Langmuir 1993,9, 2576-2582
2576
Sorption Hysteresis as a Probe of Pore Structure Hailing Liu, Lin Zhang, and Nigel A. Seaton' Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, United Kingdom Received September 30,1992. In Final Form: January 12,1993 The use of nitrogen adsorption and desorption measurementsas a probe of the structure of porous solids is discussed. The extent of sorption hysteresis reflects the connectivity of the pore network. An analysis based on percolation theory leads to an estimate of a measure of the connectivity, the mean coordination number of the network. This analysis uses only measurements that can be made with standard, commercially-available equipment.
Introduction Nitrogen sorption measurements are widely used in the characterization of porous solids, in both academic and industrial environments. The measurements can be carried out with commercially available equipment and are relatively easy and cheap to perform. Standard analyses exist for the determination of the surface area and the pore size distribution (PSD) from the adsorption isotherm. These quantities are important descriptors of the pore structure and give a rough indiction of some aspects of the performance of the solid as an adsorbent or as a catalyst support. For example, the equilibrium amount adsorbed on an adsorbent in an adsorption separation depends on the PSD. However, the surface area and pore size distribution alone do not give a sufficiently complete picture of the pore structure if one is concerned with diffusion, in predicting the adsorption rate for example; for this purpose, a measure of the connectivity (the way the pores are connected together) is required. The effective diffusion coefficient of a porous solid can be calculated using the effective medium theory of Kirkpatrick.1 This theory was originally derived for electrical conduction in a network of resistors; Burganos and Sotirchos2 adapted it to the problem of diffusion in pore networks. It is known to be accurate except for networks that are very poorly connected (i.e., are close to the percolation threshold'). The inputs to the calculation are the PSD and the mean coordination number of the pore network, 2. Figure 1gives some results of effective medium theory calculations showing the effect of varying 2 on the effective diffusion coefficient of a network of cylindrical pores, using an experimental pore size distribution measured on an alumina sample. The effective diffusion coefficient increases with 2,with the rate of increase particularly rapid for small values of 2. This can be understood in physical terms as follows. At low values of 2,the rate of diffusion within the solid is limited by the smaller pores. Although the larger pores present in the network offer relatively little resistance to diffusion, because of the low connectivity, molecules must from time to time pass through the smaller pores. On the other hand, if Z is large, there are more possible routes between any two points in the network and the smaller pores can be bypassed. In this case, diffusion is dominated by the larger
* To whom correspondence should be addressed.
(1) Kirkpatrick, S. Reu. Mod. Phys. 1973,45, 574. (2) Burganos, V. N.; Sotirchos, S. V. AIChE J. 1987,37, 1678. (3) Sahimi, M.; Gavalaa, G. R.; Tsotais, T. T. Chem. Eng. Sci. 1990,
45,1443.
0.3
+
10
0
20
Z
Figure 1. Effective diffusion coefficient (arbitrary units) as a function of the mean coordination number 2 for a network of cylindrical bonds.
I
Primary Desorption
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Figure 2. Nitrogen sorption isothermsmeasured on an alumina sample. PO is the saturation pressure.
pores. This paper deals with the determination of 2 from nitrogen sorption measurements.
Mechanisms of Sorption Hysteresis Figure 2 shows the adsorption isotherm and two desorption isotherms measured on an alumina sample. The adsorption isotherm is obtained by progressively increasing the pressure of nitrogen from a very small value up to a value close to the saturation pressure. As the pressure is increased, multilayer adsorption occurs on the walls of each of the pores, followed by capillary condensation, in which a liquidlike phase is formed. The pressure a t which this occurs (the condensation pressure) is an increasing function of the pore size. The relationship between the
0743-746319312409-2576$04.00/0 0 1993 American Chemical Society
Sorption Hysteresis as a Probe of Pore Structure
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coexistence pressure
Langmuir, Vol. 9, No. 10,1993 2577
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Figure 3. Schematic sorption isotherm for a single pore.
