Langmuir 1986, 2, 562-567
562
Multilayer Physical Adsorption on Fractal Surfaces J. J. Fripiat,*t L. Gatineau, and H. Van Damme CRSOCI, CNRS, 45071 Orl&ansCedex, and Elf-Aquitaine, Paris, France Received October 30, 1985. In Final Form: April 1, 1986
A general equation valid for multilayer adsorption on fractal surfaces is demonstrated within the framework of the classical BET theory. It is assumed that each of the i layers within slabs of i adsorbed layers has an apparent fractal dimension that is different from the mathematical fractal dimension of the surface. A two-dimensional simulation procedure suggests that within the fractal regime Didecreases as i increases. Numerical examples are computed. They show that increasing the fractal dimension of the surface affects the adsorption isotherms in the same way as decreasing the maximum number of adsorbed layers which can be built on the surface; namely, the upward curvature of the isotherms at high relative pressure is decreased. The computation of the monolayer content from the BET transform remains valid within an uncertainty less than 10% for fractal dimension lower than or equal to 2.5. Above 2.5, the upper limit of the linear BET transform shifts toward increasingly small pressures. 1. Introduction In 1916, Langmuir' postulated that when a molecule strikes an empty adsorption site of a solid at a temperature close to that of liquefaction, it remains attached to that site for some period of time and then it reevaporates. The steady state is reached when the rate of condensation equals the rate of evaporation: a&, = blsl exp(-E1/RT) (1)
= CSOX' (5) From this, one can calculate the total number of molecules on the surface which leads to the basic BET equation:
where P is the gas pressure, a. and bl are kinetic constants, depending only on temperature, and so and s1 are the bare surface area and the surface area covered with a monolayer, respectively. El is the energy of adsorption of that layer. The Langmuir model is naturally restricted to monolayer adsorption since molecules striking an occupied site are not given any chance to remain adsorbed. Also implicit in the model is the assumption that there is no energetic or geometric disorder on the surface, namely, that all the adsorption sites have the same adsorption energy and the same capture efficiency, independently of the fractional surface coverage. In order to account for multilayer adsorption, which is the standard situation in physical adsorption, Brunauer, Emmett, and Teller: in 1938, extended the steady-state equation (1)to layers of order i larger than 1: alpsl = b2s2exp(-E2/RT)
where Ns is the number of adsorbed molecules, N , the number of molecules necessary to cover the surface with a monolayer, and Ns/Nmthe surface coverage. x is nothing else but the partial pressure PIPo of the vapor at the temperature of the adsorbent. An important parameter of the BET isotherm 6 is n, the maximum number of layers which can be built on the surface. In the most classical form of the BET equation, n is assumed to be infinite, and this leads to a considerable simplification of eq 5:
al-lPsi-l= bis, exp(-E,/RT)
(2)
where siis the surface area covered by i layers. Assuming that El # E2 = E3 = ... E, = EL
and defining EL
x = Pg-l exp-
RT El y = - P expbl RT a0
(4)
a simple relationship between the si's and the bare surface area, so, is obtained by rearranging eq 1 and 2: Scientific Counsellor, Elf-Aquitaine.
