Characterisation of porous solids using integrated nitrogen sorption and mercury porosimetry

Chemical Engineering Science 59 (2004) 41 – 51

www.elsevier.com/locate/ces

Characterisation of porous solids using integrated nitrogen sorption and mercury porosimetry Sean P. Rigbya;∗ , Robin S. Fletcherb , Sandra N. Rileyb a Department b Johnson

of Chemical Engineering, University of Bath, Claverton Down, Bath BA2 7AY, UK Matthey Catalysts, P.O. Box 1, Belasis Avenue, Billingham, Cleveland TS23 1LB, UK

Received 11 June 2003; received in revised form 2 September 2003; accepted 23 September 2003

Abstract The two di-erent techniques of nitrogen sorption and mercury porosimetry, which are generally utilised completely separately, have been integrated into the same experiment to improve upon the information obtained from both methods. Nitrogen sorption isotherms have been run both before and after a mercury porosimetry experiment on the same sample. This experiment has revealed that for a particular type of sol–gel silica catalyst support the entrapped mercury is con0ned to only the very largest pores in the material. Light micrograph studies have shown that the spatial distribution of entrapped mercury is highly heterogeneous. These results suggest that mercury entrapment within the material is caused by a mechanism involving macroscopic (¿ 0:1 mm) heterogeneities in the pore structure. These 0ndings con2ict with the usual assumptions generally made in simulations of porosimetry based on random pore bond network models. The new work has shown that, in conjunction with computer simulations involving the correct mercury retraction mechanism, mercury porosimetry and nitrogen sorption can be used to study the spatial distribution of all pore sizes within a mesoporous material. A percolation analysis of the nitrogen sorption data, obtained both before and after mercury entrapment, allowed broad features of the spatial disposition of variously sized pores to be determined. The results reported here also support the use of new, semi-empirical alternatives to the Washburn Equation to analyse raw mercury porosimetry data, rather than the traditional approach. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Adsorption; Nanostructure; Porous media; Voidage; Catalyst supports; Mercury porosimetry

1. Introduction Porous materials with complex internal void structures have many uses in science and industry. For example, porous solids are used as supports for heterogeneous catalysts, absorbents of contaminants, adsorbents in gas separations, and as chromatographic media. The relative performance of different porous solids in these applications is highly dependent on the internal pore structure of each material. Therefore, in order to better understand a particular physical process taking place within a porous medium, it is necessary to have a detailed knowledge of the internal geometry and topology of the internal pore network. Techniques, such as micro-focus X-ray (MFX) tomography (Wang et al., 2001) and three-dimensional transmission electron microscopy (3D-TEM) (Koster et al., 2000), can be used to examine the full 3D structure of the void space ∗

Corresponding author. Tel.: +44-1225-384978. E-mail address: [email protected] (Sean P. Rigby).

0009-2509/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2003.09.017

of a porous material directly. With an X-ray system using 5 –20 keV hard X-rays a sample of a few mm thickness can be imaged in 3D at 1 m resolution (Wang et al., 2001). Another type of system uses soft X-rays with energies of ∼100–1000 eV to image samples with sizes of ∼1–10 m at resolutions of 50 –100 nm (Wang et al., 2001). However, unfortunately, many of the porous materials utilised 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. A resolution of ∼3 nm is possible with 3D-TEM. Hence, the pores in a mesoporous solid can be studied directly using 3D-TEM (Koster et al., 2000). However, this can only be achieved with a sample size of a few hundred nm. Characterisation studies using magnetic resonance imaging (MRI) (Hollewand and Gladden, 1993; Rigby and Daut, 2002) and MFX imaging (Rigby et al., 2002) have shown that porous catalyst-support pellets and absorbents possess heterogeneities in the spatial distribution of locally averaged porosity and pore size over macroscopic (∼0:01–10 mm)

42

Sean P. Rigby et al. / Chemical Engineering Science 59 (2004) 41 – 51

length scales. These structural heterogeneities have been shown to in2uence both steady-state and transient di-usion (Hollewand and Gladden, 1993; Rigby et al., 1996), and the deactivation of catalysts by coke deposition (Cheah et al., 1994). Therefore, in order to fully understand the performance of a typical heterogeneous catalyst pellet or absorbent, it is necessary to both possess the detailed local (for small domains of the sample) information concerning regional pore structure (porosities, pore sizes, etc.) that could be provided by 3D-TEM, but also be able to map the variations in these properties and their juxtaposition across the entirety of a macroscopic pellet (of size ¿1 mm). Unfortunately, obtaining all of this information by 3D-TEM would be both laborious and impractical, since it would necessitate the examination of ∼1011 separate microscopic samples from a single macroscopic (∼1–10 mm) catalyst pellet, while maintaining a detailed knowledge of the former locations of all of these samples within the original pellet! It is possible to produce maps of the spatial variation of local mean porosity and pore body size, averaged over voxel regions of size ∼10–100 m, within relatively pure silica and alumina catalyst supports using conventional 1 H MRI (Hollewand and Gladden, 1993; Rigby and Gladden, 1996). However, this technique is unsuitable for fully quantitative use with chemically heterogeneous materials of more general interest within catalysis, such as coked catalysts. In contrast, more conventional techniques such as nitrogen adsorption and mercury porosimetry can be used to quantitatively study these materials. In addition, MRI based on relaxometry probes variations in pore surface area-to-volume ratio, and is thus insensitive to narrow pore neck features possessing little or no volume. This problem is also shared by SAXS, thermoporometry, NMR cryoporometry and nitrogen adsorption. In contrast, since it is an invasion percolation process, mercury porosimetry is sensitive to the pore neck features of a porous solid (Rigby et al., 2001). Therefore, due to the shortcomings of the various existing techniques discussed above, there is still a need to develop a technique to determine the macroscopic spatial distribution of pore scale properties, such as pore body and neck sizes, for use with chemically heterogeneous, mesoporous materials. This lack of a suitable technique has lead Rigby and coworkers (Rigby, 2000; Rigby and Edler, 2002; Rigby et al., 2003a,b) to develop a methodology, utilising mercury porosimetry, that can address the need identi0ed above. It was proposed (Rigby, 2000) that mercury porosimetry may be used to determine the macroscopic spatial distribution of the pore-scale properties of mesoporous solids. This was because it has been discovered that, for many porous materials, the forms of the mercury intrusion and extrusion curves, are heavily in2uenced by the macroscopic properties of the porous medium. It has additionally been suggested that the macroscopic properties also completely determine the level of mercury entrapment (Rigby and Edler, 2002). These 0ndings con2ict with the usual assumptions employed in typical computer simulations of the mercury

