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Simulation of Nonwetting Phase Entrapment within Porous Media Using Magnetic Resonance Imaging Matthew J. Watt-Smith,† Sean P. Rigby,*,† John A. Chudek,‡ and Robin S. Fletcher§ Department of Chemical Engineering, UniVersity of Bath, ClaVerton Down, Bath, BA2 7AY, U.K., DiVision of Biological Chemistry and Molecular Microbiology, School of Life Sciences, UniVersity of Dundee, Dundee, DD1 4HN, U.K., and Johnson Matthey Catalysts, P.O. Box 1, Belasis AVenue, Billingham, CleVeland, TS23 1LB, U.K. ReceiVed January 16, 2006. In Final Form: March 29, 2006 Models representing the pore structures of amorphous, mesoporous silica pellets have been constructed using magnetic resonance images of the materials. Using magnetic resonance imaging (MRI), maps of the macroscopic (∼0.01-1 mm) spatial distribution of porosity and pore size were obtained. The nature and key parameters of the physical mechanism for mercury retraction, during porosimetry experiments on the silica materials, were determined using integrated gas sorption experiments. Subsequent simulations of mercury porosimetry within the structural models derived from MRI have been used to successfully predict, a priori, the point of the onset of structural hysteresis and the final levels of mercury entrapment for the silicas. Hence, a firm understanding of the physical processes of mercury retraction and entrapment in these amorphous silica materials has been established.
Introduction Determining the mechanism of the entrapment of nonwetting fluids within the void space of porous media is important in many fields, such as mercury porosimetry, enhanced oil recovery, and soil remediation. The ability to predict levels of nonwetting phase (nwp) entrapment within a particular pore structure would aid the structural characterization of a porous solid, the assessment of the likely displacement efficiency of oil from a given reservoir rock, and the estimation of organic contaminant levels in polluted soils. For example, models for the entrapment of mercury within porous media have been used as the basis of methods to determine various key statistical descriptors of pore structures, such as the pore connectivity1 and the spatial distribution of pores of different sizes.2 The withdrawal efficiency in the mercury/air system is generally thought to be controlled by various aspects of pore geometry, such as the effective pore throat size, pore throat size heterogeneity, the ratio of pore body size to neighboring pore throat size (“snap-off ratio”), and the pore coordination number (number of throats joined to a particular pore body). In general,3 the same pore geometry factors that govern mercury withdrawal efficiency also increase hydrocarbon recovery efficiency, and it is often possible to convert laboratory data on the air/mercury system to the subsurface brine/hydrocarbon system of an oil reservoir.3 The void spaces of macroporous oil-reservoir rocks and catalyst pellets have been imaged directly using X-ray microtomography.4,5 The three-dimensional (3D) visualizations of the void space thus obtained have subsequently been used to derive models on which simulations of capillary-dominated displacements and the transport of wetting and nonwetting fluid mixtures have been * Corresponding author. Telephone no.: +44 (0)1225 384978. E-mail:
[email protected]. † University of Bath. ‡ University of Dundee. § Johnson Matthey Catalysts. (1) Portsmouth, R. L.; Gladden, L. F. Chem. Eng. Sci. 1991, 46, 3023-3036. (2) Rigby, S. P.; Fletcher, R. S.; Riley, S. N. Chem. Eng. Sci. 2004, 59, 41-51. (3) Wardlaw, N. C.; McKellar, M. Powder Technol. 1981, 29 127-143. (4) Ruffino, L.; Mann, R.; Oldman, R.; Stitt, E. H.; Boller, E.; Cloetens, P.; DiMichiel, M.; Merion, J. Can. J. Chem. Eng. 2005, 83, 132-139. (5) Hazlett, R. D.; Chen, S. Y.; Soll, W. E. J. Pet. Sci. Eng. 1998, 20, 167-175.
carried out.5 The rock matrix image data was used as input to the fluid transport simulator, and the results compared with endpoint saturation images and data. With an X-ray system using 5-20 keV hard X-rays, a sample of a few millimeters thickness can be imaged in three dimensions at 1 µm resolution.6 Another type of system uses soft X-rays with energies of ∼100-1000 eV to image samples with sizes of ∼1-10 µm at resolutions of 50-100 nm.6 Unfortunately, carbonate reservoir rocks often contain pores smaller than 0.25 µm,7 and also most of the porous materials utilized as catalysts or absorbents in industry are mesoporous (with pore sizes of 2-50 nm), and thus possess pores well below the best resolution possible with X-ray tomography. However, microfocus X-ray (MFX) imaging may still be used to map the macroscopic (∼0.01-10 mm) spatial distribution of local average mesoporosity.8 Several recent studies, using MFX imaging8 and magnetic resonance imaging (MRI),9,10 have shown that many common catalyst support materials possess nonrandom, macroscopic heterogeneities in the spatial distribution of (at least) local average porosity (voidage fraction), pore size, and/or pore tortuosity. These heterogeneities have been shown to influence both steady-state11,12 and transient12 molecular diffusion and the pattern of the lay-down of coke during catalyst deactivation.13 Previous work3 on glass micromodels suggests that a degree of structural heterogeneity within a porous medium may also give rise to mercury entrapment. A wide range of work has been carried out on model porous media to determine the underlying mechanisms of the entrapment of nonwetting fluids.3,14,15 In particular, mercury porosimetry experiments on micromodels, (6) Wang, Y.; De Carlo, F.; Mancini, D. C.; McNulty, I.; Tieman, B.; Bresnahan, J.; Foster, I.; Insley, J.; Lane, P.; von Laszewski, G.; Kesselman, C.; Su, M.-H.; Thiebaux, M. ReV. Sci. Instr. 2001, 72, 2062-2068. (7) Wardlaw, N. C.; McKellar, M.; Li, Y. Carbonates EVaporites 1988, 3, 1-15. (8) Rigby, S. P.; Fletcher, R. S.; Raistrick, J. H.; Riley, S. N. Phys. Chem. Chem. Phys. 2002, 4, 3467-3481. (9) Hollewand, M. P.; Gladden, L. F. J. Catal. 1993, 144, 254-272. (10) Rigby, S. P.; Gladden, L. F. Chem. Eng. Sci. 1996, 51, 2263-2272. (11) Hollewand, M. P.; Gladden, L. F. Chem. Eng. Sci. 1995, 50, 309-326. (12) Hollewand, M. P.; Gladden, L. F. Chem. Eng. Sci. 1995, 50, 327-344. (13) Cheah, K.-Y.; Chiaranussati, N.; Hollewand, M. P.; Gladden, L. F. Appl. Catal. 1994, 115, 147-155. (14) Leonormand, R.; Zarcone, C.; Sarr, A. J. Fluid Mech. 1983, 135, 337353.
