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Use of Mercury Intrusion Data, Combined with Nitrogen Adsorption Measurements, as a Probe of Pore Network Connectivity K. L. Murray,† N. A. Seaton,*,‡ and M. A. Day§ Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, United Kingdom, School of Chemical Engineering, University of Edinburgh, King’s Buildings, Mayfield Road, Edinburgh EH9 3JL, United Kingdom, and ICI Technology, Research and Technology Centre, PO Box 90, Wilton, Middlesborough, Cleveland TS90 8JE, United Kingdom Received March 3, 1999. In Final Form: July 26, 1999 The coordination number of the pore network of a mesoporous solid can be determined from its nitrogen adsorption and desorption isotherms, using an analysis method based on percolation theory. However, measurement of these isotherms is time-consuming and problematical very close to the bulk condensation pressure of nitrogen. In this paper, we demonstrate that mercury intrusion and nitrogen desorption share the same mechanism so that the mercury intrusion curve can be used in place of the nitrogen desorption isotherm in the connectivity analysis.
Introduction The pore size distribution (PSD) of a porous solid can be obtained in the range of 20-1000 Å, approximately, from the nitrogen adsorption isotherm, using a method based on the Kelvin equation1-3 to analyze capillary condensation in the pores of the adsorbent. The connectivity of the pore network is also an important characteristic of the pore structure. Seaton and coworkers4-6 have developed an analysis which uses the hysteresis between adsorption and desorption isotherms to probe the connectivity of the pore network. The inputs to the analysis are the PSD (obtained from the adsorption isotherm), the adsorption and desorption isotherms, and data from the simulation of desorption from a model porous solid. The model porous solid is represented by an array of three-dimensional lattices, each of which represents the pore network of a microparticle within the adsorbent. Hysteresis between the adsorption and desorption isotherms is viewed as a percolation phenomenon, wherein the sharp decrease in adsorption at the “knee” of the desorption isotherm corresponds to the formation of a percolating network of vapor-filled pores. An example of an experimental sorption hysteresis loop with a welldefined desorption knee is shown in Figure 1, for sample A (an alumina). [All of the nitrogen sorption data presented in this paper were measured using a Micromeritics ASAP 2400 apparatus. Sample pretreatment comprised heating at 120 °C under nitrogen for 1 h.] The size and shape of the sorption hysteresis loop (and in particular the location of the desorption knee) are dictated by the mean coordination number of the pore network (the average number of pores which intersect at a pore junction), Z, and the †
University of Cambridge. University of Edinburgh. § ICI Technology. ‡
(1) Barrett, E. P.; Joyner, L. G.; Halenda, P. H. J. Am. Chem. Soc. 1951, 73, 373. (2) Cranston, R. W.; Inkley, F. A. Adv. Catal. 1957, 9, 143. (3) Brunauer, S.; Mikhail, R. Sh.; Bodor, E. E. J. Colloid Interface Sci. 1967, 24, 451. (4) Seaton, N. A. Chem. Eng. Sci. 1991, 46, 1895. (5) Liu, H.; Zhang, L.; Seaton, N. A. Chem. Eng. Sci. 1992, 47, 4393. (6) Liu, H.; Seaton, N. A. Chem. Eng. Sci. 1994, 49, 1869.
Figure 1. Experimental nitrogen adsorption and desorption isotherms for sample A.
size of a typical microparticle (expressed as a number of pore lengths), L. The conventional nitrogen desorption isotherm is timeconsuming to measure, typically taking from 6 to 12 h depending on the sample. Therefore it would be expedient to obtain the same structural information using a quicker and simpler method. A possible alternative to the desorption isotherm is the mercury intrusion curve, which can typically be measured in about 1 h. Figure 2 shows the mercury intrusion curve for sample A. [The mercury intrusion data presented in this paper were measured using a Micromeritics AutoPore III 9420 apparatus, with the same sample pretreatment as for the nitrogen sorption experiment.] A perhaps more significant advantage of the mercury intrusion experiment, as opposed to nitrogen sorption, arises where the solid contains pores that are larger than the range probed by the conventional nitrogen sorption experiment. Aukett et al.7 have proposed an experimental technique that involves the bulk condensation of nitrogen in the sample tube at the end of the (7) Aukett, P. N.; Jessop, C. A. In Fundamentals of Adsorption; LeVan, M. D., Ed.; Kluwer: Boston, MA, 1996; p 59.