porosimetry experiment. Over the past two decades, it has frequently (Androutsopoulos and Mann, 1979; Portsmouth and Gladden, 1991, 1992; Mata et al., 2001) been assumed that mercury entrapment is solely determined by the pore scale properties of a porous medium. For example, simulations of mercury intrusion and retraction on random pore bond network models, performed by Portsmouth and Gladden (1991, 1992), have shown that mercury entrapment is a function of both the pore network connectivity and the width of the volume-weighted pore size probability density function. The analysis of raw mercury porosimetry data using semi-empirical alternatives to the traditional Washburn (1921) Equation, conducted in previous work (Rigby and Edler, 2002), suggested that, if macroscopic scale heterogeneities are the source of entrapment, then the mercury should be con0ned exclusively to the largest pores. If, instead, the pores of various sizes are intermingled, as in random pore bond network models (Androutsopoulos and Mann, 1979; Portsmouth and Gladden, 1991, 1992; Murray et al., 1999; Mata et al., 2001), then entrapment should occur more widely in pores with a greater variety of sizes. It is the purpose of this paper to present a methodology, based on the integration of mercury porosimetry with nitrogen sorption, that demonstrates that mercury entrapment in the sol–gel silicas studied previously (Rigby and Edler, 2002) is con0ned to the largest pore sizes present, and thus con0rming the aforementioned theory of mercury entrapment. This theory provides the basis for the methodology to determine the spatial distribution of pore sizes using mercury porosimetry. The experimental method presented here also has some similarities to the variant on mercury porosimetry known as low melting-point alloy (LMPA) impregnation (Dullien, 1981; Mann et al., 1995; RuKno et al., 2001). In LMPA intrusion a molten alloy replaces mercury as the probe 2uid. The advantage of LMPA impregnation is that, simply by decreasing the temperature below the melting point of the alloy, the porosimetry experiment can be frozen at any chosen point in the intrusion/retraction cycle. The sample can then be serially sectioned and examined under an electron microscope to directly determine the size of pores etc. that are 0lled with metal at that particular point in the pressure cycle. This procedure can then be repeated on di-erent samples from the same batch to determine the location of metal at various di-erent points in the pressure cycle. The problem with this method is that it is diKcult and laborious to create uniformly polished sections and examine them by scanning electron microscopy (SEM) for macroscopic (¿1 mm) samples (Allamy et al., 2003). This means that the method is impractical for mesoporous materials. However, in the work described here, a conventional mercury impregnation experiment is frozen at the end point of the retraction curve, and the sample then subsequently examined using nitrogen sorption. In this work only one particular point in the whole intrusion/retraction cycle is studied, but the technique used

Sean P. Rigby et al. / Chemical Engineering Science 59 (2004) 41 – 51

here is suitable for use with macroscopic, mesoporous samples.

Small pores (P2)

(P2>P1)

Large pores (fill at P1)

2. Theory The void structure of a variety of di-erent porous materials is considered to consist of a three-dimensional “ patchwork” of domains with macroscopic dimensions. Contained within the con0nes of each domain is a network of similarly sized pores, whilst neighbouring domains may contain pores of more disparate sizes. The juxtaposition of di-erent pore size domains may be according to any geometric pattern with varying degrees of randomness. Wardlaw and McKellar (1981) have conducted mercury porosimetry experiments on micromodels consisting of a network of capillaries etched in glass. In one glass micromodel studied, the network was a nonrandom system, consisting of clusters of smaller pore elements occurring in isolated domains within a continuous network of exclusively larger elements. When mercury was 0rst injected into this empty model it, preferentially, 0lled the larger pore elements. As the pressure was increased further, mercury then entered the clusters of smaller pores and saturated the model. Once the pressure was subsequently decreased, mercury withdrew initially from the clusters of smaller elements. As the mercury pressure was reduced further, the mercury then withdrew from the rest of the network. In another glass micromodel, which consisted of isolated clusters of larger pore elements within a continuous network of smaller pores, no mercury entered the structure at all until the pressure had risen to the value required to intrude the smaller pore elements. Once mercury entered the model it became saturated. When the mercury pressure was subsequently reduced the mercury initially withdrew from only the smaller pore elements. However, at the stage where the pressure had been reduced below the threshold for the emptying of the clusters of larger pore elements, these had already been disconnected by snap-o- and extensive residual mercury was retained. The results of these experiments are shown schematically in Fig. 1. These experiments have shown that the levels of mercury entrapment in porous structures are sensitive to the presence of certain types of nonrandom, structural heterogeneity. It is suggested that for real porous materials with a similar type of pore structure to that described above mercury entrapment will be controlled by the nonrandom heterogeneity. Mercury entrapment leads to hysteresis between the mercury intrusion and extrusion curves. In addition, for networks of pore bodies and pore necks, it has been proposed (Wardlaw et al., 1988) that narrow pore neck sizes control mercury intrusion, whereas wider pore body sizes control retraction. These causes of hysteresis are commonly referred to as “structural hysteresis.” However, there is generally agreed to be a further source of hysteresis called “contact angle hysteresis.” This second type of hysteresis

43

Start

(a)

- Ambient pressure - Empty model

Retraction - Decrease pressure back to P1

Intrusion up to P1

Increase pressure to saturation (P2)

Large pores (P1)

(P2>P1)

Small pores (fill at P2) Start

- Ambient pressure - Empty model

Intrusion up to P2

Entrapment

(b)

Retraction - Decrease pressure back to ambient

Retraction - Decrease pressure back to P1

Fig. 1. A schematic diagram showing the results of mercury porosimetry experiments on glass micromodels obtained by Wardlaw and McKellar (1981).

