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Some remarks about capillary condensation and pore structure analysis
JOURNAL OF COLLOID AND INTERFACE SCIENCE 2 5 , 353--358
(1967)
Some Remarks about Capillary Condensation and Pore Structure Analysis S T E P H E N B R U N A U E R , R. SH. M I K H A I L , A~D E. E. B O D O R Institute of Colloid and Surface Science and Department of Chemistry, Clarkson College of Technology, Potsdam, New York 13676
Received May 1, 1967 Adsorption and desorption isotherms of nitrogen are widely used for the determination of the pore structures of adsorbents, catalysts, and other porous solids. Recently, the present authors published a paper in which they offered two new methods of pore structure analysis. These methods, like the previous ones, employ the capillary condensation region of the isotherms for the analysis--the region in which the adsorption and desorption isotherms differ from each other. The considerations advanced in the present paper attempt to answer the questions (a) why in most instances the desorption isotherm, but in certain instances the adsorption isotherm, gives better results for pore structure analysis; (b) what are the advantages of the two new methods over those presently employed; and (c) why the cylindrical pore shape idealizations used by most investigators frequently give reasonable results. INTRODUCTION Recently, the present authors published a paper with the title "Pore Structure Analysis without a Pore Shape Model" (1). All previous investigators used some pore shape model in their analyses of pore volume and pore surface distributions; the cylindrical model has been by far the most widely employed. In the earlier paper (1), the authors--in agreement with the great majority of ininvestigators of pore structures--recommended that the desorption branch of the isotherm be used for the analysis. It is shown in the present paper that for certain types of pores, the so-called ink-bottle pores, the adsorption isotherm gives better results than the desorption isotherm, if the presently employed methods of analysis are used. However, if one of the new methods is used, the desorption isotherm gives reliable results regardless of the shape of the pore. The main criterion offered by earlier investigators for the correctness of their pore structure analysis was the agreement between the cumulative surface area of the
pore walls with the B E T surface. It seemed rather surprising to the present authors, as well as to many others, how frequently the cylindrical idealization gave good agreement for pores of irregular shapes. An explanation for this is offered in this paper. CAPILLARY CONDENSATION AND HYSTERESIS 1. The hysteresis loop formed by the adsorption and desorption branches of isotherms obtained for porous adsorbents is attributed to capillary condensation. The first explanation of hysteresis was advanced by Zsigmondy (2), who assumed that during adsorption the vapor does not wet the walls of the pore completely. The Kelvin equation
for the adsorption side is given by R T In P--~ =
p8
27V co
[1]
r
where pa is observed equilibrium pressure on the adsorption branch, p8, 7, and V are the saturation pressure, surface tension, and molar volume of the liquid adsorbate, 8 is the wetting angle, and r is the radius of the cylindrical pore. At p8, however, the wetting is complete; consequently, for the desorption 353
354
BRUNAUER, MIKttAIL, AND BODOR
branch cos 0 = 1. This explains why pa, the equilibrium pressure on the desorption branch, is less than pa for a given amount adsorbed. It also follows that the desorption branch corresponds to true capillary condensation equilibrium. Subsequently, Foster (3), Cohan' (4), and others advanced other convincing arguments to explain the delay in the formation of the meniscus during the adsorption process. Foster demonstrated especially strikingly that, for the adsorbent-adsorbate systems he investigated, the desorption branch represented true capillary condensation equilibrium, and not the adsorption branch. ~ ...... It seems logical tO believe that the desorption values represent equilibrium ~t a given relative pressure (P/PO, if for nO other reason, because they are larger than the adsorption values. It is easy to accept that for some reason meniscus formation is delayed on the adsorption side, but at saturation pressure all pores are filled with liquid; consequently, there cannot be a delay in meniscus formation on the desorption side. Because of this, most investigators--ineluding the present authors--accepted the -desorption branch as the equilibrium branch (1) and based their pore structure analysis on the desorption branch. Nevertheless, the possibility exists that there is a delay in the breaking of the meniscus on the desorption side. It was suggested by Kraemer (5) and elaborated in more detail by MeBain (6) that hysteresis was Caused by ink-bottle shaped pores. On the adsorption side the narrow neck will fill at a relatively low pressure, and the wider body at a higher pressure. On the desorption side, however, the pore will empty only when the pressure is reduced so far that the liquid in the neck of the pore becomes ~unstable. If this theory were the sole and correct explanation of hysteresis, then the adsorption branch would represent true eapillary condensation equilibrium. If there is delay in meniscus formation on the adsorption side, then the adsorption branch does not represent true equilibrium even for ink-bottle pores. However, if the adsorbent does contain ink-bottle pores in not negligible pro-
