Scanning Studies on Capillary Condensation and Evaporation of Nitrogen Part 2.-Analysis of Ascending and Descending Scanning Curves within B-Type Hysteresis Loops BY JOHAN C . P. BROEKHOFF* AND WIM P. VAN BEEK Unilever Research, Vlaardingen, The Netherlands Received 17th February, 1978 The mechanism of emptying and filling of pore domains along the adsorption as well as the desorption boundary curve of the hysteresis loop was studied by the determination of a V-S curve derived from the tangents to primary scanning curves immediately after a pressure reversal at the boundaries of the hysteresis region. In all three cases studied, the V-S curve for the desorption process exhibits a discontinuity aroundp,/po = 0.5, pointing to a sudden breakdown of the capillary- condensed state, which is not directly related to the detailed domain properties of the system.This breakdown of the capillary-condensed state invalidates in a number of cases the usual procedures for obtaining pore size distributions. The filling of slit-shaped pores was found to proceed most probably by additional sorption of one to two nitrogen molecules between adjacent adsorbed layers. Hysteresis of vapour adsorption in rigid porous systems is not commonly observed below a certain typical value of the relative vapour pressure. The magnitude of this " critical " relative pressure is apparently related to the physical properties of the adsorbate rather than to the actual size distribution of the porous system. Several explanations for this curious phenomenon have been put forward. Cohan has assumed that capillary condensation in tubular pores occurs by Kelvin instability of the adsorbed layer inside the pores, while capillary evaporation results from Kelvin instability of the meniscus at the pore entrance. If it is assumed that the radius of curvature of the surface of the adsorbed layer during adsorption is equal to the pore radius minus the adsorbed layer thickness, while during desorption the radius of the curvature of the meniscus equals that of the pore, then it follows that capillary sorption becomes reversible in pores with radii smaller than about twice the thickness of the adsorbed layer.Neglect of the presence of an adsorbate layer remaining at the pore walls after capillary evaporation renders this line of reasoning invalid.Foster has included the influence of adsorption forces upon the Kelvin instability of the adsorbed film during adsorption on the basis of arguments similar to those of Cohan and thus leading to qualitatively the same conclusion. The neglect of both the remaining adsorbed layer after capillary evaporation and the influence of adsorp- tion forces upon the capillary evaporation p~ocess,~ renders this argument equally invalid [for a fuller discussion, see ref. (5)]. More relevant is the remark of Everett and Haynes that capillary condensation is only expected to occur after the adsorbed film has reached a sufficient thickness for the development of liquid-like properties, specifically of a liquid-vapour interfacial tension. This would explain the absence of capillary condensation at the lower end of the relative-pressure range.On the other hand, it has been pointed out by several authors 7* that the curvature of the liquid meniscus at the pore entrances increases 42J. C. P . BROEKHOFF AND W. P. VAN BEEK 43 along the desorption branch of the isotherm in the direction of decreasing relative pressures. This increasing curvature is accompanied by an increase in magnitude of the negative hydrostatic pressure (tension) acting on the capillary condensed phase. At a certain point, this tension may well surpass the tensile strength of the capillary- held liquid, resulting in spontaneous nucleation and subsequent evaporation. This point has been elaborated by Everett 9s10 and co-workers and by Dubinin 11*12 and co-workers who presented evidence for a relation between the estimated tensile strengths of different adsorbates and the corresponding lower closing points of the hysteresis loops.It is commonly observed for systems exhibiting B-type hysteresis loops according to the classification of De Boer l3 that the desorption branch becomes very steep at or near the “ critical ” relative pressure. This is rather suggestive of an actual forced breakdown of