10.1021/la060142s CCC: $33.50 © 2006 American Chemical Society Published on Web 04/29/2006
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consisting of pore networks etched in glass,3 have suggested that entrapment generally arises due to the “snap-off” phenomenon, where the continuity of the mercury meniscus is broken, which most often occurs within narrow pore necks between larger pore bodies or at the boundaries of larger scale heterogeneities in the spatial distribution of pore size. The glass micromodel experiments have shown that entrapment of mercury in pore bodies, arising from neighboring pore necks interspersed between them, often results in only partial filling of the pore body with residual mercury, whereas entrapment due to larger-scale heterogeneity is generally associated with a piston-type intrusion and retraction behavior, where entrapped mercury completely fills the pores. For pore body-pore neck networks, it has been observed that mercury entrapment only arises once the ratio of the sizes of neighboring pore bodies and necks exceeds a particular value. Matthews et al.16 suggest that the ratio of the sizes of neighboring large and small pore elements required to cause snap-off is greater than ∼6. For mercury retraction experiments on glass micromodels consisting of grids of pore elements of uniform size, no mercury entrapment occurred. It was also found that no entrapment occurred in glass micromodels where the sizes of individual, neighboring pore grid elements were different, but relatively similar, and were distributed at random. However, in another glass model in which isolated clusters of large pores occurred in a continuous network of smaller pores, Wardlaw and McKellar3 found that, after initially fully imbibing the void space with mercury, the mercury first withdrew from the smaller pores. However, at the stage where pressure had been reduced below the threshold for the emptying of the clusters of larger pores, these had already become disconnected by the “snap-off” of the mercury meniscus, and extensive residual mercury was retained. Therefore, one cause of mercury entrapment was found to be the presence of spatially extended structural heterogeneities. The entrapment of mercury within more structurally complex glass micromodels has also been studied. Tsakiroglou and Payatakes17 derived mathematical expressions for the critical pressure for the retraction and snap-off of mercury menisci in model porous media consisting of a network of chambers and throats. These expressions accounted for the pore aspect ratio and the effects of network topology. Tsakiroglou and Payatakes17 used their expressions to simulate mercury retraction from a chamber and throat network and compared the results with experimental data for chamber and throat networks etched in glass. Reasonable agreement between theory and experiment was obtained. However, the models used in this17 and other previous work3,14,15 had relatively regular pore structures, while the typical catalyst supports used in industry are amorphous, and are thus likely to possess much more complex pore structures. When interpreted in the light of the above findings for glass micromodels, recent experimental work18,19 on mesoporous, solgel silica catalyst support pellets has suggested that mercury entrapment within these amorphous materials arises because of the presence of pronounced macroscopic (>10 µm) structural heterogeneities. Semiempirical alternatives to the Washburn equation have been developed for the analysis of raw mercury porosimetry data.18 These new equations deconvolve contact angle hysteresis from structural effects,18 and have been validated (15) Leonormand, R.; Touboul, E.; Zarcone, C. J. Fluid Mech. 1988, 189, 165-187. (16) Matthews, G. P.; Ridgway, C. J.; Spearing, M. C. J. Colloid Interface Sci. 1995, 171, 8-27. (17) Tsakiroglou, C. D.; Payatakes, A. C. AdV. Colloid Interface Sci. 1998, 75, 215-253. (18) Rigby, S. P.; Edler, K. J. J. Colloid Interface Sci. 2002, 250, 175-190. (19) Rigby, S. P.; Fletcher, R. S.; Riley, S. N. Appl. Catal. A 2003, 247, 27-39.
