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Determining the transport diffusivity from intra-crystalline concentration profiles
Zeolites and Related Materials: Trends, Targets and Challenges Proceedings of 4th International FEZA Conference A. Gédéon, P. Massiani and F. Babonneau (Editors) © 2008 Elsevier B.V. All rights reserved.
607
Determining the transport diffusivity from intracrystalline concentration profiles L. Heinke, C. Chmelik, J. Kärger Faculty of Physics and Geosciences, Univ. of Leipzig, Linnéstr. 5, 04103 Leipzig, Germany, [email protected]
Abstract Transport properties of guest molecules are of major importance for many applications of nanoporous materials. The concentration profiles evolving during uptake and release experiments allow the determination of the transport diffusivity by different approaches with various boundary conditions and varying accuracy. We show that for one-dimensional mass transport the transport diffusivity may be directly calculated by an equation based on Fick’s 1st law. This method is independent of special boundary conditions, as required for Boltzmann’s integration method, and provides a better accuracy compared to the previously reported approach by direct application of Fick’s 2nd law. Keywords: transport diffusion, Fick’s 1st law, interference microscopy.
1. Introduction In many technical applications in catalysis and mass separation, the performance of nanoporous material is controlled by their transport properties. Over decades, relevant information about the mass transport could only be obtained by macroscopic methods, i.e., by measurements with beds of crystallites, revealing the average response curves of uptake or release of the individual molecules. These methods fail to provide detailed information on the diffusivity if large concentration ranges are covered since only the integrated uptake curve is accessible. With the introduction of interference microscopy to diffusion studies, the spatial-resolved concentration profiles of the guest molecules evolving during an uptake or release process are obtainable. These profiles contain a lot of information about the transport process. In principle, these concentration profiles may also be obtained by IR imaging, optical or fluorescence microscopy.
2. Experimental 2.1. Experimental setup The experimental setup consists of an optical cell which contains the studied zeolite crystals. The cell is connected to a vacuum system with a vacuum pump and a stock volume to execute pressure step changes at the beginning of the experiment and to keep the pressure constant at the desired value. The concentration profiles evolving in one single crystal are observed by the interference microscope (section 2.2). The analysed concentration profiles are monitored during the uptake of pure methanol at room temperature (298 K). The pressure step is from 0 to 80 mbar. 2.2. Interference microscopy Figure 1 illustrates the application of interference microscopy (IFM) to diffusion studies within nanoporous materials. It is based on an analysis of the interference pattern generated by superimposing light beams passing the nanoporous crystal and the surrounding atmosphere. Since the optical density of the crystal is a function of the
608
L. Heinke et al.
concentration of the guest molecules, changes in local concentration directly appear in corresponding changes in the interference pattern (Fig. 1 b,c). Vice versa, from the interference patterns one may deduce the corresponding concentration profiles (Fig. 1 d). The quantity directly accessible is thus the integral over the intracrystalline concentration in observation direction (x-axis in Fig. 1 a) with a spatial resolution of y × z 0.5 × 0.5 μm2. If diffusion in x-direction is excluded, interference microscopy directly yields the local concentrations c(x,y,z). If the mass transport takes place along three directions, the intracrystalline concentration cannot be directly determined from the integrated Figure 1. Schematics of interference microscopy. a) Light profiles obtained by beams passing through the crystal and through the interference microscopy. surrounding atmosphere. b) The interference microscope. Therefore, a careful c) Interference pattern caused by different optical path recalculation of the intra- lengths. d) Evolution of the concentration profiles calculated crystalline concentration from the interference pattern [1]. profiles needs to be conducted [2] 2.3. The investigated nanoporous crystals The silica-ferrierite zeolite is a cation-free zeolite with two perpendicular channel systems which intersect each other. [3]. One channel system is adjusted along ydirection and is framed by an 8-membered ring, this means they are formed by 8 oxygen and 8 silicon atoms. The other channel system is extended along z-direction and is formed by 10-membered ring channels. The outer geometry is like a cuboid with its long-side lengths in y- and z-direction (ly = 25 μm and lz = 100 μm; l denotes the half edge length) and a short-side length in x-direction (lx 10 μm). On both big side faces of the crystal (parallel to y-z) there are small roof-like parts. In previous studies [1, 4) it was found that, due to pore blocking at the channel entrances in z-direction, mass transport essentially proceeds one-dimensional (in y-direction). The ferrierite crystals are activated under high vacuum at a temperature of 673 K for 12 h.