the capillary-held phase, as it is improbable that nature should have equipped us with an abundance of porous systems all exhibiting a peak in their distribution around 2.5 nm, while the corresponding narrower pore range would be absent. We have recently demonstrated l4 that the behaviour of ascending and descending scanning curves in a system exhibiting B-type hysteresis, violates Everett’s consistency rule for ascending and descending scanning curves,l 9 while in other cases these rules were found to be closely followed.Thus, although in the former case all primary ascending curves starting from the desorption boundary curve converge towards the upper closing point of the hysteresis region, the descending scanning curves emanating from the adsorption boundary curve intersect the steep part of the desorption branch of the isotherm at an acute angle, instead of converging towards the lower closing point of the hysteresis loop. This unusual behaviour seems to suggest that this type of scanning behaviour at least in part is not determined by domain properties, but rather by some other phenomenon occurring in the capillary- condensed phase.Apparently, the run of the descending scanning curves is disrupted discontinuously around the “ critical ” relative pressure by the onset of an intrinsic instability of the capillary-held phase of the type predicted by S~hofield,~ Flood and EverettY9 rather than by the mechanism envisaged by Cohan and F ~ s t e r . ~ In the present paper, the application of the method we have published previously l7 to an analysis of the phenomena occurring around the “ critical ” relative pressure is discussed. At the same time the mechanism of filling of pores during adsorption will be studied. EXPERIMENTAL MATERIALS For this particular study, three samples were chosen from a large collection of nickel- on-silica catalysts, all three exhibiting pronounced B-type hysteresis with a very steep part in the desorption branch around pg/po = 0.5 (for nitrogen as the adsorbate, sorption hysteresis in a rigid porous sytem is hardly ever observed below pJp0 = 0.42).TABLE 1 .-PHYSICAL CHARACTERISTICS OF CATALYSTS sample surface area pore volume porosity mean pore size b mean particle size 0 code /lo3 m2 kg-1 /lO-3 m3 kg-1 /% /iO-9 m /10-9 m NS 25a 113.5 0.219 37 3.9 6.5 NP 113 200.8 0.63 1 68 6.3 3.0 NZ269 263.5 0.444 54 3.4 2.9 a Differences from the data cited for this preparation in ref. (14) are apparently to be attributed Defined here as 2 x pore volume/surface area. to heterogeneity within the bulk of the powder. C Defined here as 2/(solid density x surface area).44 CAPILLARY CONDENSATION BY SCANNING A summary of the relevant physical properties of these samples is presented in table I, NS 25 is a hydrothermally synthesized nickel silicate, apparently consisting of thin, two- dimensionally extended sheets.An electron microphotograph is presented in fig. l(a). NP 113 is a nickel silicate catalyst produced by direct precipitation of nickel hydroxide upon a support under alkaline conditions. The sheets of nickel silicate are less robust and far less extended than in the case of NS 25 [fig. l(b)]. The higher surface area indicates a finer dispersion. NZ 269 has a structure of a different morphology and was obtained by a homogeneous precipitation procedure. Small platelets may be observed [fig. l(c)], with certain fibrous elements. The high surface area indicates an even finer dispersion.RESULTS AND DISCUSSION Prior to the determination of the scanning curves, the boundary curves of the hysteresis region were traced. Sometimes a certain permanent hysteresis occurred in the low relative pressure region after the first adsorption-desorption cycle. Upon repeated cycling, the hysteresis loop usually settled down to a well-reproducible shape and position, provided the adsorption branch was always taken up to the same amount adsorbed at saturation prior to desorption. For the isotherms presented here, the adsorption branch had shifted upwards by some 5 % around ps/po = 0.5 after the first adsorption-desorption cycle, for both NS 25 and NZ 269, whereas for NP 113 the shift was negligible. The cause of this curious initial drift could not be ascertained.As there was no apparent reason for suspecting an instrumental artefact, the