condensation pressure and the pore size, which for large pores is given by the Kelvin equation, is the basis of several methods for the determination of the PSD.&I1 If, after completion of the adsorption process, the pressure is then progressively reduced from ita maximum value, the primary desorption isotherm (also $own as the desorption boundary curve) is obtained. For the sample of Figure 2, desorption is at first gradual but suddenly becomes much more rapid at a relative pressure of about 0.6 (the “knee” of the isotherm) and the desorption boundary curve converges to the adsorption isotherm. Hysteresis of this type (corresponding to the IUPAC types H1 and H2 hysteresis loops) is often observed in measurements of nitrogen sorption in mesoporous solids. If, instead of completing the adsorption process prior to desorption, the desorption experiment is begun at a pressure below the maximum value, a secondary desorption isotherm (also known as a desorption scanning curve) is obtained. A family of secondarydesorptionisotherms can be measured, with each isotherm corresponding to a different starting pressure. The isotherm has a less pronounced knee than the boundary curve and results in a smaller hysteresis loop. It is widely recognized that two mechanisms contribute to the sorption hysteresis that is observed in experimental measurements on mesoporous solids. The fiist mechanism is purely thermodynamic in origin. Because this mechanism does not rely upon the interconnectivityof the pores, it is often called the “single-pore” mechanism. As in a bulk phase transition, metastable phases can exist. Figure 3 is a schematic sorption isotherm for a single pore. The branches of the isotherm are labeled “gas” and “liquid”to emphasize the analogy with the bulk vapor-liquid transition; in the case of adsorption, the gas phase includes the adsorbed film on the pore wall. The pressure at which capillary condensation is thermodynamically favorable is the coexistence pressure, P.However, it is thermodynamically feasible for a metastable gas phase to persist beyond PC during the adsorption process. Similarly, a metastable liquid phase may persist below P during (4) Barrett, E. P.; Joyner, L. G.; Halenda, P. H. J. Am. Chem. SOC. 1951, 73, 373. (5) Cranaton, R. W.; Inkley, F. A. Adv. Catal. 1967,5, 143. (6) Brunauer, S.;Mikhail, R. Sh.;Bodor, E. E. J. ColloidZnterfaceSci. 1967,24,451. (7) Broekhoff, J. C. P.; de Boer, J. H. J. Catal. 1967,9,8; 1967,9,15. (8) Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. Carbon 1989,27,853. (9) Jeaaop, C. A.; Riddiford, S. M.;Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. In Characterization of Porous Solids II; Rodriguez-Reinoso, F., Rouqerol, J., Sing, K. S. W., E&.; Elaevier: Amsterdam, 1991; p 123. (10) Laatoskie, C. Paper presented in Kazimierz Dolny. (11) Olivier, J. Paper presented in Kazimierz Dolny.
desorption. The limit of metastability in each case is the spinodal pressure for that branch. The region indicated by the dashed line in Figure 3 is intrinsically unstable and therefore not accessible experimentally. Although metastability is thermodynamically feasible on either branch, it is reasonable to assume that in practice it occurs only on the gas branch (i.e., during the adsorption process) as the nucleation of the liquid phase involves the formation of a meniscus which in turn depends on density fluctuations. On the other hand, vaporization can occur via a receding meniscus so there is likely to be little metastability on the liquid branch. The second hysteresis mechanism is a result of the interconnectivity of the pores. During the desorption process, nitrogen can vaporize only if the pore is in direct contact with the vapor phase, i.e., if it has a meniscus. If the pore is not at the surface of the adsorbent particle, nitrogen can only vaporize if an adjacent pore contains vapor. As the condensation pressure increases with pore size, small liquid-filled pores can block the vaporization of nitrogen from adjacent larger pores. Indeed, during primary desorption, nitrogen can vaporize from a pore in the interior of the adsorbent particle only if (i) the applied pressure is below the condensation pressure for a pore of that size and (ii) there is a path from that pore to the surface of the adsorbent composed only of larger pores. The effect of this mechanism on secondary desorption is more c o m p l i c a ~this , will be addressed in the next section. The relationship between the morphology of the pore network and the extent of the hysteresis has been studied by a number of researchers.1229 The relative importance of the single-pore and network mechanisms is difficult to assess, and certainly varies from solid to solid. An upper bound on the extent of the singlepore effect can be obtained by assumingthat condensation occurs at the gas-branch spinodal pressure.w2 Ball and Evanss2compared network and single-pore effects, using a density functional theory to calculate the single-pore isotherm and representing the network by a Bethe lattice. Their results suggest that, provided the pore size distribution is not too narrow, the network effect is likely to (12) Everett, D. H. In The Solid-Gas Interface; Flood, E. A,,Ed.; Edward Arnold Ltd.: London, 1967; Vol. 2, p 1066. (13) Doe, P. H.; Haynes, J. M.In Characterization of Porous Solide, Gregg, S . J., Sing, K. S. W., Eda.; Society of Chemical Induetry: London, 1979, p 263. (14) Wall,G. C.;Brown,R.J. C. J. ColloidInterface Sci. 1981,82,141. (15) Mason,G. J. Colloid Interface. Sci. 1982,80, 36. (18) Maaon, G. Proc. R. SOC.London 1983, A390,47. (17) Mason, G. Proc. R. SOC.London 1988, A415,453. (18) Neimark, A. V. Colloid J. 1984,46, 813. (19) Neimark, A. V. Colloid J. 1984,415,1004.
