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Some remarks on pore structure analyses
Some Remarks on Pore Structure Analyses In 1968, in collaboration with E. E. Bodor, we proposed a method for the analysis of micropores in solids, and we called this method the MP-method (1). The upper limit of micropores, the boundary between micropores and mesopores, is a radius of 15-16 _~, if the pores are idealized as cylinders, or a width of 15-16 _~, if the pores are idealized as parallel plates. The dividing line was proposed by Dubinin and was adopted by IUPAC. The MP-method uses a modelless characteristic of pore sizes, the hydraulic radius, rh, which is the ratio of the volume to the surface area of a pore or pore group. If the distribution curves of pore volume and pore surface area are represented in terms of rh, the cylindrical and parallel-plate idealizations lead to the same curves for micropores. However, for cylindrical pores rh is half of the radius of the cylinder, and for slitshaped pores it is half of the width of the pore. In the original paper, the MP-method was illustrated by the analysis of a silica gel, Davidson 03, which contained only micropores (1). It was pointed out in that paper that the analysis did not apply to very small micropores, which we called ultramicropores. We did not define the boundary line of ultramicropores except by implication; using nitrogen as the absorbate for the silica gel, the mean hydraulic radius of the smallest pore group was 4.25 ~. Duhinin correctly pointed out our lack of definition of ultramicropores (2); later we defined ultramicropores as pores that have hydraulic radii smaller than one molecular diameter (3). This means that for water vapor and nitrogen the MPmethod is not applicable for pores of radii less than about 6 and 7 ~., respectively. Dubinin's volume-filling theory was applied to such adsorbents as active carbons and synthetic zeolites, having radii not exceeding 6 or 7 ~. Consequently, in his later papers, Dubinin admits--at least by implicat i o n - t h e feasibility of the MP-method for those micropores for which we did claim that the method was applicable (4, 5). Nevertheless, two differences of opinion still remain between Dubinin and us. One is not important; it is only semantics. In contrast with his earlier views, he proposes that only those pores with radii less than 6-7 .~ he called micropores, i.e., those pores that we call ultramicropores, and that the larger micropores, i.e., those to which the MP-method applies, should be called supermicropores. In view of the fact that IUPAC has already adopted Dubinin's original
proposal, i.e., defining micropores as pores with radii less than 15-16_~, we consider our nomenclature preferable to Dubinin's. The second disagreement is more serious. In our paper on the "corrected modelless" method of analysis of pores wider than micropores, we proposed two criteria for testing the correctness of the pore structure analysis: (1) the cumulative pore surface must agree with the BET surface, and (2) the cumulative pore volume must agree with v,, the volume adsorbed at p,, the saturation pressure (6). In our paper on micropore analysis, we pointed out that the surface criterion is not applicable if the adsorbent contains only micropores, but we thought that the volume criterion was applicable (1). But later Dubinin showed that the volume criterion was also inapplicable when the isotherm has a horizontal plateau (2). Subsequently, we argued in the following way. In mixed adsorbents, which contain both micropores and mesopores, the cumulative pore surface and the cumulative pore volume in the micropores cannot agree with the BET surface and v,, respectively. The total cumulative pore surface and pore volume, in the micropores plus mesopores, must agree with the BET surface and ~,, respectively. If such agreement is found by using the MP-method for the micropores and the corrected modelless method for the wider pores, it shows that both criteria are satisfied for both methods. Agreement was found for 35 adsorbents. Further, we argued that if the micropore analysis is correct in mixed-porous adsorbents, there is no reason to suppose that it is incorrect in purely microporous adsorbents. In a recent paper, Duhinin showed (by using model isotherms) that for microporous adsorbents one obtains agreement between S,(7) and SB~T even if one uses isotherms such as the Freundlich isotherm and the Dubinin-Radushkevich isotherm (5). But this does not bring us closer to agreement. Several years ago, Prof. Lee Brown of the University of Colorado proved mathematically in private correspondence to the second author that the cumulative pore volume in microporous adsorbents must agree with v,, regardless of the shape of the isotherm, if 9o is the end volume. However, for mixed adsorbents, v~is not the end volume of the micropores. So the situation now is that Dubinin still does not accept our criteria of correctness for micropores, whereas our position is that until someone shows by theoretical arguments that our criteria are
626 Journal of Colloid and Interface Science, VoI. 52, No. 3, September 1975
Copyright O 1975 by Academic Press, Inc. All rights of reproduction in any form reserved.