0743-746318612402-0562$01.50/0
s,
NS _ -
Nm
Ccix' 1=1
(6)
l+C?X' 1=1
NS _ -
cx
(7) (1 - x)[l + (C - l ) x ] N, In this '"-form", the BET isotherm has proven to be exceptionally useful for determining the surface area of solid materials, in spite of the notorious flaws of the underlying model (the most serious being to neglect lateral interactions and to consider that only the first layer has an adsorption energy different from the heat of condensation of the liquid adsorbate). As far as the geometric properties of the adsorbent surface are concerned, the BET model is best suited to ideally homogeneous planar surfaces since the number of molecules in a slab of i layers covering an area s, on the surface is simply taken as n, = islno (8) where nois the number molecules adsorbed in a monolayer per unit surface. Equation 8 is certainly not expected to hold on surfaces with important roughness. The purpose of the present paper is to extend the classical BET model to fractal surfaces (rather than to irregular surfaces in general, because the infinite morphological variety of surface irregularities that one can imagine seems hard to approach by a single treatment). In fact, the considerable body of evidence presented by Avnir, Pfeifer, and Farin3-5 in recent years suggests that (1) Langmuir, I. J. Am. Chem. SOC. 1916, 38, 2221. (2) Brunauer, S.;Emmett, P. H.; Teller, E. J. Am. Chem. SOC.1938, 6'0, 309
61986 American Chemical Society
Multilayer Physical Adsorption on Fractal Surfaces
Langmuir, Vol. 2, No. 5, 1986 563
fractal surfaces might well be a general rule for powdered materials. Since BET theory is often used to calculate monolayer contents, a reexamination of the BET model on fractals seems to be necessary. 2. Monolayer and Multilayer Adsorption on Fractal Surfaces: General Features In this section, we will try to answer an important preliminary question about the structure of a fractal multilayer: How many molecules does one need to cover a fractal surface with 1, 2, ... i molecular layers? Indeed, without a general answer to this question in terms of fractal dimension, there is little hope to derive a BET-like adsorption isotherm for fractal surfaces. Monolayer adsorption will be considered first and multilayer next. 2.1. Monolayer Adsorption. According to Pfeifer and Avnir,G the fractal dimension D* of a surface Z “extracts the intrinsic (geometric) irregularity that persists through all level of resolution”, D* being defined as
where r is the length of an isotropic yardstick used to perform the measurement and N(r) the number of yardstick molecules with their center on 2, requested for paving completely the surface, Avnir and Pfeife? suggested that measuring the monolayer coverage N , with a series of “homologous” molecules of increasing size opens an experimental technique for determining the fractal dimension of the surface since
N,
-
u-’iI2
(10)
where a is the surface area covered by one molecular species in the monolayer assembly. The monolayer coverage technique differs from the “ideal” yardstick method in two respects: (i) molecules have a finite size and (ii) they cannot cross the surface. As discussed elsewhere’ this can lead to nesting problems and to an underestimation of D*. Experimentally however, D, is the only meaningful quantity. Most real solid surfaces are not expected to be fractal at all length scales. Hence, one has to consider an inner (S,) and an outer (S? cutoff fixing the boundaries8
s, Q u Q S’
(11)
of the fractal regime in which eq 10 holds. Slcan be considered as the smallest area which can statistically generate the whole surface by application of the dilational (up to S? and translational (beyond S? symmetry operations. At length scales shorter than S1or longer than the surface is euclidean. S1and S’ are the crossover points. In fact, rather than by the a-dependence of N,, we are primarily interested in the absolute number of molecules which, for a given u, can be adsorbed on Z. On a planar surface, Zo, this number of molecules, per unit surface, is
s’,
no = u-l
(12)
Now consider the following process; we “fold” Bo in a (3)Avnir, D.; Farin, D.; Pfeifer, P. J . Chem. Phys., 1983, 79, 3566.
(4) Avnir, D.; Farin, D.; Pfeifer, P. Nature (London) 1984, 308, 261.
(5)Avnir, D.;Farin, D.; Pfeifer, P. J . Colloid. Interface Sci. 1985,103, 112. (6)Pfeifer, P.; Avnir, D. J. Chem. Phys. 1983, 79, 3558. (7) Van Damme, H.; Levitz, P.; Bergaya, F.; Alcover, J. F.; Gatineau, L.; Fripiat, J. J.; submitted for publication in J. Chem. Phys. (8)Pfeifer, P.; Avnir, D.; Farin, D. J. Stat. Phys. 1984, 36, 699.