arises because the surface tension and contact angle occurring in the Washburn (1921) Equation, relating imposed pressure to pore size, are known to depend upon the nature of the surface, the radius of the pore, and whether the mercury is advancing into, or retreating from, a sample. In order to be able to use structural hysteresis to extract information about a porous medium from the mercury intrusion and extrusion curves it is necessary to deconvolve the two di-erent types of hysteresis. This can be achieved by using semi-empirical alternatives to the Washburn (1921) Equation, that have been used in previous work (Rigby and Edler, 2002; Rigby et al., 2003a,b), to analyse the raw mercury porosimetry data. Liabastre and Orr (1978) made a study of the morphology of controlled pore glasses using electron microscopy and mercury porosimetry. These workers obtained values for the diameters of the pores in the glasses by direct observation from microscopy. They compared these values with the corresponding values obtained from mercury intrusion and extrusion porosimetry via the Washburn (1921) Equation, assuming 0xed values of contact angle and surface tension. Kloubek (1981) used these data to determine the relationships for the variation of the group  cos , that occurs in the Washburn (1921) Equation, as a function of pore radius,

44

Sean P. Rigby et al. / Chemical Engineering Science 59 (2004) 41 – 51

for both advancing and retreating mercury menisci. Kloubek (1981) obtained the expression:  cos A = −302:533 +

−0:739 r

(1)

for the variation of the product  cos  for an advancing (denoted by the subscript A) meniscus, which was valid for pore radii in the range 6 –99:75 nm, and the expression:  cos R = −68:366 +

−235:561 r

(2)

for a retreating meniscus, which was valid for pore radii in the range 4 –68:5 nm. In previous work (Rigby and Edler, 2002; Rigby et al., 2003a,b), Eq. (1), or Eq. (2), were inserted into the Washburn (1921) Equation, and the resultant expression solved for r, in order to derive a relationship between the externally imposed pressure (in MPa) and the pore radius (in nm) for mercury intrusion and retraction, respectively. For mercury intrusion the pore radius is given by √ 302:533 + 91526:216 + 1:478p r= ; (3) p while for mercury retraction the pore radius is given by √ 68:366 + 4673:91 + 471:122p : (4) r= p Since Eqs. (1) and (2) are empirical in origin, then their use leads to an experimental error in the pore sizes obtained using Eqs. (3) and (4), which is estimated (Kloubek, 1981) to be ∼4–5%. These expressions are more complicated than the Washburn (1921) Equation commonly used for this purpose, but have the advantage that they, additionally, take into account the variations that occur in the contact angle and surface tension as the radius of curvature of the liquid at the meniscus is decreased. A procedure has been developed by Seaton (1991) for determining the connectivity, Z, and lattice size, L, for a random pore bond network model of a porous solid from nitrogen sorption data. In this context, the pore connectivity is the overall average pore co-ordination number. This method can be summarised (Lee and Tsay, 1998) as follows: (i) the pore-size distribution is obtained using the method of Barrett et al. (1951) (BJH), incorporating the cylindrical pore bond model; (ii) the bond occupation probability f is obtained as a function of the probability of belonging to the percolation cluster F from a combined analysis of the nitrogen adsorption and desorption isotherms, and also using the pore-size distribution obtained above as input; and (iii) the best 0t Z and L values are obtained by 0tting the set of experimental scaling data (f; F) obtained above to a generalised scaling relation, G, between F and f. The generalised scaling relation: L= ZF = G(Zf − 3=2)L1= ;

(5)

where the critical exponents  and  take the values 0.41 and 0.88, respectively, was constructed using the simulation

data of Kirkpatrick (1979). There exists (Stau-er, 1985) a universal relationship between Z and the value of the number fraction of pores that would have emptied in a perfectly connected network at the percolation threshold, fc , which is exploited in the 0tting algorithm: Zfc = 1:5:

(6)

3. Experimental Samples for the experiments each consisted of a small number of sol–gel silica catalyst support pellets (of nominal diameter 3 mm) taken from a batch denoted G1. 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 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 pre-treatment for each particular sample was to drive o- any physisorbed water on the sample but to leave the morphology of the sample itself unchanged. A range of di-erent thermal pre-treatment procedures have been considered in the past (Rigby and Edler, 2002) in order 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 allowed to cool down to room temperature. The sample tube and its contents were then re-weighed 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 analysed using the well-known Barrett–Joyner–Halenda (BJH) (Barrett et al., 1951) method to obtain the pore size distributions. The 0lm thickness for multilayer adsorption was taken into account using the well-known Harkins and Jura equation. 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 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 0rst 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. The sample was 0rst evacuated to a pressure of 6:7 Pa in order to remove physisorbed gases from the interior of the sample. The standard equilibration time used in the experiments was 15 s.

Sean P. Rigby et al. / Chemical Engineering Science 59 (2004) 41 – 51

0.8 0.6 0.4 0.2 0.0 0.001

0.01

0.1

1 10 Pressure/MPa

100

1000

Fig. 3. Raw mercury intrusion (•) and extrusion (
) curves from an integrated nitrogen sorption and mercury porosimetry experiment (for sample 1 from batch G1).

800 700 600 500

Volume adsorbed/(cc STP/g)

600 500 400 300 200 100 0 0.0

0.2

0.4 0.6 0.8 Relative pressure

1.0

1.2

Fig. 4. Nitrogen adsorption (o) and desorption (x) isotherms obtained following mercury porosimetry during an integrated nitrogen sorption and mercury porosimetry experiment (for sample 1 from batch G1).

