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Simulation of mercury porosimetry using MRI images of porous media
Studies in Surface Science and Catalysis 160
177
P.L. Llewellyn, F. Rodriquez-Reinoso, J. Rouqerol and N. Seaton (Editors)
9 2007 Elsevier B.V. All rights reserved
Simulation of mercury porosimetry using MRI images of porous media Matthew J. Watt-Smith a, Sean P. Rigby *'a, J. A. Chudek b, and Robin S. Fletcher c
aDepartment of Chemical Engineering, University of Bath, Claverton Down, Bath, BA2 7AY, U.K. bDivision of Physical and Inorganic Chemistry, School of Life Sciences, University of Dundee, Dundee, DD 1 4HN, U.K. CJohnson Matthey Catalysts, P.O. Box 1, Belasis Avenue, Billingham, Cleveland, TS23 1LB, U.K. Abstract Models of the pore structure of pellets taken from a batch of sol-gel silica spheres have been constructed from magnetic resonance images of the macroscopic (-~0.04-1 mm), spatial distribution of spin density and spin-spin relaxation time within the material. Simulations of mercury porosimetry on these models gave rise to good predictions for the point of deviation of the intrusion and retraction curves, and the level of mercury entrapment, in agreement with those found by experiment. This finding suggested that mercury intrusion and retraction within the pellets are determined by the macroscopic structure of the material, as detected using MRI.
1. INTRODUCTION Knowledge of the mechanisms of the entrapment of non-wetting fluids within porous media is important in a number of fields of study. The mercury extrusion curve in porosimetry potentially contains useful information on the nature of a porous structure. However, in order to extract that information it is necessary to have an understanding of how the entrapment of mercury arises. Mercury porosimetry is also often used in the oil industry to evaluate reservoir rock cores. This is because the mercury recovery efficiency is expected to provide an indication of the oil recovery efficiency in a strongly water-wet system. Previous experimental work [ 1] studying the mechanisms of mercury entrapment in glass micro-models has suggested that larger scale heterogeneities in the spatial distribution of pore size give rise to entrapment. If isolated regions containing large pores are located within a continuous network of smaller pores then, on commencing retraction of mercury from a fully imbibed pore space, mercury first withdraws from the smaller pores. However, at the stage where the pressure has been reduced below the threshold for emptying the clusters of larger pores, these have already become disconnected by snap-off and extensive residual mercury is retained. More recently [2-4], some experimental evidence has suggested that the same mechanism may give rise
178
M.J. Watt-Smith et al.
to the mercury entrapment observed in commercial mesoporous, catalyst-support pellets. For various sol-gel silica and alumina pellets it was found that mercury entrapment did not occur when the porosimetry experiment was run on fragmented samples (powder size -30-100 ~tm) rather than whole pellets (size ---3-10 mm). This is what would be expected if the particle size of the fragmented material was smaller than the characteristic dimension of the isolated regions of larger pores within the materials. For transparent sol-gel silica materials, it was observed [3] that the spatial distribution of entrapped mercury within whole pellets was heterogeneous, and the isolated regions in which mercury was entrapped were macroscopic in size (0.0 l-l mm). These findings suggest that, for macroscopically heterogeneous materials, the only information that should be required to predict levels of mercury entrapment is a knowledge of the spatial distribution of porosity and pore size. This type of information can be obtained using magnetic resonance imaging (MRI) techniques [5,6], and macroscopic heterogeneities in the spatial distribution of porosity and pore size have been observed within catalyst pellets with MRI. It is the purpose of this paper to directly test the above theory of mercury entrapment within certain mesoporous catalyst support pellets by predicting the level of mercury entrapment using models of the porous media derived from MR images. MRI pore characterisation techniques are only suitable for application to model porous media which are relatively chemically homogeneous, such as pure silica and alumina pellets. However, if the theory above is shown to be correct then the form of the mercury extrusion curve and level of mercury entrapment could potentially be used to deduce information about the spatial distribution of pore sizes for relevant materials not amenable to MRi. 2. T H E O R Y 2.1. MRI
The use of MRI to probe the structure of porous media is based upon the phenomenon that the NMR relaxation rate of fluids imbibed within pores is enhanced due to molecular interactions with the pore walls. A thin layer of liquid, typically -1-2 molecular layers thick, in contact with the walls has restricted motion, and thus an increased relaxation rate. This surface layer is typically in fast diffusional exchange with the bulk fluid in the rest of the pore. Hence the observed relaxation rate is the volume-weighted mean value of the surface and bulk phases. This model is known as the two-fraction fast-exchange model [7] and the overall spin-spin (say) relaxation rate, T:, in a pore of surface area S and volume V is given by:
~2=
1-
