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Refinements of the Twofold Description of Porous Media† Vicente Mayagoitia,*,‡ Fernando Rojas,‡ Isaac Kornhauser,‡ Giorgio Zgrablich,§,| Roberto Jose´ Faccio,| Bernard Gilot,⊥ and Claude Guiglion⊥ Departamento de Quı´mica, Universidad Auto´ noma MetropolitanasIztapalapa, Apartado Postal 55-534, Me´ xico 13, D.F., 09340 Me´ xico, Centro Regional de Estudios Avanzados (CREA), 5700 San Luis, Argentina, Departamento de Fı´sica, Universidad Nacional de San Luis, 5700 San Luis, Argentina, and Ecole Nationale Supe´ rieure d’Inge´ nieurs du Ge´ nie Chimique, Institut National Polytechnique, 31078 Toulouse, France Received September 1, 1994. In Final Form: July 28, 1995X An important generalization of the “twofold description” of porous media is described. It consists of the consideration of the mutal interaction of bonds (capillaries, passages) meeting at a commnon site (cavity, antrum). Approximate and exact treatments are developed, exemplified, and compared.
Introduction description”1
The “twofold of porous media is particularly suitable to (i) conceive,2,3 represent,4 and classify5 porous network morphologies, (ii) understand and predict the mechanisms of capillary processes5-8 (capillary condensation and evaporation, mercury intrusion, imbibition, immiscible displacement, etc.), and (iii) determine the texture of porous materials.9,10 In this introduction the fundamentals of this theory, as stand up till now, are resumed. It still offers however a very crude picture of real porous media. Consequently, afterward we present a refinement of this model consisting in a more detailed analysis of the interactions between the elements of the network. Every porous network can be visualized as constituted by two kinds of alternated elements: sites (antrae, cavities) and bonds (capillaries, passages). The connectivity, C, is the number of bonds meeting at a site, while every bond is delimited by two sites. For simplicity, the size of each entity is expressed by means of a unique quantity, R, defined in the following way: for sites, considered as hollow spheres, RS is the radius of the sphere, while for bonds, idealized (due to their function of passages) as hollow cylinders open at both ends, RB is the radius of the cylinder. † Presented at the symposium on Advances in the Measurement and Modeling of Surface Phenomena, San Luis, Argentina, August 24-30, 1994. ‡ Universidad Auto ´ noma MetropolitanasIztapalapa. § Centro Regional de Estudios Avanzados. | Universidad Nacional de San Luis. ⊥ Ecole Nationale Supe ´ rieure d’Inge´nieurs du Ge´nie Chimique. X Abstract published in Advance ACS Abstracts, January 1, 1996.
(1) Mayagoitia, V.; Rojas, F.; Kornhauser, I. Langmuir 1993, 9, 2748. (2) Mayagoitia, V.; Kornhauser, I. In Principles and Applications of Pore Structural Characterization; Haynes, J. M., Rossi-Doria, P., Eds.; Arrowsmith: Bristol, 1985; p 15. (3) Mayagoitia, V.; Cruz, M. J.; Rojas, F. J. Chem. Soc., Faraday Trans. 1 1989, 85, 2071. (4) Cruz, M. J.; Mayagoitia, V.; Rojas, F. J. Chem. Soc., Faraday Trans. 1 1989, 85, 2079. (5) Mayagoitia, V.; Rojas, F.; Kornhauser, I. J. Chem. Soc., Faraday Trans. 1 1988, 84, 785. (6) Mayagoitia, V.; Gilot, B.; Rojas, F.; Kornhauser, I. J. Chem. Soc., Faraday Trans. 1 1988, 84, 801. (7) Mayagoitia, V. In Characterization of Porous Solids II; Rodrı´quezReynoso, F., Ronquerol, J., Sing, K. S. W., Unger, K. K., Eds.; Elsevier: Amsterdam, 1991; p 51. (8) Mayagoitia, V.; Rojas, F.; Kornhauser, I.; Riccardo, J. L.; Zgrablich, G. In Characterization of Porous Solids III; Rodrı´quez-Reynoso, F., Rouquerol, J., Sing, K. S. W. Unger, K. K., Eds.; Elsevier: Amsterdam, 1994; p 141. (9) Mayagoitia, V.; Rojas, F. In Fundamentals of Adsorption III; Mersmann, A. B., Scholl, S. E., Eds.; The Engineering Foundation: New York; p 563. (10) Mayagoitia, V. Catal. Lett. 1993, 22, 93.