D = 1.05 h
+
D r1.3
Figure 1. Illustration of the procedure described in section 2.1, in which a flat surface, 4, is “folded”in order to make a surface of fractal dimension D1. Each curve in this figure has-when
stretched out-the same length and the same inner cutoff, S1. It is clear that as D1increases, a smaller number of molecules can be adsorbed. certain number of elementary folds of area S,, in such a way as to make a surface Z1of fractal dimension D,,as illustrated in Figure 1. The number of molecules per unit surface which can be adsorbed on 8,in a monolayer, in the fractal regime (eq l l ) , can be written as
nl = Klu-D1/2
(13)
K1 = Sl(Dl/z-l)
(14)
More precisely, nl refers to the number of elementary folds which, if Z1were stretched out, would cover a unit surface. nl is smaller than no. As far as the evaporation-condensation equilibrium in monolayer adsorption (Langmuir model) is concerned, the direct consequence of the fractal character of Z1is that a correction factor f,,
should be applied to eq 1 in order to account fcr the effective surface area involved in adsorption and desorption (so and s1 are defined on the “unfolded”surface). However, since it applies to both sides of eq 1, it does not modify the equilibrium. 2.2. Multilayer Adsorption. In multilayer adsorption, the hull of the first adsorbed layer is the surface, Z2, on which the second layer builds up. A t large length scales, the presence of the first layer has little influence, and Z2 runs parallel to Z1.At short length scales, the fractal roughness of Z1is screened by the adsorbed molecules and Z2 is “smoothened” with respect to 8,. This screening effect is expected to end at length scales corresponding to u, the molecular cross-sectional area. Hence, the fractal regime of X 2 is no longer given by eq 11. The crossover point from Euclidean to fractal regime, i.e., the inner cutoff S2,is no longer at S1but somewhere around u (the exact
564 Langmuir, Vol. 2, No. 5, 1986
Fripiat et al.
Figure 2. Illustration of the procedure used for the measurement of the fractal dimension of a triadic Koch curve covered by 0, 1, ... i adlayers of molecules.
value depends probably on the shape of the molecules and their packing mode). Below a, the shape of Zz is still irregular, because the molecular shape, including vibrational motions, is never planar, but this roughness is not expected to be fractal. The same situation goes on as the number of adsorbed layers increases. &, the hull of the second layer, on which the third layer builds up, runs parallel to Z2 (hence, will be fractal with dimension D l ) at large length scales but will be Euclidean below the molecular size, a. The crossover point S , is again expected to be somewhere around a. And so on. Thus, a multilayer slab on a fractal surface is characterized by two well-defined regimes: D = 2 at short length scales (a but 6‘’ However, ). in order to describe accurately the growth of a BET-type multilayer, one needs to know the fractal behavior in the somewhat ill-defined crossover regime around a. In order to gain some physical insight on this, we performed a simple two-dimensional simulation. The simulation was performed on deterministic fractal curves (diadic and triadic Koch curvesg)with fractal dimensions in the range 1.05 < D < 1.50. The study was performed in two steps: (i) Adsorbed on the curve are 0,1,2, ... i layers of circular ”molecules”. The rule of the game is that each molecule should be in contact with its neighbors and with the curve, rl (or with the hull ri,for the layer of order i). The size of the circles was choosen such as to be somewhat larger than the inner cutoff (a > Sl). (ii) The fractal dimension of each of these systems is measured by covering them with a series of circles of increasing size, r. It should be stressed that this second step is not intended for simulating multilayer adsorption. Its purpose is just to measure the fractal exponents from In N, vs. In r plots ( N , is the number of yardstick circles of size r used to cover the curve). The procedure is illustrated in Figure 2 and typical results are shown in Figure 3. The main conclusion from this measurement is that the crossover region around a is much broader than expected. rz,r3,ri are indeed found to recover the fractal dimension D1 of at large length scales, but only beyond inner cutoffs, Si,much larger than a. For instance, in the example of Figure 2, S2 N 3a. S, is even larger. Interestingly, in the intermediate region, a < r < Si,the In N, vs. In r plots are quasi-linear, with a (negative) slope decreasing in absolute value as i increases. In other words, everything happens as if the set of surface Zi generated by a growing multilayer could be characterized, around a, by a family of apparent fractal dimensions Di, such as
(9) Mandelbrot, B. T h e Fractal Geometry of Nature: Freeman: San Francisco, 1982.