(20) Neimark,A.V.InCharacterizationofPorolurSolidsII:RodrigwzReinoso, F., Rouqerol, J., Sing, K. S. W., Eds.; Elaevier: Amsterdam,
1991; p 67. (21) Parlar, M.; Yortsos, Y. C. J. ColloidInterface Sci. 1988,124,162. (22) Parlar, M.; Yortsos, Y. C. J. Colloid Interface Sci. 1989,132,425. (23) Mayagoitia, V.; Rojas, F.; Kornhaueer, I. J. Chem. SOC.,Faraday Trans. 1 1988,84,786; 1988,84,801. (24) Mayagoitia, V.; C N Z ,M. J.; Rojaa, F. J. Chem. SOC.,Faraday Trans. 1 1989,85,2071; 1989,86,2079. (26)Mayagoitia, V.In CharacterizationofPorolur So1idsIl;RodriiezReiioso, F., Rouqerol, J., S i , K. 5.W., Me.; Elaevier: Amsterdam, 1991; p 51. (20) Zhdanov, V. P.; Fenelonov, V. B.; Efremov, D. K. J. Colloid Interface Sci. 1987,120,218. (27) Efremov, D. K.;Fenelonov, V. B. In Characterization of Porous Solids ZI; Rodriguez-Reiioso, F., Rouqerol, J., Sing, K. 5. W., Eda.; Elaevier: Amsterdam, 1991; p 116. (28)Zgrablich, G.; Mendioroz, S.; Dam, L.; Pajmes, J.; Mayagoita, V.; Rojaa, F.; Conner, W. C. Langmuir 1991, 7,779. (29)Mann, R.; Thomean,G. In Adsorption: Science and Technology; R o d r i i e s , A. E., et al., Eda.; Kluwer Academic: Boston, 198% p 63. (30) Seam,W. F.; Cole, M.W. Phys. Rev. B. 1975,11, 1086. (31) Ball, P. C.; EVU, R. EUrOphyS. Lett. 1987,4,715. (32) Ball, P. C.; Evana, R. Langmuir 1989,5, 714.
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2578 Langmuir, Vol. 9,No. 10,1993
nitrogen is able to vaporize from those pores that are on the surface of the microparticles, but not from pores in the interior of the microparticle. Once nitrogen has vaporized from a pore on the surface, however, it can also do so from the pores that are adjacent to that pore I (provided the applied pressure is lower than the condensation pressure for a pore of that size). As the pressure I continues to decrease, clusters of vapor-filled pores grow from the surface of the microparticle. When a critical number of pores are below their condensation pressures (i.e., they contain either vapor or metastable liquid), a percolation transition occurs in which a cluster of vaporfilled pores that spans the microparticle is formed. The percolation transition corresponds to a sharp increase in Figure 4. Two-dimensional analog of a diluted simple-cubic the amount desorbed, i.e., to the knee of the primary lattice. desorption isotherm. Once percolation has occurred the spanning, or “percolating”, cluster grows rapidly until dominate. In the development of our analysis in the next essentially all the pores have access to the vapor phase via two sections, we assume that the single-pore contribution the percolating cluster. At this point, the hysteresis loop to the hysteresis is negligible. closes. We have simulated the desorption process using the Simulation of the Desorption Process pore network model described a b o ~ e . ~ The * ~bonds ~ in If simulationsare to be used as the basis of a quantitative the simple-cubic lattice have two states: “unoccupied”, analysis of experimental adsorption data, they must be which correspondsto a pore that is above its condensation based on a model of the pore network that has the essential pressure, i.e., containing a stable liquid phase, and attributes of the real pore network. The model must allow “occupied”,which corresponds to a pore that is below its for both the geometry of the pores (via an assumed pore condensation pressure, Le., containing either vapor or shape and a PSD) and the connectivity of the network metastable liquid. At the beginning of a simulation of (via an adjustable mean coordinator number). The pore primary desorption, all the bonds are unoccupied. The network must be three-dimensional, like the real one, and desorption process is simulated by choosing bonds at should have reconnections (i.e., there should be more than random and changing their state from unoccupied to one route between any pair of pores). The Bethe lattice, occupied. The fraction of occupied bonds, X,increases which has been used by some researchers,15~16~17~1g~21~27~32 from zero to unity during the course of the simulation, does not meet the