3. Determining the transport diffusivity The mass transport of a single component is described by Fick's 1st law
j = −D
∂c ∂y
(1)
which predicts proportionality between particle flux density j and the gradient of the particle concentration c. In general, the proportionality factor, viz. the transport diffusivity D, is a function of particle concentration. Conservation of matter,
∂c ∂j , =− ∂t ∂y
(2)
means that the negative gradient of the flux density j equals the temporal change of the concentration c. By combining eq.(1) with eq.(2) integrated over y, the transport diffusivity may be directly calculated by
Determining the transport diffusivity from intracrystalline concentration profiles
D (c ) = −
∞
∂ c ( y ', t )d y ' ∂t ³y
∂c . ∂y
609
(3)
In one-dimensional systems wherein the uptake or release proceeds from two opposite faces (y = -l and y = l), due to symmetry, the upper limit of integration is substituted by the y-value in the centre (y=0). Fick’s 2nd law is obtained by inserting eq.(1) in eq.(2), 2
∂c ∂ § ∂c · ∂ 2 c ∂D § ∂c · = D = D + ¨ ¸ ¨ ¸ . ∂t ∂y © ∂y ¹ ∂y 2 ∂c © ∂y ¹
(4)
In the centre of the concentration profile, the first spatial derivative of the concentration is 0 due to symmetry. Therefore, Fick’s 2nd law (eq.(4)) can be directly used for determining the transport diffusivity in the centre,1
Dcentre ( c ) =
∂c ∂t . ∂ 2 c ∂y 2
(5)
A particularly elegant means to deduce diffusivities from transient one-dimensional concentration profiles is presented by Boltzmann’s integration method [5-7]. However, for applying this method the following conditions must be fulfilled: (i) the initial concentration is uniform all over the crystal (c(y,t=0)=c0), (ii) the evolution of the transient concentration profiles is initiated by a step change in the boundary concentration which, later on, remains constant (c(y=0,t)=ceq), and (iii) the time interval considered is small enough so that the diffusion front evolving from one side has not yet reached the opposite one. Therefore, the systems may be considered to be semi-infinitely extended (c(y=,t)=c0). By introducing a new variable η = y t and integrating over , eq.(4) transforms to
1 dη 1 dy η dc = − − ydc . ³ 2 dc c =c0 2t dc c =³c0 c
D (c ) = −
c
(6)
In this way, the diffusivity at any concentration c0 < c < ceq covered during the whole process of uptake or release may simply be determined from the respective integral
³
c
c = c0
ydc and its slope dy/dc at the given concentration.
The equations for calculating the transport diffusivity (eqs. (3), (5), (6)) can be used for one-dimensional mass transport. If the mass transport takes place along two or three directions, the quasi-one-dimensional profiles which evolve when the concentration fronts do not yet overlap can be analysed. [2]. The surface permeability or the factor by which a reduced surface permeability retards the transport process can be calculated from the intracrystalline concentration profiles too.6 Moreover, the sticking probability, which gives the probability that a gas molecule hitting the surface continues its trajectory into the crystal, is also ascertainable. [8].
4. Results and Discussion Determining the concentration-dependent transport diffusivity with this method, directly based on Fick’s 1st law, only the temporal change of the concentration integral and the spatial derivative of the concentration are required. Therewith, the entire concentration dependence of the diffusivity results already from two successive concentration profiles. But, in contrast to Boltzmann’s integration method, no restricting boundary conditions like an absence of any surface resistances or a vanishing concentration in the profile
610
L. Heinke et al.
centre (corresponding to a semi-infinite media) are needed. With this method the transport diffusivity can be calculated from any one-dimensional concentration profile. Furthermore, the results exhibit a higher accuracy compared to the previously reported approach by direct application of Fick’s 2nd law [1]. (A detailed discussion of the accuracy can be found in ref.6.) Owing to the discussed advantages, the method based on Fick’s 1st law is, in our opinion, the most powerful way of analysing the evolution of guest-concentration profiles in (quasi-) one-dimensional nanoporous host systems.