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using integrated nitrogen sorption experiments on model porous media.20 Analysis of the raw porosimetry data for whole pellet samples (approximately millimeters in size) of the silica materials using the new equations suggests that structural hysteresis and mercury entrapment are confined to only the very largest pores in the material. This suggestion has been confirmed by integrated gas sorption experiments.2 However, when the raw porosimetry data for fragmented samples (with particle sizes ∼10-100 µm) of the same batches of silica pellets are also analyzed using the new equations, no hysteresis or mercury entrapment are observed. This finding is independent of the experimental equilibration time, and thus the difference in entrapment levels between whole and fragmented samples is not simply due to the longer path to the boundary of the particle, and hence potentially slower mercury transport, for whole pellets. These findings suggest that relatively extreme macroscopic heterogeneities in the spatial distribution of pore size that are larger than the powder particle size may be responsible for mercury entrapment. In one particular batch of silica material, where pellets are transparent to visible light, the location of mercury entrapped within whole pellets was observed using light microscopy.19 The entrapped mercury was found to be concentrated into discrete, macroscopic domains surrounded by more continuous, clear regions containing no entrapped mercury. Therefore, in whole pellets, it is proposed that domains of relatively large pores are probably surrounded by a continuous network of smaller pores. As seen in the experimental micromodel results of Wardlaw and McKellar3 described above, this particular spatial arrangement of pore sizes results in mercury entrapment. Once the shielding of the regions of larger pore size by smaller pores is removed by fragmentation of the sample, then the proposed cause of mercury entrapment no longer exists, and thus no entrapment is observed. While amorphous materials undoubtedly have more complex pore space architectures than glass micromodels, the above findings suggest that the cause of mercury entrapment within some amorphous materials is still dominated solely by macroscopic structural heterogeneity. These particular macroscopic heterogeneities, in the spatial distribution of the pore surface area-to-volume ratio (related to pore size), can be detected within the silica materials using MRI.10,19 MRI is the only suitable technique that can sufficiently characterize the materials studied in this work to predict nwp entrapment. Because of the aforementioned limitations in capabilities and resolution, it is not possible to construct a fully representative model for the pore space of the macroscopically heterogeneous, mesoporous silica materials described above using X-ray tomography. While 3D transmission electron microscopy (TEM)21 is able to image the void space of mesoporous materials directly, for sample sizes of a few hundred nanometers, it would require truly astronomical numbers of images to be able to satisfactorily statistically characterize one pellet (∼1-10 mm) of a macroscopically heterogeneous material. Because of the more extreme confinement, the mechanisms of nwp entrapment within mesoporous materials may be different to those operating in macroporous samples. However, the experimental findings described above suggest that, because of the probable macroscopic mechanism for mercury entrapment within the particular mesoporous sol-gel silicas studied, MR images should contain exactly the particular information required to construct a model to predict nwp (i.e., mercury) entrapment in this type of material. In previous work,22 NMR spin-lattice relaxation time (T1) contrasted images have already been used in the construction of (20) Rigby, S. P.; Fletcher, R. S. Part. Part. Syst. Charact. 2004, 21, 138-148. (21) Janssen, A. H.; Van der Voort, P.; Koster, A. J.; de Jong, K. P. Chem. Commun. 2002, 163201633. (22) Rigby, S. P. J. Colloid Interface Sci. 2000, 224, 382-396.
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a hierarchical structural model for the interpretation of mercury porosimetry data. However, as stated in this earlier work,22 the two-dimensional (2D) NMR images obtained previously for solgel silica materials had a relatively large slice thickness (∼1 mm) which was shown to lead to a significant averaging out of the effects of heterogeneities, and hence a significant loss or blurring of structural information in the images. In this earlier work,22 the separate contributions to porosimetry hysteresis, arising from contact angle and structural effects, were not deconvolved, and thus a prediction of the onset of hysteresis was not possible. Hence, since this meant only a posteriori determinations of potential pore structure(s) from entrapment levels were possible, a priori predictions of the level of mercury entrapment were also not made. It is the purpose of the work described here to construct a model of the pore structure of a sol-gel silica material using 3D MR images, obtained with the spin-spin relaxation time (T2) contrast technique, for slices of the pellet that are thinner than the correlation length for the structure. These images will provide maps of the spatial variations in the distributions of porosity and the surface area-to-volume ratio. It will then be shown statistically that a 2D model is sufficient to accurately represent the particular physical structure of the batch of pellets studied here. Simulations of mercury retraction from the pore space of the material, using particular parameters for mercury entrapment determined using integrated nitrogen sorption and mercury porosimetry, will be performed on the models. These simulations will be used to make a priori predictions of the point of onset of (structural) hysteresis and the ultimate level of mercury entrapment in the material, and the results will be compared with experimental data. It will be seen that the simulations give rise to good predictions of the level of residual mercury, thereby directly validating the theory of entrapment and directly demonstrating that entrapment levels in certain amorphous materials are determined solely by the macroscopic structure of these complex materials.
Theory Construction of Structural Models. The structural models used in this work are constructed directly from 1H NMR images of pellets fully imbibed with water. NMR imaging using a spinspin relaxation time contrast preconditioning sequence produces a spin density map and a spin-spin relaxation time map. The pixel intensity in a spin density map is directly proportional to the number of relevant nuclei within the probe water molecules contained in the pixel volume, and thus is also proportional to the porosity in that pixel volume. The value of T2 in each of the pixels of the image may be converted to a pore surface areato-volume ratio by the adoption of a relaxation model. For a liquid imbibed in a pore space, the relaxation rate is enhanced. This is due to the interactions between the thin layer of liquid at the interface and the solid matrix increasing the relaxation rate. There is also diffusional exchange between the surfaceaffected layer and the remainder (bulk) of the liquid in the rest of the pore. In the case, as here, where the pores are several orders of magnitude smaller than the root-mean-square (rms) displacement of the probe water molecules during the course of the experiment, the “two-fraction fast exchange” model of Brownstein and Tarr23 can be used. The measured value of T2 is given by
λS 1 λS 1 1 ) 1+ T2 V T2B V T2S
(
)
(1)
where the subscripts S and B refer to the surface layer and bulk
fluid, respectively, λ is the thickness of the surface affected layer, and S/V is the pore surface area-to-volume ratio. In general, T2B . T2S, and thus S/V is given approximately by
S T2S 1 ≈ V λ T2
(2)
where S/V would be equal to 2/r for a cylindrical pore of radius r. Hence, T2 is a function of pore size. The typical value of the group λ/T2S for batch G2 is 2.1 × 10-7 m‚s-1. In this work, it is assumed that there is a monotonic relationship between the value of T2 and the characteristic pore dimension controlling the pressure of mercury intrusion and retraction. Characterization of the Spatial Geometric Arrangement of a Model Structure. The T2 maps obtained using MRI have been analyzed using the fluctuation auto correlation function, C(s), which measures the degree of correlation between f(xn) values at successive data points. Explicitly, if δfn is defined as
δfn ) f(xn) - 〈f〉
(3)
then
C(s) )
〈δfnδfn+s〉 〈δfn2〉
(4)
where the averages denoted by the brackets 〈 〉 are over the data set {xn}. In the context of the images, f(xn) is the characteristic T2 value in image pixel xn. For eq 4, successive annular shells at a distance s from each pixel are considered for each pixel in turn. The characteristic values of this function are the value of C(s ) 1) and the value of s when C(s) ) 0. The first value characterizes the degree of correlation, while the second value is known as the correlation length (ξ) and characterizes the linear extent of that correlation. Simulation of Mercury Porosimetry. As mentioned above, the MRI spin density and T2 images were used to construct a model of the void space of a porous silica catalyst support pellet. The structural representation consisted of a lattice site model where each site corresponded to a voxel in the MR images. Each lattice site corresponds to the length-scale of the in-plane image pixel resolution, that is, ∼40 µm. At the pore scale, within each lattice site, the model consists of a grid of pores of identical size. As shown by glass micromodel experiments,3 the level of entrapment is independent of the pore scale connectivity if adjacent pores have similar sizes. It will be shown below that a 2D model, created from a single slice image taken through the “equatorial” region of a spherical pellet, is sufficient to represent the pellets studied in this work because of the spherical symmetry observed in the spatial distribution of the pore sizes. The mechanisms of mercury intrusion and retraction were similar to those employed in previous work22 on abstract 2D grids. In previous work,22 the model retraction mechanism assumed that mercury will snap off and generate two fresh menisci at the boundary between any two sites in the model, so long as there was some difference in pore size between the neighboring sites. The so-called “snap-off ratio” is defined as the ratio of the sizes of neighboring pores required before a fresh meniscus can be generated at the boundary between them. Hence, in previous abstract modeling work,22 the snap-off ratio was assumed to be just above unity. However, as mentioned above, previous work16 has shown that the snap-off ratio for macroporous oil-reservoir rocks is ∼3-96. Hence, in this work, potential deviations of (23) Brownstein, K. R.; Tarr, C. E. J. Magn. Reson. 1977, 26, 17-24.
