J. Phys. Chem. 1996, 100, 7729-7734
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Evaluation of Transport Properties of Packed Beds of Microparticulate Porous and Nonporous Silica Beads by Means of Pulsed Field Gradient NMR Spectroscopy M. Hallmann and K. K. Unger Institut fu¨ r Anorganische Chemie und Analytische Chemie, Johannes Gutenberg-UniVersita¨ t, Becherweg 24, D-55099 Mainz, Germany
M. Appel, G. Fleischer, and J. Ka1 rger* Fakulta¨ t fu¨ r Physik und Geowissenschaften, UniVersita¨ t Leipzig, Linne` str. 5, D-04103 Leipzig, Germany ReceiVed: October 30, 1995; In Final Form: February 22, 1996X
The pulsed field gradient (PFG) NMR method is applied to study molecular diffusion in beds of spherical silica particles. From the observed transport properties it may be concluded that the internal pore system of the silica particles gives rise to two different modes of molecular migration: restricted motion with molecular mean square displacements of the order of 700 nm, and unrestricted motion with effective diffusivities decreasing with increasing loading over nearly 2 orders of magnitude. In PFG NMR measurements with nonporous silica, the molecular mobility in the space between silica particles and the topology of the space between these particles were investigated. In these experiments a slowly diffusing macromolecule, poly(dimethylsiloxane) (PDMS), is shown to be a most effective probe molecule for dynamic imaging.
Introduction The resolution of compounds in high-performance separation techniques such as high-performance liquid chromatography (HPLC) is essentially influenced by the time dependence of mass transfer of analytes in the interstitial and the internal volume of a packed column.1 This influence is usually expressed in terms of the theoretical plate height H as a function of the linear velocity u of the analyte. Three main terms contribute to the course of the H(u) curve: the eddy-diffusion term A, the longitudinal diffusion term B, and the mass transfer term between the mobile and stationary phase C. The magnitude of these terms for well-packed columns was determined from experimental H(u) dependencies to A ) 1, B ) 2, and C ) 0.01. Attempts have been made to assess the transport properties of analytes in a packed HPLC column from band width measurements.2,3 The drawback of such measurements, however, lies in the following facts: (i) The total band width of an analyte at the column outlet also includes contributions from outside the column (injection volume, connections, detector cell volume). (ii) Even when these contributions are negligibly small compared with the dispersion of analyte in the column itself, the total plate height represents an integral parameter and does not reflect local distortions (differences) across and along the column. (iii) Furthermore, one is only able to monitor the total mass transport characteristics and cannot discriminate between the eddy diffusion, molecular diffusion, and the adsorption and desorption kinetics. In order to shed more light into this area pulsed field gradient (PFG) NMR spectroscopy was applied to segments of columns which were packed similar to those of traditional HPLC columns. In order to be able to distinguish between mass transport properties in the internal (pore) volume and the interstitial volume of the packed bed, porous and nonporous spherical silica particles were employed having an average particle diameter in the same order of magnitude. X
Abstract published in AdVance ACS Abstracts, April 15, 1996.