portion, then the desorption does not represent true capillary condensation equilibrium either. 2. An ink-bottle pore is any pore with a narrow entrance and a wider body. Such pores are likely to exist in any porous body, and in some solids they may represent the major fraction of the pores, iViikhail, Copeland, and Brunauer (7) demonstrated beyond question that in certain hardened portland eement pastes a large fraction of the pores have wide bodies with very narrow entrances. For adsorbents having such pore systems, the adsorption branch of the isotherm comes closer to representing capillary condensation equilibrium than the desorption branch. Mikhail, Copeland, and Brunauer analyzed the pore structures of hardened portland cement pastes by the method of Cranston and Inkley (8), using the adsorption branches of their nitrogen isotherms for the analysis. All investigators prior to Cranston and Inkley used the desorption branches of nitrogen isotherms for pore structure analysis, and they offered as the main criterion of the eorreetness of their analysis the agreement between the cumulative surfaee area of the pore walls and the BET surface. Cranston and Inkley, however, used both the adsorption and desorption branches in their analysis, and found that for most of their adsorbents the adsorption branch gave cumulative surfaces in better agreement with the BET surface. Mikhail, Copeland, and Brunauer also used both the adsorption and desorption branches in their analysis, and they found that the desorption branch gave much higher cumulative surface than the BET area, whereas the adsorption branch gave good agreement with the BET surface. They discarded therefore the desorption results and used the adsorption results. As stated above, these investigators also found that th6ir adsorbents had predominantly wide pores with narrow entrances. They did 'not realize the connection between this fact and the fact that the adsorption branch gave better results than the desorption branch, but now the connection is clear. For adsorbents which have ink-
CAPILLARY CONDENSATION AND PORE STRUCTURE ANALYSIS bottle pores in negligible quantities, the desorption branch should give better resuits. If agreement between cumulative pore surface and B E T surface is accepted as the criterion for choosing the adsorption or the desorption branch for pore structure analysis, the first step is to perform analysis for both branches. This doubles the work of analysis. One may then find that the surface values obtained from the two branches are quite close to each other, but the pore strueture curves are quite different; or one may find poor agreement with the B E T surface for both branches. Even if one branch gives a considerably better agreement with the B E T surface, there is no assurance that the pore structure curve obtained from that branch is the right one. Further complications arise if one attempts to compare the structure curves of two adsorbents, one of which gives better agreement with the B E T surface on the basis of the adsorption branch, and the other, on the basis of the desorption branch. The weight of the evidence presented in the literature is strongly on the side of the desorption branch's representing true capillary condensation equilibrium in most instances. I t is recommended, therefore, that the desorption branch always be used for pore structure analysis. It will be shown next that for an ink-bottle pore the desorption branch gives grossly distorted values for the volume and surface of the pore, but a method of analysis presented by the authors earlier (1) gives results very near to the true values. AN
ILLUSTRATIVE
EXAMPLE
i. Let the ink-bottle pore be two cylinders: the neck having a radius of 26.5 A and a length of 50.0 A, the body having a radius of 53.1 A and a length of i00.0 A. The volume of the pore is 995,000 A 3, and the surface is 41,570 A 2. For simplicity, the surfaces of the bottom and the top of the body of the ink bottle are neglected and only the two cylindrical surfaces are considered. The hydraulic radius of the pore, i.e., the volume of the pore divided by its surface, is 23.9 A. The hydraulic radius of a cylinder is half of the radius of the cylinder; thus, the by-
355
draulie radius of the pore is somewhat smaller than that of the body of the ink bottle. On the desorption branch of a nitrogen isotherm, this pore will empty at p/p~ = 0.60. An adsorbed film having a thickness of 7.8 A will remain on the pore walls. This value was taken from the t-curve of Cranston and Inkley (8). A t-curve is a plot of the statistical thickness of the film adsorbed on nonporous adsorbents against p / p s . Several t-curves for nitrogen can be found in the literature, but that of Cranston and Inldey is the most useful because it is based on a wide variety of adsorbents. The Kelvin radius for nitrogen at p/p~ = 0.60 is 18.7 A; the sum of the Kelvin radius and the thickness of the adsorbed film remaining on the pore walls is equal to the radius of the neck of the pore, 26.5 A. The part of the pore that remains e m p t y after desorption will be called the core (1). The volume of the core after desorption at p/p8 --- 0.60 is 699,000 A s, the surface of the core is 34,000 A 2, and the hydraulic radius of the core is 20.4 A. In most presently employed methods of pore structure analysis, it is assumed that the pore is a eylinder having a radius equal to r~ + t, where rk is the Kelvin radius and t is the thickness of the adsorbed film remaining on the wall. The calculated volume of the pore will be
v~ = vo (r~ + t) 2
[2]
7~k2
where V v and Vc are the volumes of the pore and core, respectively. The volume of the core (the volume desorbed at p/p~ = 0.60) is 699,000 A3; consequently, the calculated volume of the pore will be 1,398,000 A 3. This is 40 % larger than the true volume of the pore. The calculated surface of the pore will be Sp-
2Vp r~ + t '
[3]
which is 105,500 A 2. This is 2.54 times the true surface of the pore. The calculated hydraulic radius is 13.25 A, which is the hydraulic radius of the neck of the ink bottle.