Nonwetting Phase Entrapment within Porous Media
the snap-off ratio from just above unity will be considered. Fragmented samples of pellets from batch G2 (the silica material studied here), with particle sizes less than (at least) 300 µm (found by porosimetry and laser diffraction), show no internal mercury entrapment following porosimetry.18 Hence, the ratios of the characteristic dimension of any shielded pore to the corresponding dimension of any potentially shielding pore located within (at least) ∼300 µm must all be less than the snap-off ratio for G2. This finding suggests that, to sufficiently characterize G2 using MRI to predict entrapment levels, the image pixel/ voxel resolution (and slice thickness) should be less than ∼300 µm; that way, any structural heterogeneity causing mercury entrapment will not be averaged out in the imaging process. The retraction mechanism employed in the simulations is based upon previous3 experimental observations of the retraction of mercury from glass micromodels with a heterogeneous spatial distribution of pore size. It is assumed that the characteristic pore dimension determining the pressure at which mercury intrudes or retracts from a given region of the sample was proportional to the value of T2 in the image voxel volume corresponding to that region (since T2 is proportional to pore size from eq 2). To simulate mercury intrusion, the value of a cutoff in T2, above which mercury was deemed to be able to penetrate, was gradually lowered in small steps to mimic the stepwise increase in pressure in a real experiment. The cutoff initially commenced at a value of T2 above any value present in the image, and was subsequently lowered until all sites (voxels) in the model were penetrated with mercury. The intruded pore volume in any model site (voxel) was taken as being proportional to the spin density for the corresponding image voxel. In simulations of mercury retraction, the value of the T2 cutoff was raised steadily in small steps. Mercury was allowed to retract back from a given model site if the T2 value in the corresponding image voxel was below the cutoff value, and, up to this retraction step, a path existed between the site under consideration and the edge of the model which involved only “stepping” on intermediate sites still filled with mercury. The retraction continued until the initial value of the T2 cutoff was re-reached. Hence, in this case, the snap-off ratio permitted is effectively determined by the choice of the T2 step size in the simulation. A difference in T2 (and thus pore size) between two neighboring sites of less than the simulation step size will not be “resolved”, and the two sites will behave as if they had the same pore size. The mechanism employed thus permits the occurrence of multiple snap-off phenomena. This is thought plausible for the material studied here for the following reasons. Theoretical considerations17 and micromodel studies14,15 suggest that the thinning of the mercury thread required to lead to snap-off and the subsequent generation of two new menisci requires a seed of wetting phase. Micromodel14,15,17 studies suggest that, for mercury intrusion into noncylindrical and irregularly shaped pores, pockets of wetting phase are always retained within all pores, and thus many seed sites for snap-off exist. The material studied here is amorphous, thus it is highly unlikely that its pores consist of regular cylinders, and it has also been shown to possess some nanoscopic surface roughness.24 An experimental measurement of the snap-off ratio (or at least an upper bound on its possible value) of a material may be obtained from integrated gas sorption and mercury porosimetry experiments.2,20 These experiments consist of a sequence of sorption and porosimetry experiments conducted in series on the same sample. This is facilitated by freezing entrapped mercury in place before commencing the next gas sorption experiment. Previous (24) Watt-Smith, M. J.; Edler, K. J.; Rigby, S. P. Langmuir 2005, 21, 22812292.