Figure 2. (a) Experimental concentration profiles of methanol in ferrierite for a gas pressure step from 0 to 80 mbar obtained by IFM.7 (b) Molecular diffusivity as a function of concentration evaluated by several methods.
5. Conclusion In the present work, three different ways of determining the transport diffusivity from transient concentration profiles are presented. If the prerequisites of Boltzmann’s integration method are fulfilled this method is very accurate and enables the analysis of the entire dependence of diffusivity from one single profile. The method based on Fick’s 1st law is also very precise and can be generally used without any limitations to special constraints. The differential application of Fick’s 2nd law, however, shows a significant scattering of the results.
Acknowledgement The authors thank Pavel Kortunov and Despina Tzoulaki for many stimulating discussions. Financial support by the Studienstiftung des deutschen Volkes and by the DFG- and NWO-sponsored International Research Training Group “Diffusion in Porous Materials” is gratefully acknowledged.
References [1] J. Kärger, P. Kortunov, S. Vasenkov, et al., Angew. Chem. Int. Ed. 45 (2006) 7846 [2] L. Heinke, P. Kortunov, D. Tzoulaki, et al., Europhys. Lett. 81 (2008) 26002 [3] C. Baerlocher, L. B. McCusker, and D. H. Olson, Atlas of Zeolite Framework Types (Elsevier, Amsterdam, 2007). [4] P. Kortunov, L. Heinke, S. Vasenkov, et al., J. Phys. Chem. B, 110 (2006) 23821. [5] L. Boltzmann, Wiedemanns Ann. d. Physik 53 (1894) 959 [6] L. Heinke and J. Kärger, New Journal of Physics 10 (2008) 23 [7] P. Kortunov, L. Heinke, and J. Kärger, Chem. Mater. 19 (2007) 3917 [8] L. Heinke, P. Kortunov, D. Tzoulaki, et al., Phys. Rev. Lett. 99 (2007) 228
607
Determining the transport diffusivity from intracrystalline concentration profiles L. Heinke, C. Chmelik, J. Kärger Faculty of Physics and Geosciences, Univ. of Leipzig, Linnéstr. 5, 04103 Leipzig, Germany, [email protected]
Abstract Transport properties of guest molecules are of major importance for many applications of nanoporous materials. The concentration profiles evolving during uptake and release experiments allow the determination of the transport diffusivity by different approaches with various boundary conditions and varying accuracy. We show that for one-dimensional mass transport the transport diffusivity may be directly calculated by an equation based on Fick’s 1st law. This method is independent of special boundary conditions, as required for Boltzmann’s integration method, and provides a better accuracy compared to the previously reported approach by direct application of Fick’s 2nd law. Keywords: transport diffusion, Fick’s 1st law, interference microscopy.
1. Introduction In many technical applications in catalysis and mass separation, the performance of nanoporous material is controlled by their transport properties. Over decades, relevant information about the mass transport could only be obtained by macroscopic methods, i.e., by measurements with beds of crystallites, revealing the average response curves of uptake or release of the individual molecules. These methods fail to provide detailed information on the diffusivity if large concentration ranges are covered since only the integrated uptake curve is accessible. With the introduction of interference microscopy to diffusion studies, the spatial-resolved concentration profiles of the guest molecules evolving during an uptake or release process are obtainable. These profiles contain a lot of information about the transport process. In principle, these concentration profiles may also be obtained by IR imaging, optical or fluorescence microscopy.