10.1021/la990250x CCC: $18.00 © 1999 American Chemical Society Published on Web 09/23/1999
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to the nitrogen desorption process. In this paper, we perform a detailed comparison between mercury intrusion and nitrogen desorption. We demonstrate experimentally that they share the same mechanism and that with a suitable transformation of the experimental data the nitrogen desorption isotherm and the mercury intrusion curve are in quantitative agreement. In view of this conclusion, we then demonstrate that the mercury intrusion curve can be used to replace the nitrogen desorption isotherm in the connectivity analysis, to obtain values for the connectivity parameters Z and L. Finally, an approach for estimating the contact angle in mercury intrusion is proposed. Comparison of the Mercury Intrusion and Nitrogen Sorption Data Figure 2. Experimental mercury intrusion curve for sample A.
adsorption stage of the experiment, prior to the desorption stage. Murray et al.8 recently presented modified analyses for the PSD and the connectivity for this experiment. Unlike the standard nitrogen sorption experiment, which can be carried out using automated equipment, an experiment involving bulk condensation requires careful manual intervention;8 this is undesirable in a routine characterization method. Consider Figure 3, which shows two schematics of a section of the pore network as it would appear at an intermediate stage during (i) nitrogen desorption and (ii) mercury intrusion. In this simple illustration, the pore network section contains only two pore sizes, which we will refer to as “large” and “small”. At the start of the nitrogen desorption experiment (i), all pores are full of liquid. As the relative pressure is reduced below the condensation pressure of the large pores, the nitrogen in those large pores which are connected to the surface either directly or via other large pores (termed “accessible” pores) can vaporize. We can visualize this process as vaporized nitrogen “penetrating” the large pores. However, the large pore on the left of the network is only connected to the surface via small pores, which still contain liquid nitrogen. Thus vaporized nitrogen cannot penetrate this pore, and the liquid nitrogen within it is trapped. Only when the relative pressure is reduced to the condensation pressure of the small pores, and vaporized nitrogen penetrates these pores, will the large pore become accessible. The process of mercury intrusion in (ii) is analogous, except that in this case the experiment begins at very low pressures at which point the network is full of mercury vapor. As the pressure is increased above the intrusion pressure of the large pores, those large pores which are accessible will be penetrated by liquid mercury. However, mercury vapor (and residual gas) will be trapped in the large pore on the left until the pressure is increased to the intrusion pressure of the small pores. Therefore, nitrogen desorption and mercury intrusion appear to share the same mechanism and should, in principle, be equivalent in the way they probe the pore structure. Percolation theory has been applied both implicitly and formally to modeling of mercury porosimetry by several workers.9-14 However, no direct comparison has been made (8) Murray, K. L.; Seaton, N. A.; Day, M. A. Langmuir 1999, 15, 6728. (9) Androutsopoulos, G. P.; Mann, R. Chem. Eng. Sci. 1979, 34, 1203. (10) Wall, G. C.; Brown, R. J. C. J. Colloid Interface Sci. 1981, 82, 141. (11) Larson, R. G.; Morrow, N. R. Powder Technol. 1981, 30, 123. (12) Chatzis, I.; Dullien, F. A. L. Int. Chem. Eng. 1985, 25, 47.