S0022-3654(95)03188-1 CCC: $12.00
In the last couple of years, NMR spectroscopy has proved to be a rather efficient tool for the in situ study of molecular transportation and reaction in heterogeneous media.4 Under favorable condition, NMR microscopy5,6 is able to provide structural information with a spatial resolution of the order of 10 µm. However, since the diameter of the adsorbent particles applied in HPLC is typically only of the order of 5 µm, the benefit to be expected from this technique for elucidating the elementary processes during HPCL is presently still limited. As an alternative, we have therefore applied the PFG NMR method which is known to provide information about both the intrinsic mobility and structural confinements within the sample.6,7 The latter technique which has been termed dynamic imaging8,9 is based on the concept of the average propagator10 and results from the similarity of the theoretical framework of PFG NMR and neutron scattering.11,12 Experimental Section Materials. Two types of microparticulate spherical silica particles were employed in this study: LiChrospher Si 60 (E. Merck, Darmstadt, Germany) of an average particle diameter of dP ) 5 µm and nonporous silica beads (MICRA Scientific Inc., Norwalk, IL) of dP ) 4 µm. The specific surface area, aS(BET), the specific pore volume, VP, and the mean pore diameter, pd, of the LiChrospher Si 60 sample were determined from the nitrogen sorption isotherm at 77 K13 using an ASAP 2000, Micromeritics, Neuwied, Germany, with the calculation procedures implemented in the software of the equipment. The values were aS(BET) ) 690 m2/g, Vp ) 0.87 cm3/g, pd ) 5 nm. Figure 1 shows a scanning electron micrograph of the LiChrospher Si 60 particles at two different magnifications. A micrograph of the nonporous particles is shown in Figure 2. The samples are prepared by weighing the desired amount of silica particles and liquid into NMR tubes and flame-sealing of the tubes. A dense packing of the tubes was achieved by subsequent centrifugation of the samples. PFG NMR Measurements. The PFG NMR self-diffusion measurements were carried out by means of the home-built © 1996 American Chemical Society
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Figure 1. Scanning electron micrographs of porous silica particles of type LiChrospher Si 60. In (b) the internal pore structure becomes perceptible.
NMR pulse spectrometer FEGRIS 40012 operating at a proton resonance frequency of 400 MHz. The spectrometer allows the application of magnetic field gradient pulses with amplitudes up to 25 T/m which permit the measurement of molecular displacements of as small as 100 nm.14 The measurements were carried out in the conventional way (cf., e.g. refs 6 and 7) by monitoring the attenuation of the NMR signal (the “spin echo”) as a function of the intensity of the applied field gradient pulses. For normal diffusion within a homogeneous region the spinecho attenuation obeys the relation
Ψ ) exp[-γ2δ2g2D(∆ - δ/3)]
(1)
where δ, g, and ∆ denote the width, amplitude, and separation of the field gradient pulses, respectively. γ stands for the magnetogyric ratio (2.67 × 108 T-1 s-1 for protons), and D is the self-diffusion coefficient. In homogeneous systems D is related to the mean square displacement 〈z2(t)〉 in the direction of the applied field gradients by the Einstein equation
D ) 〈z2(t)〉/(2t)
(2)
Equation 1 may also serve as a satisfactory approximation for the spin-echo attenuation in the case of restricted diffusion. In this case, however, the quantity D has to be understood as an effective diffusivity being defined by eq 2 with the more general understanding that now molecular propagation may deviate from
Figure 2. Scanning electron micrographs of nonporous silica particles of type MICRA.
ordinary diffusion being subjected to structural confinement within the sample.6,7 The NMR spin echo has been generated by applying either the π/2-π rf pulse sequence (primary spin echo, in general for observation times of less than 10 ms) or the π/2-π/2-π/2 rf pulse sequence (stimulated echo). All measurements were carried out at room temperature. Results and Discussion Figure 3, a and b, shows the logarithmic representations of the NMR spin-echo amplitude for cyclohexane in LiChrospher Si 60 at relative pore filling of 10% and 100% as a function of the quantity γ2g2δ2(∆ - δ/3). This is exactly the quantity appearing in the exponent of eq 1. Therefore, for diffusion in a homogeneous medium, in such a representation any experimental point should lie on a straight line whose slope is proportional to the self-diffusivity of the given sample. The obtained patterns reveal at least three different messages: (i) The cyclohexane molecules are subjected to two different states of mobility (“phases”), which appear in the difference in the slopes in each of the attenuation curves at small and large values of γ2g2δ2(∆ - δ/3). During the observation time (10 ms maximum) there is no fast exchange between these two states. (ii) The relative amount of molecules contributing to the phase of higher mobility (first, steep decay in the attenuation) is decreasing with increasing loading.
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Figure 4. Time dependence of the effective diffusivities of cyclohexane in LiChrospher Si 60 as resulting from the best fit of the spin echo attenuations by two exponentials of the type of eq 1 for different loadings. Open symbols refer to the mobile phase (first, steep decay in Figure 3), full symbols refer to the second decay in Figure 3, representing restricted diffusion.