possibility of irreversible swelling in the stacking of nickel silicate sheets during the first sorption cycle cannot be excluded. Primary descending scanning curves were pursued from their origin on the adsorption boundary curve down to beyond the lower closing point of the hysteresis loop. The return curve then provides a check upon the reproducibility of the adsorption branch during the scanning process. This was found to be within experimental accuracy in most cases. Ascending scanning curves were not pursued all the way up to saturation, but traced up to a relative pressure exceeding 0.9. In previous studies it had been found that the exact position of the desorption boundary curve of the hysteresis loop is sometimes dependent on the actual value of the volume taken up at saturation (this points to the presence of very large pore domains which are only accessible through relatively narrow entrances).In the present study, process control is based on relative pressure, which makes it difficult to reproduce the exact uptake at saturation. For our purpose, it is sufficient to determine the initial slope of the ascending scanning curves. After direction reversal these scanning curves were found to return exactly to their origin at the desorption branch of the isotherm, although a secondary hysteresis loop with the primary ascending curve pointed to some domain filling at the high relative pressure end.It has not been possible to record both primary ascending and primary descending scanning curves in a single run, due to the necessity of mechanical maintenance or to the occurrence of an apparatus breakd0~n.l~ In general, care was taken to use the same individual sample for different runs, but sometimes the sample was lost during handling operations, and another sample of the same batch of material had to be used. NS 25 Ascending and descending scanning curves are presented in fig. 2(a) and (b). The corresponding t plot transforms are presented in fig. 2(c) and (d). The part of the primary ascending scanning curves in the direct vicinity of a direction reversal point could be successfully fitted to eqn (1) of Part l . 1 7 Typically, five to tenh 9, [To face page 44(4 FIG.1.-Transmission electron microphotos of catalysts used in the scanning study : (a) NS 25 ; (b) NP 113 ; (c) NZ 269.J . C . P . BROEKHOFF AND W. P . VAN BEEK consecutive experimental points, including the direction reversal point, were found to satisfy this relation. The relative pressure range for which eqn (I) was found to hold varied from 0.07 for scanning curves situated near the closing points of the hysteresis loop, to 0.25 for scanning curves traversing the full width of the hysteresis region. Correlation coefficients for the selected set of points for all of the scanning curves of NS 25 were between 0.99 and 1.0, lending support to the adopted analytical procedure. This satisfactory behaviour was not unexpected, as the t plot of the adsorption boundary curve does not show any upward deviation, suggesting that both appreciable reversible capillary pore filling and appreciable curvature of the pore walls are absent.It is remarkable in fig. 2(c) that, although the initial part of the t plot could satisfactorily be fitted to a straight line, a distinct upward curvature can often be discerned before the scanning curve approaches the t plot transform of the adsorption 0.2t I 1 s ~ " " ' * ' ' 0 02 0 4 06 0.8 1 .o (4 PgIPo 1.0r (6) :L PgIPo 0.8 1 .o FIG. 246 5 r I CAPILLARY CONDENSATION BY SCANNING &/lo3 m2 kg-' FIG. 2.-contd.J . C . P. BROEKHOFF AND W . P. VAN BEEK 1.0 - 47 0 0.2 0.4 0.6 Q8 1.0 0 PglPo FIG. 2.-Scanning behaviour on NS 25: (a) primary ascending scanning curves; (6) primary descending scanning curves ; (c) t plots of primary ascending curves ; (d) t plots of primary descendlng curves ; (e) characteristic V-S curves from ascending and descending scanning curves ; (f) sizes of domains emptying or filling along the boundary curves of the hysteresis region as a function of relative pressure. As a reference the thickness of the adsorbed layer on a free surface is also presented.boundary curve. Such a curvature is usually taken as an indication of capillary condensation processes, which are unexpected for systems where the straight-line behaviour of