In physical terms, mercury intrusion is a penetration process, whereas nitrogen desorption is a phase equilibrium process. In effect, the mercury intrusion curve measures the accessibility of the pore network to a meniscus of a particular size. In contrast, nitrogen desorption measures the fraction of the pore network within which a stable meniscus of a given size can form. The two processes can be compared directly only if the experimental data from both are presented in the same terms. One way to achieve this is to transform the mercury intrusion data into the coordinates of sorption isotherms, as described below. The PSD is obtained from nitrogen adsorption data using the method of Barrett et al.,1 termed the “BJH” analysis. In this analysis, the adsorption isotherm gives the number of moles adsorbed in each pressure interval, Ni, from which the incremental pore volume against pore size data, i.e., the PSD, are calculated. Nitrogen adsorbs in a film on the pore walls at pressures below the condensation pressure and thus each data point on the sorption isotherms may contain contributions from capillary condensation in pores which are above their condensation pressure and also from adsorption on the walls of pores which are below their condensation pressure. The calculations in the BJH analysis take into account these two contributions to adsorption in order to extract the PSD. The intrusion curve measures the volume of pores which can be penetrated at a series of increasing pressures. In contrast to nitrogen sorption, mercury enters a pore only when the intrusion pressure of that pore is attained, so there is no adsorbed film to consider. If the pore network were perfectly connected, so that each pore had direct access to the mercury, the intrusion curve would be simply a cumulative PSD. This would, in principle, be equal to the output of the BJH analysis of the nitrogen adsorption isotherm. For such a solid, there would be no connectivity effect in the nitrogen sorption experiment and the adsorption and desorption isotherms would be coincident. However, substantial sorption hysteresis is a characteristic feature of mesoporous solids (see Figure 1), indicating that connectivity has an effect on nitrogen desorption. [In addition to the hysteresis due to finite connectivity, there is also a “single-pore” contribution to the hysteresis, due to a delay in capillary condensation during the adsorption process. Measurements on MCM-41 materials (see, e.g., ref 15), where the pores are unconnected, show that this (13) Tsakiroglou, C. D.; Payatakes, A. C. J. Colloid Interface Sci. 1990, 137, 315. (14) Ioannidis, M. A.; Chatzis, I. J. Colloid Interface Sci. 1993, 161, 278. (15) Ravikovitch, P. I.; Wei, D.; Chueh, W. T.; Haller, G. L.; Neimark, A. V. J. Phys. Chem. B 1997, 101, 3671.
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Figure 3. Schematic of an intermediate stage during (i) nitrogen desorption and (ii) mercury intrusion in a pore network, where dark shading represents liquid and light shading represents vapor.
contribution is much smaller than that found in mesoporous solids with interconnected pores, such as sample A.] If nitrogen desorption and mercury intrusion shared the same mechanism, the mercury intrusion curve would in fact be equivalent to a “PSD” (reflecting both the true PSD and the connectivity effect) calculated from the nitrogen desorption isotherm. Therefore the calculation of the mercury intrusion curve in sorption isotherm coordinates is essentially the reverse of the BJH analysis. In other words, the mercury intrusion curve provides the incremental pore volume against pore size data, from which we calculate the equivalent number of moles of nitrogen which would be adsorbed at each pressure on an equivalent nitrogen desorption isotherm. We call this representation of the mercury intrusion data the “mercury desorption isotherm”, although of course it does not refer to a real mercury sorption experiment. The procedure for transforming the mercury data is described below, assuming in this illustration a cylindrical pore shape. For mercury porosimetry, the relationship between intrusion pressure (pHg) and pore radius (r) is given by the Young-Laplace equation
r)
-2γHg cos θ pHg
-2γNVl RT ln (P/P0)
t ) 3.54
(
-5 ln(p/p0)
)
1/3
(3)
where t is in Angstroms. The following values are substituted for the constants in equations (1) and (2): θ ) 140°,17 γHg ) 0.485 N m-1, γN ) 8.84 × 10-3 Nm-1, Vl ) 3.468 × 10-5 m3 mol-1,1 R ) 8.314 J mol-1 K-1, T ) 77.4 K. Equations 1-3 are then combined to give
(
-5 -9.53 + 0.354 ln(p/p0) ln(p/p0)
)
1/3
-
7.43 × 109 ) 0 (4) pHg
where pHg is in N m-2. This equation gives the relationship between the real pressure measured during the mercury intrusion experiment and the equivalent relative pressure p/p0 of the mercury desorption isotherm. Figure 2 shows the mercury intrusion curve for sample A. The incremental volume of mercury intruded in each interval i in the mercury intrusion curve, VHg,i is converted to an equivalent number of moles of nitrogen, Ne,i, according to
(1) Ne,i )
where γHg is the surface tension of mercury and θ is the contact angle between mercury and the solid surface. The relationship between pressure and pore radius for nitrogen sorption is given by the Kelvin equation
r)
using the empirical Halsey equation for nitrogen16
(2)
where p is the system pressure, p0 is the saturation vapor pressure of nitrogen, γΝ and Vl are the surface tension and molar volume, respectively, of liquid nitrogen, R is the gas constant, T is the temperature at which the isotherm is measured (77.4 K, in our case, for the usual experiment carried out at atmospheric pressure), and t is the thickness of the adsorbed film just prior to capillary condensation in the pore. In this work, t was determined
VHg,i Vl
(5)
where Vl is the molar volume of liquid nitrogen. In this calculation, the intervals are counted from the highpressure end of the mercury desorption isotherm, corresponding to the low-pressure end of the experimental mercury intrusion curve. Because of the presence of an adsorbed film on the pore walls prior to capillary condensation (and after vaporization), Ne,i is not yet the amount adsorbed in the mercury desorption isotherm. The thickness of the adsorbed films at the beginning of each pressure interval, calculated using the Halsey eq 3, is denoted by ti. In the first interval, Ndes,1 is given by (16) Halsey, G. J. Chem. Phys, 1948, 16 (10), 931. (17) Gregg, S. J.; Sing, K. S. W. Adsorption, surface area and porosity; Academic Press: New York, 1982.
