Langmuir 1993,9, 2499-2503
2499
Connectivity of Heterogeneous Porous Structures and Catalytic Deactivation R. J. Faccio,t A. M. Vidales,? G. Zgrablich,*ptJ and V. P. Zhdanovs Departamento de Flsica, Universidad Nacional de San Luis, 5700 San Luis, Argentina, Centro Regional de Estudios Avanzados (CREA), Gob. de la Provincia de San Luis, C.C. 256,5700 San Luis, Argentina, and Institute of Catalysis, Novosibirsk 630090, Russia Received October 20,1992. In Final Form: January 8, 199P Amore realisticrepresentationof heterogeneousporous structuresand its applicationsto the phenomenon of catalytic deactivation under kinetic control conditions is considered. Pore space is considered as a correlated network of voids (sites) interconnected by necks (bonds) following the so-called construction principle. Percolation theory is used to show that the connectivity of the pore space is a fundamental concept for understanding the catalytic behavior. Results for different types of porous structures me presented.
Introduction Catalyst deactivation has been the subject of intensive research activity because of its great technological importance.' The deactivation of porous catalysts as a result of coke formation includes coverage of active sites simultaneously with pore blockage. An analysis of this process is a very difficult problem because one has not only to describe reaction and diffusion in a porous medium, but it is also necessary to take into account interconnections between pores. Early modeh of catalytic deactivation have used simplified representations of the catalyst's porous structure, such as a bundle of nonintersecting parallel tubes, but they fail in determining the effect of interconnectivity of the pore system. In an attempt to describe catalyst deactivation by site coverage and pore blockage, more recent studies by Froment and co-workers2" have employed statistical representations of the pore space. These models represent an improvement over the previous ones, because they explain for the first time why the catalytic activity of a given catalyst undergoing deactivation will vanish much earlier than one would expect. However, the models are still unsuited to take into account the effect of the porous topology on the deactivation process. Catalytic deactivation due to coke formation presents an abrupt change in the properties of the system. The sudden transition from the activated into the deactivated state is characteristic of a percolative phase transition. The study of the deactivation process as a percolation one was first undertaken by Sahimi and Tsotsis? In that work the catalyst's porous matrix is represented by a threedimensional network of interconnected pores, and percolation theory and Monte Carlo simulation are used to ~~~
~
* To whom correspondence should be addressed.
+ Universidad Nacional de Sam Luis. CREA. Institute of Catalysis. Abstract published in Advance ACSAbstracts, August 15,1993. (1) See Catalyst Deactiuation; Delmon, B., Froment, G. F., Us.; Elsevier: Amsterdam, 1980. (2) Breeckman, J. W.; Froment, G. F. Ind. Eng. Chem. Fundam. 1979, 18,245; Chem. Eng. Sci. 1980,35,805; Ind. Eng. Chem. Fundam. 1982, 21, 243.
(3) Froment, G. F. NATOAdu. Study Zmt.,Ser. E 1982,54,103;Actas Simp. Iberoam. Catal. 1984, I , 80. (4) Marin, G. B.; Breeckman, J. W.; Froment, G. F. J. Catal. 1986,97, 416. (5) Nam, I. 5.; Froment, G. F. J. Catal. 1987, 108, 271. (6) Sahimi, M.; Tsotais, T. T. J. Catal. 1986,96,652.
0143-1463/93/2409-2499$04.00/0
describe the effects of the porous topology on the deactivation process. In this paper we consider the porous medium as a correlatednetwork of voids (sites)interconnectadby necks (bonds). Void and neck sizes are statistically represented by their distribution functions and overlapping between them is allowed. In order to construct the lattice we follow the construction principle: for a given void ita size must always be greater or at most equal to the size of any of ita necks. Percolation theory is then applied on a Bethe lattice of correlated voids and necks and this allows us to investigate the effect of porous structure on the catalytic deactivation process under kinetic control. The use of Bethe lattices is suitable because these types of lattices admit approximate analytic solutions for site and bond percolation probabilities.