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studies25 have shown that the shape of the isotherm in the capillary condensation region for pores that do not themselves become filled with mercury is unaffected by the presence of mercury. This is probably due to the shielding effect of the nitrogen multilayer at the interface with filled pores. In many materials, entrapped mercury completely fills the larger pores in which it has been retained.2 The particular sizes of the pores in which mercury becomes entrapped can be determined from the bins in the nitrogen adsorption BarrettJoyner-Halenda pore size distribution (BJH PSD) that show a difference (decrease) in pore volume between the before and after porosimetry data sets. In contrast, the BJH PSD obtained from the nitrogen desorption isotherm is weighted by the volume of void space shielded by a particular pore size, rather than just the particular volume of void made up of pores of that size, as in adsorption. Hence, the pore sizes in the desorption PSD, for which there is an apparent decrease in specific volume following porosimetry, are the smallest pore sizes encountered in the paths joining the pores in which entrapment of mercury actually happened and the surface of the sample. Since snap-off generally occurs at the margins between the larger pores and the smaller pores, the snap-off ratio can be estimated from the ratio of the pore size in which mercury actually becomes entrapped (from the adsorption isotherm) and the smaller pore size shielding that pore (from the desorption isotherm). By using mercury porosimetry scanning loops, it is, in principle, possible to only marginally increase the amount of mercury entering the sample and thus only fill a small number of pores with entrapped mercury between two gas sorption experiments. Hence, scanning loops can be used to obtain more accurate values of the snap-off ratio for specific pores. However, in practice, there is a lower limit on the size of the scanning loop that can be employed to obtain a sufficiently high signal-to-noise ratio in the data and thus a limit on the accuracy of the estimate of the snap-off ratio. Analysis of Raw, Experimental Mercury Porosimetry Data. Traditionally, raw mercury porosimetry data is analyzed using the Washburn equation.26 Raw porosimetry data is typically characterized by the presence of hysteresis. This hysteresis is generally acknowledged3 to arise from either, or both, contact angle hysteresis and/or structural hysteresis. Recently, building on earlier work on controlled pore glasses,27,28 Rigby and Edler18 proposed semiempirical alternatives to the Washburn equation for the analysis of raw mercury porosimetry data that deconvolves the contact angle component of the hysteresis from the structural. This allows the structural contribution to hysteresis to be used to characterize the pore structure of a particular porous material. For mercury intrusion, the pore radius (nm) is given by
r)
302.533 + x91526.216 + 1.478p p
(5)
where p is the applied pressure (in MPa), while, for mercury retraction, the pore radius is given by
r)
68.366 + x4673.91 + 471.122p p
(6)
Since eqs 5 and 6 are empirical in origin, their use leads to an experimental error in the pore sizes obtained, which is estimated28 to be ∼4-5%. A consideration of previous work20,24 suggests that the degree of contact angle hysteresis is affected by differing (25) Rigby, S. P.; Fletcher, R. S. J. Phys. Chem. B 2004, 108, 4690-4695. (26) Washburn, E. W. Phys. ReV. 1921, 17, 273-283. (27) Liabastre, A. A.; Orr, C. J. Colloid Interface Sci. 1978, 64, 1-18. (28) Kloubek, J. Powder Technol. 1981, 29, 63-73.
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surface chemistry, but not by differing degrees of surface roughness. Equations 5 and 6 can be used to completely superimpose the intrusion and extrusion curves for various batches of fragmented sol-gel silica pellets that have been shown,24 by small-angle X-ray scattering (SAXS) and gas sorption, to have different surface fractal dimensions and thus differing degrees of nanoscopic surface roughness. However, a different version of eq 6 is required to remove contact angle hysteresis for alumina materials.20 Experimental Section The main material studied in this work is a batch of sol-gel silica spheres, denoted G2, with a typical pellet diameter of ∼3 mm and a nitrogen Brunauer-Emmett-Teller (BET) surface area of ∼99 m2‚g-1. MRI. Samples were prepared by impregnation with deionized water under ambient conditions for 24 h. Previous work9 has shown that this technique leads to complete pore filling. The values of the specific pore volume obtained independently from the ultimate intruded mercury volume (below), and gravimetrically following water impregnation, were found to be identical within experimental error. This finding suggests that mercury and water probe the same void space features. MRI experiments were carried out on a Bruker AVANCE NMR System with a static field strength of 7.05 T, yielding a resonance frequency of 300.05 MHz. All samples were placed within a 10 mm Birdcage coil. Spin-spin relaxation time (T2) and spin density maps (which probe porosity) were acquired together using the Bruker sequence “m_msme”, and employed 90° selective and 180° nonselective pulses. A T2 preconditioned imaging sequence with an echo time of 7 ms was used. 3D images were acquired using the “m_se3d” sequence. Data acquisition, initial data transformation, 2D and 3D data processing, and workup was performed on an SGI O2 5000 workstation, running the Paravision suite of software (Bruker Analitische Messtechnik Gmbh, Karlsruhe, Germany). The in-plane pixel resolution was 40 µm, and the slice thickness was 250 µm. 3D spin-echo data was subsequently worked up using AMIRA software. Nitrogen Sorption and Mercury Porosimetry. Samples for the experiments each consisted of a small number of catalyst support pellets. Nitrogen sorption experiments were carried out at 77 K using a Micromeritics ASAP 2400 apparatus. The sample tube and its contents were loaded into the degassing port of the apparatus and initially degassed at room temperature until a vacuum of 0.27 Pa was recorded. A heating mantle was then applied to the sample tube, and the contents were heated, under vacuum, to a temperature of 623 K. The sample was then left under vacuum for 14 h at a pressure of 0.27 Pa. The purpose of the thermal pretreatment for each particular sample was to drive off any physisorbed water on the sample but leave the morphology of the sample itself unchanged. A range of different thermal pretreatment procedures have been considered in the past18 to determine whether the experimental results were sensitive to the temperature or time period used. For all samples, at this point, the heating mantle was removed, and the sample was allowed to cool to room temperature. The sample tube and its contents were then reweighed to obtain the dry weight of the sample before being transferred to the analysis port for the automated analysis procedure. The sample was then immersed in liquid nitrogen at 77 K before the sorption measurements were taken. The adsorption and desorption isotherms obtained were analyzed using the well-known BJH29 method to obtain the PSDs. The film thickness for multilayer adsorption was taken into account using the well-known Harkins and Jura equation.30 In the Kelvin equation, the adsorbate property factor was taken as 9.53 × 10-10 m, and it was assumed that the fraction of pores open at both ends was 0.0 for both adsorption and desorption. It was therefore assumed that capillary condensation (29) Barrett, E. P.; Joyner, L. G.; Halenda, P. H. J. Am. Chem. Soc. 1951, 73, 373-380. (30) Rouquerol, F.; Rouquerol, J.; Sing, K. Adsorption by Powders and Porous Solids: Principles, Methodology and Applications; Academic Press: London, 1999.