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(
Ndes,1 ) Ne,1
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)
(r1 - t2)2 r12
(6)
where the term in brackets is the fraction of the total volume of the pores in this size interval which would actually empty in the first pressure interval and r1 is the radius of these pores. The calculation for subsequent intervals is more complicated because the adsorbed films in pores that have already emptied must be taken into account. The amount of adsorption on the pore walls decreases as the pressure is reduced, and this desorption must be added to the amount desorbed from newly opened pores in these pressure intervals. The general calculation for each interval i is given by
(
Ndes,i ) Ne,i
) ((
(ri - ti+1)2 ri2 i-1
Figure 4. Experimental adsorption (open circles) and desorption (solid circles) isotherms and mercury desorption isotherm (squares) for sample A.
+
Ne,j ∑ j)1
)
(rj - ti+1)2 - (rj - ti)2 rj2
(7)
After the volume desorbed in the last interval, n, has been calculated, the volume of the adsorbed film remaining in all pores at the end of that interval, Ndes,n+1, is given by n
Ndes,n+1 )
Ne,i ∑ i)1
(ri2 - (ri - tn+1)2) ri2
(8)
(This is the volume which would be desorbed between the last pressure point and p ) 0.) The mercury desorption isotherm is given by n+1
NHg,i )
Ndes,j ∑ j)i
(9)
Comparison between the Mechanisms of Nitrogen Desorption and Mercury Intrusion Figure 4 shows the conventional adsorption and desorption isotherms and the mercury desorption isotherm for sample A. There is very good agreement between the mercury desorption and the nitrogen desorption isotherms in the vicinity of the desorption knee. This supports the proposition that the two processes share the same mechanism. The agreement in the low-pressure region, which is less important from the point of view of the connectivity analysis because it is well beyond the percolation threshold, is somewhat poorer. The likely reason for this is discussed later. There is a small disparity between the maximum amount adsorbed on the mercury desorption isotherm and that on the conventional sorption isotherms in Figure 4, indicating that the total pore volume is smaller in the former case. A similar disparity was found in another six samples studied, to a varying degree but always in the same direction as that for sample A. At the maximum pressure used for the intrusion experiments, mercury cannot penetrate pores smaller than about 36 Å, indicating that the presence of many pores smaller than this size could perhaps account for the difference in pore volumes. However, a t-plot analysis17 of the adsorption isotherm did not detect significant microporosity in any of the
Figure 5. Experimental adsorption (open circles) and desorption (solid circles) isotherms and adjusted, mercury desorption isotherm (squares) for sample A.