Figure 3. Spin-echo attenuation in PFG NMR experiments with LiChrospher 60 loaded with cyclohexane up to pore filling factors of 10% (a) and of 100% (b). The lines are biexponential fits.
(iii) There is a pronounced dependence on the observation time for the phase of lower mobility (second, less steep decay), indicating a restriction in the propagation of the molecules contributing to this phase. In all considered cases the observed spin echo attenuation may be approximated by a superposition of two exponentials of the type of eq 1. Figure 4 shows the corresponding effective diffusivities. It is remarkable that for the lowest loading the diffusivities in the mobile phase are distinctly above those for the free liquid (1.4 × 10-9 m2 s-1 15). Such a behavior may in fact be rationalized on the basis of an enhanced mobility in the not yet filled pores or free space between the spheres for sufficiently high gas phase concentrations.16,17 Correspondingly, the mobility of the molecules in the mobile phase is found to decrease continuously with increasing loading. Since the diffusivities of this phase are independent of the observation time they may be expected to reflect genuine translational mobilities. By contrast, the diffusivities deduced for the less mobile phase are found to depend on the observation time. The results of Figure 4 indicate that there is a linear relationship between the effective diffusivity and the reciprocal value of the observation time. On the basis of eq 2 such a behavior is found to be equivalent to the statement that the molecular mean square displacement is independent of the observation time. Quantitative analysis yields a value of 0.7 ( 0.1 µm for the root mean square displacement. Therefore, one has to conclude that the molecules contributing to the second decay are confined to ranges with a mean radius of the order of 0.7 µm. Since even for the shortest observation time of ∆ ) 1 ms no deviation from
the proportionality D ∝ t-1 is perceptible, the diffusivity within these ranges of molecular confinement can only be estimated to be larger than the observed upper value of 10-10 m2 s-1 of the apparent diffusivity. It is remarkable that the extension of the confining ranges remains unaffected by a variation of the loading. The confinement of a certain fraction of the cyclohexane molecules cannot be explained by the effect of the individual pores, since their diameters (∼5 nm) are much smaller than the diameters of the confining regions (∼800 nm). There is also no indication of a peculiarity in the pore distribution function which might be correlated with the existence of two different types of pores giving rise to the observed different modes of molecular propagation. As a possible explanation of the remarkable difference in the propagation patterns one might speculate that the pore network consists of regions of enhanced pore density separated from each other by layers of lower pore density. In such a case the molecules may quite easily exchange between neighboring cavities within one region, while an exchange between different regions is much more difficult. Therefore, during the observation time of the PFG NMR experiments (e10 ms) the molecules might essentially be found to remain within a particular one of these regions. The different regions may be assumed to be connected by a few “bottleneck”-type pores which are open at small molecular concentrations and which are eventually closed with increasing loading. Hence, at sufficiently low loadings, at least over a part of the sample, molecular propagation may proceed through the still open bottlenecks over much larger distances, giving rise to the observed phase of high mobility. With increasing loading the relative amount of freely moving molecules is decreasing until, when all “bottlenecks” are closed, all molecules are eventually confined to the individual regions. The properties of the packed bed depend on the pore systems within the particles as well as on the free space between the individual particles (interstitial volume). For elucidating the structural peculiarities of the latter space we have considered nonporous silica particles. In this way it could be ensured that the observed transport properties are exclusively determined by the free space between the silica particles. Figure 5 shows the spin echo attenuation in a compacted bed of nonporous silica particles of 4 µm average diameter saturated
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Figure 5. Spin-echo attenuation in PFG NMR experiments with cyclohexane in a compacted bed of nonporous silica particles of type MICRA. Different symbols refer to different observation times, which were varied between 3 and 643 ms.