356
BRUNAUER, MIKHAIL, AND BODOR
This greatly exaggerated example shows how the cylindrical idealizations used at present distort the volume and surface of an ink-bottle pore. Cranston and Inkley, using the desorption branch of a nitrogen isotherm, obtained for a silica-alumina adsorbent a cumulative pore surface that exceeded the BET surface by 57%, and the cumulative pore volume exceeded the total pore volume by 13 %. The situation becomes worse if one considers that their analysis neglects the surface in the small pores, the pores in which capillary condensation does not take place; and it neglects the volume in the large pores, the pores with radii greater than 150 A. 2. In the earlier paper (1), two new methods of pore structure analysis were proposed by the authors. One of these methods uses no model at all for the pore shape. This "modelless" method gives reasonably good approximations for the volumes and surfaces of the cores and excellent values for their hydraulic radii. The other method is not entirely modelless. The main core surface and volume terms are calculated without assuming any pore shape model, but then certain correction terms are applied. These correction terms are identical for cylindrical and for parallel-plate pores. It is possible that for pores of other shapes the correction terms would be slightly different, but such slight differences can introduce only very small errors into the calculated values of core surfaces and volumes, and practically no error into the calculated core hydraulic radii. The completely modelless method is well suited for most practical purposes; it is especially suitabIe for comparing similar adsorbents and catalysts. However, for accurate work the "corrected modelless" method should be used. The great superiority of this method over the presently employed cylindrical or parallel-plate idealizations lies in this: the present idealizations are based on one value obtained from the isotherm, the volume of the core (the volume desorbed), but the hydraulic radius and surface of the core are hypothetical; whereas the new method is based on two values obtained from the isotherm, the volume and the surface of the core, which automatically gives the hydraulic radius of the core.
The advantages of the increased information will be illustrated by using the core parameters calculated previously for the ink-bottle pore. It will be assumed that the corrected modelless method obtains the following results for the core of a pore of unkno~m shape: V~ = 699,000 Aa, S~ = 34,300 As, r~ = 20.4 A. First it will be assumed that the pore has a cylindrical shape. The volume, surface, and hydraulic radius of core and pore are related by the equations V ~ _ ( 2 r ~ + t ) 2 Sp _ 2 r ~ + t V~ (2r~)~ 'So 2r~ ' andrerc
[4]
2r~ + t 2r~
The volume calculated for the pore is 991,000 A~, which is 0.4% smaller than the true volume; the calculated pore surface is 40,900 As, which is 1.6% smaller than the true surface; and the calculated pore hydraulic radius is 24.2 A, which is 1.3% larger than the true hydraulic radius of the ink-bottle pore. Next it will be assumed that the pore walls are parallel plates. The volume, surface, and hydraulic radius of core and pore are related by the equations Vp gc
r~+t rc
,Sp = & ,
[5] andrerc
rcq-t rc
The volume calculated for the pore is 965,000 Aa, which is 3 % smaller than the true volume; the calculated surface is 34,300 A=, which is 17.4% smaller than the true surface; and the hydraulic radius is 28.2 A, which is 18.0 % larger than the true hydraulic radius. Finally it will be assumed that the pore is spherical. The volume, surface, and hydraulic radius of core and pore are related by the equations V ~ _ (3re + t ) a S T _ (3re + t) 2 (3r~)2 Vo (3r~) 3 ' S~
[6]
and r~, _ 3ro q- t re 3r¢ The volume calculated for the pore is
CAPILLARY CONDENSATION AND PORE STRUCTURE ANALYSIS
357