- -t T28 Fr2s
~
Fr2s-
(])
where 2 is the thickness of the surface layer, the subscripts B and S refer to the bulk and surface layer, respectively. In general T2B>>T2s, and hence the measured T2 is directly proportional to the volume-to-surface area ratio of a pore (- radius/2 for a cylindrical pore). In order to obtain a characteristic pore dimension an assumption concerning the pore geometry must be made. However, in this work, it is only necessary to assume that T2 is directly proportional to the characteristic dimension(s) of a pore that determine(s) the pressure at which mercury intrusion and extrusion occur during porosimetry. The MRI technique allows this measurement to be made
179
Simulation of mercury porosimetry
spatially resolved over length-scales >-0.01 mm. More detail conceming MRI can be found elsewhere [5]. MRI, using spin-spin relaxation time pre-conditioning, has been used here to simultaneously obtain maps of the spatial distribution of both spin density (proportional to porosity) and I"2. The 7"2 maps obtained using MRI have been analysed using the fluctuation auto correlation function, C(s), which measures the degree of correlation between f(x n) values at successive data points. Explicitly, defining 6fn as:
8so : s(..)-(s) then:
(3) 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. (3) 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=l) and the value of s when C(s)-O. The first value characterises the degree of correlation, while the second value is known as the correlation length ({) and characterises the linear extent of that correlation.
2.2. Simulation of mercury porosimetry The MRI spin density and 7"2 images were used to construct a model of the porous pellet. The structural representation consisted of a lattice site model where each site corresponded to a voxel in the MR images. The mechanisms of mercury intrusion and retraction were the same as those employed in previous work [2]. As mentioned above it was 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 7'2 in the image voxel volume corresponding to that region of the sample. In order to simulate mercury intrusion, the value of a cutoff in T:, 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 ~ cutoff was raised steadily in small steps. Mercury was allowed to retract back from a given model site if the 7"2 value in the corresponding image voxel was below the cutoff value, and 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 T: cutoff was rereached.
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2.3. Analysis of raw, experimental mercury porosimetry data In general, mercury porosimetry data is analysed using the Washburn equation [8]. In addition, raw porosimetry data is also generally characterised by the presence of hysteresis. This hysteresis is generally acknowledged [1] to arise from either, or both, contact angle hysteresis and structural hysteresis. Recently, building on earlier work on controlled pore glasses [9,10], Rigby and Edler [2] proposed semi-empirical 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 characterise the pore structure of a particular porous material. For mercury intrusion the pore radius (nm) is given by: 302.533
+ x/91526.216
+ 1.478p
r -
,
(4)
where p is the applied pressure (in MPa), while for mercury retraction the pore radius is given by:
r =
68.366 + x/4673.91 + 471.122p
.
(5)
P Since Equations (4) and (5) are empirical in origin, then their use leads to an experimental error in the pore sizes obtained, which is estimated [10] to be --4-5 %. 3. EXPERIMENTAL The 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 BET surface area of-~99 mE.gl.
3.1. MRI Samples were prepared by impregnation with de-ionised water under ambient conditions for 24 h. 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 l0 mm Birdcage coil. Spin-spin relaxation time (T:) and spin density maps (which probe porosity) were acquired together using the Bruker sequence "m_msme ", and employed 90 ~ selective and 180 ~ non-selective pulses. A T2 pre-conditioned 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, two and three dimensional data processing, and workup was handled on an Aspect X32 workstation, running the Paravision 9 suite of software (Bruker Analitische Messtechnik Gmbh, Karlsruhe, Germany). The in-plane pixel resolution was 40 ~tm and the slice thickness was 250 ~tm. 3D spin-echo data was subsequently worked up using AMIRA software.
Simulation of mercury porosimetry
181
3.2. Mercury porosimetry 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 in order 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 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). 4. RESULTS AND DISCUSSION
4.1. Image analysis Figure 1 shows examples of two T2-contrasted images of different slices through the centre of a spherical pellet from batch G2. 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 analysed using the auto-correlation function. The variation of the values of C(s=l) and ~ 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, say, are more concentrated towards the centre of the sphere and decrease in concentration in weakly-defined bands located progressively further from the centre 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 shown in Figure 3, the image would slice through several different bands and would detect the correlation in I"2 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 image 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.