Figure 1. T2 images of perpendicular 2D slices through the center of a pellet from batch G2. The pixel resolution is 40 µm, and the slice thickness is 250 µm. commenced at the closed end of a pore to form a hemispherical meniscus, and the process of evaporation also commenced at a hemispherical meniscus. Following the first nitrogen sorption experiment, the sample was allowed to reach room temperature (298.9 K) and then transferred to the mercury porosimeter still under nitrogen. Mercury porosimetry experiments were performed using a Micromeritics Autopore IV 9420. Samples typically consisted of ∼3-4 pellets. The sample was first evacuated to a pressure of 6.7 Pa to remove physisorbed gases from the interior of the sample. The standard equilibration time used in the experiments was 15 s. Following the experiments, the pellets were left for several days under ambient conditions, and no further mercury was observed to leave the sample. X-ray images have revealed that, even several days after the experiments, mercury entrapped within the central core region of the pellets was still there (further details of X-ray imaging experiments of entrapped mercury will be described in a subsequent publication). If movement of mercury was taking place, it is highly unlikely that this would be toward the center of the pellet, and thus it was concluded that entrapped mercury remained stationary at the end of the experiment.
Results and Discussion Image Analysis. Figure 1 shows examples of two T2-contrasted images of different slices through the center of a spherical pellet from batch G2. The typical value of T2 for a sample from batch G2 was ∼27 ms. This batch was chosen because of its relatively high value of T2, which allows a higher signal-to-noise ratio to be achieved for a given number of scans. Separate images were obtained of each of the slices in the stack from the top to the bottom of an individual pellet sample taken from batch G2. The individual T2-contrasted images of slices through pellets taken from batch G2 were analyzed using the autocorrelation function. The variation in the values of C(s ) 1) and ξ with position within the pellet for images taken of different slices through a sample of a typical pellet taken from batch G2 is shown in Figure 2. It can be seen that the values of both parameters of the correlation function peak within the central region of the pellet. As shown schematically in Figure 3, this is the result that would be expected if the distribution of T2 through the pellet possessed spherical symmetry arising from a correlated structure. It is supposed that the pellet consists of a structure where higher T2 values, for example, are more concentrated toward the center of the sphere and decrease in concentration in weakly defined bands located progressively further from the center of the pellet. If a 2D slice MR image were taken of this structure, such that the plane of the image passed through the central zone of the pellet, then, as
Nonwetting Phase Entrapment within Porous Media
Figure 2. Variation with position of the correlation length (() and the degree of correlation (×) for T2 maps of parallel slices of a pellet from batch G2. The central region of the pellet corresponds to slice 6.
shown in Figure 3, the image would slice through several different bands and would detect the correlation in T2 values. Hence, the correlation function would show a relatively high value of C(s) at shorter distances (corresponding to one pixel). However, if the plane of the 2D MR image were taken closer to the top (“pole”) of the pellet, then it would slice through only one or two bands. The T2 values within one particular band are envisaged to be closer together than those in different bands, and hence the image taken nearer the top of the pellet would be relatively more dominated by the (unavoidable) noise in the image. Hence, the correlation function would be of a form closer to that expected for a completely random arrangement of pixel intensities (a horizontal line along C(s) ) 0). The experimental data shown in Figure 2 is consistent with this scenario. Therefore, it is proposed that the structure of pellets from batch G2 has some similarities to the type of structure shown schematically in Figure 3, and possesses some sort of spherical symmetry. Hence, it is reasonable, in this case, to use a 2D model constructed from a single central MR image for the structure of pellet G2. Figure
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Figure 4. Plot of the reduced spin density against the corresponding value of reduced T2 for the same pixel location for all the pixels in a typical set of spin density and T2 images for a central slice through a pellet from batch G2. The parameters have been reduced by dividing by the relevant mean value within the image.
4 shows a plot of the value of (reduced) T2 against the corresponding value of (reduced) spin density for each pixel location in a set of images of an equatorial plane of a pellet taken from batch G2. It can be seen that there is a marked inverse correlation between spin density (corresponding to porosity) and T2 (corresponding to pore size). Hence, regions of void space containing larger pores tend to have lower overall porosity (voidage fraction). Determination of Model Parameters for Mercury Porosimetry Simulations. Experimental Determination of the SnapOff Ratio. As mentioned above, batch G2 was selected for the imaging studies because of its relatively high value of average T2. However, this means that, while being mesoporous, the average size of pores within G2 is toward the upper end of the mesopore size range, where the nitrogen sorption apparatus used experiences some difficulty obtaining precise values of relative pressure. Hence, batch G2 was, itself, unsuitable for determining the
Figure 3. Schematic diagram illustrating the expected variation in the form of the correlation function with the position of the T2 image slice if the pellet possessed a spherically symmetric spatial distribution of T2.
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Figure 5. Cumulative BJH pore diameter distributions for P123t derived from nitrogen adsorption (b and solid line) and desorption (9 and dashed line) isotherms from both before (lines) and after (symbols) a mercury porosimetry experiment to 414 MPa. The ultimate pore volume obtained after porosimetry was renormalized to that obtained before porosimetry.