1,000,000 A 3, which is 0.5 % larger than the tween a completely curved pore (spherical) true volume; the surface is 43,500 A 2, which and a completely fiat pore (parallel plate). is 4.6 % larger than the true surface; and the For most adsorbents, the shapes of the hydraulic radius is 23.0 A, which is 3.8 % pores are unknown. It is clear from the above equations that the best procedure in smaller than the true hydraulic radius. It is clear from the above example that as such cases is to convert core data into long as one has good values for the volume, cylindrical pore data. If the shape of the surface, and hydraulic radius of the core, pore approximates a sphere, less error is one earmot go too far wrong in calculating committed if it is considered a cylinder the pore parameters, no matter what pore than if it is considered a parallel-plate pore. shape is assumed. Calculations similar to Likewise, if the walls of a pore of unknown those offered above were made for another ink-bottle pore with the same neck and body shape approximate parallel plates, less error radii, but the neck was made twice as long, is committed if the pore is considered cylina n d the body was made five times as long. drical than if it is considered spherical. An intensive study was made by de Boer In these calculations, the surfaces of the bottom and top of the body of the ink bottle (9) of the relationship between the shape of were not neglected. The results were quite the pore and the shape of the hysteresis similar: (a) the core data converted to loop. If the shape of the hysteresis loop cylindrical pore data gave a volume that supplies adequate information about the was 1.6 % smaller than the true volume, the shapes of the pores, the core parameters surface was 4.5% too small, and the hycan be converted into parameters of pores draulic radius was 3.2% too large; (b) the of that shape. Equations similar to [7], core data converted to parallel plate pore [81, and [9] can be deduced for pores of any data gave a volume 4.0% too small, a geometrical shape. surface 19.4% too small, and a hydraulic By far the most pore structure analyses radius 19.2 % too large; and (c) the core data published to date were based on the assumpconverted to spherical pore data gave a tion that the pores have cylindrical shapes. volume 0.4 % too small, a surface 1.8 % too The reason for using the cylindrical shape large, and a hydraulic radius 2.1% too small. was that for a pore of such shape the Kelvin RELATION BETWEEN CORE AND PORE equation has a simple form. It is clear from PARAMETERS the preceding considerations that this was a The relation between the volume, surface, fortunate choice. Pores of irregular shapes and hydraulic radius of pore and core for may have flat, partly curved, or completely spherical, cylindrical, and parallel-plate curved surfaces in random distribution. The agreement between cumulative pore surface pores is given by areas and BET areas published in the literature appeared surprisingly frequent to the V , _ (3to + t~ ~. (2ro + ! y . / ) ' present authors, as stated earlier. It does [7] not appear to be so surprising now.
-~-/. so
ACKNOWLEDGMENT
[8] re
The authors gratefully acknowledge their indebtedness to the National Science Foundation for Grant GP 5612, which supported the work presented in the paper.
/
REFERENCES
r~ _ 3r~ + t. 2re -}- t. rc + t re 3re ' 2re ' ro
[9]
The cylindrical pore is intermediate be-
1. B~uNAu~, S., MIKHAIL, R. SH., AND BODOR, E. E., J. Colloid and Interface Sci., To be published.
358
BRUNAUER, MIKHAIL, AND BODOR
2. ZSIGMONDY, ~., Z. Anorg. Allgem. Chem. 71, 356 (1911). 3. FOSTER, A. G., Proc. Roy. Soc. (London) A147, 128 (1934); ibid. A150, 77 (1935). 4. COHAN,L. It., J. Am. Chem. Soe. 60,433 (1938). 5. ~41RAEMER, •. O., I n H. S. Taylor, ed., "A Treatise on Physical Chemistry," Chapter XX, p, 1661. Van Nostrand, New York, 1931. 6. McBAIN, J. W., J. Am. Chem. Soc. 57,699 (1935).
7. MIKHA!L, R. SH., COPELAND, L. ]~., AND BRUNAL~ER, S., Can. J. Chem. 42, 426 (1964). 8. CRANSTON, R. W., AND INKLEY, F. A., Advan. Catalysis 9, 143 (1957). 9. DE BOER, J. H., "The Shapes of Capillaries," in D. H. Everett and F. S. Stone, eds., "The Structure and Properties of Porous Materials," p. 68. Butterworths, London, 1958.