0743-7463/96/2412-0211$12.00/0
Instead of only considering the size distribution of voids without regard to the kind of element (site or bond) to which they belong, it would seem more appropriate to establish a double distribution of sizes. For this, FS(R) and FB(R) are the normalized size distribution functions, for sites and bonds, on a number of elements basis, so that the probabilities of finding a site or a bond having a size R or smaller are, respectively:
S(R) )
∫0R FS(R) dR;
B(R) )
∫0R FB(R) dR
(1)
From the very definitions of “site” and “bond” emerges, in a natural way, the following “construction principle”: the size of any bond is always smaller than the size of the site to which it leads. Two self-consistency laws guarantee the fulfillment of the above principle. The “first law” establishes that bonds must be sufficiently small as to accommodate together with the sites belonging to a given size distribution:
first law
B(R) g S(R) for every R
(2)
A “second law” is still required since when there is a considerable overlap between the size distributions, there appear topological size correlations between neighboring elements. Thus, the events of finding a size RS for a site and a size RB for a given one of its bonds are not independent. The probability density for this joint event to occur is
F(RS∩RB) ) FS(RS) FB(RB) φ(RS,RB)
(3)
The “second law” can be expressed as:
second law
φ(RS,RB) ) 0
for RS < RB
(4)
If the randomness in the topological assignation of sizes is raised up to a maximum, while complying with the restriction imposed by the construction principle, the “most verisimilar” form of φ for the correct case RS g RB is obtained3
(
S(R ) dS ∫S(R ) B - S)
exp φ(RS,RB) )
(
S
B(RS) - S(RS)
B(R ) dB ∫B(R ) B - S)
exp -
B
)
S
B
B(RB) - S(RB)
(5)
Topological size correlations promote a “size segregation effect”, in that sites and bonds of a bigger size join together © 1996 American Chemical Society
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a
b
Figure 1. (a) A case of impossibility for a site to link bonds of his own size. (b) The site is depicted as a sphere x2 times bigger than the initially proposed size, and having the appearance of a “diving helmet”.
to form regions of large elements, while elements of the smaller size reunite to constitute alternated regions of small entities. This effect is more important the greater the overlap. Its consequences on the development of capillary processes are of the utmost importance. Recently,5 we have proposed a classification of porous structures based upon the relative position of the size distributions of sites and bonds. Five types of structures have been recognized, each one exhibiting a characteristic behavior during capillary condensation and evaporation. Geometric characteristics of pores play a major role in the development of adsorption hysteresis. Nevertheless, the energetic heterogeneity of the surface of pores has probably some influence, the onset of the ascending boundary curve being certainly the most sensitive one to this effect. At higher uptakes, however, the multimolecular adsorbed layer would considerably reduce that initial influence. Bond Interactions All along the previous development, even if the meeting of C bonds at every site had been foreseen, site-bond interactions were considered indeed as if only one bond would meet each site. Here, previously neglected bond interactions will be taken into account. For the sake of simplicity, we assume throughout this treatment that bonds meet sites radially. The key concept in our former description was the construction principle, preventing that the size of any bond be bigger than the size of the site to which it leads. However, if multiple bonds meet at a unique site, it would seem that a much more restrictive condition arises, as bonds could not be allowed to approach so close to the site size. An explanatory example is now provided. Suppose that four bonds are linked to a site, and all these five elements are of the same size, as depicted in Figure 1a. This figure clearly shows that, in fact, the site cannot keep the same size as its bonds, as the inscribed circle (dashed line) becomes x2 times bigger than the initially proposed size RS (full line). If the originally proposed site size were maintained, the shaded areas
Figure 2. (a) Bonds are confined into “reserved domains”. (b) A bond invades the reserved domains of its neighbors.