Basic Equations In the analysis of the deactivation process to be developed in this section we make the following assumptions: (i) The pore space of a catalyst particle is treated a a lattice of voids (spheres) connected by necks (cylinders). The catalytic surface area mainly belongs to the voids of the lattice. (ii) Active sites are uniformly distributed over the pore space with initial concentration Ct. (iii) During the course of the catalytic reaction, active sitesare covered (poisoned)by depositiohof a contaminant, andsimultaneouslypart of the pore volume may be blocked by the growth of contaminant whiskers on each site. This deactivation process occurs under kinetic control. (iv) The conditions at the catalyst-fluid interface do not change with time. The process is isothermal. If the rate of disappearance of active sites can be described by a first-order law, then
dC,/dt = k(Ct - C,) (1) where C,is the concentration of covered sites (which are uniformly distributed in the pore space) and k is the deactivation rate constant. The available (not occupied by deposits) pore volume is given by Va(n) = ur:L - 2urnLC,b = ur:L(l- 2C,b/r,)
(2)
for necks of cylindrical shape with radius rn and length L, and 0 1993 American Chemical Society
Faccio et al.
2500 Langmuir, Vol. 9, No. 10, 1993
Va(‘) = 4 / 3 ~ r ,-3 4.lrr;CBb = 4/37rr,,?1 - 3C,b/rv) (3) for voids of spherical shape with radius r,. Here b is a pore average deposit volume occupying a single site. Following the analysis of ref 6, we can define effective radii rneffand rvefffor necks and voids, respectively, in such a way that
va(,)= r(r,sffl2L
(neck)
(4)
Va(v) = 4/3?r(r:ff)3
(void)
(5)
Hence
rneff = r , [ l -
cy,
g(~)/r,l~/~
(6)
r:f = rv[l- c y , g ( ~ ) / r ~ ~ ’ / ~ (7) where: = 2Ctb, cys = 3Ctb, 8 = kt (dimensionless time), and g(0) = 1 - e-@. Equations 6 and 7 demonstrated that if &n,v < r , , , then rn,veff > 0 at all times, but if a n , v 2 rn,v, one has rn,veff = 0 after a fiiite plugging time 8,. Then a t any given time 0, every neck or void with initial radius rn,v will be plugged as long as rn,v < r,!,v(8), where riJ8) = a,#@ (8) Le., the pores with radii r,,v such that the brackets in (6) and (7) become zero or negative. If a porous catalyst is considered as formed by a collection of independent pores, the previous analysis shows that the porous structure would never be plugged as long as the average pore radius is greater than a. It is important to state that in such a model the pores are plugged as a whole, not only their mouths. Thus, in this case and under the assumption of local or global kinetic control,pore blockage has no effect on the catalyticactivity, but the activitydecreasesas a result of the gradual coverage of the catalytic active sites and finally vanishes only when all active sites have been poisoned. However, it is found that the time for total deactivation of actual catalyst is always finite. This discrepancyis due to the fact that the pores that have partially reacted, but are not yet plugged, may be surrounded by plugged ones and become inaccessible. Thus, this evokes the picture of the formation of larger and larger islands of unplugged pores in the network that do not contribute to the reaction. Then we may expect that at some critical value of the fraction of unplugged pores the global connectivityof the pore space is lost. These ideas constitute the basis of the percolationtheory of disorderedsystemswhich are essentialto take in account the network characteristics and connectivity. However, first of all a model for a correlated network of voids and necks must be formulated.
Porous Network Model Following the model of porous materials developed by Mayagoitia and co-worker~,~.~ these may be represented by a network of interconnected pores, with two main components, sites (voids) and bonds (necks), of effective radii Rs and Re, respectively. The number of bonds emerging from a site is the connectivity of the network, while every bond is supposed to link only two nearestneighbor sites. Site and bond sizes can be statistically represented by their frequency distribution functions: (7) Mayagoitia,V.; Cruz, M. J.; Rojas, F. J. Chem.Soc., Faraday Trans. 1 1989,85,2071. (8)Cruz, M. J.;Mayagoitia,V.;Rojas, F. J. Chem. Soc.,Faraday Trans. 1 1989,85, 2079.
R Figure 1. Geometrical interpretationof the rule by which bonds are assigned to sites.