latter requirement. Finally, the model corresponding in experimental terms to the reduction of network should reflect the bidisperse nature of most the pressure from near saturation to close to zero. The industrial adsorbents and catalyst supports, which are occupied bonds that belong to clusters of bonds that are aggregatesof smaller microparticles, perhaps held together in contact with the surface of the lattice are said to be by a binder. Only the pores in the microparticles can be “accessible”, and correspond to pores that contain vapor. effectivelyprobed by pitrogen sorption as most of the pores The fraction of accessible bonds (Le., the number of formed by the interstices between the microparticles are accessible bonds divided by the total number of bonds) is too large for capillary condensation to occur in them. We called the “accessibility”,XA.The accessibility contains have used3335 a simple cubic lattice, the two-dimensional a contribution from the clusters of occupied bonds that analog of which is illustrated in Figure 4, to represent are in contact with the surface of the lattice (the “surface each of the microparticles, so that the adsorbent or catalyst clusters”) and, above the percolation threshold, from the support itself is represented by an array of such lattices. percolating cluster. The occupied bonds that do not belong In this model, each bond represents a pore and each node to clusters in contact with the surface correspond to pores represents a pore junction. A pore size distribution is containing metastable liquid. At each stage in the assigned to the bonds, and the mean coordination number, simulation, the accessible bonds are identified using a 2, is varied from its original value of six by “diluting” the variant of the widely-used algorithm of Hoshen and network, Le., by randomly deleting bonds until the desired Kopelman.36 The simulation is repeated many times using value of 2 is reached. The linear dimension of the network, different sequencesof random numbers. Each realization L,corresponds to the size of a typical microparticle in the of the lattice corresponds to a microparticle within the real adsorbent, expressed as a number of pore lengths. solid, and the average over many realizations of the lattice The role of the connectivity of the pore network in the corresponds to the adsorption behavior of the solid itself. desorption process is shown schematically in Figure 5 Figures 6 and 7 show the effect of the two parameters (again in two dimensions). We assume for the moment of the model, Z and L, on the accessibility of the lattice that all the pores within the microparticle contain liquid as a function of the fraction of occupied bonds. In both nitrogen at the end of the adsorption process. This is the figures, the close connection between the simulations and case for solids having hysteresis loops of the IUPAC types the real desorption process is emphasized by plotting the H1 and H2 (of which the samplewhose isotherms are shown as the fraction of pores containing liquid (either graphs in Figure 2 is an example),in which the adsorption isotherm versus the fraction of pores stable or metastable), 1- XA, is almost flat at high pressure, indicating that no concontaining stable liquid, 1 - X. Equivalently, 1- X* is densation occurs in this region. As the pressure is reduced the fraction of pores actually containing liquid during the from the maximum pressure reached during the adsorption desorption process, and 1- X is the fraction of pores that process, the liquid nitrogen in some of the pores becomes would contain liquid during the desorption process if metastable with respect to the vapor phase. Initially, vaporization could occur from all the pores when they reach their condensation pressure. On these axes, the (33) Seaton, N.A. Chem. Eng. Sei. 1991,46,1896.
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(34) Liu, H.;Zhang, L.; Seaton, N. A. Chem. Eng. Sei. 1992,47,4393. (36) Liu, H.;Zhang, L.; Seaton,N. A. J. Colloidlnterfaee Sei., in press.
(36) Hoshen, J.; Kolpemann, R. Phys. Reo. 1976,B14, 3438.