would correspond to regions of interpenetrating bonds where the construction principle, set in its original form, would turn out to be insufficient. Let us now consider the spatial situation of six bonds meeting at a site, all of these elements being equally sized (Figure 1b). Now the originally proposed site is totally hidden inside the 6-fold star of bonds. Under these conditions it could seem better to consider as the “site” a sphere (broken line) x2 times bigger than the previous one, its outer surface (shaded areas) having the appearance of a “dividing helmet”. Approximate Treatment. In order to avoid interpenetration of neighboring bonds, a crude method can be proposed, simply consisting of dividing the surface of the site into C equally-sized “reserved domains”, each confining one of the C bonds, Figure 2a. This method is interesting since, for the first time in the context of a twofold description, both connectivity and dimensionality, which determine the maximum size RBmax of a bond, are taken into account as essential parameters for the quantitative description of the network. If we define R as the ratio RBmax/RS, then, for example in two-dimensional networks, R ) sin(π/C). According to this draconian method, the most relevant aspects of the twofold description are readily modified as follows: Construction Principle. The size of any bond is always smaller or at most equal to R times the size of the site to which it leads.
first law
B(RR) g S(R)
for every R
(6)
so that the highest possible overlap between the site and bond size distributions is controlled by R (the bond-size distribution is displaced to the left).
second law
φ(RS,RB) ) 0
for RRS < RB
(7)
As RRS is now taking the role of the restriction previously assigned to RS (for the original case of noninteracting bonds), it is straightforward to find from (5), in the event of RB e RRS, that
Description of Porous Media
(
exp φ(RS,RB) )
Langmuir, Vol. 12, No. 1, 1996 213
S(RR ) dS ∫S(R ) B - S) S
B
B(RRS) - S(RRS)
)
(
exp -
B(RR ) dB ∫B(R ) B - S) S
B
B(RB) - S(RB)
(8)
However, this oversimplified procedure is unable to manage some events which are likely to occur, essentially that some bonds invade the reserved domains of their neighbours; see Figure 2b. Then this approximate treatment constrains too much the sizes of bonds. An exact treatment is now to be envisaged, in which C bonds meeting at a site display all possible sizes and interact on the same footing, i.e. sharing alike the randomness involved in these events. Exact Treatment for C Bonds Meeting at a Site Consequently with the ideas exposed in the previous paragraphs, the construction principle has to be seriously reformulated (at least when sites are conceived as hollow spheres, and bonds as hollow cylinders) as follows: in an ensemble of C bonds delimiting a given site, these must be small enough as to allow a proper linking. The hyperspace of Bond Sizes. In order to determine which bond sizes should be properly assigned to a site of size RS, a hyperspace of bond sizes will be established, i.e. a C-dimensional space of coordinates RB1‚‚‚RBc. Therein each point represents an event of having C bonds of sizes RB1‚‚‚RBc linked to a common site. If the site has the size RS, this event possesses a density F(RS∩RB1‚‚‚RBc) to occur. Note that the value of F to be assigned to each point RB1‚‚‚RBc is then dependent on RS. The twofold (or, more properly, the manifold (C + 1)) description of a porous medium consists precisely in determining F in terms of RS and RB1‚‚‚RBc. The “incumbent volume” for a certain RS is to be defined as the volume in the hyperspace of bond sizes where the construction principle, given the mutual interaction of the C bonds, is fulfilled. F(RS∩RB1‚‚‚RBc) ) 0 outside the incumbent volume. In order to clarify this concept, several examples are provided: (i) (a) Bond interactions are absent if merely (see Figure 3a.1) one bond (in white) is linked to each site (in black). Of course, it is imposible to form a continuous network in this case, but this formalism corresponds, for instance, to the topic of assigning activation energies (playing the role of bonds) to independent adsorption sites, each one endowed of an adsorption energy well. In this case, the incumbent volume degenerates into a line going from zero to RS (Figure 3a.2). (b) Even when several bonds meet at a site, these are unable to disturb each other by virtue of the geometry of their linking. In Figure 3b some examples corresponding to multiple bonds without interaction are presented. (c) Two cylindrical bonds (in white) attack coaxially and by opposite sides a spherical site (in black) and, due to this fact, are unable to interact (Figure 3b.1). (d) In HeleShaw cells,12 which are the spaces between rugged-parallel plates (Figure 3b.2), element sizes correspond to the local perpendicular spacing between the plates. This characteristic precludes any interaction among the multiple bonds meeting at the common site. (e) The site (see Figure 3b.3) is conceived as a square object (in black) and the bonds (in white), due to their mutual orientation and disposition around the site, meet without interaction. (11) Mayagoitia, V.; Kornhauser, I. In Fundamentals of Adsorption IV; Suzuki, M., Ed.; Kodansha: Tokyo, 1993; p 421. (12) Paterson, L. J. Fluid Mech. 1981, 113, 513.