1.2 Cumulative volume/(cc/g)

An example of a complete set of raw data for an integrated nitrogen sorption and mercury porosimetry experiment is shown in Figs. 2–4. Figs. 2 and 4 show the nitrogen sorption isotherms run before and after, respectively, the mercury porosimetry experiment. Fig. 3 shows the data from the mercury porosimetry experiment itself. The raw data from the mercury porosimetry experiments on two samples of whole pellets from batch G1 have been analysed using the conventional Washburn (1921) Equation, and Eqs. (3) and (4), and examples of the results are shown in Figs. 5 and 6, respectively. As has been observed in previous studies (Rigby and Edler, 2002; Rigby et al., 2003b), when analysed using the new, semi-empirical alternatives to the Washburn (1921) Equation, the mercury intrusion and extrusion curves become superimposed (within the error present in Eqs. (3) and (4)) at the lowest pore sizes. A deviation between the intrusion and extrusion curves, that is larger than the error present in Eqs. (3) and (4), arises at an occupied pore volume of ∼0:35 ml=g. It can be seen, in the data in Fig. 3, that

Volume adsorbed/(cc STP/g)

1.0

700

4. Results

1.0 0.8 0.6 0.4 0.2 0.0

400

1

10

100

1000

10000

Pore radius/nm

300 200 100 0 0.0

1.2 Cumulative volume/(cc/g)

However, separate experiments were conducted using di-erent equilibration times in the range between 15 and 3600 s. Since it is diKcult to thoroughly clean entrapped mercury from a given sample after an experiment for re-use in a further mercury porosimetry experiment, each particular set of experiments was, instead, repeated on several samples from the same batch. Where the results were analysed using the conventional approach using the standard Washburn (1921) Equation both the advancing and retreating contact angles were taken to be 140◦ , and the surface tension was taken as 0:485 N=m (485 dynes=cm). Following mercury porosimetry the sample was transferred back to the nitrogen sorption apparatus. The sample was then cooled to 77 K to freeze the mercury in place. Following the cooling of the sample, it was evacuated to vacuum and the next nitrogen sorption experiment commenced.

45

0.2

0.4 0.6 0.8 Relative pressure

1.0

1.2

Fig. 2. Nitrogen adsorption (o) and desorption (x) isotherms obtained before the mercury intrusion stage of an integrated nitrogen sorption and mercury porosimetry experiment (for sample 1 from batch G1).

Fig. 5. The cumulative pore size distributions obtained from an analysis of the raw mercury intrusion (•) and extrusion (
) curves given in Fig. 3 using the conventional approach described in the text utilising the Washburn (1921) Equation.

there is a small amount of mercury intrusion at pressures of 0.001–0:01 MPa, followed by a long, horizontal plateau at large pore sizes before signi0cant intrusion starts at

46

Sean P. Rigby et al. / Chemical Engineering Science 59 (2004) 41 – 51

visible surrounded by neighbouring clear zones containing little or no mercury. Fig. 7(b) shows a close-up view of this region. The close up view of the sphere in Fig. 7(b) clearly demonstrates that the distribution of mercury entrapment is very heterogeneous and is concentrated into particular domains within the sample. The images in Figs. 7(a) and (b) show that the overall order of the length scales of the domains containing entrapped mercury is ∼100–1000 m. It is noted that the pore size distribution obtained for pellets from batch G1, given in Fig. 6, suggests that the sizes of the pores present in G1 are in the range ∼10–80 nm. The pores themselves are thus well below the resolution possible with light microscopy. Hence the domains of entrapped mercury appear as blobs trapped within the clear silica, as in Fig. 7(a). The presence of ganglia of entrapped mercury possessing distinctive morphologies, such as the one present in Fig. 7(b), enables the long-term stability of the locations of entrapped mercury to be monitored. By studying the mercury ganglion evident in Fig. 7(b) on repeated occasions over several days, it was possible to determine that the entrapped mercury did not move once it had become entrapped because the morphology and location of the ganglion remained constant. The nitrogen adsorption isotherms, obtained both before and after mercury intrusion for two samples from batch G1, have been analysed using the BJH algorithm (Barrett et al., 1951) to determine the volume distribution of pores with diameters in the range 1.7–50 nm. Plots of the cumulative pore size distributions thus obtained are shown in Fig. 8. The total speci0c pore volumes for pores with diameters in the range 1.7–50 nm, both before and after mercury intrusion, for both samples studied, are shown in Table 1. Also reported in Table 1 are the reduced standard deviations (standard deviation divided by the mean, =) for the pore size distributions shown in Fig. 8. As noted by Seaton (1991), there is an upper limit beyond which apparatus used for nitrogen adsorption is not able to precisely achieve capillary condensation in pores with critical relative pressures close to unity. This upper limit is ∼50 nm. Also shown in Table 1

Cumulative volume/(cc/g)

1.2 1.0 0.8 0.6 0.4 0.2 0.0 10

1

100 Pore radius/nm

1000

10000

Fig. 6. The cumulative pore size distributions obtained from an analysis of the raw mercury intrusion (•) and extrusion (
) curves given in Fig. 3 using Eqs. (3) and (4), respectively.

pressures of ∼10–100 MPa. Due to the apparent corresponding pore size (¿104 nm) at which it occurs, it is suggested that the small amount of initial intrusion is due to inter-pellet penetration of mercury within the gaps in-between pellets. This initial intrusion can thus be deducted from the ultimate, cumulative intrusion volume to determine the speci0c intra-pellet pore volume. The corrected, intra-pellet speci0c pore volume for batch G1 determined by mercury porosimetry is shown in Table 1. The corrected, 0nal level of mercury entrapment is also given in Table 1. Fig. 7 shows an example of a light micrograph image of a typical sample from batch G1 following a full primary intrusion and retraction cycle. The dark zones within the sphere in Fig. 7(a) correspond to regions containing large quantities of entrapped mercury, whereas the more transparent and paler regions correspond to areas with no, or much less, entrapped mercury. The image in Fig. 7(a) clearly shows that the spatial distribution of entrapped mercury is very heterogeneous, over macroscopic length scales. It can be seen that, towards the right-hand limb of the image in Fig. 7(a), an apparently isolated domain of entrapped mercury is clearly

Table 1 Pore structural parameters obtained from nitrogen sorption and mercury porosimetry for samples taken from batch G1 Sample

Mercury porosimetry

Nitrogen adsorption BJH Before intrusion

1 2

After intrusion

Speci0c pore volume, V0 (cm3 =g)

Entrapment (cm3 =g)

BJH pore volume, (1.7–50 nm dia.) b VBJH (cm3 =g)

Volumeweighted mean pore radius (nm)

=

BJH pore volume, (1.7–50 nm dia.) a VBJH (cm3 =g)

Volumeweighted mean pore radius (nm)

a V0 − VBJH (cm3 =g)

1.05 1.05

0.11 0.12

1.005 1.012

6.91 7.08

0.37 0.34

0.951 0.940

6.47 6.67

0.10 0.11

Renormalised cumulative volume/(cc/g)

Sean P. Rigby et al. / Chemical Engineering Science 59 (2004) 41 – 51

1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

5

10

15

20

25

20

25

Pore radius/nm

(a) Renormalised cumulative volume/(cc/g)

47

1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

5

10

15

Pore radius/nm

(b) Fig. 8. Cumulative pore size distributions, for sample 1 (a) and sample 2 (b) taken from batch G1, obtained using a BJH analysis of the nitrogen adsorption isotherms measured both before (solid line) and after (•) mercury porosimetry experiments.

