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Multiplicity in Adsorption on Heterogeneous Surfaces† A. Co´rdoba,* M. L. Basco´n, and M. C. Lemos Departamento de Fı´sica de la Materia Condensada, Universidad de Sevilla, Sevilla, Spain Received November 3, 1995. In Final Form: June 6, 1996X In adsorption-desorption phenomena, interaction among adatoms can cause multiple steady states and, consequently, hysteresis loops. Starting from a master equation and applying a mean field approximation to obtain the kinetic equations, adsorption on a heterogeneous surface with two types of randomly distributed adsorbent sites is characterized by different activation energies. Moreover, interaction among adsorbates contributes to the total activation energy. In general, depending on the values of the kinetic parameters and the heterogeneity degree, up to three stable steady states can exist for values of the parameter J, which measures the interaction energy, greater than the critical one, Jc. Once the other parameters have been fixed and considering different heterogeneity degrees, the highest Jc is obtained for the case of a 50-50% site distribution. Mobility of adatoms on the surface disfavors multistability, and interactions stronger than those corresponding to a case without mobility are required to cause multiplicity of stable steady states. On the other hand, a dimerization reaction catalyzed by a surface like the one above described does not change essentially the number of stable steady states. Thus, for the case considered, multiplicity is a result due to the adsorption process and does not depend on the reaction one. Multistability causes different reaction rates for given conditions, depending on the initital states. When three or four types of adsorbent sites are considered, multistability does not appear. Also, when different continuous distributions of types of adsorbent sites are considered, multistability is also destroyed. Therefore, it can be concluded that an “excess” of surface heterogeneity prevents the occurrence of multistability in adsorption with interacting adsorbates.
Introduction In adsorption-desorption phenomena interaction among adatoms can cause multiple stable steady states1-3 for given values of the parameters characterizing the problem. In the case of multistability, the system evolves toward a particular stable steady state depending on the initital conditions. Thus multistability leads to hysteresis loops. A key aspect of multistability is the nonlinearity of the rate equations governing the process. Particularly, for only one type of adsorbed particles, with adsorption probability in the Arrhenius form and interaction between nearest neighbors, one or three steady states can be obtained (for the latter case only two states are stable). For competitive adsorption of two types of particles on a homogeneous surface the result becomes more complicated, with up to seven steady states, three of them being stable.4-5 In this paper we deal with adsorption of a single species on a lattice, but now this adsorbent lattice is heterogeneous. In general, heterogeneity increases the number and the complexity of the rate equations and, as a consequence, their nonlinearity degree. Thus a more complicated behavior of the system can be expected as heterogeneity increases. To gain insight into this problem, a number of cases have been analyzed: lattices with two types of sites distributed in a random or regular way, lattices with more than two types of adsorbent sites, and lattices with continuous distributions. Mobility of adatoms on the lattice and a dimerization reaction have been considered as well as processes which can influence multistability. The analysis of the following sections has made clear †
Presented at the Second International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland/Slovakia, September 4-10, 1995. X Abstract published in Advance ACS Abstracts, September 15, 1996. (1) Adamson, A. W. Physical Chemistry of Surfaces, 3rd ed.; WileyInterscience Publishers: New York, 1976. (2) Nitzam, A.; Ortoleva, P.; Deutch, J.; Ross, J. J. Chem. Phys. 1974, 61, 1056. (3) Co´rdoba, A.; Luque, J. J. Phys. Rev. B 1986, 33, 5836. (4) Co´rdoba, A.; Lemos, M. C.; Luque, J. J. J. Chem. Phys. 1990, 92, 5636. (5) Co´rdoba, A.; Lemos, M. C. J. Chem. Phys. 1993, 99, 4821.
S0743-7463(95)01001-8 CCC: $14.00
that only for certain cases is the number of stable steady states greater than that of the homogeneous lattice, whereas for others multistability is destroyed. Model and Results Let us consider a lattice with coordination number c, with N sites, constituted by r types of adsorbent sites, called 1, 2, ..., r, where Ni (i ) 1, 2, ..., r) denotes the r fraction of sites i (∑i)1 Ni ) 1). One particle at most can be adsorbed on or desorbed from a site. A configuration of the lattice can be described by a set of variables {sk} (k ) 1, ..., N); each variable sk includes two values describing the type of adsorbent site i (1, ..., r) and its state (full or empty). Then the time evolution of the system can be described by a master equation
∂P({si},t) )∂t
∑ W({si} f {si}′) P({si},t) +
{si}′
∑ W({si}′′ f {si}) P({si}′′,t)
(1)
{si}′′
where P({si}, t)denotes the probability of finding the system in the configuration {si} at the time t and W({si} f {si}′) is the transition probability from configuration {si} to {si}′. The probabilities for adsorption-desorption are taken in the Arrhenius form, and it is assumed that the activation energy for the site i depends, on one hand, on the type of site i and, on the other, on interaction with nearest neighbors. Thus the transition probabilities can be written
Win ) Ai exp[-(Ei + nE0)/kT] ) aiDn ai ) Ai exp(-Ei/kT)
D ) exp(-E0/kT)
(2)
for adsorption and
W′in ) A′i exp[-(E′i + nE′0)/kT] ) a′i D′n a′i ) A′i exp(-E′i/kT) for desorption. © 1997 American Chemical Society
D′ ) exp(-E′0/kT) (3)
Multiplicity in Adsorption on Heterogeneous Surfaces
Figure 1. Diagram of steady states in the plane a′-J for a linear chain and different values of N1 (0.99, 0.8, 0.6, and 0.5); b ) 1, b′ ) 500 are fixed. The region including the lowest values of J corresponds to one steady solution; the other region corresponds to three steady solutions. For N1 ) 0.5 there is a region of five steady solutions surrounded by three regions of three steady solutions.