(1967)
Some Remarks about Capillary Condensation and Pore Structure Analysis S T E P H E N B R U N A U E R , R. SH. M I K H A I L , A~D E. E. B O D O R Institute of Colloid and Surface Science and Department of Chemistry, Clarkson College of Technology, Potsdam, New York 13676
Received May 1, 1967 Adsorption and desorption isotherms of nitrogen are widely used for the determination of the pore structures of adsorbents, catalysts, and other porous solids. Recently, the present authors published a paper in which they offered two new methods of pore structure analysis. These methods, like the previous ones, employ the capillary condensation region of the isotherms for the analysis--the region in which the adsorption and desorption isotherms differ from each other. The considerations advanced in the present paper attempt to answer the questions (a) why in most instances the desorption isotherm, but in certain instances the adsorption isotherm, gives better results for pore structure analysis; (b) what are the advantages of the two new methods over those presently employed; and (c) why the cylindrical pore shape idealizations used by most investigators frequently give reasonable results. INTRODUCTION Recently, the present authors published a paper with the title "Pore Structure Analysis without a Pore Shape Model" (1). All previous investigators used some pore shape model in their analyses of pore volume and pore surface distributions; the cylindrical model has been by far the most widely employed. In the earlier paper (1), the authors--in agreement with the great majority of ininvestigators of pore structures--recommended that the desorption branch of the isotherm be used for the analysis. It is shown in the present paper that for certain types of pores, the so-called ink-bottle pores, the adsorption isotherm gives better results than the desorption isotherm, if the presently employed methods of analysis are used. However, if one of the new methods is used, the desorption isotherm gives reliable results regardless of the shape of the pore. The main criterion offered by earlier investigators for the correctness of their pore structure analysis was the agreement between the cumulative surface area of the
pore walls with the B E T surface. It seemed rather surprising to the present authors, as well as to many others, how frequently the cylindrical idealization gave good agreement for pores of irregular shapes. An explanation for this is offered in this paper. CAPILLARY CONDENSATION AND HYSTERESIS 1. The hysteresis loop formed by the adsorption and desorption branches of isotherms obtained for porous adsorbents is attributed to capillary condensation. The first explanation of hysteresis was advanced by Zsigmondy (2), who assumed that during adsorption the vapor does not wet the walls of the pore completely. The Kelvin equation
for the adsorption side is given by R T In P--~ =
p8
27V co
[1]
r
where pa is observed equilibrium pressure on the adsorption branch, p8, 7, and V are the saturation pressure, surface tension, and molar volume of the liquid adsorbate, 8 is the wetting angle, and r is the radius of the cylindrical pore. At p8, however, the wetting is complete; consequently, for the desorption 353
354
BRUNAUER, MIKttAIL, AND BODOR
branch cos 0 = 1. This explains why pa, the equilibrium pressure on the desorption branch, is less than pa for a given amount adsorbed. It also follows that the desorption branch corresponds to true capillary condensation equilibrium. Subsequently, Foster (3), Cohan' (4), and others advanced other convincing arguments to explain the delay in the formation of the meniscus during the adsorption process. Foster demonstrated especially strikingly that, for the adsorbent-adsorbate systems he investigated, the desorption branch represented true capillary condensation equilibrium, and not the adsorption branch. ~ ...... It seems logical tO believe that the desorption values represent equilibrium ~t a given relative pressure (P/PO, if for nO other reason, because they are larger than the adsorption values. It is easy to accept that for some reason meniscus formation is delayed on the adsorption side, but at saturation pressure all pores are filled with liquid; consequently, there cannot be a delay in meniscus formation on the desorption side. Because of this, most investigators--ineluding the present authors--accepted the -desorption branch as the equilibrium branch (1) and based their pore structure analysis on the desorption branch. Nevertheless, the possibility exists that there is a delay in the breaking of the meniscus on the desorption side. It was suggested by Kraemer (5) and elaborated in more detail by MeBain (6) that hysteresis was Caused by ink-bottle shaped pores. On the adsorption side the narrow neck will fill at a relatively low pressure, and the wider body at a higher pressure. On the desorption side, however, the pore will empty only when the pressure is reduced so far that the liquid in the neck of the pore becomes ~unstable. If this theory were the sole and correct explanation of hysteresis, then the adsorption branch would represent true eapillary condensation equilibrium. If there is delay in meniscus formation on the adsorption side, then the adsorption branch does not represent true equilibrium even for ink-bottle pores. However, if the adsorbent does contain ink-bottle pores in not negligible pro-