mercury snap-off ratio directly from integrated gas sorption experiments. However, previous work18 has shown that mercury intrusion and retraction within G2 occurs by a piston-type mechanism.1 It was found18 that no mercury becomes entrapped within a powdered sample of G2, while entrapment does arise in whole pellet samples. As mentioned above, piston-type intrusion and retraction behavior and the lack of entrapment for fragmented pellets are associated with entrapment arising from macroscopic structural heterogeneities. A number of other materials, some of which are also silicas made by the same general sol-gel technique used for G2, have also been studied previously31,32 using integrated gas adsorption and mercury porosimetry experiments and were found to have piston-type intrusion and retraction behavior; hence, mercury snap-off is likely to arise from the same mechanism as that found with G2. The materials studied were a templated porous silica material, synthesized (as described in earlier work32) using Pluronic P123 polymer and denoted as P123t, W1 (a sol-gel silica), and F1 (an Fe2O3/ Fe3O4/Cr2O3 catalyst). Figure 5 shows the BJH PSDs derived from both the adsorption and desorption isotherms obtained before and after a mercury porosimetry experiment32 to an ultimate pressure of 414 MPa on a sample of P123t. The ultimate pore volume obtained after porosimetry has been renormalized to the initial value obtained before porosimetry to enable differences to be more clearly discerned. From Figure 5, it can be seen that, in the adsorption PSD, pores ∼4-7 nm in size are lost following porosimetry, while, in the desorption PSD, pores ∼3.5-4.5 nm in size are apparently lost. Thus, the snap-off ratio for P123t is ∼1.1-2 or less, as this is an upper bound. Mercury porosimetry scanning loops with integrated gas sorption experiments have been conducted on a sol-gel silica material, W1.33 The entire set of raw nitrogen sorption data obtained before, between, and after 276 and 414 MPa porosimetry scanning loops has been presented elsewhere.33 The BJH PSDs obtained from the original adsorption and desorption isotherms are also presented elsewhere.33 However, the difference distributions between the differential PSDs obtained between each scanning loop are shown in Figure 6. In Figure 6 it can be seen that there is a difference in the size distribution of the pores in which mercury becomes entrapped following the 276 MPa (31) Rigby, S. P.; Evbuomwan, I. O.; Watt-Smith, M. J.; Edler, K. J.; Fletcher, R. S. Part. Part. Syst. Charact., in press. (32) Rigby, S. P.; Beanlands, K.; Evbuomwan, I. O.; Watt-Smith, M. J.; Edler, K. J.; Fletcher, R. S. Chem. Eng. Sci. 2004, 59, 5113-5120. (33) Rigby, S. P.; Watt-Smith, M. J.; Fletcher, R. S. Adsorption 2005, 11, 201-206.
Figure 6. Differential PSDs for batch W1 obtained for the difference between pairs of cumulative BJH PSDs for either nitrogen adsorption (a) or desorption (b) isotherms. The pairs of isotherms for which the difference distributions were obtained were, first, a fresh sample and following a mercury porosimetry scanning loop to 276 MPa ((), and, second, between following a mercury porosimetry scanning loop to 276 MPa and following a 414 MPa scanning loop (9).
scanning loop, compared with the distribution of pores that additionally become filled with entrapped mercury following the 414 MPa scanning loop. It can also be seen that the large pores (∼6-14 nm in size) that become filled with entrapped mercury following the 276 MPa scanning loop are shielded by relatively larger small pores that are ∼5-6 nm in size, giving a snap-off ratio of ∼1-2.8, while the pores ∼6-12 nm in size that are additionally filled by the 414 MPa porosimetry experiment are shielded by pores that are ∼4.5-5.5 nm in size, giving a snap-off ratio of ∼1.1-2.2. Figure 7 shows the BJH PSDs derived from both the adsorption and desorption isotherms obtained after two porosimetry scanning loops to 103 and 165 MPa on a sample from F1. For both types of isotherms, the ultimate pore volume in the cumulative PSD obtained after mercury intrusion to 165 MPa has been renormalized such that it is the same as that obtained after mercury intrusion to 103 MPa. From the adsorption isotherms, it can be seen that pores ∼2.25-4 nm in size are lost because of the mercury entrapment within them. From the desorption isotherms, pores ∼1.75-2.25 nm in size are apparently lost. Therefore, the snap-off ratio for F1 is ∼1-1.8. The estimates for the upper bounds on the snap-off ratio, found above, for the three very different materials studied here are all much smaller than the values of ∼3-96 previously obtained16 for macroporous rocks and are all close to unity. Therefore, it is proposed that it is reasonable to assume that the snap-off ratio in the simulations is close to unity. It is also noted that previous31 estimates of the snap-off ratio for mesoporous materials in which
Nonwetting Phase Entrapment within Porous Media
Figure 7. Cumulative PSDs obtained between porosimetry scanning loops to 103 MPa (lines) and 165 MPa (symbols) and derived, using the BJH algorithm, from nitrogen adsorption (solid line and 9) and desorption (dashed line and 2) isotherms for a sample from batch F1.
retraction is not by a piston-type mechanism have also been less than 3 and closer to unity. Determination of the Appropriate T2 Step Size. While a large number of scans (240) was used to obtain the NMR images employed in this work, the presence of some random noise within the images is unavoidable. Therefore regions of the pellet, corresponding to image voxels, that may actually contain exactly the same pore size may appear to have different values of T2 simply because of noise. Hence, it is necessary to choose a step size in the simulations of mercury retraction that is larger than the typical value of noise within an image pixel. However, previous22 simulations on abstract (i.e., computer-generated) model grids have shown that, if the step size is too large relative to the spread in pore sizes present in the grid, then the predicted entrapment is too low. This is because a large step size fails to distinguish between pores of different but relatively similar sizes, and thus the grid behaves as if it is of smaller overall size than it actually is. Experiments (with different pressure step sizes) on the real material have suggested that the degree of mercury entrapment is independent of pressure step size, and this is also what was found for simulations on abstract model grids provided the step size was below a threshold size. Therefore, there are two apparently conflicting constraints on the step size for simulations of mercury retraction on model grids constructed from NMR images. To determine the appropriate values of the T2 step size to determine the correct level of mercury entrapment, simulations were conducted with a range of step sizes. An example of a set of results for one image model is shown in Figure 8. Also shown in Figure 8 are fits of the data over selected ranges to straight lines. The parameters of the fitted straight lines are shown in Table 1. From Figure 8 and Table 1, it can be seen that there are three ranges in the data with (statistically) significantly different slopes. For the largest step sizes, as the step size is decreased, the level of entrapment increases, as was found for abstract grids in previous work.22 However, for step sizes in the range of ∼1.1-1.5 ms, there is a plateau in the predicted entrapment at ∼40%, where the entrapment level becomes virtually independent of step size (the error range of the slope of the fitted line encompasses zero). For step sizes less than ∼1-1.1 ms, the entrapment level increases with decreased step size. It is noted that a typical value for the error in the fitted value of T2 for a pixel in the image used to generate the model employed in the simulations for Figure 8 is ∼1.1 ms. Hence, the step sizes in the range less than ∼1.1 are smaller than the typical noise
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Figure 8. Variation in the level of mercury entrapment with T2 step size for simulations of mercury porosimetry on a structural model derived from NMR images of an equatorial slice through a typical pellet from batch G2. The lines shown are the results of separate linear regressions of the data in the ranges 0-1.05, 1.10-1.46, and 1.50-2.12 ms. Gradients of these lines are listed in Table 1. The inset shows the data for the intermediate plateau region (1.1-1.5 ms). The line in the inset represents the mean entrapment level (39.8%) for step sizes in the range 1.10-1.46 ms. Table 1. Gradients of the Lines of Best Fit Shown in Figure 8 range of T2 step sizes (ms)
gradient of fitted line (% ms-1)
0-1.05 1.10-1.46 1.50-2.12
-5.18 ( 0.21 -0.68 ( 1.65 -5.40 ( 1.55
Figure 9. Variation in fractional occupied volume on the extrusion curve against the corresponding fractional occupied volume on the intrusion curve for simulations of porosimetry on a typical model created from MR images of a pellet from batch G2.