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Figure 3. Diagram of steady states in the plane a′-J for a hexagonal lattice and N1 ) 0.99, 0.6, and 0.5; b ) 1 and b′ ) 500 are fixed. The definitions of the different regions are similar to those of Figure 1.
(i ) 1, ..., r; 0 e n e c). Ai and A′i are frequency factors, Ei and E′i are activation energies due to the adsorbent substrate, E0 and E′0 are the contributions to the activation energy made by the neighbor adatoms, k is the Boltzmann constant, and T is the temperature. To obtain the rate equations for the coverage of the different types of sites, θi, we have applied a mean field approximation. At this point information about the distribution of the types of adsorbent sites is required. First we have assumed that the types of sites are randomly distributed. Then the resulting rate equations are
dθi ) ai(Ni - θi)(1 + Jθ)c - aiθi(1 + J′θ)c dt
(4)
where r
θ) Figure 2. Diagram of steady states in the plane a′-b′ for a linear chain. N1 ) 0.5, b ) 1, and J ) 8 are fixed. The exterior of the close curve corresponds to one steady solution; in the loop three regions of three steady solutions surround a region of five solutions.
i denotes the type of site where the process takes place, and n, the number of nearest neighbors full
θi, ∑ i)1
J ) D - 1, J′ ) D′ - 1
Assuming only two types of sites, the rate equations are
dθ1 ) (N1 - θ1)[1 + J(θ1 + θ2)]c dt a′θ1[1 + J′(θ1 + θ2)]c ) f1(θ1,θ2;a′,J,J′,N1,c)
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Figure 4. Diagram of steady states in the plane a′-J for a square lattice and N1 ) 0.99, 0.6, and 0.5; b ) 1, b′)500 are fixed. The definitions of the different regions are similar to those of Figure 1.
Figure 5. Diagram of steady states in the plane a′-J for a triangular lattice and N1 ) 0.99, 0.6, and 0.5; b ) 1, and b′ ) 500 are fixed. The definitions of the different regions are similar to those of Figure 1.
dθ2 ) b(N2 - θ2)[1 + J(θ1 + θ2)]c dt b′θ2[1 + J′(θ1 + θ2)]c ) f2(θ1,θ2;b,b′,J,J′,N1,c) (5)
From one to five steady solutions result, depending on the values of the parameters. For J > Jc multiplicity results. Jc depends on the coordination number, c, of the lattice, once other parameters have been fixed. As c increases, Jc decreases. In Figures 1-5 typical diagrams are shown, where different regions in the framework of the parameters correspond to different behaviors of the system (different number of steady states). In the set of steady solutions that has been found, actually only stable solutions have physical meaning. When there are several stable solutions for given values of a′, b, b′, J, and N1, it is possible that the system evolves to different final steady states, depending on the initial conditions. The stability analysis is done in the standard way.6 When there is one steady solution, it is a stable node; when there are three steady solutions, two stable nodes and one saddle point result; when there are five steady solutions, three stable nodes and two saddle points result. Thus heterogeneity can increase the multiplicity degree (up to three stable steady states) in comparison with the homogeneous case (up to two stable steady states) when there are two types of adsorbent sites. The greatest multiplicity results for values of N1 close to 0.5. As a consequence of multiplicity, hysteresis can appear in adsorption-desorption processes. As an illustration we have analyzed a case of temperature-programmed desorption and later readsorption. We have assumed a linear variation of temperature versus time T ) T0 + Rt. A typical result is shown in Figure 6. As can be seen there, multistability causes a hysteretic cycle because a sudden change in the total adsorption degree θ takes place
including a1 in the time, denoting a′ ) a′1/a1, b ) a2/a1, and b′ ) a′2/a1, and taking into account N1 + N2 ) 1. To obtain the steady solution, we need to solve the nonlinear algebraic equation set
f1(θ1,θ2; a′,J,J′,N1, c) ) 0 f2(θ1,θ2;b,b′,J,J′,N1,c) ) 0
(6)
depending on the set of parameters a′, b, b′, J, J′, N1, and c. We have assumed that interaction plays opposite roles in adsorption and that desorption and E′0 ) -E0 (then J′ can be written in terms of J as J′ ) -J/(1 + J)). We have performed an extensive sweeping in the framework of the parameters to obtain steady solutions, solving eqs 6. On account of the nonlinearity of eqs 6, in general several solutions can exist. The number of solutions depends on the value of the parameters a′, b, b′, J, N1, and c. Once c is fixed, the framework of axes a′, b, b′, J, and N1 can be considered, and the space in this framework becomes divided into different regions, each of them corresponding to a different number of solutions of eqs 6 and, therefore, to a different behavior of the system. To visualize them, projections of the hypersurface separating these regions can be represented in the planes J-a′, b-a′, and so on, the other parameters having been fixed.