Fig. 7. (a) Light micrograph image of a sample from batch G1 following a full primary mercury intrusion and retraction cycle. The overall pellet diameter is 3 mm. (b) Close-up view of a region of (a) containing an apparently isolated, entrapped mercury ganglion.

are estimates of the volume-weighted mean pore radius, derived from the BJH pore size distribution, obtained both before and after mercury intrusion, for both samples studied. It can be seen, from the data shown in Table 1, that there is a signi0cant reduction in the mean pore radius for the pore volume accessible to nitrogen following mercury intrusion. It should be noted that, in Fig. 8, the total speci0c pore volume for a cumulative pore size distribution obtained after mercury intrusion has been renormalised to the same total pore volume as that for the pore size distribution obtained for the same sample before mercury intrusion. Due to the decrease in pore volume accessible to nitrogen following mercury entrapment, this leads to a 2attening out to the horizontal of the cumulative nitrogen pore size distribution obtained after mercury intrusion before it reaches the pore size axis. Displayed in this manner, the nitrogen cumulative pore size distributions for the samples following mercury intru-

sion appear to have lost all of the largest pores previously present in the distribution. It is proposed that this manner of reporting the cumulative pore size distributions shown in Fig. 8 is the most appropriate for the following reasons. As stated above, the data shown in Table 1 indicate that there has been a decrease in the volume-weighted mean pore radius following mercury intrusion. This is what would be expected if nitrogen is excluded from the largest pores present in the sample following mercury porosimetry because entrapped mercury is now occupying those pores. It can also be noted from Fig. 8 that the cumulative pore size distributions, obtained using nitrogen adsorption before and after mercury porosimetry, are negligibly di-erent for pore radii less than ∼12 nm for sample 1, and ∼11 nm for sample 2. Hence mercury porosimetry does not appear to have a-ected the accessibility of nitrogen to pores of radii less than ∼11–12 nm. In addition, for example, it can be noted that for sample 2 the di-erence in the BJH surface areas for pores of radii greater than 11 nm for the sample before and after mercury intrusion is 5:2 m2 =g. This value is almost exactly equal (within likely errors) to the di-erence of 4:4 m2 =g between the BJH cumulative adsorption surface areas of pores with diameters between 1.7 and 50 nm for the sample

48

Sean P. Rigby et al. / Chemical Engineering Science 59 (2004) 41 – 51

before and after mercury intrusion. This suggests that the loss in accessible surface area is occurring exclusively in pores with diameters greater than ∼22 nm. The nitrogen sorption isotherms obtained both before and after mercury porosimetry have been analysed, using the method based on percolation theory suggested by Seaton (1991), to determine the pore network connectivity and apparent lattice size. These parameters can be used to characterise the shape of the nitrogen sorption hysteresis loop. An example of a 0t of the nitrogen sorption data to the universal scaling relation (Eq. (5)) is given in Fig. 9. The results of this analysis are shown in Table 2. It can be seen that there is little di-erence between the values obtained for pore connectivity and lattice size before and after mercury intrusion. For a particular sample from batch G1, the value of pore connectivity obtained using the Seaton (1991) analysis (Table 2), and the value of the reduced standard deviation of the pore size distribution obtained from the BJH analysis (Barrett et al., 1951) (Table 1), can be used to predict the expected value of mercury entrapment. The level of the mercury entrapment can be predicted using the above two parameters and the results of the simulations of mercury intrusion and retraction in random pore bond networks conducted by Portsmouth and Gladden (1991). The model of Portsmouth and Gladden (1991) predicts a level of mercury entrapment of at least 40% for samples 1 and 2. This

L^(0.41/0.88)ZF

9 8 7 6 5 4 3 2 1 0 -15

-10

-5

0 5 10 (Zf-3/2)L^(1/0.88)

15

20

25

Fig. 9. A 0t of the nitrogen sorption data (•), obtained for sample 1 from batch G1 following mercury porosimetry, to the universal scaling relation (Eq. (5)) (solid line).

Table 2 The results of the Seaton (1991) percolation analysis of the nitrogen sorption data for samples 1 and 2 taken from batch G1

Sample Sample Sample Sample

1 1 2 2

before porosimetry after porosimetry before porosimetry after porosimetry

Pore connectivity, Z (±0:1)

Lattice size, L (±0:1)

4.6 4.8 4.3 4.8

8 9 8 9

prediction is signi0cantly higher than the ∼11–12% that was actually observed by experiment (see Fig. 3). It can be seen from Table 1 that the total speci0c pore volume determined by mercury porosimetry is greater than the total speci0c pore volume for pores in the diameter range 1.7–50 nm determined by nitrogen adsorption before mercury intrusion. Since the lower pore diameter limit for mercury intrusion is ∼3 nm then this di-erence must indicate the presence of pores in the sample with diameters ¿50 nm. This 0nding is in agreement with the previous proposal (Rigby et al., 2003b) of the presence of pores in the size range of 50 –80 nm in pellets from batch G1. This proposal was suggested as a result of data obtained from mercury retraction and re-injection curves. It can also be noted from Table 1 that the decrease in accessible pore volume for pores in the diameter range 1.7–50 nm following mercury intrusion is less than the volume of mercury entrapped in the structure. However, it can also be seen from Table 1 that the di-erence between the total speci0c pore volume from mercury intrusion and the volume of pores with diameters in the range 1.7–50 nm obtained from nitrogen adsorption following mercury intrusion, is identical, within likely errors, to the volume of entrapped mercury. This result indicates that mercury is also becoming entrapped in the pores that are too large to detect precisely using nitrogen adsorption. In addition, this 0nding suggests that the contraction of mercury, following cooling to 77 K, has had no e-ect on the results, since there is an insigni0cant change in the volume of the entrapped mercury. A t-plot analysis of the data from the nitrogen adsorption isotherms obtained before porosimetry suggests that pellets from batch G1 possess negligible microporosity. This 0nding is in agreement with the results of previous t-plot studies of pellets from this batch (Rigby, 1999). In addition, t-plot analyses of the data obtained from nitrogen adsorption following porosimetry, for both samples studied, showed that G1 still had negligible microporosity. This result suggests that the thermal contraction of mercury on cooling does not create microporous “cracks” of suf0cient size into which nitrogen can 0t. Hence, the above 0ndings suggest that the in2uence of the thermal contraction of mercury, on the results of the nitrogen sorption run following porosimetry, is negligible. This is unsurprising, since it is anticipated that the thermal contraction of mercury would be very slight under the experimental conditions described above. The reduction in the volume occupied by entrapped liquid mercury between the temperature at which the porosimetry experiments were carried out (298:9 K) and 263:2 K is 0.6% (Weast et al., 1988). The freezing point of mercury is 234:3 K. The linear expansion coeKcient for solid metals is typically ∼10−5 K −1 and thus the variation in the volume occupied by solid mercury between 234 and 77 K is expected to be miniscule. The total pore volumes obtained by single point measurements from the isotherms before mercury intrusion at relative pressures of 0.997 and 0.9997 for samples 1 and 2, respectively, are 1.05 and 1:06 cm3 =g, respectively. Since