level in the image. In previous work,22 the value of entrapment obtained for abstract model grids with a completely random spatial arrangement of pore sizes was ∼50%. Therefore, the upward trend in the entrapment levels with decreasing step sizes below the typical T2 error size is probably caused by progressively increasing effects of noise, as the trend is toward the entrapment value expected for a completely random grid. These findings suggest that, as the step size is reduced relative to the spread in T2 values in the image, the level of entrapment tends to plateau, as found22 for abstract grids, but, at the smallest step sizes, the degree of entrapment is affected by the noise level. Hence, it is appropriate to use the step sizes for the intermediate plateau region (∼1.1-1.5 ms) to determine the correct entrapment level for the image model grids. Given the typical value of T2 in the images of pellets from batch G2, this step size range corresponds to values for the snap-off ratio in the range ∼1.04-1.06. This size of snap-off ratio is similar to, or within, the range of values
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Table 2. Point of Separation for the Mercury Intrusion and Extrusion Curves, and Level of Entrapment, for Both Simulations on Image-Derived Models and Experimental Data for Batch G2a simulation
a
experiment
sample
entrapment (%)
fractional occupied volume at point of separation of curves
entrapment (%)
fractional occupied volume at point of separation of curves
1 2 3
39.8 43.1 41.6
0.70 0.69 0.70
41.9 39.4 42.4
0.66 0.67 0.71
Note that the data shown is for several samples, but the image and porosimetry sample numbers do not correspond.
estimated experimentally above for other materials with pistontype intrusion and retraction of mercury, similar to G2. Comparison of Simulation with Experiment. Simulations of mercury intrusion and retraction, according to the mechanisms described above, were performed on 2D structural models constructed from spin density and T2-weighted images of central slices through pellets taken from batch G2. An example of a typical set of data for one particular model is shown in Figure 9. Figure 9 shows a plot of the fractional occupied volume for the extrusion curve against the fractional occupied volume for the intrusion curve at the same T2 value obtained from simulations of the mercury porosimetry experiment on a model created from MR images. The choice of variables for this figure makes it very clear when the deviation between intrusion and extrusion commences in the simulations, independent of the form of the probability density function of T2 values (or critical pore sizes). It can be seen that the curves separate at an occupied volume fraction of ∼0.7, and the entrapment is ∼40%. Data from simulations on models constructed from different images of separate pellet samples are shown in Table 2. Previous simulations18 on abstract model grids have shown that different spatial arrangements of pore sizes lead to different combinations of the point of deviation of the intrusion and extrusion curves and level of entrapment. It is noted that the level of mercury entrapment for the models generated from all MR images is significantly different from the value (of 50%) expected for a completely random arrangement of pixel intensities.22 Figure 10 shows the mercury intrusion and retraction curves, analyzed using eqs 5 and 6, respectively, for a sample of pellets from batch G2. It can be seen that, at smaller pore sizes, within the error in eqs 5 and 6, the intrusion and extrusion curves overlay each other while the point of separation of the intrusion and retraction curves occurs at a fractional occupied volume of ∼0.67, and that the final level of mercury entrapment is ∼40%. Table 2 shows the points of onset of the structural hysteresis and mercury entrapment levels for several samples taken from batch G2. From Table 2, it can be seen that the experiments are repeatable, and the values for the point of onset of hysteresis and entrapped mercury for each sample agree well with those predicted from the models derived from the MR images above. Hence, these
Figure 10. Experimental mercury intrusion (×) and extrusion (0) curves, analyzed using eqs 5 and 6, for a sample from batch G2. The solid lines are to guide the eye.
results validate the structural model employed and the mechanisms of mercury intrusion, retraction, and entrapment in the material G2.
Conclusions The snap-off ratio has been measured, using integrated nitrogen sorption and mercury porosimetry, for a number of amorphous, mesoporous materials and has been found to be close to unity, and is thus much lower than that found previously for macroporous materials. The characteristic pore dimension measured using NMR relaxometry also controls the pressures of mercury intrusion and retraction. The point of onset of structural hysteresis and the exact level of mercury entrapment within an amorphous, mesoporous material has been predicted by simulations of mercury intrusion and retraction in structural models created using MR images. The fact that this is possible suggests that the model used is a good structural representation of the material, and the mechanisms of mercury retraction and entrapment are now better understood. Acknowledgment. S.P.R. and M.J.W.-S. thank the EPSRC for financial support (under Grant No. GR/R61680/01). LA060142S