the adsorption boundary curve is commonly interpreted as the absence of pore wall curvature and capillary condensation phenomena. Fig. 2(d) demon- strates rather dramatically the acute intersection between the descending scanning curves and the steep part of the desorption boundary curve.The descending scanning curves near the lower closing point of the hysteresis loop are very limited in extent. In the others a distinct downward curvature can be discerned, pointing to a certain degree of pore emptying due to capillary evaporation at a decrease in relative pressure. Massive capillary evaporation does not set in before a certain, apparently critical, relative pressure has been reached. The relation between Vp and S, derived from the respective intercepts and slopes of the tangents to the descending and the ascending scanning curves (the V-S curve) is given in fig. 2(e). The size of the crosses indicates the 90 % confidence intervals resulting from independent estimations of slope and intercept.l Quite remarkable is the discontinuity in the V-S curve obtained from the ascending scanning curves.No corresponding discontinuity is found in the V-S curve derived from descending scanning curves. This distinction in behaviour is even more dramatically demon- strated by a plot of the slope of the V-S curves according to eqn (2) of ref. (17) against the relative pressure of the corresponding origins of each primary scanning curve situated at the boundary curves of the hysteresis region [fig. 2(f)]. Around a pe/po value of 0.495 (the relative pressure at the start of the very steep part in the desorption branch) very large formal domain diameters are derived from the V-S curve. A further decrease in relative pressure results in the emptying of domains48 CAPILLARY CONDENSATION BY SCANNING with progressively smaller dimensions, until finally around pg/po = 0.42 the size of the domains emptying just equals the thickness t of the adsorbed layer at that relative pressure. Apparently, over a quite restricted range of relative pressures, emptying occurs of all domains which were previously filled with a capillary-condensed phase, apart from the permanent presence of an adsorbed layer.This seems to happen to a first approximation independently of the size of the domain, but it is clear that the largest-size domains empty first, followed by increasingly smaller domains as the relative pressure is lowered towards the final closing point of the hysteresis loop.This behaviour is highly suggestive of a mechanism for hysteresis breakdown caused by surpassing of the tensile strength of the capillary-condensed phase below a certain critical relative pressure, as predicted by Schofield,’ Flood * and Everett. According to simple thermodynamics,s 9 the formal hydrodynamic pressure in the capillary condensed phase at a distance t from the nearest pore wall can be written as : (1) PL = 1/ VLrPg VL - RT In (PO/PA + Wl where VL is the liquid molar volume and F(t) is a correction term for the influence o adsorption forces emanating from the pore walls, for the sake of simplicity taken here to be independent of the pore geometrya4* According to this equation pL equals - 12.9 MPa (- 127 atm) at pg/po = 0.5 in very wide pores and -12.5 MPa (-123 atm) at ps/po = 0.4 in pores with a radius of 1.0 nm, if F ( t ) is approximated by the relation F(t) = 2.3 RT (0.14/t2-0.034), a fairly accurate representation of the De Boer-Lippens t curve.4* Thus, the stabilising effect of adsorption forces may at least qualitatively account for the sequential breakdown of the capillary condensed state in increasingly narrower pores with decreasing relative pressures.In narrower pores, disturbances in adsorbate packing may also lead to changes in VL, as well as in tensile strength, so we may not expect more than a qualitative agreement with eqn (1). The presence of very wide domains which empty around pg/po = 0.5, in itself is no more than an indication that certain wide domains exist, which are only connected to their environment through narrower-sized pathways (e.g.as in the classical bottle- neck theory of hysteresis) and thus are unable to empty at relative pressures corres- ponding to their proper size. Rather the fact that the emptying of these wide domains is closely followed by