r
I 4
3
5
Ln r
Figure 3. Determination of the (apparent) fractal dimensions D1, D,, and D,of the Koch curve of Figure 2 covered by 0,1, and 2 layers of molecules, respectively. The Si’sare the inner cutoffs. u is the cross-sectional area of the molecules.
\i M
s3
1.0
0
I
1 2 3 number of adlayers
Figure 4. i dependence of the apparent fractal dimension of a triadic Koch curve covered by i adlayers, in the crossover regime u < r < Si(i = 1, 2 and 3).
This is nothing else than a progressive smoothing, or screening, of Z1 by its molecular cover, as shown in Figure 4. This progressive smoothing is directly apparent in Figure 5, in which up to 14 layers of circles have been “adsorbed” on a curve rl with fractal dimension D1 = 1.25. As the number of layers increases, an increasingly large fraction of the porosity generated by the convolutions of rl is filled. Increasingly large features of the curve are “rubbed out”, and the length of the ri’sdecreases steadily. In other words, the number of molecules, Nmi,in the layer of order i decreases as i increases. The numerical data (Figure 6) show clearly that the smoothing of is faster in the first layers and suggest that Nmi (or ni, if one refers to unit surface on rl)tends toward a constant value for large i.
Multilayer Physical Adsorption on Fractal Surfaces 0.
Langmuir, Vol. 2, No. 5, 1986 565
,
Figure 7. In-In plot of the i dependence of the ratio ni/nl, for i 3 2, in the simulation of Figure 5. Table 1. Experimental Parameters of the Empirical Equation ( 1 9 ) O D. K a 1-D. Figure 5. Illustration of a few steps of a simulation in which 14 layers of circles were "adsorbed" on a fractal curve of dimensionality D,= 1.25. The orientation of the edges of the multilayer were drawn as if the c w e was part of a larger curve with the same fractal dimension.
Figure 6. i dependence of the number of molecules in the layer of order i in the simulation of Figure 5.
A direct way to evaluate the smoothing or pore-filling efficiency of each layer is to calculate the correction factors defined as
f; = Nm;/Nml= s / n l
(18)
for i 3 2. Figure 7 shows that-at least in the example which is shown-f; obeys a remarkably simple expression f; = Ki-
for i 3 2
(19)
where K and a are two constant parameters. K is very close to 1(0.96) and a is very close to D,- 1. In fact, this seems to he a general behavior since the same result was
1.05 1.25 1.40
1.0 0.96 1
-0.045 -0.19
4.38
-0.050 -0.20 4.40
" f , = Xi" resulting for simulated adsorption procedure as illustrated in Figure 5.