Langmuir, Vol. 9,No. 10,1993 2579
Sorption Hysteresis as a Probe of Pore Structure liquid-filledpore
/
nitrogen begins to vaporize from pores near the surface
percolation threshold; a spanning cluster is formed
all pores have access to vapor phase; hysteresis loop closed
Decreasing hessure b
Figure 5. Schematic illustration of primary desorption.
1 .xA
OM
05s
om
ou
om
on
ow
08(
ow
095
IW
-x Figure 6. Fraction of pores containing liquid nitrogen (1 XA) versus the fraction of pores containing stable liquid (1- X)for various valuea of L:L = 10 (solidsquares);L = 20 (open squares); L = 60 (solid circles);L = (open circles). 1
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Figure 7. Fraction of pores containing liquid nitrogen (1- XA) versus the fraction pores containing stable liquid (1- X)for 2 = 3 (squares) and Z = 6 (circles).
adsorption process is represented by the straight line XA = X so that Figures 6 and 7 are effectively simulated primary adsorption and desorption isotherms, plotted on unconventionalaxes. Figure 6showsthe effect of changing the size of the lattice on the desorption process. As the lattice size increases, the percolation transition becomes more pronounced and the knee of the primary desorption
isotherm becomes sharper. In an infinite system, the surface clusters are insignificant and no desorption occurs until the percolating cluster is formed at the percolation threshold. This behavior is approximated in Figure 6 by taking the results for L = 60 and ignoring the contribution due to the surface clusters. Figure 7 shows the effect of varying the mean coordination number, 2,by randomly deleting bonds from the lattice. As 2 is reduced, the number of routes between a bond in the interior of the lattice and the surface is reduced so that at any value of X the accessibility XAis less (and the extent of hysteresis greater) than for the undiluted lattice. In practice results for percolation on a diluted lattice can be obtained by scaling the results for the complete simple cubic lattice." At the beginning of the secondary desorption process, those pores that were below their condensation pressure at the end of the preceding adsorption experiment contain nitrogen vapor. The existence of such pores is indicated in Figure 2 by the fact that the adsorption isotherm is still increasing rapidly at the maximum pressure of the secondary desorption isotherm. During secondary desorption, nitrogen can vaporize into these pores as well as into pores connected to the surface of the microparticle. This is illustrated in Figure 8. Secondary desorption is simulated by designating a fraction of the bonds as occupied at the start of the simulation. These bonds, which we call "seed bonds", correspond to the vapor-filled pores present at the start of secondary desorption. The simulation proceeds as for the case of primary desorption except that clusters of occupied bonds containing one or more of the seed bonds are counted as contributing to the accessibility, XA, as well as the percolating and surface clusters. Figure 9 shows the effect of the fraction of occupied bonds at the start of the simulation, Xv,on the accessibility XA. A striking feature of these results is that even small values of Xv have a significant effect on XA. The hysteresis loop decreases rapidly with increasing XV and almost disappears for Xv 1 0.1. Determination of the Mean Coordination Number 2 and the Characteristic Length L The procedure for the determination of these quantities from the experimental sorption data can be summarized as follows:33~" (1)The PSD is obtained from the primary adsorption isotherm. (2)The accessibility XAis obtained as a function of X from the primary adsorption and desorption isotherms, using the PSD as an input. (3) 2 and L are calculated by fitting the simulation data for XA as a function of X (a subset of which is shown in Figures
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2580 Langmuir, Vol. 9, No. 10,1993
nitrogenixgins to vaporize from pores neighbouring seed pores and surface pores
start of secondary desorption
percolation threshold, spanning cluster formed
ail pores have access to vapor phase, hysteresis loop closed
Decreasing Pressure Figure 8. Schematic illustration of secondary desorption.