Sean P. Rigby et al. / Chemical Engineering Science 59 (2004) 41 – 51

these values are similar to the total cumulative pore volumes obtained from mercury porosimetry, these results suggest that the pores larger than 50 nm present in G1 would be 0lled with liquid nitrogen at the top of the isotherms. Therefore these larger pores will not be able to act as “seed” sites for the growth of the vapour phase during desorption. 5. Discussion Previous work (Rigby and Edler, 2002) has shown that the analysis of raw mercury porosimetry data for the sol– gel silica material G1 with the new, semi-empirical alternatives to the traditional Washburn equation reveals that the hysteresis and mercury entrapment present for whole samples does not arise when the samples are fragmented to particle sizes of ∼50–100 m. In addition, this particular result was also obtained for several other di-erent silica materials. This 0nding suggests that the hysteresis removed by Eqs. (3) and (4), for the fragmented silica samples studied, was purely contact angle hysteresis, and not structural hysteresis. This is because di-erent amorphous silica materials, each exhibiting di-ering nitrogen sorption isotherms, are unlikely to all possess identical pore body to pore neck size ratios at all pore body sizes. More recently (Rigby, 2002), similar results have also been found for many different (but opaque) alumina catalyst supports formed using various di-erent methods (e.g. extrusion, compaction, etc.). These 0ndings suggest that the theories discussed here may be wider in scope than just for sol–gel silica materials. The above light microscopy studies have revealed that the mercury entrapped within whole samples of the sol–gel silica G1 is concentrated into particular regions of the sample. The hetereogeneities in the spatial distribution of entrapped mercury are ∼0:1–1 mm in size. In the light of the porosimetry experiments on heterogeneous glass micromodels conducted by Wardlaw and McKellar (1981), a particular interpretation of these results has been proposed (Rigby et al., 2003b). It is suggested that the mercury entrapment arising in the sol–gel silicas occurs because of the presence of isolated, spatially extended domains of similarly-sized large pores that are completely surrounded by a continuous network of smaller pores. It is mercury residing in these domains that is visible in the light micrographs. The fragmentation of the sample into particles of ¡100 m in size removes the “shielding” e-ect by the smaller pores that gives rise to the entrapment. Hence, the entrapment disappears on fragmentation of the sample. The studies reported here have demonstrated that the entrapped mercury is concentrated within the very largest pores present in the material. This 0nding is consistent with each of the macroscopic, spatially extended domains only containing pores of a similar size. All of the above 0ndings are also consistent with the results of MRI studies (Rigby and Gladden, 1996) which have indicated that there are macroscopic, nonrandom 2uctuations in local average pore size across pellets from batch G1. As

49

the comparison of the above experimental results with the predictions of random pore bond network models shows, a larger amount of mercury than actually observed would have been expected to have become entrapped if the pores of various sizes were more intermingled. These studies con0rm that mercury porosimetry can be used in accordance with the method proposed in earlier work (Rigby and Edler, 2002) to determine the macroscopic spatial distribution of pore sizes. As proposed previously (Rigby et al., 2002), the actual 3D spatial distribution of entrapped mercury may be obtained experimentally using MFX imaging and this information, together with the sizes of pores containing entrapped mercury obtained using integrated nitrogen sorption, can be used to validate a particular geometric pattern for the arrangement of all pore sizes obtained using simulations of mercury porosimetry (Rigby and Edler, 2002). An analysis (Rigby and Edler, 2002) of the small-angle X-ray scattering data for pellets from batch G1 has suggested that the typical radius of a pore body in this material is ∼5 nm and the typical pore neck radius is ∼4 nm. These sizes are very similar to the modal pore size in the distribution obtained from an analysis of the raw mercury porosimetry data using Eqs. (3) and (4) (see Fig. 6). The integrated nitrogen sorption and mercury porosimetry method also permits a test of the validity of the results of using Eqs. (3) and (4) to analyse raw mercury porosimetry data. G1 is a good material on which to carry out this test because of the marked di-erence between the predictions of the Washburn (1921) Equation and the new semi-empirical alternatives. As hysteresis commences very early (at a mercury void occupancy of ∼99%) when the raw porosimetry data for G1 is analysed using the standard approach (described above) with the Washburn (1921) Equation (see Fig. 5), then this method would be consistent with mercury entrapment occurring in a wide range of pore sizes. However, as seen in Fig. 6, the use of Eqs. (3) and (4) leads to a more restricted prediction of the size of pores that should contain entrapped mercury. The absence of hysteresis for smaller pores in Fig. 6 suggests that the intrusion of the mercury is completely reversible for the sections of the void space 0lled when levels of mercury occupancy are ¿∼35%. If the presence of any entrapped mercury (and thus loss of accessible volume) within the smallest ∼65% of pores had been detected using nitrogen adsorption, then this would have meant that the interpretation of the raw data using the new equations was de0nitely incorrect. It is thus noted that the con0nement of entrapped mercury to the largest pores present in G1 is consistent with the results of the analysis of the mercury intrusion and extrusion curves using the new, semi-empirical alternatives to the Washburn (1921) Equation. The results of this work and previous work (Rigby et al., 2003b) have also discounted potential competing explanations of the results described above. It is conceivable that mercury entrapment in G1 could be caused by irreversible, pore structural collapse, rather than macroscopic structural