Figure 3. Several cases of noninteractive bonds (geometries and incumbent volumes): (a) a lonely bond; (b) multiple noninteracting bonds.
These sorts of structures have a practical interest: they correspond to the micromodels conceived and studied by Lenormand,13 consisting in equal depth-square capillary channels (bonds) intersecting at square sites of the same depth (rectangular cross-sectioned sites would require a bivariate distribution of site sizes, each site having two characteristic sizes). Another example of these noninteracting bond systems, while out of the scope of porous structures, is related to the twofold description of heterogeneous adsorbent surfaces.14,15 In all these cases, the classical interpretation of the construction principle is still good enough. The incumbent volume, Figure 3b.4, is the square region where RB1 e RS, RB2 e RS, ..., RBc e RS. Then the relationships previously defined for porous1-10 and surface11,14,15 structures outlined in the introduction keep all their validity for these systems. (ii) Two illustrative situations of strong bond interaction are depicted in Figure 4: (a) The site (in black) is a strip and two bonds (in white), running parallel, share the site length, Figure 4a. So that RS g RB1 + RB2 and the incumbent volume corresponds to an equilateral triangle determined by the above condition (obviously, radial symmetry is not applied to this particular example). (b) The site is a sphere (in black), and two interacting bonds (in white) meet perpendicular to each other, Figure 4b. The condition delimiting the incumbent volume is now RS2 g RB12 + RB22. From these primary examples, the interaction of, e.g. six cylindrical bonds meeting at a spheric site, can be readily imagined. B(R), the proportion of bonds having a size R or smaller is obtained simply by means of integrating FB(R) dR from 0 to R, eq 1. Had the same quantity been calculated from (13) Lenormand, R. The`se de Docteur e`s Sciences (Institut National Polytechnique de Toulouse, 1981). (14) Mayagoitia, V.; Rojas, F.; Pereyra, V. D.; Zgrablich, G. Surf. Sci. 1989, 221, 394. (15) Mayagoitia, V.; Rojas, F.; Riccardo, J. L.; Pereyra, V. D.; Zgrablich, G. Phys. Rev. B 1990, 41, 7150.
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a
b
Figure 4. Several cases of strongly interactive bonds (geometries and incumbent volumes): (a) parallel bonds; (b) perpendicular bonds.
the hyperspace of bond sizes, this would be associated, in principle, with the integration, over the incumbent volume of R, i.e. of FB(RB1) ... FB(RBc) dRB1 ... dRBc. When bond interactions are absent, this operation renders B(R)c instead of the expected value, B(R), no matter the particular form of the bond size distribution and the procedure of integration. Then, every element FB(R) dR within the bond size distribution corresponds to (FB(RB1) ... FB(RBc) dRB1 ... dRBc)1/C inside the hyperspace of bond sizes. First Law. Again, two self-consistency laws guarantee the fulfillment of the construction principle. The first law deals with the whole size distributions and states: bonds must be kept small enough as to be accommodated together with a set of sites having a given size distribution. Being primarily interested in having enough bonds to link all sites of size RS or smaller, it is necessary to consider first that these bonds lie precisely in the incumbent volume of RS. The proportion of sites smaller than or equal to RS is S(RS). The proportion of bonds of size RS or smaller is given by:
∫R
BC(RS) ) {
S
...