samples studied indicating that this explanation is unlikely to be correct. Another possible explanation for the disparity in volumes is that the presence of a small number of micropores, occupying a very small volume themselves, might block the entry of mercury into some sections of the pore network. This could cause a portion of the pore network to be inaccessible to mercury, while accessible to condensing nitrogen. If the inaccessible pores had the same PSD as the rest of the network, this inaccessible volume could be accounted for by adjusting each data point on the mercury isotherm as follows
( )
N/Hg,i ) NHg,i
Nmax NHg,1
(10)
where N/Hg,i is the adjusted amount adsorbed on the mercury isotherm, Nmax is the maximum amount adsorbed on the nitrogen isotherm and, as before, NHg,1 is the maximum amount adsorbed in the calculated mercury isotherm. While there is no direct evidence for this conjecture, eq 10 gives the mercury intrusion and nitrogen desorption processes the same total pore volume, enabling a direct comparison to be made between the two processes. Figure 5 shows the conventional nitrogen adsorption and desorption isotherms and (using eq 10) the adjusted mercury desorption isotherm for sample A. The agreement
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Table 1. Comparison between the Connectivity Parameters Obtained Using Nitrogen Desorption and Mercury Intrusion Data, for Various Samples nitrogen data
Mercury data
sample
material
mean pore size (Å)
ZN
LN
ZHg
LHg
deviation (%)a
A Bb Cb D E F G
alumina mixed oxide on alumina silica silica silica-alumina alumina alumina
145 344 1206 296 142 138 122
7.3 7.6 22 7.2 4.9 4.6 6.1
55 101 62 71 120 50 40
7.5 7.8 20 6.9 4.0 5.5 8.4
37 90 70 81 101 32 28
2.7 2.6 -9.1 -4.2 -18 20 38
a
[(ZHg - ZN)/ZN] × 100. b Samples with some VL pores, which use data from the BC experiment.
between the nitrogen desorption and the mercury intrusion curves is excellent in the region of p/p0 ) 0.8-1.0. The small disparity between the curves at low pressures can be accounted for by small errors due to the assumptions used in eqs 2 and 3, as follows. The Kelvin equation, eq 2, assumes the following: (i) the molar volume of the condensed phase and the surface tension within the pores are equal to those of bulk liquid adsorptive; (ii) there is no interaction between the adsorbed films on opposing pore walls. Both of these assumptions become less accurate as the pores become smaller. Further, the Halsey equation, eq 3, is based on a standard nitrogen adsorption isotherm measured on a nonporous solid. However, in practice the degree of adsorption in the adsorbed film is likely to be sample-dependent. Therefore calculation of the film thickness is an approximation, which will cause greater inaccuracies for smaller pores where the contribution of the adsorbed film to the calculation of pore size is greater. From the discussion above, the assumptions in the calculations of adsorption will be less valid for smaller pores. Any errors arising from these assumptions will be more apparent at lower pressures on the mercury desorption isotherm, but should not, therefore, significantly affect the position of the desorption knee (which occurs at higher pressures). The most important assumption in the Young-Laplace equation, eq 1, is the value of the contact angle. Uncertainty arising from this parameter will apply to the entire pore size range and any error in its value would result in the mercury isotherm being shifted to lower or higher pressures. The good agreement seen in the vicinity of the desorption knee in Figure 4 indicates that 140° is a good estimate for the contact angle in sample A. Use of Mercury Intrusion Data in the Connectivity Analysis The connectivity analysis of Seaton and co-workers,4-6 which uses the desorption isotherm, was applied to seven different mesoporous samples, including silicas, aluminas, a silica-alumina, and a mixed oxide on an alumina support, to obtain values for the connectivity parameters Z and L. The analysis was then repeated using the adjusted mercury desorption isotherm (using the correction of eq 10) in place of the nitrogen desorption isotherm. Table 1 shows a comparison between the parameters obtained from the connectivity analysis using both nitrogen desorption and mercury intrusion data for these seven samples. Samples B and C have some “very large” (VL) pores (defined as >1000 Å in width), which remain filled with vapor at the maximum pressure on the adsorption isotherm. These vapor-filled pores interfere with the percolation threshold on the desorption isotherm, causing the knee to be ill-defined, and the isotherm cannot be