4.2. Simulations of mercury porosimetry Simulations of mercury intrusion and retraction, according to the mechanisms described above, were performed on 2D structural models constructed from spin density and Te-weighted images of central slices through pellets taken from batch G2. An example of a typical set of data
182
Fig. 1. T2 images of perpendicular 2D slices through the centre of a pellet from batch G2. The pixel resolution is 40 ~m and the slice thickness is 250 lam.
M.J. Watt-Smith et al.
Fig. 2. Variation with position of the correlation length (O) and degree of correlation (x) for T2 maps of parallel slices of a pellet from batch G2.The central region of the pellet corresponds to slice 5.
for one particular model is shown in Figure 4. Figure 4 shows a plot of the fractional occupied volume for the extrusion curve against the fractional occupied volume for the intrusion curve at the same 7'2 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 pore size probability density function of T2 values (or critical pore sizes). It can be seen that the curves separate at an occupied volume fraction of-~0.65, and the entrapment is ---40 %. Data from simulations on models constructed from different images of separate pellet samples are shown in Table 1. Previous simulations on abstract model grids [2] 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 MR images is significantly different to the value (of 50 %) expected for a completely random arrangement of pixel intensities [2].
4.3. Experimental mercury porosimetry data Figure 5 shows the mercury intrusion and retraction curves, analysed using Eqs. (4) and (5), respectively, for a sample of pellets from batch G2. It can be seen that at smaller pore sizes, within the error in eqs. (4) and (5), 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 1 shows the mercury entrapment levels for several samples taken from batch G2. From Table 1, it can be seen that the experiments are repeatable, and the value of entrapped mercury agrees well with that predicted from the models derived from MR images above.
Simulation of mercury porosimetry
183
Fig. 3. Schematic diagram illustrating the expected variation of 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. 1.0
1.2
0.9 0.8
1.0-
"~ 0.7
0
0.8-
.~ 0.6 ~9 0.5
0.6-
0.4
O
g "~ 0.3
o= 0.4-
..~
0.2 0.1
~, 0.2-
0.0
!
0.0
0.2
|
|
!
0.4 0.6 0.8 Fractional intrusion volume
0.0
1.0
Fig. 4. 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.
........
1
i
10
........
i
........
1O0
I
1000
........
10000
Pore radius/nm Fig. 5. Experimental mercury intrusion (x) and extrusion (El) curves, analysed using eqs. (4) and (5), for a sample from batch G2. The solid lines are to guide the eye.
184
M.J. Watt-Smith et al.
Table 1 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 G2. (N.B. The data shown is for several samples but image and porosimetry sample numbers do not correspond). Sample Simulation ~Experiment Entrapment Fractional occupied Entrapment Fractiona] occupied (%) volume at point of (%) volume at point of separation of curves separatio n of curves 43.8 0.70 41.9 0.66 43.9 0.69 39.4 0.67 45.6 0.71 42.4 0.71 5. CONCLUSIONS It has been found that models for the structure of sol-gel silica spheres, constructed from spin density and spin-spin relaxation time images, give rise to good predictions for the point of separation of mercury intrusion and retraction curves, and the level of mercury entrapment, found for experimental porosimetry data for the same material. This finding suggests that the MR images contain sufficient information to determine the level of mercury entrapment. Hence, this result supports the view that mercury intrusion and retraction within this material is determined by macroscopic (0.01-1 mm) heterogeneities in the spatial distribution of porosity and pore size. REFERENCES
[1] [2] [31 [4] [5] [6] [7] [8] [9] [10]
N.C. Wardlaw and M. McKellar, Powder Technol. 29 (1981 ) 127. S.P. Rigby and K.J. Edler, J. Colloid Interface Sci. 250 (2002) 175. S.P. Rigby, R.S. Fletcher, and S.N. Riley, Appl. Catal. A 247 (2003) 27. S.P. Rigby, R.S. Fletcher, and S.N. Riley, Chem. Engng Sci. 59 (2004) 4 I. M.P. Hollewand and L.F. Gladden, J. Catal. 144 (1993) 254. S.P. Rigby and L.F. Gladden, Chem. Engng Sci. 51 (1996) 2263. K.R. Brownstein and C.E. Tarr, J. Magn. Reson. 26 (1977) 17. E.W. Washburn, Phys. Rev. 17 (1921 ) 273. A.A. Liabastre and C. Orr, J. Colloid Interface Sci. 64 (1978) 1. J. Kloubek, Powder Technol. 29 (1981 ) 63.