observed with other curves in the range 1.05 < D < 1.50 (Table I). A priori, it is not unexpected that the smoothing or pore-filling properties of a multilayer are governed (via a power law) by an exponent a = D,- 1rather than by D,. Indeed, D, - 1 is a dimensionality which characterizes directly the scaling of the porosity generated by the wiggleness of r,. This comes from the rule for the intersection of fractal sets proposed by Mandelbrot (ref 9, p 365), which states that if two sets, Saand Sb, with fractal dimensions D. and Db, are such that D,+ Db 3 E (E is the dimension of the embedding Euclidean space), then D. + D b- E is the fractal dimension of the intersection Sa n Sb.This rule, applied to the intersection of a fradal m e of fractal dimension D, = D,and an arbitrary straight line (D,= 1)drawn through it, in a plane (E = 2), leads to D (S, n Sb)= D , - ( E - 1) = D,- 1. The intersection is a dust. Each point of this dust is on a wall of one of the many pores of many sizes generated by the fractal curve. D,(E - l),the fractal dimension of this dust, characterizes the degree of concavity of the curve and the scaling of the pores which are progressively filled as the multilayer thickens. The natural extension of this is that, in the case of a fractal surface, XI, covered by i layers of molecules, the exponent a should he equal to a = D,- (E - 1) = D,- 2, and the expression for the corection factors should he
f, = n;/n, = KiF-*) for i B 2 (20) and K should be -1. Equation 20 permits the calculation of the total number of molecules in a multilayer slab on a fractal surface, knowing the number of molecules in the very first layer of the slab. It is an empirical equation, buh the evidence obtained from the simulation experiment and the rationale which is behind it seem to be strong enough for using it-at least as a first approximation-in a BET-like theory of multilayer adsorption on fractal surfaces. Together with eq 15 for the first layer, it allows for the calculation of the number of molecules in a mul-
Fripiat et al.
566 Langmuir, Vol. 2, No. 5, 1986
tilayer slab, knowing only the intrinsic parameters of the surface, namely, the fractal dimension and the inner cutoff. Finally, it should be noted that, just as for monolayer adsorption, the correction factors (i 3 2) do not modify the evaporation-condensation equilibrium in eq 2 , since the same correction has to be applied to both the right-hand the left-hand sides (rhs and lhs) (the same Z i is involved in the evaporation and in the condensation process). 3. The Adsorption Isotherm In what follows, we will stick to the fundamental kinetic and energetic assumptions of the BET theory (eq 3 ) , and we will only consider the geometric modifications brought about by the fractal character of the surface. Let A be the area of the Euclidan surface Zo and A , that of the "folded" fractal surface Z1, measured using a molecule of cross-sectional area u. A, is, of course, the only experimentally meaningful quantity. The monolayer content, Nm, of 2 , is
N m = Aau-' = nOAa
51
I I
(21)
,
,
,
0.5
L
1
P/P,
and
.
=x
(22)
Figure 8. Adsorptions isotherms obtained for n = 10, C = 100, and 2 d D1 d 3.
in agreement with eq 15. According to the BET logic, A is divided in fractions so, sl, ...s, covered by 0, 1, ...i layers of adsorbate such as
It can be seen easily that Ns converges more rapidly for D , > 2 than for D , = 2. By combination of eq 24 and 19,
Aa = flA
n
A =
n
Csi and A,
i=O
= flCsi
(23)
i=O
Since the equations accounting for the evaporation-condensation equilibrium are not modified (section 11),the si's are still related to so by the same equation as in BET theory (s, = C S G ~eq , 5 ) , and
the reduced adsorption isotherm, N s / N m = f ( x ) ,is obtained:
1Ym
1
+ Cik= l x i
n
N , = nof,so(l + C C x ' )
(24)
i=l
On the other hand, the total number of molecules on the surface is
where Ni is the number of molecules in the slab of i layers covering the area fisi on Z1. From the definition of f i (eq 18), Ni can be expressed as N i = nosi(fi + f i f i + f J 3 +
(26)
or n
Ni = nosifl(l +
Cfi)
(27)
i=2
This equation applies as long as S1 < u. From eq 5 (si = Csoxi) for si, NS
= not,cso[%ci i=,
5xi(?fj)1
+ ,=2 ,=,
(28)
Thanks to the semiempirical eq 20 (with K = 1 ) for f i , a useful simplification occurs: n
n
Ns = n o f l C s o ~ : i 2 - D ~ ~ x ~ j=i
(29)
which is a short-hand writing for N s = noflCso[(x+ x2 + . . . x ~ ) ~+~ (x2 -~+ I x3 +
+
. . . x " ) ~ ~ - ~. .I .