2. Experimental 2.1. Experimental setup The experimental setup consists of an optical cell which contains the studied zeolite crystals. The cell is connected to a vacuum system with a vacuum pump and a stock volume to execute pressure step changes at the beginning of the experiment and to keep the pressure constant at the desired value. The concentration profiles evolving in one single crystal are observed by the interference microscope (section 2.2). The analysed concentration profiles are monitored during the uptake of pure methanol at room temperature (298 K). The pressure step is from 0 to 80 mbar. 2.2. Interference microscopy Figure 1 illustrates the application of interference microscopy (IFM) to diffusion studies within nanoporous materials. It is based on an analysis of the interference pattern generated by superimposing light beams passing the nanoporous crystal and the surrounding atmosphere. Since the optical density of the crystal is a function of the
608
L. Heinke et al.
concentration of the guest molecules, changes in local concentration directly appear in corresponding changes in the interference pattern (Fig. 1 b,c). Vice versa, from the interference patterns one may deduce the corresponding concentration profiles (Fig. 1 d). The quantity directly accessible is thus the integral over the intracrystalline concentration in observation direction (x-axis in Fig. 1 a) with a spatial resolution of y × z 0.5 × 0.5 μm2. If diffusion in x-direction is excluded, interference microscopy directly yields the local concentrations c(x,y,z). If the mass transport takes place along three directions, the intracrystalline concentration cannot be directly determined from the integrated Figure 1. Schematics of interference microscopy. a) Light profiles obtained by beams passing through the crystal and through the interference microscopy. surrounding atmosphere. b) The interference microscope. Therefore, a careful c) Interference pattern caused by different optical path recalculation of the intra- lengths. d) Evolution of the concentration profiles calculated crystalline concentration from the interference pattern [1]. profiles needs to be conducted [2] 2.3. The investigated nanoporous crystals The silica-ferrierite zeolite is a cation-free zeolite with two perpendicular channel systems which intersect each other. [3]. One channel system is adjusted along ydirection and is framed by an 8-membered ring, this means they are formed by 8 oxygen and 8 silicon atoms. The other channel system is extended along z-direction and is formed by 10-membered ring channels. The outer geometry is like a cuboid with its long-side lengths in y- and z-direction (ly = 25 μm and lz = 100 μm; l denotes the half edge length) and a short-side length in x-direction (lx 10 μm). On both big side faces of the crystal (parallel to y-z) there are small roof-like parts. In previous studies [1, 4) it was found that, due to pore blocking at the channel entrances in z-direction, mass transport essentially proceeds one-dimensional (in y-direction). The ferrierite crystals are activated under high vacuum at a temperature of 673 K for 12 h.
3. Determining the transport diffusivity The mass transport of a single component is described by Fick's 1st law
j = −D
∂c ∂y
(1)
which predicts proportionality between particle flux density j and the gradient of the particle concentration c. In general, the proportionality factor, viz. the transport diffusivity D, is a function of particle concentration. Conservation of matter,
∂c ∂j , =− ∂t ∂y
(2)
means that the negative gradient of the flux density j equals the temporal change of the concentration c. By combining eq.(1) with eq.(2) integrated over y, the transport diffusivity may be directly calculated by
Determining the transport diffusivity from intracrystalline concentration profiles
D (c ) = −
∞
∂ c ( y ', t )d y ' ∂t ³y
∂c . ∂y
609
(3)
In one-dimensional systems wherein the uptake or release proceeds from two opposite faces (y = -l and y = l), due to symmetry, the upper limit of integration is substituted by the y-value in the centre (y=0). Fick’s 2nd law is obtained by inserting eq.(1) in eq.(2), 2
∂c ∂ § ∂c · ∂ 2 c ∂D § ∂c · = D = D + ¨ ¸ ¨ ¸ . ∂t ∂y © ∂y ¹ ∂y 2 ∂c © ∂y ¹
(4)
In the centre of the concentration profile, the first spatial derivative of the concentration is 0 due to symmetry. Therefore, Fick’s 2nd law (eq.(4)) can be directly used for determining the transport diffusivity in the centre,1
Dcentre ( c ) =
∂c ∂t . ∂ 2 c ∂y 2
(5)
A particularly elegant means to deduce diffusivities from transient one-dimensional concentration profiles is presented by Boltzmann’s integration method [5-7]. However, for applying this method the following conditions must be fulfilled: (i) the initial concentration is uniform all over the crystal (c(y,t=0)=c0), (ii) the evolution of the transient concentration profiles is initiated by a step change in the boundary concentration which, later on, remains constant (c(y=0,t)=ceq), and (iii) the time interval considered is small enough so that the diffusion front evolving from one side has not yet reached the opposite one. Therefore, the systems may be considered to be semi-infinitely extended (c(y=,t)=c0). By introducing a new variable η = y t and integrating over , eq.(4) transforms to
1 dη 1 dy η dc = − − ydc . ³ 2 dc c =c0 2t dc c =³c0 c
D (c ) = −
c
(6)
In this way, the diffusivity at any concentration c0 < c < ceq covered during the whole process of uptake or release may simply be determined from the respective integral
³
c
c = c0
ydc and its slope dy/dc at the given concentration.