Figure 6. Spin-echo attenuation in PFG NMR experiments with PDMS with a molar mass Mw of 5.930 g/mol in a compacted bed of nonporous silica particles of type MICRA. The lines are biexponential fits.
with cyclohexane. In contrast to Figure 3, the spin echo attenuation is found to be a sole function of the quantity γ2g2δ2(∆ - δ/3). Any time dependence of the effective diffusivities may therefore be excluded. It turns out that the spin echo attenuation may be satisfactorily represented by the superposition of exponentials of the type of eq 1 with effective diffusivities independent of the observation time. The first fast decay roughly reproduces the values of the diffusivity in the free liquid.15 Such a behavior may be expected for molecules within larger internal cavities or on the outer surface of the bed of particles where molecular diffusion should proceed without essential restrictions. The diffusivity determined from the slope at the highest values of the quantity γ2g2δ2(∆ - δ/3) is substantially smaller and reflects the effect of the bed of particles on molecular propagation. The reduction of the rate of molecular propagation may be easily rationalized as a consequence of the tortuosity of the bed of packed particles and the corresponding enhancement of the diffusion paths within the fluid. The reduction factor of 3 is within the range of typical tortuosity factors.18 In refs 19 and 20, the spin-echo attenuation plot vs the pulsed field gradient “wave vector” q ) (2π)-1γδg for diffusion of a fluid around packed particles was found to pass a relative maximum for wavevectors of the reciprocal value of the distance between adjacent “compartments” formed by the free space between the particles. Such a behavior may be rationalized by interpreting PFG NMR as a generalized scattering experiment. In the present study, we varied wave vector q only up to values which were about 1 order of magnitude smaller than necessary to reveal the formation of relative maxima in the spin-echo attenuation curve. As a consequence of the high molecular mobility of cyclohexane it was impossible to reduce the molecular displacement during the PFG NMR experiments to values comparable with the characteristic diameters of the free space between the individual silica particles. Sinc any dynamic imaging must be based on the measurement of molecular propagation over displacements comparable with the characteristic dimensions of the structures, for this purpose molecules with a much smaller translational mobility had to be selected. The feasibility of lowtemperature experiments for reducing the mobility of the cyclohexane molecules turned out to be limited by the dramatic increase in the transverse nuclear magnetic relaxation rates.
Owing to their internal flexibility, the decrease in the translational mobility of macromolecules is not necessarily accompanied by enhanced transverse nuclear magnetic relaxation rates and a corresponding deterioration of the measuring conditions.11,21 Macromolecules are therefore excellent candidates for the measurements of small molecular displacements. For our purpose, poly(dimethylsiloxane) (PDMS) turned out to provide most appropriate measuring conditions.22 Figure 6 shows the PFG NMR spin-echo attenuation observed for PDMS with a molar mass Mw of 5930 g/mol in a compacted bed of nonporous silica particles. The obtained attenuation plots may be approximated by a superposition of two exponentials of the type of eq 1. The diffusivities resulting from the best fit and the molecular mean square displacements corresponding to them on the basis of eq 2 are shown in Figure 7. It turns out that the mobility of one part of the PDMS molecules essentially coincides with that of the pure PDMS phase.22,23 As in the previously discussed measurements with cyclohexane, this fraction of the probe molecules is most likely to be found outside of the bed or within larger cavities. This diffusivity is clearly independent of the observation time (Figure 7a) corresponding to a linear increase of the mean square displacement with increasing observation time (Figure 7b). The second part of the PDMS molecules is characterized by a distinct time dependence of the effective diffusivity. At sufficiently small observation times the obtained values appear to be independent of the observation time, being of the order of or slightly smaller than the values for the pure polymer. With increasing observation time, however, the effective diffusivity decreases, eventually approaching reciprocal proportionality with the observation time (Figure 7a). According to eq 2 this means that the molecule mean square displacement approaches an upper limit (Figure 7b). Thus, it turns out that during the observation time of the PFG NMR experiments, PDMS remains within confined spaces. With 〈r2〉 ≈ 〈x2〉 + 〈y2〉 + 〈z2〉 ) 3〈z2〉 the mean radius of the confining regions is found to be of the order of 700 nm. It is the same value that has been observed for the confining regions within the LiChrospher Si 60 particles. However, since it is impossible to assign the origin of the confinement in these two cases to the same structural origin, this agreement appears to be only incidental. In the present case, the space of confinement may be easily attributed to the free volume comprised by neighboring particles. It is most interesting to note that in contrast to the cyclohexane molecules,
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Figure 8. Time dependence of the molecular mean square displacements 〈z2〉 of PDMS with a molar mass of 22.530 g/mol in a compacted bed of nonporous silica particles of type MICRA.