50

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heterogeneities. The collapse of pores within silica xerogels during mercury porosimetry experiments resulting in apparent mercury entrapment has been observed by Pirard et al. (1997) and Alie et al. (2001). However, the light micrographs discussed above clearly show the presence of real entrapped mercury in pellets from batch G1. In addition, in previous work (Rigby et al., 2003b), primary and secondary retraction curves from mercury porosimetry experiments on batch G1 have shown that relatively large pores of sizes 60 –100 nm can survive repeated applications of pressures up to ∼412 MPa. The similarity in the shapes of primary intrusion and mercury re-injection (secondary intrusion) curves (for the same sample) at the highest pressures have shown that small pores in G1 can also survive the imposition of high pressure. It is also conceivable that subsequent re-organisation and/or 2ow of mercury following initial entrapment may be the reason why no entrapped mercury is observed for fragmented samples, or why entrapped mercury is concentrated in the largest pores in whole pellets. Initially entrapped mercury may be able to migrate to the surface in smaller particles and then leave the material. However, previous results (Rigby et al., 2003b) have indicated that this is not the case for batch G1. If the entrapped mercury were migrating over the time-scale of the experiment itself, then it might be expected that the extrusion curves for whole pellets obtained with longer equilibration times would show less entrapment because some entrapped mercury near the periphery of the pellets would have had enough time to reach the edge of the pellet. Mercury porosimetry experiments on whole pellets have been conducted (Rigby et al., 2003b) with equilibration times in the range of values between 15 and 3600 s. Varying the equilibration time over two orders of magnitude was found to make no di-erence to the shape of the intrusion and extrusion curves, and the level of mercury entrapment. This result suggests that migration of entrapped mercury does not occur over the time-scale of the experiment itself. The periodic observations of the location and morphology of entrapped mercury ganglia mentioned above has also shown that mercury entrapped within pellets from batch G1 does not migrate over the time-scale of a few days either. The very small di-erence in the percolation parameters obtained before and after mercury intrusion is consistent with the existence of a large continuous network of smaller pore elements surrounding domains of larger pores. This is because, when occupied with entrapped mercury, these domains containing larger pores obviously do not signi0cantly interrupt the formation of the percolating cluster during nitrogen desorption from smaller pores. In contrast, if all pores were intermingled in a random network, then, following mercury entrapment, it would be expected that the connectivity should decrease by ∼5–7%, whilst only the lattice size remained constant. If the entrapped mercury was preferentially located around the periphery of the pellets, then it would be expected that the apparent lattice size would

certainly increase markedly, and the connectivity may decrease, following porosimetry, due to the greatly decreased accessibility of the interior of the pellet to the invading vapour phase. The results obtained here suggest that neither of these scenarios is correct. However, a consideration of these scenarios does illustrate how a study of the signi0cant changes (or not, in this case) of the percolation parameters Z and L before and after entrapment allows a broad overview of the main features of a pore structure to be deduced without the involved statistical analysis that would be necessary with a full 3D reconstruction of the pore space obtained from tomography. In general, the presence of entrapped mercury might be expected to reduce the accessibility of the remaining open pore network. As described above, the lattice size would be expected to increase, whilst the connectivity would be expected to decrease following mercury entrapment. However, the results for sample 2 from batch G1 show a slight increase in apparent connectivity following mercury entrapment. This apparently paradoxical result is thought to arise because, as proposed above, the (originally open) larger mesopores occur together within particular domains that are surrounded by a continuous network of smaller mesopores. In this particular spatial arrangement, the larger mesopores would only become accessible to the invading vapour phase when liquid nitrogen had desorbed from the surrounding smaller pores. This nonrandom spatial arrangement mimics the behaviour of a very poorly connected random lattice, where a percolation path to a larger pore is likely to be constrained to pass through much smaller pores. Hence the original pellet appears to have a low connectivity. Once the larger mesopores are 0lled with entrapped mercury, the fraction of the network remaining open is relatively more accessible than the original network as a whole. In conjunction with mercury scanning loops (partial intrusions) this approach, using a percolation analysis of the nitrogen sorption data, could be used to deduce the broad spatial distribution of di-erent fractions of the pore size distribution (particularly in samples with higher overall levels of mercury entrapment than G1). More complex modelling of the results from these types of experiments will be the subject of a future work.

6. Conclusions The use of nitrogen sorption as an integrated part of a mercury porosimetry experiment has shown that the entrapment of mercury in a particular type of sol–gel silica, catalyst support pellet only occurs in the largest pores present. Together with data obtained in previous work, this result shows that the most likely cause of mercury entrapment in this material is the presence of macroscopic heterogeneities in the spatial distribution of pore size. The observed location of the entrapped mercury is also consistent with the