The equations for calculating the transport diffusivity (eqs. (3), (5), (6)) can be used for one-dimensional mass transport. If the mass transport takes place along two or three directions, the quasi-one-dimensional profiles which evolve when the concentration fronts do not yet overlap can be analysed. [2]. The surface permeability or the factor by which a reduced surface permeability retards the transport process can be calculated from the intracrystalline concentration profiles too.6 Moreover, the sticking probability, which gives the probability that a gas molecule hitting the surface continues its trajectory into the crystal, is also ascertainable. [8].
4. Results and Discussion Determining the concentration-dependent transport diffusivity with this method, directly based on Fick’s 1st law, only the temporal change of the concentration integral and the spatial derivative of the concentration are required. Therewith, the entire concentration dependence of the diffusivity results already from two successive concentration profiles. But, in contrast to Boltzmann’s integration method, no restricting boundary conditions like an absence of any surface resistances or a vanishing concentration in the profile
610
L. Heinke et al.
centre (corresponding to a semi-infinite media) are needed. With this method the transport diffusivity can be calculated from any one-dimensional concentration profile. Furthermore, the results exhibit a higher accuracy compared to the previously reported approach by direct application of Fick’s 2nd law [1]. (A detailed discussion of the accuracy can be found in ref.6.) Owing to the discussed advantages, the method based on Fick’s 1st law is, in our opinion, the most powerful way of analysing the evolution of guest-concentration profiles in (quasi-) one-dimensional nanoporous host systems.
Figure 2. (a) Experimental concentration profiles of methanol in ferrierite for a gas pressure step from 0 to 80 mbar obtained by IFM.7 (b) Molecular diffusivity as a function of concentration evaluated by several methods.
5. Conclusion In the present work, three different ways of determining the transport diffusivity from transient concentration profiles are presented. If the prerequisites of Boltzmann’s integration method are fulfilled this method is very accurate and enables the analysis of the entire dependence of diffusivity from one single profile. The method based on Fick’s 1st law is also very precise and can be generally used without any limitations to special constraints. The differential application of Fick’s 2nd law, however, shows a significant scattering of the results.
Acknowledgement The authors thank Pavel Kortunov and Despina Tzoulaki for many stimulating discussions. Financial support by the Studienstiftung des deutschen Volkes and by the DFG- and NWO-sponsored International Research Training Group “Diffusion in Porous Materials” is gratefully acknowledged.
References [1] J. Kärger, P. Kortunov, S. Vasenkov, et al., Angew. Chem. Int. Ed. 45 (2006) 7846 [2] L. Heinke, P. Kortunov, D. Tzoulaki, et al., Europhys. Lett. 81 (2008) 26002 [3] C. Baerlocher, L. B. McCusker, and D. H. Olson, Atlas of Zeolite Framework Types (Elsevier, Amsterdam, 2007). [4] P. Kortunov, L. Heinke, S. Vasenkov, et al., J. Phys. Chem. B, 110 (2006) 23821. [5] L. Boltzmann, Wiedemanns Ann. d. Physik 53 (1894) 959 [6] L. Heinke and J. Kärger, New Journal of Physics 10 (2008) 23 [7] P. Kortunov, L. Heinke, and J. Kärger, Chem. Mater. 19 (2007) 3917 [8] L. Heinke, P. Kortunov, D. Tzoulaki, et al., Phys. Rev. Lett. 99 (2007) 228