emptying of all other ones, even those which on the basis of a Kelvin-relation would be expected to empty at significantly lower relative pressures, is the most powerful indication of the occurrence of a sudden breakdown of the capillary condensed state as such. In this respect, it is particularly significant that below pg/po = 0.5 the slope of the characteristic curve in fig. 2(f) does not return to the extrapolation of the first branch of the curve, but definitely intersects it. The filling of pore domains along the adsorption boundary curve of the hysteresis loop can only be followed with the present technique in the region above the closing point of the hysteresis loop.Thus, on the basis of the foregoing discussion, we should not expect any discontinuity in the corresponding characteristic curve, as indeed is found. Remarkably, the relation between slope of the characteristic curve and the relative pressure of domain filling [fig. 2(f)] closely follows the t curve for the thickness of the adsorbed layer as a function of relative pressure, but is shifted upward over a distance of 0.2 to 0.4 nm. The basis of the t and MP method for assessing the size of micropores, is formed by the implicit assumption that pores should fill if the thickness of the adsorbed layer equals half the pore diameter for slit-shaped pores.In that case, the slope of the characteristic curve at all relative pressures wouId coincide with the t curve. The most simple interpretation of the present results is, that at pressures around pn/po = 0.5 pore filling occurs as soon asJ. C. P. BROEKHOFF AND W. P. VAN BEEK 49 one nitrogen molecule fits between opposite adsorbed layers, whereas in wider pores, and thus at higher relative pressures, filling occurs if two nitrogen molecules can bridge the distance between the adsorbed layers. The results obtained here are particularly significant in this context as the electron micrograph suggests that we are indeed dealing with sheetlike structures, where the absence of pore wall curvature will prevent the premature occurrence of capillary condensation.Nevertheless, this proof of the occurrence of additional sorption between adjacent adsorbed layers cannot be regarded as definite, as the presence of tapered pore structures might lead to the same behavi0ur.l In any case, there is ample reason to review the quantitative basis of the t and MP method. Returning to the analysis of the V-S curve from the desorption branch of the hysteresis loop, it is furthermore remarkable that the variation of the slope of the characteristic curve with relative pressure in the region above pg/po = 0.5 is slight and definitely less than is predicted by relations of the Kelvin type. Thus the quantitative validity of the Kelvin equation or any of its sophistications 5 9 l6 for the present case, is not confirmed.Whether this is the rule or an exception, can only be ascertained on the basis of more extended experimental work. Qualitatively, domains empty sequentially in the direction of smaller diameters at lower relative pressures, as is predicted by the Kelvin theory of capillary evaporation. The interference of swelling and shrinking phenomena of the porous system during the adsorption-desorption cycle cannot be completely excluded. However, the perfect reproducibility of the final adsorption and, under certain restrictions, the desorption branches of the system, even after partially filling or emptying, and the straightness of the t plots of the ascending scanning curves over a fairly large range of relative pressures, is in our opinion an indication that extensive swelling does not occur.In a swelling system, the volume adsorbed in completely filled domains would increase with relative pressure according to a capillary condensation mechanism rather than remain constant or follow the t curve relation as found in the present study. NZ 269 AND NP 113 The ascending and descending scanning behaviour in the system NZ 269 [fig. 3(a)- (f)] essentially confirms the findings for NS 25. Above a relative pressure of 0.65, I 02 04 06 0.8 1 .o50 5- 4 - 3- 2 - 1- CAPILLARY CONDENSATION BY SCANNING I I I I 1 , , i I , I , , , ( # ID- 0.0 - - 0.6 - s c \ - 0.4 - 0 0.2 0.4 0.6 0.e 1.0 PglPo "I Ll I ' 1 1 I " L 0 0.2 0.4 0.6 0.8 1.0 1.2 ' 1.4 1.6 (d) t/= FIG. 3.