distribute properly sets of values RB1 ... RBc among sites of all sizes. This treatment follows very closely those for the deduction of similar functions φ playing a role in the topological structuralization of porous media and adsorbent surfaces, when mutual bond interactions are absent.3,11,14 We consider first the sites of smaller sizes, and then assign to them a set of values RB1 ... RBc of randomly chosen bonds within the particular incumbent volume of RS. If the first law has been observed, this is always possible. The procedure follows in such a way as to continue the exhaustion of sites, each time of bigger and bigger sizes. This method would seem very arbitrary at first sight. This point will be discussed later, when an alternative procedure will be outlined. An intermediate stage of such a process is accounted as follows: According to eq 11 the conditional probability density of finding a set of values RB1 ... RBc for a site of a given size RS is
F(RB1 ... RBc/RS) ) FB(RB1) ... FB(RBc) φ(RS,RB1 ... RBc) (13) Sets of bonds of fixed values RB1 ... RBc are progressively exhausted as sites with higher and higher RS (starting with those sites with RS equal to RC, the minimal size of a site suitable to accommodate this precise set of RB1 ... RBc bonds) are undergoing assignation of bonds. The “exhaustion function”, F (RB1 ... RBc,RS), of values RB1...RBc being exhausted for sites of size RS or smaller, grows differentially along the process of assignation according to two factors: (i) the fraction of new sites, dS(RS) and (ii) the provision that the sites have in fact this particular set of bonds, F(RB1...RBc/RS):
dF (RB1...RBc,RS) ) F(RB1...RBc/RS) dS(RS) ) FB(RB1)...FB(RBc) φ(RS,RB1...RBc) dS(RS) (14) Taking into account the restriction imposed by eq 12, the construction principle may be expressed as
∫R
S
∫0FB(RB1) ...
‚‚‚
∫0 φ(RS,RB1...RBc) FB(RB1) ... FB(RBc) dRB1 ... dRBc ) 1 (15)
FB(RBc) dRB1 ... dRBc}1/c (9) This integral runs over the incumbent volume of RS. So that the first law now presents as:
first law
BC(RS) g S(RS)
for every RS
(10)
Second Law. A second law is still required in order to fulfill locally the construction principle. If the probability density for the joint event RS∩RB1 ... RBc to occur is
F(RS∩RB1 ... RBc) ) FS(RS) FB(RB1) ... FB(RBc) φ(RS,RB1 ... RBc) (11) The second law is expressed as
second law
φ(RS,RB1 ... RBc) ) 0 outside the incumbent volume (12)
In order to develop an expression for φ inside the incumbent volume, let us propose a particular method to
The fraction of exhausted bonds of all sizes inside the incumbent volume when sites having sizes between RS and RS + dRS are linked is determined by performing an integration along RB1...RBc while keeping RS constant. From (14) and (15) we obtained
∫R
{
S
...
∫0 dF dRB1...dRBc}1/c ) dBC(RS)
(16)
During a differential step of assignation, the ratio between the fraction of bonds dF dRB1...dRBc (all of a specific set RB1...RBc, i.e. at a fixed point inside the incumbent volume), and the total fraction of bonds dBC(RS) (allowing for all the possible sets, i.e. for the whole incumbent volume), both assigned to sites of size RS, is equal to the ratio between available bonds RB1...RBc (those which have not yet been assigned to any site), [FB(RB1) ... FB(RBc) - F (RB1 ... RBc,RS)] dRB1...dRBc, and the fraction of all the available bonds of all possible sets (the free part of the incumbent volume, containing the sets not yet assigned to any site), BC(RS)-S(RS)
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dF (RB1...RBc,RS) dS(RS)
) FB(RB1)...FB(RBc) - F (RB1...RBc,RS) BC(RS) - S(RS)
(17)
This expression can be integrated, while keeping RS constant, between the following limits: at the start of the assignment of the particular set of bonds, we deal with sites of size RC and F ) 0, while the assignment proceeds to a stage characterized by the conditions RS > RC and 0 < F < FB(RB1)...FB(RBc). Since the set RB1...RBc has been kept constant, FB(RB1)...FB(RBc) also adopts a fixed value. Then, the integration yields:
F (RB1...RBc,RS) ) FB(RB1) ...
[
(
FB(RBc) 1 - exp -
]
S(R ) dS ∫S(R ) B - S) C S
c
(18)
From eqs 14, 17, and 18, a final expression for φ is obtained:
(
S(R ) dS ∫S(R ) B - S) C S
exp φ(RS,RB1...RBc) )
Figure 5. Maximum allowed overlap of the twofold distribution according to three methods of assessing bond interactions (see text).