used in the connectivity analysis. These samples were analyzed with the modified connectivity analysis of Murray et al.,8 using data from the bulk condensation (BC) experimental method of Aukett.7 At the start of the desorption experiment (i.e., at the end of the adsorption experiment) the relative pressure is increased to a value slightly greater than unity. This causes nitrogen to condense in the bulk of the sample container and ensures that all of the pores are full of liquid nitrogen at the start of the desorption experiment. The pressure is then reduced to slightly below unity, the bulk condensed nitrogen vaporizes, and the primary desorption isotherm, or the BC desorption isotherm, can then be measured. The BC desorption isotherm, which has a well-defined knee, is then used in the BC connectivity analysis. In Table 1, the nitrogen data results for samples B and C were obtained using the BC connectivity analysis. Table 1 shows very good agreement in both Z and L values for samples A-C and D, which have the largest mean pore sizes. [For sample C, which has many VL pores, the estimates of Z seem unphysically high, suggesting that our “intersecting capillaries” pore structure model might not be appropriate for this sample.] This good agreement indicates that the assumption of a small portion of the pore network being inaccessible to mercury, according to eq 10, is reasonable, at least in terms of the connectivity analysis. For samples E-G, the agreement is not as good. In the cases of samples F and G, this may partly reflect their relatively small network sizes, which means that the desorption knee is less well defined and Z values cannot be assigned as accurately. In addition, the increased inaccuracy of the Kelvin equation for smaller pores would have a stronger influence as the proportion of small pores increases. However, these inaccuracies are unlikely to have a significant enough impact on the position of the percolation threshold to cause the degree of deviation of ZHg from ZN seen in the three samples. A more likely explanation for the deviation is a poor estimate of the contact angle in eq 1. This possibility is investigated in the next section. Estimation of Contact Angle The value of θ measured on a plane surface has been observed to vary with both the amount of surface roughness and the type of material (ref. 18 and references therein). Therefore it is reasonable to assume that the value of θ inside a pore will also vary from one type of mesoporous solid to another. From eq 1, the effect of reducing θ from 140° is to reduce the values of r corresponding to a given pressure. The values of p/p0 determined from eq 5 will also be lower as condensation pressure decreases with pore size. Thus the mercury (18) Kloubek, J. Powder Technol. 1981, 29, 63.
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with values of θ of between 134° and 150°. These results suggest that the appropriate value of θ varies from one mesoporous solid to another. Clearly, caution needs to be exercised in inferring fundamental information from this indirect observation of the contact angle. The contact angle presumably varies from pore to pore, even within a particular sample, so that these values should be regarded as effective values for the solid as a whole, rather than true, local contact angles. Finally, although one might expect the contact angle to depend to some extent on the chemical composition of the solid, such behavior is not evident in Table 1; alumina samples show both the largest and (almost) the smallest deviation between ZHg and ZN.
Figure 6. Experimental adsorption (open circles) and desorption (solid circles) isotherms and adjusted, mercury desorption isotherm using θ ) 140° (squares) and θ ) 134° (diamonds) for sample G.
desorption isotherm will be shifted to lower pressures relative to the isotherm obtained for θ ) 140°. This in turn will result in a lower value of ZHg when the new isotherm is used in the connectivity analysis. A large deviation between the two estimates of the coordination number, ZHg and ZN, therefore suggests a substantial error in the chosen contact angle of 140°, and therefore that an estimate of the true contact angle for that sample might be obtained by fitting the mercury desorption isotherm to the nitrogen desorption isotherm. From Table 1, Sample G has the largest deviation between ZHg and ZN. Figure 6 shows a comparison between the original and optimized (using θ ) 134°) mercury desorption isotherms, and the experimental nitrogen adsorption and desorption isotherms. The agreement between the mercury and nitrogen isotherms is dramatically improved when θ is reduced from 140° to the optimal value of 134°. Similar improvements were obtained for the other samples
Conclusions Good agreement between nitrogen desorption isotherms and transformed mercury intrusion curves in the region of the desorption knee confirms that nitrogen desorption and mercury intrusion share the same mechanism. For samples with a sufficiently large mean pore size, the mercury intrusion curve can be used in place of the desorption or BC desorption isotherm, as appropriate, in both the conventional and BC connectivity analyses. (Our results, in Table 1, suggest that acceptable accuracy is obtained for mean pore sizes greater than about 150 Å.) The use of mercury intrusion data is particularly attractive as an alternative to the BC desorption experiment, which is more complex than the standard desorption experiment. The agreement between ZHg and ZN worsens as the mean pore size decreases. This might be caused by assuming an incorrect value of the contact angle, θ. A small alteration in the contact angle can dramatically improve the agreement between ZHg and ZN. A comparison of the mercury desorption isotherm with the experimental nitrogen desorption isotherm offers a route to the estimation of θ. Acknowledgment. The authors thank ICI for use of their experimental equipment. K.L.M. thanks Merck and Co. for their support through their doctoral study program. LA990250X