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Figure 6. Coverage versus temperature in a desorption and later readsorption process with a programmed linear temperature variation. A1/R ) 1 K-1, A2/R ) 2.683 × 105 K-1, A′1/R ) 2.4365 × 102 K-1, A′2/R ) 1.3417 × 108 K-1, E1/k ) 0, E′1/k ) 103 K, E2/k ) 5 × 103 K, E′2/k ) 5 × 103 K, E0/k ) -9.25 × 102 K, T0 ) 370 K, θ1(0) ) 0.499, and θ2(0) ) 0.484.
at a temperature more elevated for desorption than for readsorption. Next we have considered the cases where there are three and four types of sites randomly distributed. The number of rate equations increases, and the complexity of the problem increases as well. However, for sites clearly different, i.e. characterized by ai and a′i parameters with values not very close to those of the other sites, multiplicity does not appear. Next we have considered continuous distributions for the parameters a and a′, and two cases have been analyzed: a uniform distribution ranging between two given values and a Gaussian distribution. Again we have found that there is no multistability. Turning again to the case of only two types of adsorbent sites, we have considered a square lattice with N1 ) 0.5, but now distribution of sites is not random but regular because sites 1 and 2 alternate in the lattice. Then, for sites 1 and 2, which are clearly different, multiplicity does not exist. Mobility and Dimerization Reaction We consider again a lattice with two types of sites randomly distributed, and next we are going to analyze the influence of adatom mobility on multistability. We have assumed that an adatom can hop to a vacant adjacent site. Transition probabilities are chosen in a similar form to adsorption-desorption ones using the parameters mi instead of ai, assuming the same value for mobility toward any direction on the lattice, and with an interaction parameter L similar to J. Then we must add (6) Nicolis, G.; Prigogine, I. Self-Organization in Nonequilibrium Systems; Wiley-Interscience Publishers: New York, 1977.
Figure 7. Diagrams of steady states in the plane a′-J for a linear chain and m ) 0.1 and 1. N1 ) 0.5, b ) 1, b′ ) 500, and L ) 0 are fixed. The definitions of the different regions are similar to those of Figure 1.
a mobility term to the rate equations. Thus for a linear chain the following term must be added to the rate equation for θ1
2[1 + L(θ1 + θ2) - L(θ1 + θ2)2][m2θ2(N1 - θ1) m1θ1(N2 - θ2)] and the same term with a minus sign must be included in the rate equation for θ2. For simplicity, we have assumed m1 ) m2 ) m. Two steady state diagrams are shown in Figure 7 for different values of m. Mobility does not change the system behavior qualitatively. As mobility increases, Jc increases as well. Therefore a stronger interaction between nearest neighbors is required for multiplicity. Then mobility disfavors multistability but does not preclude it. Finally we have considered that a dimerization reaction on two adjacent sites 1 and 2 is possible, the product being immediately desorbed. A term -qθ1θ2 is added to the rate equations for θ1 and θ2, where q is a parameter characterizing the reaction. There is no essential change in the result, but the fact that multiplicity in the adsorption process produces multiplicity in reaction rates is remarkable. Thus, for this dimerization reaction, different re-
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action rates can be obtained for the same values of the parameter in the rate equations, depending on the initial conditions. Conclusions We can summarize the results obtained as follows: (i) In a lattice with two types of adsorbent sites with interaction between nearest neighbors, multiplicity exits and up to three stable steady states can be obtained. So heterogeneity increases the multiplicity degree (up to three stable steady states versus up to two stable stady states in the homogeneous lattice). (ii) Mobility of adatoms disfavors multiplicity but does not preclude it. (iii) A
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dimerization reaction does not change the result essentially. But as a consequence of multiplicity in the adsorption process, multiple reaction rates are obtained. (iv) In a regular alternate lattice with sites clearly different, multiplicity does not appear. (v) An “excess” of heterogeneity (three or more types of different sites and continuous distributions) removes multiplicity. Acknowledgment. This work was partially supported by Contract EV5V-CT94-0537 of the European Community and Grants SEC95-1105-CE and PB94-1439 of the CICYT of the Spanish Government. LA951001L