Fs(Rs) and FB(RB)in such a way that Fs(Rs)dRs (FB(RB) ~ R Bis)the probability of finding a site (bond)with radius between Rs and Rs + dRs (RBand RB + ~ R B ) . It is evident that the definitions of site and bond lead to the following “construction principle”: for a given site its size must always be greater than or at most equal to the size of any of its bonds. Overlap between the frequency curves is allowed, meaning that there exist some bonds with sizesgreater than that of some sites but not connected to them. Early studies on network modeling have assumed that the arrangement of these two types of entities to form the network is completely at random. However, the degree of randomness is usually limited in natural materials by correlations. The fact that sites and bonds are not randomly interconnected can be taken into account by assuming that site-bond connected pairs are statistically described by the joint probability density (7,8) F(Rs,RB)dRs ~ R BFs(Rs)FB(RB)~,(Rs,RB) a s~ R B (9) where F(Rs,RB)dRs ~ R isB the probability of finding a site whose radius is in the range (Rs,Rs + dRs), connected to a bond whose radius is in the range (RB,RB+ ~RB), and d,(Rs,RB)is a correlationfunction characterizingthe porous medium. If d, = 1for all values of Rs and RB,the events of finding the sizes Rs and RB would be independent and the network is constructed completely at random. d, # 1means that these events are correlated. The best structure (in statistical sense) is the one obtained by allowing the maximum degree of randomness that is compatible with the “construction principle”. The correlation function assigns to sites whose radii are in the range (Rs,Rs dRs), area a in Figure 1,bonds whose radii ) , a’ in Figure 1. Such are in the range (0,RB ~ R B area a correlation function is given by
+
+
for Rs < RB
lo where
In the simple case where site and bonds are distributed uniformly l/A f o r s I R s 5 s + A Fs(Rs) =( 0 otherwise FB(RB)
l/A f o r b I R B < b + A otherwise
=( 0
(11)
(12)
Connectiuity of Heterogeneous Porous Structures F(R)
Langmuir, Vol. 9, No. 10, 1993 2501
(a)
is probably small, but at some defined value of p s @B) an infiiite cluster entirelyconnecting the network will appear. This defined value of p s @B) is called the site (bond) percolation threshold, psc @gC).Exact values for psc and pBChave been foundonly for a fewtwo-dimensional lattices and for the Bethe lattices (which are infinitely ramified lattices without closed loops). Perhaps the simplest generalization of classical percolation is the random site-bond percolation, in which both sites and bonds on a lattice are randomly occupied with probabilities p s and p ~ . ’ ~ An infinite cluster of connected sites and bonds will appear when both p s and PB become large. How large each one must be depends on the other, thus psc and p~~ are interdependent. The line on the p s - p ~space formed with all the pairs @sC, P B ~ from ) which an infinite cluster appears is the boundary between the percolative and nonpercolative phases. In the correlated site-bond percolation problem, sites and bonds are not occupied at random. It is possible to obtain the percolation probabilities of the correlated porous structure, but firat we must define when a site or bond is considered to beoccupied. We choosethe following definition which replaces “occupiedn by “open”: a site (bond) is considered to be open if ita radius Rs (RB)is greater than some critical radius cs (cB). Then the following probabilities are defined for the general size distributions, Fs(Rs) and FB(RB),of sites and bonds
4 1 R
b*A r*A
Figure 2. (a) Uniform size dmtributions of sites and bonds showing the overlapping fl between them. (b) Bethe lattice of coordination number z.
d@~&) = exp[-(Rs - s)/(l - Q)]/(l- Q) exp[-Q/(l- Q)I/(l - Q) exp[-(R, - RB)/(l- a)]/ (1- Q) exp[-(b + A - RB)/(l- Q)]/ (1- Q)
{
for RB Is, R, Ib + A forRBIs,Rs > b + A for RB > S, Rs Ib + A for RB > 6,Rs > b + A
PB
~BFB(R,)
(13)
Here, Q is the overlapping between the curves Fs(Rs) and FB(RB),represented by the shaded area in Figure 2a. This parameter carries valuable information on the physical characteristics of the porous network. For example, a high degree of overlapwas found in consolidated porous materials and a low degree of overlap exists in regular packing8 of uniform spheres. An intermediate degree of overlap was observed for a random packing of nonuniform sphere^.^ It is our objective to study how this parameter affects the catalytic deactivation process. In the next section, the percolative properties of correlated sits-bond networks are presented.