-contd.J . C. P . BROEKHOFF AND W.P. VAN BEEK &/lo3 m2 kg-I , I 0.2 0.4 0.6 0.8 1.0 PgIPo 51 FIG. 3.-Scanning behaviour on NZ 269 : (a) primary ascending scanning curves ; (b) primary descending scanning curves; (c) t plots of primary ascending curves; (d) t plots of primary descending curves ; (e) characteristic V-S curves from ascending and descending scanning curves ; (f) sizes of domains emptying or filling along the boundary curves of the hysteresis region as a function of relative pressure. As a reference the thickness of the adsorbed layer on a free surface is also presented.52 CAPILLARY CONDENSATION BY SCANNING the t plot of the adsorption boundary curve shows a small but distinct upward curvature, suggesting that pore wall curvature may not be completely negligible in this case.Nevertheless, the initial part of all scanning curves could be fitted in eqn (1) of Part 1 with correlation coefficients of 0.99 or better. Also here, there is a clearly distinguishable discontinuity just below pg/p0 = 0.5. In this case, the breakdown occurs near the lower closing point of the hysteresis loop, corresponding to the second steep part of the desorption boundary curve. The onset of a first steep part in this desorption curve, which occurs aroundpg/po = 0.61, is not reflected as a discontinuity in the V-S curve, and thus is caused by the presence in this particular structure of a large number of pores with a certain size emptying by straightforward capillary desorption around pg/po = 0.61, and not by a breakdown of the capillary condensed phase.This seems to reinforce strongly the point of view presented in the previous section, that final breakdown is related to relative pressure and not to details in the domain structure of the system. As to the relation between the slope of the V-S curve derived from the descending curves and the relative pressure of their original state, it is remarkable that this line virtually coincides with the one obtained for NS 25, and thus seems to be insensitive to the details of the porous structure. This may be considered as a reinforcement of 0.e I I c 2 0.6 - 1 s - 0.4 - 0.2 - i 0 0.2 0.4 0.6 0.8 I .o (4 PglPo (b) t/nm FIG. 4J . C. P. BROEKHOFF AND W. P . VAN BEEK 53 II 4.0 - - z 0" - 3.0 - Fl c; s 20- a a U 1.0 - &/lo3 m2 kg-l 0.2 0.4 06 00 1.0 (4 PgIPo FIG.4.-Scanning behaviour on Np 113 : (a) ascending scanning curves ; (b) t plots of ascending curves ; (c) characteristic V-S curve from ascending curves ; (d) size of domains emptying along the desorption branch of the hysteresis loop.54 CAPILLARY CONDENSATION BY SCANNING the view that during pore filling, additional sorption between adjacent sorbed layers occurs rather than capillary condensation in a tapered slit-shaped porous structure. As to NP 113, we have succeeded in obtaining data of sufficient accuracy only for the behaviour of the ascending scanning curves. In the case of the descending scanning curves, difficulties were encountered in obtaining sufficient reproducibility of the adsorption branch upon repeated cycling, either due to experimental difficulties or to some intrinsic property of the system (swelling?).The data for the ascending scanning curves are presented in fig. 4(a)-(d), because the V-S curve shows the same discontinuity around ps/po = 0.5 as observed for the other two systems, and thus is a confirmation of the (probably general) trend, for the slit-shaped porous systems. CONCLUSIONS A V-S curve for the relation between pore size, pore volume and pore surface area for the domains either emptying along the desorption boundary curve or filling along the adsorption one, can be obtained from a careful estimation of the position equation of the tangent lines to either ascending or descending primary scanning curves immediately after a pressure reversal at the boundary curves of the hysteresis region.In practice, this is possible for systems with sufficiently small curvature of the domain walls, such as slit-shaped pore systems, by application of the well-known t method. In the three cases studied, the characteristic curve for the desorption boundary curve showed a very pronounced discontinuity around ps/po = 0.5, corresponding to the commonly observed steep part of the desorption boundary curve. Steep parts in the desorption