584
The Journal of Physical Chemistry, Vol. 82, No. 5, 1978
M. Zamora and A. Cbrdoba
A Study of Surface Hydroxyl Groups on y-Alumina M. Zamora and A. CBrdoba" Departamento de Termolog?a,Universidad de Sevi//a, Sevila, Spain (Received July 7, 1976) Publicatlon costs assisted by Universidad de Sevilla
A statistical-mechanicstudy of the surface dehydration of y-alumina is made. The surface monolayer is considered as a monomer-asymmetrical dimer system on a square lattice, so that only nearest neighbors interact. The evolution of a wide set of surface ionic groups is studied. The results explain successfully the experimental data of thermogravimetricanalysis, infrared spectra, and mobility of surface ions at T > 600 "C. The contributions to the activation energy of the different surface interactions have been estimated, giving eoo N 2 kcal/mol (oxide-oxide interaction), eHO = 1 kcal/mol (hydroxyl-oxide interaction), and e" N 0 (hydroxyl-hydroxyl interaction). The adsorbed water molecule energy is a surface coverage function and falls between -26 and -89 kcal/mol, within the expected range.
Introduction The first y-alumina surface model for different stages of dehydration was proposed by Peril and was based on a Monte Carlo method. There is fair agreement between this model and the surface reactivity and infrared spectroscopy experimental data, except for some details. Since its publication the model has been widely applied, in spite of the fact that its assumptions are very speculative and not very accurate. Consequently, it would seem justifiable to design a model which would delve deeper into the nature of the problem, using explicitly its energetic bases.
Theoretical Formulation of the Model The y-alumina (100) cleavage plane structure was specified in Peri's paper.l At a medium stage of dehydration, it can be considered as a system composed of monomers H (hydroxyl groups) and asymmetrical dimers 0-V (oxide ion-vacant site pairs) on a square lattice; the role of the underlying solid is similar to that of an external field and the water vapor is considered as a particle reservoir, whose chemical potential, pW, is known (it is a temperature and vapor pressure function). We assume that only nearest neighbors interact. The pair interaction energies of nearest neighbors are considered constant, namely, e", eHO, eHV,em, eov,and ew. The energies due to the external field are denominated eH, eo, and ev, and the chemical potentials pH, p o , and pv. We also define a quantity E = 2 e -~ eo - ev. Actually the dehydration process is irreversible, that is to say we work with a system which is not in equilibrium. The validity of the equilibrium statistical mechanic methods could be questioned in this situation. However, Hoffman2 has shown that in the dissociative adsorption of a diatomic gas, a similar situation to the one we are dealing with here, the distribution functions for irreversible adsorption are of equilibrium form; the interaction energies of the adsorbed molecules are replaced by the contribution to the activation energy of the molecule which is being adsorbed due to its interaction with the molecules which are on the surface. From now on we have to understand the energies e", eHO, etc. in this sense. This allows us to apply equilibrium statistical mechanic methods in the study of these phenomena. The study of such a system follows the guidelines of the two-dimensional Ising model.3 Several author^^-^ have treated the symmetrical dimer system with different approximate methods, considering only the interaction of the hard-core repulsion. Because of the characteristics of the proposed model we must consider the interaction 0022-3654/78/2082-0584$0 1.OO/O