Figure 7. Time dependence of the effective diffusivities (a) and the corresponding molecular mean square displacements (b) determined from the spin-echo attenuation plots shown in Figure 6. The two values result from the best fit of the experimental data by a superposition of two exponentials of the type of eq 1.
whose mobility is reduced by no more than a factor of 3, PDMS turns out to remain completely captured within the “compartments” formed by the free space between the nonporous silica particles during the observation time of the PFG NMR experiments. Obviously, as a consequence of the reduced mobility of the PDMS molecules in the vicinity of the particle surface24 and their mutual topological hindrances, the diameters of the “effective” orifices between adjacent compartments are significantly reduced, leading to the observed inhibition in molecular propagation. It results from Figure 7b that during the maximum observation time the mean square displacement of PDMS in the pure polymer is by 2 orders of magnitude larger than within the space of confinement. Since at sufficiently short observation times the diffusivities are essentially found to approach those of the pure polymer, this reduction must in fact be explained by an enhanced confinement rather than by a reduction in the intrinsic mobility. This conclusion has been confirmed by measurements with PDMS with a molar mass Mw of 22 530 g/mol. Owing to the enhanced chain length and the resulting lower mobility, in this case a further reduction of the molecular displacements during the PFG NMR experiment has become possible. Figure 8 shows that now, for sufficiently short observation times, there is a complete agreement between the diffusivities of the pure polymer and the polymers in the “compartments” between the nonporous silica particles, and, as a result of the lower diffusivity of this PDMS, the inhibition of molecular propagation occurs at later times. Conclusion PFG NMR diffusion measurements have been carried out to investigate the transport properties of porous and nonporous
silica particles. Using cyclohexane as a probe liquid, the mesoporous silica (LiChrospher Si 60) was found to give rise to two different modes of intrinsic mobility. During the observation time of the NMR experiment, a certain fraction of molecules was found to migrate freely through the pore system, while a second part appears to be confined to regions with mean radii of the order of 700 nm. With increasing concentration the relative amount of molecules contributing to the second phase is increasing at the expense of the first phase. In parallel with the reduction in the relative amount, also the mobility of the first phase is decreasing over 2 orders of magnitude, from values roughly 1 order of magnitude above the values for the free liquid. The size of the regions of confinement is found to be unaffected by the loading. The intrinsic diffusivity within these regions can only be estimated to be at least of the order of 10-10 m2 s-1. In beds of nonporous particles, the cyclohexane diffusivity is found to be reduced by a factor of about 3 in comparison with the free liquid. Since the diffusion paths were much larger than the mean distances between neighboring particles, in this way any probing of the free space between the individual particles (“dynamic imaging”) turned out to be impossible. As a more suitable molecular species, we have applied poly(dimethylsiloxane) (PDMS). Owing to the dramatically reduced translational mobility of PDMS, it has become possible to follow molecular displacements of the order of and even below the characteristic diameters of the “compartments” between the silica particles. Moreover, the PDMS molecules are found to give rise