branch at higher relative pressures do not lead to such discontinuity. The present work seems to confirm the notion that around a certain critical relative pressure complete breakdown of the capillary condensed state occurs due to surpassing of the tensile strength of the capillary condensed liquid. Evidence for the hypothesis was hitherto mostly of an indirect nature.l0-l2+ 22 Confirmation of the quantitative validity of the Kelvin mechanism for the desorption process from the V-S curves, could not be obtained and should be a subject for further study.The sudden breakdown of the capillary condensed state around aps/po value of 0.5 for nitrogen at 78 K, invalidates the usual procedures for obtaining pore size distri- butions from the desorption branch of the hysteresis loop foi all samples which exhibit hysteresis in this pg/po region. In practical terms, this means that the peak in the pore distribution often observed in the region of 2.5 nm sizes has to be considered as an artefact of the method. Along the adsorption branch of the hysteresis loop, the dimensions of the pores filling at any relative pressure exceed the statistical thickness of the adsorbed layer on their pore walls by one or two adsorbate molecular diameters.This is suggestive of a filling mechanism by way of adsorption of one or two additional adsorbate molecules in between adjacent adsorbed layers, rather than a simple merging of adjacent adsorbed layers at the appropriate statistical thickness. In principle, the determination of the V-S curve from the study of ascending and descending primary scanning curves, should enable assessment of the structure of a porous system as well as study of the mechanism of capillary filling and emptying of pores. A stimulating discussion with Prof. D. H. Everett, University of Bristol, on the properties of scanning curves within the hysteresis loop is gratefully acknowledged.J .C. P . BROEKHOFF AND W . P. VAN BEEK 55 LIST OF SYMBOLS correction term for the influence of the adsorption forces emanating from the pore walls equilibrium gas pressure formal hydrodynamic pressure in the capillary- condensed phase saturation vapour pressure gas constant cumulative surface area from t plots thickness of the adsorbed layer temperature adsorbed volume liquid molar volume total pore volume reference for scaling purposes adsorbed volume at pg/po 3 1 J mol-I Pa Pa Pa J mol-' K-I m2 kg-' nm K m3 kg-I m3 mol-' m3 kg-I m3 kg-I m3 kg-l L. H. Cohan, J. Amer. Chem. Soc., 1944,66,98. D. H. Everett and J. M. Haynes, J. Colloid Interface Sci., 1972, 38, 125. A. G. Foster, J. Chem. Soc., 1952, 1806. B. V. Deryagin, Acta Physicochim. U.R.S.S., 1940, 12,139. J. C. P. Broekhoff, 27zesis (University of Technology, Delft, 1969), chap. 1 and 3. D. H. Everett and J. M. Haynes, Colloid Science (Spec. Period. Rep., Chem. SOC., London, 1973), vol. 1, p. 136. E. A. Flood, The Solid-Gas Interface, ed. E. A. Flood (Marcel Dekker, New York, 1969, vol. 1, chap. 1. D. H. Everett, The Solid-Gas Interface, ed. E. A. Flood (Marcel Dekker, New York, 1967), vol. 2, p. 1055. ' R. K. Schofield, Disc. Fiaday SOC., 1948,3, 105. l o C. G. V. Burgess and D. H. Everett, J. Colloid Interface Sci., 1970, 33, 611. 'l M. M. Dubinin, Pure and Appl. Chem., 1965, 10, 309. l2 0. Kadlec and M. M. Dubinin, J. Colloid Interface Sci., 1968, 31,479. l3 J. H. De Boer, The Structure and Properties of Porous Materials, ed. D. H. Everett and F. S. l4 J. C. P. Broekhoff, L. F. Brown and W. P. van Beek, Proc. Int. Symp. on Pore Structure and l5 D. H. Everett, Trans. Faraday Soc., 1954, 50, 1077. l6 D. H. Everett and F. S. Smith, Trans. Faraday Soc., 1954,50, 187. '' J. C. P. Broekhoff and W. P. van Beek, J.C.S. Farahy I, 1979,75, 36. l 8 F. S. Acton, in Andysis of Straight-line Data (Dover, New York, 1959). Stone (Butterworth, London, 1958), p. 68. Properties of Materials, ed. S . Modry (Prague, 1973), vol. IV, C-85. S. J. Gregg and K. S. W. Sing, Adsorption, Porosity and Surface Area (Academic Press, London, 1967), chap. 4. 2o J. H. De Boer, B. G. Linsen, Th. van der Plas and G. J. Zondervan, J. Catalysis, 1965,4, 649. 21 R. G. Mikhail, S. Brunauer and E. E. Bodor, J. Colloid Interface Sci., 1969, 26, 45. 22 R. G. Avery and J. D. F. Ramsay, J. Colloid Interface Sci., 1973, 42, 597. (PAPER 8 /282)