portion, then the desorption does not represent true capillary condensation equilibrium either. 2. An ink-bottle pore is any pore with a narrow entrance and a wider body. Such pores are likely to exist in any porous body, and in some solids they may represent the major fraction of the pores, iViikhail, Copeland, and Brunauer (7) demonstrated beyond question that in certain hardened portland eement pastes a large fraction of the pores have wide bodies with very narrow entrances. For adsorbents having such pore systems, the adsorption branch of the isotherm comes closer to representing capillary condensation equilibrium than the desorption branch. Mikhail, Copeland, and Brunauer analyzed the pore structures of hardened portland cement pastes by the method of Cranston and Inkley (8), using the adsorption branches of their nitrogen isotherms for the analysis. All investigators prior to Cranston and Inkley used the desorption branches of nitrogen isotherms for pore structure analysis, and they offered as the main criterion of the eorreetness of their analysis the agreement between the cumulative surfaee area of the pore walls and the BET surface. Cranston and Inkley, however, used both the adsorption and desorption branches in their analysis, and found that for most of their adsorbents the adsorption branch gave cumulative surfaces in better agreement with the BET surface. Mikhail, Copeland, and Brunauer also used both the adsorption and desorption branches in their analysis, and they found that the desorption branch gave much higher cumulative surface than the BET area, whereas the adsorption branch gave good agreement with the BET surface. They discarded therefore the desorption results and used the adsorption results. As stated above, these investigators also found that th6ir adsorbents had predominantly wide pores with narrow entrances. They did 'not realize the connection between this fact and the fact that the adsorption branch gave better results than the desorption branch, but now the connection is clear. For adsorbents which have ink-
CAPILLARY CONDENSATION AND PORE STRUCTURE ANALYSIS bottle pores in negligible quantities, the desorption branch should give better resuits. If agreement between cumulative pore surface and B E T surface is accepted as the criterion for choosing the adsorption or the desorption branch for pore structure analysis, the first step is to perform analysis for both branches. This doubles the work of analysis. One may then find that the surface values obtained from the two branches are quite close to each other, but the pore strueture curves are quite different; or one may find poor agreement with the B E T surface for both branches. Even if one branch gives a considerably better agreement with the B E T surface, there is no assurance that the pore structure curve obtained from that branch is the right one. Further complications arise if one attempts to compare the structure curves of two adsorbents, one of which gives better agreement with the B E T surface on the basis of the adsorption branch, and the other, on the basis of the desorption branch. The weight of the evidence presented in the literature is strongly on the side of the desorption branch's representing true capillary condensation equilibrium in most instances. I t is recommended, therefore, that the desorption branch always be used for pore structure analysis. It will be shown next that for an ink-bottle pore the desorption branch gives grossly distorted values for the volume and surface of the pore, but a method of analysis presented by the authors earlier (1) gives results very near to the true values. AN
ILLUSTRATIVE
EXAMPLE
i. Let the ink-bottle pore be two cylinders: the neck having a radius of 26.5 A and a length of 50.0 A, the body having a radius of 53.1 A and a length of i00.0 A. The volume of the pore is 995,000 A 3, and the surface is 41,570 A 2. For simplicity, the surfaces of the bottom and the top of the body of the ink bottle are neglected and only the two cylindrical surfaces are considered. The hydraulic radius of the pore, i.e., the volume of the pore divided by its surface, is 23.9 A. The hydraulic radius of a cylinder is half of the radius of the cylinder; thus, the by-
355
draulie radius of the pore is somewhat smaller than that of the body of the ink bottle. On the desorption branch of a nitrogen isotherm, this pore will empty at p/p~ = 0.60. An adsorbed film having a thickness of 7.8 A will remain on the pore walls. This value was taken from the t-curve of Cranston and Inkley (8). A t-curve is a plot of the statistical thickness of the film adsorbed on nonporous adsorbents against p / p s . Several t-curves for nitrogen can be found in the literature, but that of Cranston and Inldey is the most useful because it is based on a wide variety of adsorbents. The Kelvin radius for nitrogen at p/p~ = 0.60 is 18.7 A; the sum of the Kelvin radius and the thickness of the adsorbed film remaining on the pore walls is equal to the radius of the neck of the pore, 26.5 A. The part of the pore that remains e m p t y after desorption will be called the core (1). The volume of the core after desorption at p/p8 --- 0.60 is 699,000 A s, the surface of the core is 34,000 A 2, and the hydraulic radius of the core is 20.4 A. In most presently employed methods of pore structure analysis, it is assumed that the pore is a eylinder having a radius equal to r~ + t, where rk is the Kelvin radius and t is the thickness of the adsorbed film remaining on the wall. The calculated volume of the pore will be
v~ = vo (r~ + t) 2
[2]
7~k2
where V v and Vc are the volumes of the pore and core, respectively. The volume of the core (the volume desorbed at p/p~ = 0.60) is 699,000 A3; consequently, the calculated volume of the pore will be 1,398,000 A 3. This is 40 % larger than the true volume of the pore. The calculated surface of the pore will be Sp-
2Vp r~ + t '
[3]
which is 105,500 A 2. This is 2.54 times the true surface of the pore. The calculated hydraulic radius is 13.25 A, which is the hydraulic radius of the neck of the ink bottle.