0
5
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Figure 9. Fraction of pores containing liquid nitrogen (1 - XA) versus the fraction of pores containing stable liquid (1- X)for various values of XV:XV = 0 (open circles);XV = 0.01 (solid circles);XV = 0.06 (open squares);XV = 0.1 (solid squares). 6 and 7) to the experimental accessibility data. These steps are now described in detail, using as an illustration the analysis of the sorption data shown in Figure 2, which was measured on an alumina sample. A number of methods exist for the determination of the PSD of mesoporous solids from the analysis of the primary adsorption isotherm.&l' All the methods use the phenomenon of capillary condensation as a probe of pore size, but they differ in the model used to describe adsorption in individual pores. In the methods of Barrett et al.? Cranston and I n k l e ~and , ~ Brunauer et the condensation pressure is related to the pore size by the Kelvin equation, and the adsorbed layer on the pore walls at pressures below the condensation pressure is considered to have the same thickness as on a nonporous solid at the same pressure. The method of Broekhoff and de Boer' also uses the Kelvin equation, but takes into account the increase in the thickness of the adsorbed layer prior to condensation due to the curvature of the liquid-vapor interface. These methods are suitable for solids with little microporosity. As capillary condensation does not occur in micropores, they are unsuitable for solids with substantial microporosity. In this case, the PSD analysis should be based on a statistical-mechanical model of nitrogen adsorption. Such models give a more accurate description of adsorption in micropores and small mesopores than the Kelvin equation-based models, while reducing to the Kelvin equation for large pores. Seaton et al.8 have developed a method based on a density
functional theory of nitrogen adsorption in porous solids (see also Jessop et alaB).More recently, methods based on improved versions of the density functional theory have been presented by Lastoskie et al.1° and 0livier.l' Whichever PSD method is used, a pore shape must be assumed. In practice, either cylindrical or slit-shaped pores are used. The most widely used PSD method is that of Barrett et al.? which is usually implemented with a cylindrical pore shape, as in the original derivation of the method. For some solids, such as carbons, the pores are more like slits than cylinders so an analysis based on slit-shaped pores is more appropriate; this is the geometry used in the statistical-mechanical methods of Seaton et ale8 and Lastoskie et al.lo It is worth pointing out that the calculated PSD depends strongly on the choice of pore shape. For example, in the limit of large pores the Kelvin equation indicates that nitrogen condenses in a cylindrical pore of diameter d at the same pressure as in a slit-shaped pore of width d/2. We have used3w6 both the Kelvin equation-based method of Barrett et aL4 and the statistical-mechanical method of Seaton et al.8 to calculate the PSD, using both slit and cylindrical geometries. In our illustration of the procedure, we use the method of Barrett et al. with cylindrical pores. Figure 10shows the PSD obtained from the primary adsorption isotherm of Figure 2. This PSD is the "pore volume distribution", u(d),Le., the probability density function for pore volume as a function of diameter. (Note that u(d) is normalized to the total pore volume rather than unity.) In the present model, it is the number of pores of a given size, rather than their volume, that is of interest. n(d), the probability density function for
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1.x
A
0.0 0.0
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01
0.3
0.4
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0.6
0.7
0.8
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1.0
PiPO
Figure 11. Illustration of the calculation of the ratio X A / X . pore number as a function of diameter, is related to u(d) by
44 n(d) = -
(r/4)d21 where 1is the pore length, whichwe assume to be the same for all pores. The next task is to transform the experimental primary adsorption and desorption isotherms to values of X A as a function of X so that they can be compared with the simulation data. At pressure P, the fraction of occupied bonds, X,is the fraction of pores that are below their condensation pressure, i.e., that contain vapor or metastable liquids. We have assumed that adsorption is an equilibrium process so X is also the fraction of proes that contain vapor at pressure P during the adsorption process. X* is the fraction of pores that contain vapor at pressure P during the desorption process. The PSDs of the pores that contribute to X and to XAare the same, so
vol of pores containg vapor at P during desorptn vol of pores containg vapor at P during adsorptn (2) Here the pressure dependence of XA and X is made explicit. It is straightforward to relate the pore volumes appearing in eq 2 to the primary adsorption and desorption isotherms.33 X*(P)/X(P)= EF/EG (3) where EF and EG are the lengths of the lines shown in Figure 11. X ( P ) is related to n(d) by
X ( P ) = [Jd:n(d) ddll[Kn(d) ddl
(4)
where d* is the size of the pores in which nitrogen condenses at pressure P. Equations 1, 3, and 4 and the PSD u ( d ) (shown in Figure 10) together convert the primary adsorption and desorption isotherms of Figure 2 to the accessibility data shown in Figure 12. (Note that it is not necessary to assign a value to the pore length 1 in eq 1 as this variable appears in both the numerator and the denominator of eq 4.) The simulation data are then fitted to the transformed experimental data by varying 2 and L, interpolating within the set of simulation data as required. The fit between the simulation and experimental data, yielding for this sample 2 = 4.9 and L = 120, is also shown in Figure 12. Of these two parameters, 2 is likely to be the more useful in most applications because of its
0.1
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Figure 12. Fit of the simulationresulta (line)to the experimental data (circles)for the accessibility XA. The parameter values are Z = 4.9 and L = 120. direct role in controlling the rate of diffusion. However,
L is also of some interest as the size of the pore network in a typical microparticle (in terms of the number of pores it contains) determines the circumstances under which diffusion and reaction in the microparticle can be described by a continuum model.37 Also, if the approximate physical size, A, of the microparticles is known, AIL is an estimate of the pore length.