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results of the analysis of the raw porosimetry data using new, semi-empirical alternatives to the traditional Washburn Equation. These 0ndings con0rm that the macroscopic spatial arrangement of pore sizes can be obtained from mercury porosimetry using the method proposed in earlier work (Rigby and Edler, 2002). References Alie, C., Pirard, R., Pirard, J.P., 2001. Mercury porosimetry: applicability of the buckling-intrusion mechanism to low density xerogels. Journal of Non-Crystalline Solids 292, 138–149. Allamy, A., Mann, R., Holt, A., 2003. Modelling of catalyst particle skin e-ects using a 3-D pore network model and quantitative microscopy. Chemical Engineering Science 58, 1989–2000. Androutsopoulos, G.P., Mann, R., 1979. Evaluation of mercury porosimeter experiments using a network pore structure model. Chemical Engineering Science 34, 1203–1212. Barrett, E.P., Joyner, L.G., Halenda, P.H., 1951. The determination of pore volume and area distributions in porous substances-I. Computations from nitrogen isotherms. Journal of the American Chemical Society 73, 373–380. Cheah, K.-Y., Chiaranussati, N., Hollewand, M.P., Gladden, L.F., 1994. Coke pro0les in deactivated alumina pellets studied by NMR imaging. Applied Catalysis 115, 147–155. Dullien, F.A.L., 1981. Wood’s metal porosimetry and its relation to mercury porosimetry. Powder Technology 29, 109–116. Hollewand, M.P., Gladden, L.F., 1993. Heterogeneities in structure and di-usion within porous catalyst support pellets observed by NMR imaging. Journal of Catalysis 144, 254–272. Kirkpatrick, S., 1979. In: Balian, R., Mayward, R., Toulouse, G. (Eds.), Ill-condensed Matter. North-Holland, Amsterdam, p. 321. Kloubek, J., 1981. Hysteresis in porosimetry. Powder Technology 29, 63–73. Koster, A.J., Ziese, U., Verklejj, A.J., Janssen, A.H., De Jong, K.P., 2000. Three-dimensional transmission electron microscopy: a novel imaging and characterization technique with nanometre scale resolution for materials science. Journal of Physical Chemistry B 104, 9368–9370. Lee, C.K., Tsay, C.S., 1998. Pore connectivity of alumina and aluminium borate from nitrogen isotherms. Journal of the Chemical Society, Faraday Transactions 94, 573–577. Liabastre, A.A., Orr, C., 1978. An evaluation of pore structure by mercury penetration. Journal of Colloid and Interface Science 64, 1–18. Mann, R., Allamy, A., Holt, A., 1995. Visualised porosimetry for pore structure characterisation of nickel/alumina reforming catalysts. Transactions of the Institution of Chemical Engineers Part A 73, 147–153. Mata, V.G., Lopes, J.C.B., Dias, M.M., 2001. Porous media characterization using mercury porosimetry simulation. 1. Description of the simulator and its sensitivity to model parameters. Industrial and Engineering Chemistry Research 40, 3511–3522. Murray, K.L., Seaton, N.A., Day, M.A., 1999. Use of mercury intrusion data, combined with nitrogen adsorption measurements, as a probe of network connectivity. Langmuir 15, 8155–8160. Pirard, R., Heinrichs, B., Pirard, J.P., 1997. Mercury porosimetry applied to low density xerogels. In: McEnaney, B., Mays, T.J., Rouquerol, J., Rodriguez-Reinoso, F., Sing, K.S.W., Unger, K.K. (Eds.), Characterisation of Porous Solids IV. Royal Society of Chemistry, Cambridge. Portsmouth, R.L., Gladden, L.F., 1991. Determination of pore connectivity by mercury porosimetry. Chemical Engineering Science 46, 3023–3036.

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Portsmouth, R.L., Gladden, L.F., 1992. Mercury porosimetry as a probe of pore connectivity. Transactions of the Institution of Chemical Engineers 70A, 63–70. Rigby, S.P., 1999. NMR and modelling studies of structural heterogeneity over several lengthscales in amorphous catalyst supports. Catalysis Today 53, 207–223. Rigby, S.P., 2000. A hierarchical structural model for the interpretation of mercury porosimetry and nitrogen sorption. Journal of Colloid and Interface Science 224, 382–396. Rigby, S.P., 2002. New methodologies in mercury porosimetry. Studies in Surface Science and Catalysis 144, 185–192. Rigby, S.P., Daut, S., 2002. A statistical model for the heterogeneous structure of porous catalyst pellets. Advances in Colloid and Interface Science 98, 87–119. Rigby, S.P., Edler, K.J., 2002. The in2uence of mercury contact angle, surface tension and retraction mechanism on the interpretation of mercury porosimetry data. Journal of Colloid and Interface Science 250, 175–190. Rigby, S.P., Gladden, L.F., 1996. NMR and fractal modelling studies of transport in porous media. Chemical Engineering Science 51, 2263–2272. Rigby, S.P., Cheah, K.-Y., Gladden, L.F., 1996. NMR imaging studies of transport heterogeneity and anisotropic di-usion in porous alumina pellets. Applied Catalysis A 144, 377–388. Rigby, S.P., Fletcher, R.S., Riley, S.N., 2001. Characterization of macroscopic structural disorder in porous media using mercury porosimetry. Journal of Colloid and Interface Science 240, 190–210. Rigby, S.P., Fletcher, R.S., Raistrick, J.H., Riley, S.N., 2002. Characterisation of porous solids using a synergistic combination of nitrogen sorption, mercury porosimetry, electron microscopy and micro-focus X-ray imaging techniques. Physical Chemistry Chemical Physics 4, 3467–3481. Rigby, S.P., Barwick, D., Fletcher, R.S., Riley, S.N., 2003a. Interpreting mercury porosimetry data for catalyst supports using semi-empirical alternatives to the Washburn equation. Applied Catalysis A 238, 303–318. Rigby, S.P., Fletcher, R.S., Riley, S.N., 2003b. Determination of the cause of mercury entrapment during porosimetry experiments on sol-gel silica catalyst supports. Applied Catalysis A 247, 27–39. RuKno, L., Mann, R., Oldman, R.J., Rigby, S.P., Allen, S., 2001. Using low melting point alloy intrusion to quantify pore structure: studies on an alumina catalyst support. Studies in Surface Science and Catalysis 133, 155–162. Seaton, N.A., 1991. Determination of the connectivity of porous solids from nitrogen sorption measurements. Chemical Engineering Science 46, 1895–1909. Stau-er, D., 1985. Introduction to Percolation Theory. Taylor & Francis, London. 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., 2001. A high-throughput X-ray microtomography system at the advanced photon source. Review of Scienti0c Instruments 72, 2062–2068. Wardlaw, N.C., McKellar, M., 1981. Mercury porosimetry and the interpretation of pore geometry in sedimentary rocks and arti0cial models. Powder Technology 29, 127–143. Wardlaw, N.C., McKellar, M., Li, Y., 1988. Pore and throat size distributions determined by mercury porosimetry and direct observation. Carbonates and Evaporites 3, 1–15. Washburn, E.W., 1921. The dynamics of capillary 2ow. Physical Review 17, 273–283. Weast, R.C., Astle, M.J., Beyer, W.H., 1988. CRC Handbook of Chemistry and Physics, 69th Edition. CRC Press Inc., Boca Raton, FL.