356
BRUNAUER, MIKHAIL, AND BODOR
This greatly exaggerated example shows how the cylindrical idealizations used at present distort the volume and surface of an ink-bottle pore. Cranston and Inkley, using the desorption branch of a nitrogen isotherm, obtained for a silica-alumina adsorbent a cumulative pore surface that exceeded the BET surface by 57%, and the cumulative pore volume exceeded the total pore volume by 13 %. The situation becomes worse if one considers that their analysis neglects the surface in the small pores, the pores in which capillary condensation does not take place; and it neglects the volume in the large pores, the pores with radii greater than 150 A. 2. In the earlier paper (1), two new methods of pore structure analysis were proposed by the authors. One of these methods uses no model at all for the pore shape. This "modelless" method gives reasonably good approximations for the volumes and surfaces of the cores and excellent values for their hydraulic radii. The other method is not entirely modelless. The main core surface and volume terms are calculated without assuming any pore shape model, but then certain correction terms are applied. These correction terms are identical for cylindrical and for parallel-plate pores. It is possible that for pores of other shapes the correction terms would be slightly different, but such slight differences can introduce only very small errors into the calculated values of core surfaces and volumes, and practically no error into the calculated core hydraulic radii. The completely modelless method is well suited for most practical purposes; it is especially suitabIe for comparing similar adsorbents and catalysts. However, for accurate work the "corrected modelless" method should be used. The great superiority of this method over the presently employed cylindrical or parallel-plate idealizations lies in this: the present idealizations are based on one value obtained from the isotherm, the volume of the core (the volume desorbed), but the hydraulic radius and surface of the core are hypothetical; whereas the new method is based on two values obtained from the isotherm, the volume and the surface of the core, which automatically gives the hydraulic radius of the core.
The advantages of the increased information will be illustrated by using the core parameters calculated previously for the ink-bottle pore. It will be assumed that the corrected modelless method obtains the following results for the core of a pore of unkno~m shape: V~ = 699,000 Aa, S~ = 34,300 As, r~ = 20.4 A. First it will be assumed that the pore has a cylindrical shape. The volume, surface, and hydraulic radius of core and pore are related by the equations V ~ _ ( 2 r ~ + t ) 2 Sp _ 2 r ~ + t V~ (2r~)~ 'So 2r~ ' andrerc
[4]
2r~ + t 2r~
The volume calculated for the pore is 991,000 A~, which is 0.4% smaller than the true volume; the calculated pore surface is 40,900 As, which is 1.6% smaller than the true surface; and the calculated pore hydraulic radius is 24.2 A, which is 1.3% larger than the true hydraulic radius of the ink-bottle pore. Next it will be assumed that the pore walls are parallel plates. The volume, surface, and hydraulic radius of core and pore are related by the equations Vp gc
r~+t rc
,Sp = & ,
[5] andrerc
rcq-t rc
The volume calculated for the pore is 965,000 Aa, which is 3 % smaller than the true volume; the calculated surface is 34,300 A=, which is 17.4% smaller than the true surface; and the hydraulic radius is 28.2 A, which is 18.0 % larger than the true hydraulic radius. Finally it will be assumed that the pore is spherical. The volume, surface, and hydraulic radius of core and pore are related by the equations V ~ _ (3re + t ) a S T _ (3re + t) 2 (3r~)2 Vo (3r~) 3 ' S~
[6]
and r~, _ 3ro q- t re 3r¢ The volume calculated for the pore is
CAPILLARY CONDENSATION AND PORE STRUCTURE ANALYSIS
357