to a “self-confinement” within the space between the silica particles; i.e., it is by the mere existence of this particular diffusant that the exchange rate between adjacent compartments is dramatically reduced. In contrast to cyclohexane, whose diffusivity through the bed of nonporous silica particles is reduced by no more than a factor of 3 (corresponding to a reduction of the molecular mean square displacement by the same factor), the molecular mean square displacement of PDMS is found to be by at least 2 orders of magnitude smaller. This reduction in the rate of molecular transportation cannot be explained by a reduction of the intrinsic mobility, since for sufficiently short observation times the diffusivities of PDMS within the bed of particles and in the pure polymer system are found to be essentially the same. As to our knowledge, the present study involves the first experiments of PFG NMR dynamic imaging with macromolecular compounds, purpose-selected on the basis of their transport properties in the bulk phase. Only in this way a
7734 J. Phys. Chem., Vol. 100, No. 18, 1996 determination of the extension of the compartments between the investigated unporous silica particles has become possible. Acknowledgment. The authors thank E. Merck, Darmstadt, and MICRA Scientific, Inc., Norwalk, IL, for providing the porous and nonporous silica samples. The authors are also obliged to U. Lorenz, GKSS Geesthacht, for the scanning electron micrographs and to Dr. G. Meier, MPI fu¨r Polymerforschung Mainz, for providing us with the two fractions of PDMS used in the experiments. Financial support by the Deutsche Forschungsgemeinschaft (SFB 294) is gratefully acknowledged. References and Notes (1) Unger, K. K. Packings and Stationary Phases in Chromatographic Techniques; Marcel Dekker: New York, 1983. (2) Bristow, P. A.; Knox, J. H. Chromatographia 1976, 10, 279. (3) Hallmann, M. Ph.D. Thesis, Johann-Gutenberg-Universita¨t Mainz, 1995. (4) Bell, A. T., Pines, A., Eds. NMR Techniques in Catalysis; Marcel Dekker: New York, 1994. (5) Blu¨mich, B., Kuhn, W., Eds. Magnetic Resonance Microscopy; VCH: Weinheim, Germany, 1992. (6) Callaghan, P. T. Principles of Nuclear Magnetic Resonance Microscopy; Clarendon Press: Oxford, U.K., 1991. (7) Ka¨rger, J.; Ruthven, D. M. Diffusion in Zeolites and Other Microporous Solids; Wiley: New York, 1992.
Hallmann et al. (8) Coy, A.; Callaghan, P. T. J. Chem. Phys. 1994, 101, 4599. (9) Cory, D. G.; Garroway, A. N. Magn. Reson. Med. 1990, 14, 435. (10) Ka¨rger, J.; Heink, W. J. Magn. Reson. 1983, 51, 1. (11) Fleischer, G.; Fujara, F. NMRsBasic Principles Prog. 1994, 30, 159. (12) Ka¨rger, J.; Ba¨r, N.-K.; Heink, W.; Pfeifer, H.; Seiffert, G. Z. Naturforsch. 1995, 50a, 186. (13) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: London, 1982. (14) Heink, W.; Ka¨rger, J.; Seiffert, G.; Fleischer, G.; Rauchfuss, J. J. Magn. Reson. A 1995, 114, 101. (15) Holz, M.; Weinga¨rtner, H. J. Magn. Reson. 1991, 92, 115. (16) Tabony, J.; Cosgrove, T. Chem. Phys. Lett. 1979, 67, 103. (17) Ka¨rger, J. Mol. Phys. 1981, 43, 1189. (18) Satterfield, C. N. Mass. Transfer in Heterogeneous Catalysis; MIT Press: Cambridge, MA, 1970. (19) Callaghan, P. T.; Coy, A.; MacGowan, D.; Packer, K. J.; Zelaya, F. O. 1991, 351, 467. (20) Callaghan, P. T.; Coy, A.; Halpin, T. P. J.; MacGowan, D.; Packer, K. J.; Zelaya, F. O. J. Chem. Phys. 1992, 97, 651. (21) Ka¨rger, J.; Fleischer, G. Trends Anal. Chem. 1994, 13, 145. (22) Appel, M.; Fleischer, G.; Ka¨rger, J.; Fujara, F.; Chang, I. Macromolecules 1994, 27, 4274. (23) Appel, M.; Fleischer, G. Macromolecules 1993, 26, 5520. (24) Kirst, K. U.; Kremer, F.; Litvinov, V. M. Macromolecules 1993, 26, 975.
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