177
P.L. Llewellyn, F. Rodriquez-Reinoso, J. Rouqerol and N. Seaton (Editors)
9 2007 Elsevier B.V. All rights reserved
Simulation of mercury porosimetry using MRI images of porous media Matthew J. Watt-Smith a, Sean P. Rigby *'a, J. A. Chudek b, and Robin S. Fletcher c
aDepartment of Chemical Engineering, University of Bath, Claverton Down, Bath, BA2 7AY, U.K. bDivision of Physical and Inorganic Chemistry, School of Life Sciences, University of Dundee, Dundee, DD 1 4HN, U.K. CJohnson Matthey Catalysts, P.O. Box 1, Belasis Avenue, Billingham, Cleveland, TS23 1LB, U.K. Abstract Models of the pore structure of pellets taken from a batch of sol-gel silica spheres have been constructed from magnetic resonance images of the macroscopic (-~0.04-1 mm), spatial distribution of spin density and spin-spin relaxation time within the material. Simulations of mercury porosimetry on these models gave rise to good predictions for the point of deviation of the intrusion and retraction curves, and the level of mercury entrapment, in agreement with those found by experiment. This finding suggested that mercury intrusion and retraction within the pellets are determined by the macroscopic structure of the material, as detected using MRI.
1. INTRODUCTION Knowledge of the mechanisms of the entrapment of non-wetting fluids within porous media is important in a number of fields of study. The mercury extrusion curve in porosimetry potentially contains useful information on the nature of a porous structure. However, in order to extract that information it is necessary to have an understanding of how the entrapment of mercury arises. Mercury porosimetry is also often used in the oil industry to evaluate reservoir rock cores. This is because the mercury recovery efficiency is expected to provide an indication of the oil recovery efficiency in a strongly water-wet system. Previous experimental work [ 1] studying the mechanisms of mercury entrapment in glass micro-models has suggested that larger scale heterogeneities in the spatial distribution of pore size give rise to entrapment. If isolated regions containing large pores are located within a continuous network of smaller pores then, on commencing retraction of mercury from a fully imbibed pore space, mercury first withdraws from the smaller pores. However, at the stage where the pressure has been reduced below the threshold for emptying the clusters of larger pores, these have already become disconnected by snap-off and extensive residual mercury is retained. More recently [2-4], some experimental evidence has suggested that the same mechanism may give rise
178
M.J. Watt-Smith et al.
to the mercury entrapment observed in commercial mesoporous, catalyst-support pellets. For various sol-gel silica and alumina pellets it was found that mercury entrapment did not occur when the porosimetry experiment was run on fragmented samples (powder size -30-100 ~tm) rather than whole pellets (size ---3-10 mm). This is what would be expected if the particle size of the fragmented material was smaller than the characteristic dimension of the isolated regions of larger pores within the materials. For transparent sol-gel silica materials, it was observed [3] that the spatial distribution of entrapped mercury within whole pellets was heterogeneous, and the isolated regions in which mercury was entrapped were macroscopic in size (0.0 l-l mm). These findings suggest that, for macroscopically heterogeneous materials, the only information that should be required to predict levels of mercury entrapment is a knowledge of the spatial distribution of porosity and pore size. This type of information can be obtained using magnetic resonance imaging (MRI) techniques [5,6], and macroscopic heterogeneities in the spatial distribution of porosity and pore size have been observed within catalyst pellets with MRI. It is the purpose of this paper to directly test the above theory of mercury entrapment within certain mesoporous catalyst support pellets by predicting the level of mercury entrapment using models of the porous media derived from MR images. MRI pore characterisation techniques are only suitable for application to model porous media which are relatively chemically homogeneous, such as pure silica and alumina pellets. However, if the theory above is shown to be correct then the form of the mercury extrusion curve and level of mercury entrapment could potentially be used to deduce information about the spatial distribution of pore sizes for relevant materials not amenable to MRi. 2. T H E O R Y 2.1. MRI
The use of MRI to probe the structure of porous media is based upon the phenomenon that the NMR relaxation rate of fluids imbibed within pores is enhanced due to molecular interactions with the pore walls. A thin layer of liquid, typically -1-2 molecular layers thick, in contact with the walls has restricted motion, and thus an increased relaxation rate. This surface layer is typically in fast diffusional exchange with the bulk fluid in the rest of the pore. Hence the observed relaxation rate is the volume-weighted mean value of the surface and bulk phases. This model is known as the two-fraction fast-exchange model [7] and the overall spin-spin (say) relaxation rate, T:, in a pore of surface area S and volume V is given by:
~2=
1-
- -t T28 Fr2s
~
Fr2s-
(])