energies between nearest neighbors and dimer asymmetry. Of the approximate methods available, we have chosen the "exact finite method' (EFM),7because of the clarity of its hypothesis and since it allows us to make a complete study of the finite range correlations without making previous assumptions about them. We have modified some points of the usual formulism to make it more appropriate for our problem. The EFM considers the lattice as a strip with N X M sites on it in which N 00 and M is finite. The orientation of the strip is indicated in Figure 1. In each row and in each column, both infinite in number, there are L sites (in Figure 1,L = 8). Any configuration of the system may be decomposed in N states s, as shown in Figure 1;there are L' = L / 2 sites in each state. To avoid boundary effects, toroidal topology is imposed on the lattice. Each dimer is decomposed into two particles 0 and V. There are five kinds of particles: H (monomers), O1 and V2 (forming part of a dimer 0-V), and V1 and O2 (forming part of a dimer V-0). Each OIVz and V102 pair must fit in appropriately. There are d(L) independent states, which we shall identify by m, (i = 1,...,d(L)). We use the grand partition function, E , for which In Z lim - - In h l
-
N-m
N
X1 being the dominant eigenvalue of a matrix P, expressed by N H j + NHj +
2
0=
l / k T , k being the Boltzmann constant and T the temperature. N H , and NH, are the number of H particles in states m, and m,, respectively; No,, No,, Nvt,and Nv, have a similar meaning, cCIbeing the compatibility between the states m, and ml (zero if they are incompatible and one if they are compatible) and E , is the interaction energy between the m, and m1 states. All thermodynamic information is contained in XI. Since Pn is a positive matrix, n being a finite positive integer, X1 always exists and is univalued, nondegenerated, and positive. It is well known that in the EFM the states can be packed in equivalence classes using the properties of
0 1978 American Chemical Society
The Journal of Physical Chemistry, Vol. 82, No. 5, 1978 585
Surface Hydroxyl Groups on y-Alumina
symmetry. DLJois the symmetry group and p ( L ) I d(L) is the number of classes. Instead of operating with P it is simpler to use a p(L) X p(L) reduced matrix A whose dominant eigenvalue X1 is the same as the one in P. A's left- and right-hand side eigenvectors, s and r, allow us to reproduce the equivalents, w and v, of P. This fact greatly simplifies the calculations. Dehydration may be represented by the reaction 2H(s)
.-+
\:,, , if\:
' 2
, 5
O(s) t V(s) + H,O(v) (s, surface; v, vapor)
,' ,,f' ,,e' , ,',,/'
/'
,
s+l
whose equilibrium equation is
2PH
= PO
+ PV + p w
(3)
Because po = pv in all cases (po and pv are the densities of particles 0 and V, respectively), then aA,/al-co = a A l / a P v
Figure 1. Strip chosen in the lattice. s and
s 4- 1 are two adjacent
and X1 = X1(po + pv), that is to say hl is a function of yo pv and not of ko and pv separately. Therefore we may define an arbitrary relation between yo and pv; e.g., po = pv. In this case, it is possible to define a matrix Q, so that
+
hydroxyl ion o x i d e ion vacant s i t e
E,
~ H H ~, H O ~, H Veoo
, eov evv , T) 3
(43
We may operate with Q instead of P to calculate the thermodynamic functions and state correlations. Matrix Q is specified if pressure, temperature, and surface energies are known. No valid criterion exists to fix the values of these energies, although it is to be expected that they are bounded by known values. Some supplementary assumptions in the model seem unavoidable. Because of this we have chosen the trial values eij (i, j H, 0, V) and E so that they agree with the experimental thermogravimetric analysis. Comparison of the estimated results with the experimental data of infrared spectroscopy a t elevated temperature has led to greater accuracy in the values of eij. The densities of the different particles, pH, pot and pv, are
-
n
L-
U
U
C"
C'"
Flgure 2. Groups associated to the infrared bands. The groups B, and B, are considered equivalent and the same band B is attributed to them.
