed based on analysis of the rates of uptake or release rather than of the uptake curve itself. The measured concentration profiles show that diffusion occurs predominantly by radial diffusion with an activation energy of approximately 34 kJ/mol. The diffusivity increases strongly with loading but, within the accuracy of the measurements, this effect appears to be accounted for by the thermodynamic factor and the corrected diffusivity (D0 ) appears to be essentially constant (at a given temperature). This pattern of behavior is typical of a system in which the molecule in the transition state (in the window between adjacent cages) is severely constrained. The simple Langmuirian model (Eq. 10) with constant D0 (estimated from the desorption rate measurements) provides an excellent representation of the transient concentration profiles at all three temperatures. The center-line profiles at short times suggest a small contribution from axial diffusion, presumably arising from structural defects in the actual crystals. However, the data suggest that the axial diffusivity is at least an order of magnitude smaller than the radial diffusivity and so, although its effect is detectable, it is too small to measure with any accuracy and any modification of the overall profiles is minimal. Although the motivation for this study was mainly academic it also has practical relevance since the pore size of DDR is such that it
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Fig. 11. Desorption curves for the central axis of the crystal showing the comparison between the experimental curves and the theoretical curves predicted from both the constant diffusivity and variable diffusivity models using the parameters given in Table 1 (left side). On the right side a “zoom-in” view of the data points within the gray oval is shown. In this short-time regime the experimental data points are slightly higher than the simulated ones, thus indicating the influence of diffusion in axial direction.
provides a very clean molecular sieve separation between propene and propane.
Notation c co cs C D Do Dz Jα(x) L mt m1 p r R t
λ τ
sorbate concentration in crystal/[mol/kg] initial or final (equilibrium) value of c/[mol/kg] saturation capacity (Langmuir model)/[mol/kg] dimensionless concentration c/co diffusivity/[m2/s] limiting (or thermodynamically corrected) diffusivity (Eq. (6))/[m2/s] axial diffusivity/[m2/s] Bessel functions of the first kind of order α crystal thickness/[mm] mass of sorbate adsorbed or desorbed at time t/[mol/kg] mass of sorbate adsorbed or desorbed as t-1/[mol/kg] equilibrium (partial) pressure of sorbate/[bar] radial coordinate/[m] crystal radius/[m] time/[s] co/cs Dot/R2(dimensionless time variable)
Acknowledgment The research presented in this study has been funded by the German Science Foundation. Furthermore, the authors express their gratitude for the continuous support by the Alexander von Humboldt Foundation.
Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.ces.2015.07.029.
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