where 2 is the thickness of the surface layer, the subscripts B and S refer to the bulk and surface layer, respectively. In general T2B>>T2s, and hence the measured T2 is directly proportional to the volume-to-surface area ratio of a pore (- radius/2 for a cylindrical pore). In order to obtain a characteristic pore dimension an assumption concerning the pore geometry must be made. However, in this work, it is only necessary to assume that T2 is directly proportional to the characteristic dimension(s) of a pore that determine(s) the pressure at which mercury intrusion and extrusion occur during porosimetry. The MRI technique allows this measurement to be made
179
Simulation of mercury porosimetry
spatially resolved over length-scales >-0.01 mm. More detail conceming MRI can be found elsewhere [5]. MRI, using spin-spin relaxation time pre-conditioning, has been used here to simultaneously obtain maps of the spatial distribution of both spin density (proportional to porosity) and I"2. The 7"2 maps obtained using MRI have been analysed using the fluctuation auto correlation function, C(s), which measures the degree of correlation between f(x n) values at successive data points. Explicitly, defining 6fn as:
8so : s(..)-(s) then:
(3) 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. (3) 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=l) and the value of s when C(s)-O. The first value characterises the degree of correlation, while the second value is known as the correlation length ({) and characterises the linear extent of that correlation.
2.2. Simulation of mercury porosimetry The MRI spin density and 7"2 images were used to construct a model of the porous pellet. The structural representation consisted of a lattice site model where each site corresponded to a voxel in the MR images. The mechanisms of mercury intrusion and retraction were the same as those employed in previous work [2]. As mentioned above it was 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 7'2 in the image voxel volume corresponding to that region of the sample. In order to simulate mercury intrusion, the value of a cutoff in T:, 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 ~ cutoff was raised steadily in small steps. Mercury was allowed to retract back from a given model site if the 7"2 value in the corresponding image voxel was below the cutoff value, and 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 T: cutoff was rereached.
180
M.J. Watt-Smith et al.
2.3. Analysis of raw, experimental mercury porosimetry data In general, mercury porosimetry data is analysed using the Washburn equation [8]. In addition, raw porosimetry data is also generally characterised by the presence of hysteresis. This hysteresis is generally acknowledged [1] to arise from either, or both, contact angle hysteresis and structural hysteresis. Recently, building on earlier work on controlled pore glasses [9,10], Rigby and Edler [2] proposed semi-empirical 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 characterise the pore structure of a particular porous material. For mercury intrusion the pore radius (nm) is given by: 302.533
+ x/91526.216
+ 1.478p
r -
,
(4)
where p is the applied pressure (in MPa), while for mercury retraction the pore radius is given by:
r =
68.366 + x/4673.91 + 471.122p
.
(5)
P Since Equations (4) and (5) are empirical in origin, then their use leads to an experimental error in the pore sizes obtained, which is estimated [10] to be --4-5 %. 3. EXPERIMENTAL The 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 BET surface area of-~99 mE.gl.
3.1. MRI Samples were prepared by impregnation with de-ionised water under ambient conditions for 24 h. 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 l0 mm Birdcage coil. Spin-spin relaxation time (T:) and spin density maps (which probe porosity) were acquired together using the Bruker sequence "m_msme ", and employed 90 ~ selective and 180 ~ non-selective pulses. A T2 pre-conditioned 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, two and three dimensional data processing, and workup was handled on an Aspect X32 workstation, running the Paravision 9 suite of software (Bruker Analitische Messtechnik Gmbh, Karlsruhe, Germany). The in-plane pixel resolution was 40 ~tm and the slice thickness was 250 ~tm. 3D spin-echo data was subsequently worked up using AMIRA software.
Simulation of mercury porosimetry
181
3.2. Mercury porosimetry 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 in order 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 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). 4. RESULTS AND DISCUSSION
4.1. Image analysis Figure 1 shows examples of two T2-contrasted images of different slices through the centre of a spherical pellet from batch G2. 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 analysed using the auto-correlation function. The variation of the values of C(s=l) and ~ 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, say, are more concentrated towards the centre of the sphere and decrease in concentration in weakly-defined bands located progressively further from the centre 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 shown in Figure 3, the image would slice through several different bands and would detect the correlation in I"2 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 image 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.