(5)
w and v are normalized so that w-v = 1. Expressions 5 enable us to obtain explicit formulae for the correlations. In fact, (mi) = w;u, may be considered the probability of finding state mi in the lattice; ( mim,) = w i P i j u j / X 1 is the probability that both states, mi and mi, occupy adjacent sites in the lattice and, in general
indicates the probability of finding states mi,, ,.., minin n
adjacent positions. The number of any surface ionic group, a t a given pressure and temperature, may be calculated through eq 6. The surface groups of greatest interest are shown in Figure 2 and their connection with the infrared spectra has been pointed out by P e r i l Other interesting groups are the 0-0 and V-V pairs and the 0-0-0 and the V-V-V triplets forming a right angle. In fact, any ionic group creating surface irregularities may play an important role in the catalytic activity of y-alumina. A wide diversity of surface defects could be considered. One would expect their densities to depend strongly on lateral interaction energies which have very inaccurate values. Therefore, such a calculation would not be very useful. We have only calculated the groups shown in Figure 2 and the pairs and triplets mentioned previously. Results and Discussion Calculations have been made with a Univac-1108. L has been given the value of 6. The calculation of X1 and the associated eigenvectors is by the usual procedure in the EFM. Calculations of the degree of hydroxylation and the density of the ionic groups were continued until four significant figures were obtained. The memory required
M. Zamora and A. CBrdoba
The Journal of Physical Chemistry, Vol. 82, No. 5, 1978
586
'O0l 801
'Ol
v\
0
5> 'm
a)--
:
ci
2
2
0.6-
4 e)
-----
0.4-
f
0.2-
Y
200
400
800
600
-
200
100
200
300
400 Temperature
500
600
700
800
1°C)
Flgure 3. Dehydration of alumina: (0)Haldeman and Emmet;" (A) Hindin and Weller;" (V)Whalley and Winter;12 (A) Klo~sterziel;'~ (0) Boreskov et a1.;I4 ( 0 )Peri,' plate I; (m)Peril, plate 11; (A)Peril, plate 111; (V)Peril, deuterium exchange. Solid line, this work: A = 63, B = 89, eHO= 1, em = 2 kcal/mol (lines adjusted for different values of interaction energies are very close).
600
400
Temperature 1 ° C )
Figure 4, Evolution of the isolated hydroxyls: (a) A = 61, B = 88, ,e, = 0,eHO= 0; (b) A = 61, B = 87,,e, = 5, e H O = 0; (c)A = 51, B = 82,,e, = 5, eHo= -1; (d) A = 63,B = 89, e,, = 2, = 1; (e) A = 55, B = 85,,e, = 0,e,,, = -1. All energies are given in kcal/mol.
d)--
for the program was 22K and it took between 35 and 39 min to carry out calculations for 71 different temperatures. Values of eLjhave been established as indicated e" has been considered null according to the assumption proposed by Peri.l eoo is expected to be repulsive and lower than 5 kcal/mol; calculations have been made for eo0 values from 0 to 5 kcal/mol. eHO is expected to be weak, as we infer from the changes observed in infrared bands; calculations have been made for values of eHO contained between -2 and 2 kcal/mol. The interaction energies of vacant sites have been considered null. These values are trial values and similar calculations have been made for all of them; by confrontation with experimental data it is possible to choose the best values. Only the results relative to a part of these trial values are shown here. In the first place, we have proceeded to give E a value according to the thermogravimetric analysis experimental (see Figure 3); these data are very heterogeneous and experimental conditions are often specified in little detail, in particular pressure. This has been fixed as Torr, a value which is probably less than those used to obtain some of the data; this fact may introduce some error in estimated E. Calculations a t several pressures have been made, obtaining curves which are slightly shifted (1 A p ~ l I5% for a pressure ratio p ' / p = 10). From the former calculations it was apparent that E must be a function of pH, and that a linear form E = A p H - B ( A > 0, B > 0) was quite a good approximation. The value of E is expected to be close to the heat of adsorption of water on y-alumina. Experimental data are very scarce and diverse; most of them fall between 10 and 30 kcal/mol, but values up to 105 kcal/mol have been obtained a t p H close to The lowest values have been attributed to adsorbed molecular water and the most elevated ones to the existence of localized surface strains which cause these high energies. In fact, it has been suggested by several authors18-20that a t elevated temperature some type of strain becomes operative on the surface. The E values obtained, according to the interaction energy values, fall between a minimum of 26 and 31 kcal/mol and a maximum of 82 and 89 kcal/mol. They are included within the above-mentioned range for adsorption heats without reaching the boundary values, and, therefore, the result seems acceptable. However, we should take into account that this model probably leads to values of E which are greater than the real ones, because under experimental
800
('C)
Temperature
I
i 1
A *