Percolation in Correlated Site-Bond Lattices Percolation theory was developed for the description of disordered systems, and has been the subject of intensive research due both to its theoretical importance and to its many applications for describing a diversity of phenomena.l0-I4 In the classical percolation theory two fundamental problems exist: the random site percolation problem and the random bond percolation problem. In random site (bond) percolation the sites (bonds) of and infinite network are either randomly occupied with probability p s @B) or vacant with probability 1- ps(1p ~ ) Two . sites (bonds) are connected, if there exists at least one path of occupied sites (bonds) between them. A set of connected sites (bonds) surrounded by vacant ones is called a cluster. If p s @B) is small, the size of any cluster ~~~~
p s = iBFs(Rs)dRs = probability of finding an open site
~~
(9) Yanuka, M. J . Colloid Interface Sci. 1990, 134, 198. (10) Shante, V. K. S.; Kirkpatrick, S. Adv. Phys. 1971,20, 326. (11) Essam, J. W. In Phase Tramitiom and Critical Phenomena; Domb, C., Green, M. S., Eds.; Academic Press: New York, 1972. (12) Kirkpatrick, S. Rev. Mod. Phys. 1973,45, 674. (13) Stauffer, D. Phys. Rep. 1979,54, 1. (14) Essam, J. W. Rep. Prog. Phys. 1980,43,833.
= probability of finding an open bond
pSB= probability that an open site is connected to an open bond PSBS’
=
~dRS~,aSr~.
min(Rs&)
FS(&,) FB(RB) Fs(Rs?
4(RS&) 4(RSf$B) d R B = probability that an open bond is connected to two open sites PBSB’
=~
B ~ S L : ~ B J ~ F ~ ( R QFB(RB) ) FB(RB~)
$(RS&) cb(RS&)) a B j = probability that an open site is connected to two open bonds and so on for greater and greater clusters. All these probabilities are easily calculable in terms of cs, CB, and Q for the case of uniform size distributions, eqs 11,12, and 13. The site (bond)percolation probability P ( P )is defied as the probability that an arbitrary chosen site (bond), supposed to be open, belongs to the infinite cluster. We can bring up approximate solutions for these percolation probabilities only in the case of Bethe lattices. Let us consider a Bethe lattice of connectivity z,Figure 2b. From a given site there are z directions to follow, but there are only two from a given bond. If Qs (QB) is the probability that starting at an open site (bond) a walk of open elements (sites and bonds) in a given direction be of finite length, then (15) Zallen, R. The Physics of Amorphous Solids; John Wiley and Sons: New York, 1983.
Faccio et ai.
2502 Langmuir, Vol. 9, No. 10,1993
PS
t
0.5
0
0.5
1
PB
Figure3. Percolative phase transitiondiagram for the correlated Bethe lattice of coordination 3 showing the transition lines for some values of the overlapping, where QZ > QI.
PS=l-Q," PB=l-Q; (14) In the simplest case of z = 3, and taking into account only the correlations up to triplets, the following approximate analytical solutions are obtained16 (a) for ps I p B + 6/10(1- 8 ) (15)
and (16)
I1
otherwise
and
This allows the calculation of site and bond percolation probabilities,P and PB,respectively, and the percolation threshold as functions of ps, PB, and 8. Typical results are presented in Figures 3 and 4. The phase diagram for the percolation transition, i.e. when the threaholdspscandpBCfrom which the percolation probabilitiesp a n d P do not become zero, is represented in Figure 3. As can be seen, the percolation region of the uncorrelated lattice (52 = 0) expands as 52 increases up to the whole space of the fully connected lattice (8 = 1). Thus a porous structure with a high degree of correlation, such as in sandstones, may present a very low percolation threshold." The effect of overlappingon the percolationprobabilities P and P is displayed in Figure 4. Only the normal behavior of the percolation probabilities (monotonically (16) Faccio, R. J.; Zgrablich, G.; Mayagoitia, V. J. Phys., in press. (17) Thompson,A. H.;Katz, A. J.; Krohn, C . E.Ado. Phys. 1987,36, 625.