4.2. Simulations of mercury porosimetry Simulations of mercury intrusion and retraction, according to the mechanisms described above, were performed on 2D structural models constructed from spin density and Te-weighted images of central slices through pellets taken from batch G2. An example of a typical set of data
182
Fig. 1. T2 images of perpendicular 2D slices through the centre of a pellet from batch G2. The pixel resolution is 40 ~m and the slice thickness is 250 lam.
M.J. Watt-Smith et al.
Fig. 2. Variation with position of the correlation length (O) and degree of correlation (x) for T2 maps of parallel slices of a pellet from batch G2.The central region of the pellet corresponds to slice 5.
for one particular model is shown in Figure 4. Figure 4 shows a plot of the fractional occupied volume for the extrusion curve against the fractional occupied volume for the intrusion curve at the same 7'2 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 pore size probability density function of T2 values (or critical pore sizes). It can be seen that the curves separate at an occupied volume fraction of-~0.65, and the entrapment is ---40 %. Data from simulations on models constructed from different images of separate pellet samples are shown in Table 1. Previous simulations on abstract model grids [2] 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 MR images is significantly different to the value (of 50 %) expected for a completely random arrangement of pixel intensities [2].
4.3. Experimental mercury porosimetry data Figure 5 shows the mercury intrusion and retraction curves, analysed using Eqs. (4) and (5), respectively, for a sample of pellets from batch G2. It can be seen that at smaller pore sizes, within the error in eqs. (4) and (5), 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 1 shows the mercury entrapment levels for several samples taken from batch G2. From Table 1, it can be seen that the experiments are repeatable, and the value of entrapped mercury agrees well with that predicted from the models derived from MR images above.
Simulation of mercury porosimetry
183
Fig. 3. Schematic diagram illustrating the expected variation of 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. 1.0
1.2
0.9 0.8
1.0-
"~ 0.7
0
0.8-
.~ 0.6 ~9 0.5
0.6-
0.4
O
g "~ 0.3
o= 0.4-
..~
0.2 0.1
~, 0.2-
0.0
!
0.0
0.2
|
|
!
0.4 0.6 0.8 Fractional intrusion volume
0.0
1.0
Fig. 4. 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.
........
1
i
10
........
i
........
1O0
I
1000
........
10000
Pore radius/nm Fig. 5. Experimental mercury intrusion (x) and extrusion (El) curves, analysed using eqs. (4) and (5), for a sample from batch G2. The solid lines are to guide the eye.
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M.J. Watt-Smith et al.
Table 1 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 G2. (N.B. The data shown is for several samples but image and porosimetry sample numbers do not correspond). Sample Simulation ~Experiment Entrapment Fractional occupied Entrapment Fractiona] occupied (%) volume at point of (%) volume at point of separation of curves separatio n of curves 43.8 0.70 41.9 0.66 43.9 0.69 39.4 0.67 45.6 0.71 42.4 0.71 5. CONCLUSIONS It has been found that models for the structure of sol-gel silica spheres, constructed from spin density and spin-spin relaxation time images, give rise to good predictions for the point of separation of mercury intrusion and retraction curves, and the level of mercury entrapment, found for experimental porosimetry data for the same material. This finding suggests that the MR images contain sufficient information to determine the level of mercury entrapment. Hence, this result supports the view that mercury intrusion and retraction within this material is determined by macroscopic (0.01-1 mm) heterogeneities in the spatial distribution of porosity and pore size. REFERENCES
[1] [2] [31 [4] [5] [6] [7] [8] [9] [10]
N.C. Wardlaw and M. McKellar, Powder Technol. 29 (1981 ) 127. S.P. Rigby and K.J. Edler, J. Colloid Interface Sci. 250 (2002) 175. S.P. Rigby, R.S. Fletcher, and S.N. Riley, Appl. Catal. A 247 (2003) 27. S.P. Rigby, R.S. Fletcher, and S.N. Riley, Chem. Engng Sci. 59 (2004) 4 I. M.P. Hollewand and L.F. Gladden, J. Catal. 144 (1993) 254. S.P. Rigby and L.F. Gladden, Chem. Engng Sci. 51 (1996) 2263. K.R. Brownstein and C.E. Tarr, J. Magn. Reson. 26 (1977) 17. E.W. Washburn, Phys. Rev. 17 (1921 ) 273. A.A. Liabastre and C. Orr, J. Colloid Interface Sci. 64 (1978) 1. J. Kloubek, Powder Technol. 29 (1981 ) 63.