Discussion The experimental inputs to the analysis method described here are the primary adsorption and desorption isotherms, both of which can be measured using standard, commercially available equipment. The determination of the pore size distribution, which forms part of this analysis, can be carried out using any one of a range of methods, including those that are generally provided with commercial equipment for nitrogen sorption measurements. The method is thus suitable for routine use in an industrial environment. In common with other characterization methods, the present method is based on some assumptions about the nature of the porous solid, and about the physical process that is being used to characterize it. In order to have confidence in the analysis, it is necessary to consider the effect of these assumptions on the results. The two principle assumptions here are (i) the choice of pore shape and (ii) the neglecting of the single-pore contribution to hysteresis. We have considered in detail elsewhere33p%the effect of the assumed pore shape on the calculated value of the mean coordination number 2. Of the two simple shapes we have considered-slits and cylinders-the latter geometry always gives a larger value of 2.% As these two shapes represent extremes of low and high curvature, at least for pores of simple shapes with a constant crosssection, this suggeststhe followinginformal bound Z(s1its) < 2 < Z(cy1inders). In practice, it is often not the connectivity itself but rather the effective diffusion coefficient that is of interest. We have predicted the effective diffusion coefficients for a number of samples,% for both slit-shaped and cylindrical pores, using the effective medium theory of Kirkpatrick.' We find that although the effectivediffusion coefficientspredicted using the two pore shapes are significantly different, they are (37) Zhang, L.; Seaton, N. A. Manuscript in preparation.
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0.30 0.35
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0.60 0.65
0.70 0.75
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PIP0
Figure 13. Comparison of predicted and experimental desorption isotherms. The experimental data are shown as pointa and the predictions as thin lines: desorption boundary curve (diamonds);scanning curve l (XV= 0.0028) (circles);scanning curve 2 (XV= 0.0124) (triangles); scanning curve 3 (XV= 0.0639) (squares). The experimental adsorption isotherm is shown as a thick line.
correlated so that if a set of samples is ordered by diffusion coefficient,the order is similar regardless of the pore shape used. We have attempted to assess the relative contributions of the single-pore and network hysteresis mechanisms by predicting secondary desorption isotherms using simulation results for secondary desorption (some of which are presented in Figure 9).35 The simulation results are transformed to the amount adsorbed as a function of pressure using the PSD,and the values of ZandL obtained
from the analysis of the primary desorption isotherm. The fraction of vapor-filled pores at the start of the secondary desorption process, Xv,is obtained by integrating the PSD between the size of pore in which nitrogen condenses at the starting pressure of the secondaryadsorption isotherm and infinity. The comparison between the predictions and experiment is shown in Figure 13. The first two secondary desorption isotherms (counting from the highpressure end) are quite well predicted by the simulation results, although the prediction becomes poorer as the pressure decreases. However, the third experimental isotherm is well above the simulation isotherm. We have also found significant hysteresis in experimental secondary desorption isotherms at lower pressures (although for clarity these are not shown in Figure 13), while the corresponding simulated isotherms show almost no hysteresis. This comparison indicates that the simulations fail to give a good account of the desorption process when a large number of the pores have access to the vapor phase, Le., toward the low-pressure end of the isotherm, or when XV is large (as it is for secondary desorption isotherms starting at low pressure). The single-pore effect (which is ignored in the simulations) should therefore be taken into account in a full description of sorption hysteresis. However, because the single-pore effect is masked by the network effect when few pores have access to the vapor phase, the use of the network model to determine Z from the primary desorption isotherm appears to be justified.
Acknowledgment. The authors would like to thank
P.Aukett, C. Jessop, D.MacGowan, and S.M. Riddiford of BP Research for useful discussions and for providing samples in support of this work.