0' 0
0.2
0x
I
I
0.6
,
J
0.8
1.o
Ps
Figure4. Classical behavior of the site percolation probability P as a function of ps, for a fiied value of pe and QZ > QI. The behavior of P for a fixed value of p s is similar. increasing functions), that will be used in the discussion of the catalytic deactivation, is shown. However, for certain values Ofps, pB, and 8,these probability functions exhibit anomalous properties that cannot be explained from the classical percolation theory.I6 The model of correlatedporous networks and the results for the percolationprobabilities presented here have been used recently in a qualitative way to explain the properties of the hysteresis loops observed in mercury porosimetry and in the capillary condensation of nitrogen for a great variety of porous materials.l* Even though the present results were obtained for a Bethe lattice, which does not present closed loops, Monte Carlo simulationsperformed on a two-dimensionalsquare lattice reveal the same behavior.
Results and Discussion The deactivation of a porous catalyst by site coverage and pore blockage can be expressed in terms of a relative activity, A, which is the ratio of the rate of the catalytic reaction at time t , to the rate at time t = 0. We consider the catalyst as a Bethe latticeof correlatedsites and bonds. We also assume that the catalytic surface area mainly belongs to the sites of the lattice. Then, using the assumptions (i-iv), the relative activity is given by A = P S ( C , , C B , ~ ~ ) Xexp(-kt) (19) where P(cs,cB,~) is the site percolation probability in the Bethe lattice, which is a function of the site and bond critical radii, cs and CB, and also of the overlapping 52, and x is the fraction of porous surface belonging to the open sites
S,;R:Fs(Rs) a s X'
(20) lm~:~s(~s) as
Since not every open site (that contributes to x ) is connected to the infinite cluster, Psx then representa the accessible fraction of porous surface, i.e. the fraction of porous surface that is really percolating. When PS (18) Zgrablich, G.;Mendiroz, S.;Daza, L.; Pajares, J.; Mayagoitia, V.; Fbjas, F.;Conner, W.C. Langmuir 1991, 7,779.
Connectivity of Heterogeneous Porous Structures
Langmuir, Vol. 9, No.10, 1993 2503
dn-0 dv.0
A
lo\
50i \ 0
0
20
4.0
6.0
80 Kt
Kt
Figure 5. Relative activity as a function of the dimensionless time, for a, = 0 and a. = 0.
vanishes, the porous network loses its global connectivity and this corresponds to the catalyst’s deactivation point. For the special case of uniform size distributions, we can expressthe time dependence of the critical radii, using eq 8, as follows
+ a,[l - exp(-kt)l CB = b + a,[l- exp(-kt)] cs = s
Figure 6. Relative activity as a function of the dmeneionleee time, for several values of overlapping a, and a, = 1, a. = */a. A
(21)
(22)
where av = 3Ctb and a, = 2Ctb. In Figures 5-7 we present the dependenceof the relative activity on the dimensionless time kt, for some values of the overlapping a. As can be seen, the time for complete deactivation increases as increases, and this effect is more pronounced for smaller values of avand a, (Le. for small injtialconcentrationof active sites, Ct, or smallvalues of the average deposit volume on a single active site, b). For small values of avand an,the catalytic deactivation is due principally to the poisoning of active sites, but the more correlated the porous structure, the more resistant the catalyst will be to deactivation; see Figure 6. In the extreme case of av= a, = 0, the poisoning of active sites is so slow that the correlations have no effect on the deactivation process. Obviously,this is an ideal situation because the deactivation will never end (kt ---c a);see Figure 5.
For large values of av and an, the catalyst is mainly deactivated by pore blockage, and again the most corre-
~t.10
Figure 7. Relative activity as a function of the dimensionless time, for several values of overlapping n, and a, = 3, a. = 2.
lated structure showsthe highest resistanceto deactivation, but here the effect is less severe; see Figure 7. For very large values of avand a,, the pore blockage is so fast (kt -0) that even the presence of a highly connectedstructure would have no significant effect. We conclude that the introduction of correlated porous network model, combined with the site-bond percolation theory, gives a reasonable description of the phenomenon of catalytic deactivation in a great variety of porous materials. Future work will be directed to the analysis of experimental data in the light of the model presented here.