627
LETTERS TO THE EDITORS invalid for mixed adsorbents, we consider that our criteria are valid and that the MP-method gives a correct analysis for micropores larger than ultramicropores. Dubinin accepts our method of wide pore analysis, but we disagree on one important point. In our method, the core parameters--volume, surface, and hydraulic radius--are determined for each pore group. The core is the empty space that fills up by capillary condensation or empties by capillary evaporation. To obtain the pore parameters, one must add the thickness of the film adsorbed on the pore walls prior to condensation or after evaporation. This thickness is different for cylindrical and parallel-plate pores. Dubinin argues that, for example, for cylindrical pores, one obtains the same surface and volume distribution curves whether one uses our "corrected modelless" method or one of the older methods (4). This is incorrect. Dubinin would be right, if in the adsorption ideal thermodynamic equilibrium existed, including thermal, chemical, and mechanical equilibrium. In the isotherms, there is thermal and chemical equilibrium, but not mechanical equilibrium. This is shown by the existence of the hysteresis loops. One must find out by various methods whether the adsorption or the desorption branch represents equilibrium, or the equilibrium is somewhere between them. Let us consider now a pore group that fills up between p/p, = 0.40 and 0.45. In the old methods, these two relative pressures immediately determine the mean radius of the core group, by means of the Kelvin equation. The amounts adsorbed at the two relative pressures determine the volume of the core group. The surface is determined by these two data; consequently, it is independent of the shape of the isotherm between the two pressures. In our method, the volume is determined in the same way as in the older methods, but the surface is not. Our core surface is dependent on the shape of t h e isotherm between the two pressures as seen from the equation s = - - (1/-/)
f 040. ao.~5RTln(p/p~)da
[17
where s is the surface area of the core group; q, is the surface tension; p/p, is the relative pressure; and a0.40 and ao.45 are the number of moles of adsorbate adsorbed at relative pressures of 0,40 and 0.45, respectively. The
hydraulic radius is the volume divided by the surface, and the cylindrical radius is twice the hydraulic radius. Thus, in the older methods and our method, only one parameter, the volume, is the same; the surface area and the mean radius of the core group are different. The points on the adsorption or desorption isotherms, if either of these are equilibrium points, can be considered metastable equilibrium points. This justifies the use of the Kelvin equation in the older methods, or Eq. [-1], which is a modification of Kiselev's generalized theory of capillary condensation (8). Naturally, if the core parameters of our method differ from those of the older methods, the pore parameters will also be different, since the correction for the thickness of the adsorbed film is the same in both cases. REFERENCES 1. MIKHAIL, R. S., BRUNAUER, S., AND BODOR, E. E., Y. Colloid Interface Sci. 26, 45 (1968). 2. DuBIIqlI~, M. M., in "Surface Area Determination, Proceedings of the International Symposium, Bristol, 1969," p. 75, Butterworths, London, 1970. 3. BRUNAUER,S., SKALNY,7., AND ODLER, I., in "Pore Structure and Properties of Materials," (S. Modry, Ed.), Vol. 1, p. C-3, Academia, Prague, 1973. 4. DIJBINII~, M. M., in "Pore Structure and Properties of Materials," (S. Modry, Ed.), Academia, Prague, 1973. 5. D ~ B I ~ , M. M., Y. Colloid Interface Sci. 46, 351 (1974). 6. BRUNAUER, S., MIKHAIL, R. S., AND BODOR, E. E., J. Colloid Interface Sci. 24, 451 (1967). 7. LIPPENS, B. C., AND DE BOER, J. H., J. Catalysis 4, 319 (1965). 8. KISELEV, A. V., Usp. Khlm. 14, 367 (1945). RAOUF SH. MIKHAIL STEPHEN BRUNAUER
Institute of Colloid and Surface Science and Department of Chemistry, Clarkson College of Technology, Potsdam, New York 13676 Received February 28, 1975; accepted May 29, 1975
Journal of Colloid and Interface Science, Vol. 52, No. 3, September 1975
proposal, i.e., defining micropores as pores with radii less than 15-16_~, we consider our nomenclature preferable to Dubinin's. The second disagreement is more serious. In our paper on the "corrected modelless" method of analysis of pores wider than micropores, we proposed two criteria for testing the correctness of the pore structure analysis: (1) the cumulative pore surface must agree with the BET surface, and (2) the cumulative pore volume must agree with v,, the volume adsorbed at p,, the saturation pressure (6). In our paper on micropore analysis, we pointed out that the surface criterion is not applicable if the adsorbent contains only micropores, but we thought that the volume criterion was applicable (1). But later Dubinin showed that the volume criterion was also inapplicable when the isotherm has a horizontal plateau (2). Subsequently, we argued in the following way. In mixed adsorbents, which contain both micropores and mesopores, the cumulative pore surface and the cumulative pore volume in the micropores cannot agree with the BET surface and v,, respectively. The total cumulative pore surface and pore volume, in the micropores plus mesopores, must agree with the BET surface and ~,, respectively. If such agreement is found by using the MP-method for the micropores and the corrected modelless method for the wider pores, it shows that both criteria are satisfied for both methods. Agreement was found for 35 adsorbents. Further, we argued that if the micropore analysis is correct in mixed-porous adsorbents, there is no reason to suppose that it is incorrect in purely microporous adsorbents. In a recent paper, Duhinin showed (by using model isotherms) that for microporous adsorbents one obtains agreement between S,(7) and SB~T even if one uses isotherms such as the Freundlich isotherm and the Dubinin-Radushkevich isotherm (5). But this does not bring us closer to agreement. Several years ago, Prof. Lee Brown of the University of Colorado proved mathematically in private correspondence to the second author that the cumulative pore volume in microporous adsorbents must agree with v,, regardless of the shape of the isotherm, if 9o is the end volume. However, for mixed adsorbents, v~is not the end volume of the micropores. So the situation now is that Dubinin still does not accept our criteria of correctness for micropores, whereas our position is that until someone shows by theoretical arguments that our criteria are
626 Journal of Colloid and Interface Science, VoI. 52, No. 3, September 1975
Copyright O 1975 by Academic Press, Inc. All rights of reproduction in any form reserved.