(30)
X ~ ~ ~ - ~ I ]
In this form, it does not contain f,, the correction factor for the first layer (f,is a function of S,, the inner cutoff of the fractal surface, and is not an easily accessible parameter). Equation 31 is the BET equation on a surface of fractal dimension D,.It has a general character as long as the semiempirical eq 20 applies. The only additional parameter with respect to the classical BET equation (eq 6 ) is D1.As anticipated, eq 31 turns into eq 6 for Dl = 2. It can also be seen easily that it reduces to a Langmuir isotherm for n = 1. Indeed, in that case N s --- c x _ N, 1+Cx
For higher values of n, N s / N mon a fractal surface will be always smaller than on an Euclidean surface as illustrated in Figures 8-10 which represent adsorption isotherms computed for C = 100 and 2 and 2 S D < 3. Increasing D1 affects the adsorption isotherms in a way which is similar to that obtained by decreasing n in the classical BET theory. The noticeable modification of the shape and specially of the curvature may be expected to influence the measurement of N , from the linear BET transform obtained by plotting x / ( N s / N , ) ( l - z) vs. x. Indeed for D1 2.8 the lack of linearity is evident. For D1 < 2.5 the scattering of the "experimental data" is not too pronounced and an observer who would not be aware of the fractal character of the surface would assign it to "experimental uncertainty". He would calculate N s / N , = 0.95 for D , = 2.2 and N s / N , = 0.89 for D1 = 2.5 instead of N s / N , = 1for D 1= 2. Thus for D1 S 2.5, N , can be obtained within 10% in the classical manner and an In N , vs. In u plot can be used in order to determine D,. For D , > 2.5 the range
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Langmuir, Vol. 2, No. 5, 1986 567
Figure 9. Adsorption isotherms obtained for n = 10, C = 2, and 2 d D1d 2.8.
of linearity of the BET transform shrinks to the extent that the computation of N , becomes questionable. On the other hand, the modification of the curvature of the adsorption isotherms belonging to Brunauer types I1 and I11 is indicative of the fractal character but this information is ambiguous since it may be confused with a limitation of the maximum number of layers which can be built on an Euclidean surface. 4. Discussion The physical fact which accounts for the effect of the fractal character of a surface upon multilayer adsorption is the filling of the (micro)pores generated by the surface, according to a scaling process. This can also be described as a progressive smoothing of the surface or, alternatively, as a decrease of the apparent fractal dimension of the surface as the thickness of the multilayer cover increases. Empirically, it is found that this smoothing can be quantitatively described in terms of a remarkably simple power law (eq 20) relating ni,the number of molecules in the layer of order i within a multilayer slab, to nl, the number of molecules in the first layer. When used within the framework of the classical kinetic and energetic assumptions of the BET model, eq 20 leads to a general n-layer form of a BET-like isotherm for fractal surfaces, which invariably predicts that the buildup of the
Figure 10. Adsorption isotherms obtained for n = 100,C = 100, and 2 d D, 6 3.
multilayer coverage (i.e., NS/N,) will be slower on a fractal surface than on a flat surface. An important question is the relationship between n, the maximum number of layers which can be adsorbed on the surface, and D1, the fractal dimension of the surface. Indeed, one might a priori expect n to decrease as D 1increases, because of the increasing space-filling character of the surface, and this could still more restrict the growth of the multilayer. In fact, this seems to be a misleading problem because adsorption on a fractal is intrinsically a scaling process. The shape of the surface generates a broad distribution of pore sizes,6and each type of pore is filled with a different number of layers. What the decrease of ni with i implies is that as the multilayer thickens, an increasingly small hull is available for condensation of the next layer. This does not correspond to a limitation of n over the whole surface of the adsorbant but to a reduction of the surface over which n can still increase.
Acknowledgment. We thank the numerous anonymous referees whose suggestions have been most fruitful. Discussions with Professors Pfeifer and Avnir were enlightnening.