C
BC(RS) - S(RS)
(19)
Had bonds been taken from the largest to the smallest ones, while applying in each case the proper site sizes (randomly but still respecting the construction principle), this inverse procedure would lead to the completely equivalent expression for φ
(∫
exp φ(RS,RB1...RBc) )
)
dBC BC - S
BC(RS)
BC(RC)
BC(RC) - S(RC)
(20)
If the function φ were always equal to unity for any combination of values RS, RB1...RBc, the porous network should structuralize totally at random. This can occur only when the incumbent volume contains all the spectrum of bond sizes for any value of RS, i.e. BC(R) ) 1 for every value of R (a zero overlap between the site and bond size distributions is now revealed as an insufficient condition for having a fully random character of the network). Conversely, φ * 1 means that these values are correlated. Since the above expressions for φ have been obtained for the maximum randomness, the resultant morphology of the porous network can be termed as the most verisimilar, i.e. that corresponding to the minimal number of constraints (the reformulated construction principle being the only restriction introduced). Or, in other words, this model is the closest that corresponds to a real system in the absence of any other information (additional constraints) about the particular physicochemical nature of its genesis or any other relevant aspect of it. Some kinds of possible constraints that could affect the former expressions are mentioned elsewhere.2 Discussion In order to estimate the effect that bond interactions introduce in the morphology of networks, now we propose to compare the value of F(RB/RS) ) FB(RB)φ(RS,RB) as a function of RB for a set of values of RS. The geometry to be considered in this example is very simple, that depicted in Figure 4b, consisting of two cylindric bonds meeting normally to each other at a spherical site. The following
Figure 6. Probability densities for a bond of a given size to be linked to a site of size corresponding to the maximum of the site size distribution, according to three methods of assessing bond interactions (see text).
cases will be visualized: (i) under the hypothesis of noninteraction, φ(RS,RB) will be simply obtained by means of eq 5, (ii) using the approximate treatment, φ(RS,RB) will be derived from eq 8 with R ) 1/x2, and finally (iii) with the exact procedure, and noting that φ(RS,RB) (concerning the precise size, RB, of a given bond, while the size of the second bond, RB2, may adopt any value in accordance with RB and RS), is related to φ(RS,RB1,RB2) by
F(RB/RS) ) FB(RB) φ(RS,RB) )
∫
FB(RB) 0xRS2-RB2 φ(RS,RB,RB2)FB(RB2) dRB2 (21) φ(RS,RB,RB2) inside the above integral is calculated in this case with successive values of RC ) (RB2 + RB22).1/2. First of all, in order to propose a convenient twofold distribution, sufficiently overlapped as to promote important size correlations while still fulfilling all three corresponding versions of the first law given by eqs 2, 6, and 10, it is necessary to investigate what is the maximum overlap allowed by each of the three methods. Figure 5
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shows that, starting from a given Gaussian bond size distribution, the closest site-size distribution results to be (i) obviously the same (full overlap) for the first method, (ii) a curve much more disperse and distant, with low overlap for the second method, and (iii) an intermediate situation of dispersion, location, and overlap for the exact method. The bond distribution is then kept fixed while the site distribution can now be chosen in a convenient position. In Figure 6, F(RB/RS) ) FBφ is plotted against RB for a constant value of RS (corresponding to the maximum of the size distribution). It is clearly shown that (i) φ is almost unity for the first method (since the coincidence between the bond distribution curve and FBφ is very high), so that no correlations would appear if bond interactions were absent, (ii) the approximate treatment greatly overestimates the effect of bond interactions, FBφ is very important for low values of RB, vanishing naturally at a value of RB ) RS/x2, and finally (iii) the exact method reveals important interactions, while all possible bond sizes are still allowable to link to sites, only provided that they be smaller than RS.
Mayagoitia et al.
Conclusions The successive incorporation of restrictions of increasing order of complexity to describe porous media reveals that (i) the twofold description formalism seems to have a remarkable property of adaptation, as revealed by the similarity of forms of the correlation function φ, (ii) oversimplified alternative methods appear to be very inexact, and (iii) the generalizations involved in the consideration of new effects provide a better understanding of the original theory. Acknowledgment. This work was supported and made possible by the National Council of Science and Technology of Me´xico (CONACyT) under Project No. 5387 - E, as well as under the joint Project: “Cata´lisis, Fisicoquı´mica de Superficies e Interfases Gas - So´lido” by CONICET (Argentina) and CONACyT (Me´xico). Our special thanks are due to Miss Irma D. Franco Morales, for her help in performing the calculations. LA940704K