627
LETTERS TO THE EDITORS invalid for mixed adsorbents, we consider that our criteria are valid and that the MP-method gives a correct analysis for micropores larger than ultramicropores. Dubinin accepts our method of wide pore analysis, but we disagree on one important point. In our method, the core parameters--volume, surface, and hydraulic radius--are determined for each pore group. The core is the empty space that fills up by capillary condensation or empties by capillary evaporation. To obtain the pore parameters, one must add the thickness of the film adsorbed on the pore walls prior to condensation or after evaporation. This thickness is different for cylindrical and parallel-plate pores. Dubinin argues that, for example, for cylindrical pores, one obtains the same surface and volume distribution curves whether one uses our "corrected modelless" method or one of the older methods (4). This is incorrect. Dubinin would be right, if in the adsorption ideal thermodynamic equilibrium existed, including thermal, chemical, and mechanical equilibrium. In the isotherms, there is thermal and chemical equilibrium, but not mechanical equilibrium. This is shown by the existence of the hysteresis loops. One must find out by various methods whether the adsorption or the desorption branch represents equilibrium, or the equilibrium is somewhere between them. Let us consider now a pore group that fills up between p/p, = 0.40 and 0.45. In the old methods, these two relative pressures immediately determine the mean radius of the core group, by means of the Kelvin equation. The amounts adsorbed at the two relative pressures determine the volume of the core group. The surface is determined by these two data; consequently, it is independent of the shape of the isotherm between the two pressures. In our method, the volume is determined in the same way as in the older methods, but the surface is not. Our core surface is dependent on the shape of t h e isotherm between the two pressures as seen from the equation s = - - (1/-/)
f 040. ao.~5RTln(p/p~)da
[17
where s is the surface area of the core group; q, is the surface tension; p/p, is the relative pressure; and a0.40 and ao.45 are the number of moles of adsorbate adsorbed at relative pressures of 0,40 and 0.45, respectively. The
hydraulic radius is the volume divided by the surface, and the cylindrical radius is twice the hydraulic radius. Thus, in the older methods and our method, only one parameter, the volume, is the same; the surface area and the mean radius of the core group are different. The points on the adsorption or desorption isotherms, if either of these are equilibrium points, can be considered metastable equilibrium points. This justifies the use of the Kelvin equation in the older methods, or Eq. [-1], which is a modification of Kiselev's generalized theory of capillary condensation (8). Naturally, if the core parameters of our method differ from those of the older methods, the pore parameters will also be different, since the correction for the thickness of the adsorbed film is the same in both cases. REFERENCES 1. MIKHAIL, R. S., BRUNAUER, S., AND BODOR, E. E., Y. Colloid Interface Sci. 26, 45 (1968). 2. DuBIIqlI~, M. M., in "Surface Area Determination, Proceedings of the International Symposium, Bristol, 1969," p. 75, Butterworths, London, 1970. 3. BRUNAUER,S., SKALNY,7., AND ODLER, I., in "Pore Structure and Properties of Materials," (S. Modry, Ed.), Vol. 1, p. C-3, Academia, Prague, 1973. 4. DIJBINII~, M. M., in "Pore Structure and Properties of Materials," (S. Modry, Ed.), Academia, Prague, 1973. 5. D ~ B I ~ , M. M., Y. Colloid Interface Sci. 46, 351 (1974). 6. BRUNAUER, S., MIKHAIL, R. S., AND BODOR, E. E., J. Colloid Interface Sci. 24, 451 (1967). 7. LIPPENS, B. C., AND DE BOER, J. H., J. Catalysis 4, 319 (1965). 8. KISELEV, A. V., Usp. Khlm. 14, 367 (1945). RAOUF SH. MIKHAIL STEPHEN BRUNAUER
Institute of Colloid and Surface Science and Department of Chemistry, Clarkson College of Technology, Potsdam, New York 13676 Received February 28, 1975; accepted May 29, 1975
Journal of Colloid and Interface Science, Vol. 52, No. 3, September 1975