1,000,000 A 3, which is 0.5 % larger than the tween a completely curved pore (spherical) true volume; the surface is 43,500 A 2, which and a completely fiat pore (parallel plate). is 4.6 % larger than the true surface; and the For most adsorbents, the shapes of the hydraulic radius is 23.0 A, which is 3.8 % pores are unknown. It is clear from the above equations that the best procedure in smaller than the true hydraulic radius. It is clear from the above example that as such cases is to convert core data into long as one has good values for the volume, cylindrical pore data. If the shape of the surface, and hydraulic radius of the core, pore approximates a sphere, less error is one earmot go too far wrong in calculating committed if it is considered a cylinder the pore parameters, no matter what pore than if it is considered a parallel-plate pore. shape is assumed. Calculations similar to Likewise, if the walls of a pore of unknown those offered above were made for another ink-bottle pore with the same neck and body shape approximate parallel plates, less error radii, but the neck was made twice as long, is committed if the pore is considered cylina n d the body was made five times as long. drical than if it is considered spherical. An intensive study was made by de Boer In these calculations, the surfaces of the bottom and top of the body of the ink bottle (9) of the relationship between the shape of were not neglected. The results were quite the pore and the shape of the hysteresis similar: (a) the core data converted to loop. If the shape of the hysteresis loop cylindrical pore data gave a volume that supplies adequate information about the was 1.6 % smaller than the true volume, the shapes of the pores, the core parameters surface was 4.5% too small, and the hycan be converted into parameters of pores draulic radius was 3.2% too large; (b) the of that shape. Equations similar to [7], core data converted to parallel plate pore [81, and [9] can be deduced for pores of any data gave a volume 4.0% too small, a geometrical shape. surface 19.4% too small, and a hydraulic By far the most pore structure analyses radius 19.2 % too large; and (c) the core data published to date were based on the assumpconverted to spherical pore data gave a tion that the pores have cylindrical shapes. volume 0.4 % too small, a surface 1.8 % too The reason for using the cylindrical shape large, and a hydraulic radius 2.1% too small. was that for a pore of such shape the Kelvin RELATION BETWEEN CORE AND PORE equation has a simple form. It is clear from PARAMETERS the preceding considerations that this was a The relation between the volume, surface, fortunate choice. Pores of irregular shapes and hydraulic radius of pore and core for may have flat, partly curved, or completely spherical, cylindrical, and parallel-plate curved surfaces in random distribution. The agreement between cumulative pore surface pores is given by areas and BET areas published in the literature appeared surprisingly frequent to the V , _ (3to + t~ ~. (2ro + ! y . / ) ' present authors, as stated earlier. It does [7] not appear to be so surprising now.
-~-/. so
ACKNOWLEDGMENT
[8] re
The authors gratefully acknowledge their indebtedness to the National Science Foundation for Grant GP 5612, which supported the work presented in the paper.
/
REFERENCES
r~ _ 3r~ + t. 2re -}- t. rc + t re 3re ' 2re ' ro
[9]
The cylindrical pore is intermediate be-
1. B~uNAu~, S., MIKHAIL, R. SH., AND BODOR, E. E., J. Colloid and Interface Sci., To be published.
358
BRUNAUER, MIKHAIL, AND BODOR
2. ZSIGMONDY, ~., Z. Anorg. Allgem. Chem. 71, 356 (1911). 3. FOSTER, A. G., Proc. Roy. Soc. (London) A147, 128 (1934); ibid. A150, 77 (1935). 4. COHAN,L. It., J. Am. Chem. Soe. 60,433 (1938). 5. ~41RAEMER, •. O., I n H. S. Taylor, ed., "A Treatise on Physical Chemistry," Chapter XX, p, 1661. Van Nostrand, New York, 1931. 6. McBAIN, J. W., J. Am. Chem. Soc. 57,699 (1935).
7. MIKHA!L, R. SH., COPELAND, L. ]~., AND BRUNAL~ER, S., Can. J. Chem. 42, 426 (1964). 8. CRANSTON, R. W., AND INKLEY, F. A., Advan. Catalysis 9, 143 (1957). 9. DE BOER, J. H., "The Shapes of Capillaries," in D. H. Everett and F. S. Stone, eds., "The Structure and Properties of Porous Materials," p. 68. Butterworths, London, 1958.