Multiple steady states on an adsorbent surface

PHYSICAL REVIE%' 8VOLUME 33, NUMBER 8 Multiple steady states on an adsorbent surface 15 APRIL 1986 A. Cordoba and J. J. Luque Departamento de Terrnoiogla, Facultad de Fi'siea, Universidad de Sevilla, Sevilla, Spain (Received 22 July 1985) The occurrence of multiple steady states in asystem formed by monomers adsorbed on alattice, saith attractive interaction between nearest adatoms, is demonstrated. This fact causes hysteresis in the adsorption-desorption or desorption-readsorption processes. On the other hand, the coordination number of the adsorbent lattice and the corresponding degree of nonlinearity of the rate equation do not essentially influence the results obtained. I. INTRODUCTION Interactions between adatoms have helped explain adsorption energies which are functions of coverage. These interactions influence the rate of surface reactions, making them quicker or slower, thus producing "autocatalytic" mechanisms. As is well known, multiple steady states can exist in chemical reactions including autocatalytic steps. 'A steady state may be stable or unstable according to whether asmall deviation from it decays to the original steady state or evolves away from it. In several previous papers' we have dealt with the kinetics of various adsorption processes from an extension of the Ising-Glauber mode15 where, starting from amaster equation and suitably chosen transition probabilities, the macroscopic equations governing the time evolution of the system were obtained. %hen interactions between adatoms exist, the kinetic equations involve nonlinear terms corresponding to these interactions. As shall be demonstrated, multiple steady states, for arange of the parameters characterizing the problem, can exist. Several of these steady states would be stable, Thus, initial conditions essentially determine the final steady state reached by the system and hysteresis phenomena can take place. The simplest case is the adsorption-desorption of amonomer, with aconstant interaction energy existing only between nearest neighbors. II. THE MODEL %e consider aparticle which can be adsorbed on avacant site of alinear chain (later we shall extend the results to the case of alattice with coordination number z). Conversely, an adsorbed particle can be desorbed, thus creating avacant site. Schematically, the process may be represented as W(v)+ V(s) =X(s), where A, X, and Vdenote, respectively, the particles of the vapor, the adatoms, and the vacancies on the adsorbent lattice. When interactions between adatoms exist, the above simple scheme is not valid, because the interactions favor or retard the adsorption-desorption mechanism, and in general more than one partic1e is invo1ved in each elemental process. Thus, this reaction is autocatalytic and the rate equations are non}inear. This fact suggests the existence of multiple steady states. %e assume that the process takes place as follows: The transition probability of each elemental process is chosen in the Arrhenius form, and the activation energy is decomposed into two additive terms, corresponding to the action of the substrate and to the interaction between nearest adatoms, respectively. If the plus sign denotes afilled site and the minus sign avacant one, the elemental processes taking place on the lattice and their respective probabilities are as follows. Adsorption: +—+—+ + +, 8'++ -A exp[ —(E+2E;)/kT] +——++— ~W+ =Aexp[ —(E+E;)/kT ] ——+—++, W+-A exp[ — (E+E;)/kT] —————+—, W' =Aexp( — E/kT) Desorption: ++++—+, W++ =A'exp[ — (E'+2E,")/kT] + + —+——, IY+ =A'exp[ — (E'+EI')/kT] —++ ——+, W' =A'exp[ — (E'+E )/kT] —+—————, W' =A' exp( — E'/kT) Here, Aand A'are constant factors, Eand E' are the contribution of the substrate to the total activation energy, E; and Eare the contribution to the total activation energy due to interaction between two neighbor adatoms, kis the Boltzmann constant, and Tis the temperature. An attractive interaction facilitates adsorption and makes desorption difficult, and therefore E; &0and E&0. On the contrary, arepulsive interaction facilitates desorption and makes adsorption difficult, and therefore E& &0and E&0. Thus E; and Ehave different signs. For the sake of simplicity, we have assumed that all the preexponential factors are the same for the different eases (A for adsorption and A' for desorption) . Starting from this model and following the guideline of Ref. 2, we obtain ahierarchy of rate equations. The first equation of this hierarchy is dn+ =n+ +8'+++2n+ 8'+ +n 8' dt — n++ ~W++ — 2n++ W+ — n~W', (1) where n; and njk (i,j,k+, — )denote the fractions of singlets and triplets in iand i-j -k states, respectively. 33 5836 1986 The American Physical Society In order to close this hierarchy at afinite level, it is necessary to use some approximation method. %e have chosen the Bragg-Williams approximation, so that n&k p1 'pljElk and we retain only the first equation of the hierarchy. Introducing this approximation into Eq. (2), we obtain — (1— 8)(J8+1)z— h8(J'8+ 1)z=P3(0) .(2) d(Hr) where 8-n+ denotes the coverage, J=exp( — E;/kT) — 1, J'-exp( — E/kT) — 1, H=& exp( — E/kT), and k=(A'/ A)exp[ — (E' — E)/kT]. III. RESULTS AND DISCUSSION To calculate the steady values of 0,0„we consider d8/d(Ht)-P3(8) =0, which is an algebraic equation of the third degree, and therefore has three roots. It can easily be demonstrated that there is always one real root in the range 0&0&1, and it is also possible that three roots exist in the above range. In this case, two roots correspond to stable states, both of which are separated by an unstable one. The number of steady states depends on the values of the parameters J, J', and h. We assume IE'I =IEI and, therefore, J'= — J/(J+1). Thus the behavior of the system is determined by only two parameters, Jand h. The stability diagram in the J-h plane is sho~n in Fig. 1, where 1and 3denote the number of steady states (for case 3, only two steady states are stable). The aand biines, separating the different regions of the diagram, coincide at a "critical" point, C, which is defined by hc -9and Jc=2. For J&Jc or h&h~ there is only one steady state. %e can conclude that, for arepulsive or aslightly attractive lateral interaction such that E; &— kTln3, multiplicity of steady states does not exist. This multiplicity is possible only for E; &— kT ln3 and avalue of hin the suitable range (between the aand blines). As Jincreases, the range of h giving multiple steady states becomes larger and larger. The value of E; necessary to cause multiplicity of steady states is quite reasonable. Thus, at room temperature, ~E; ~=1 kcal/mol is enough for this phenomenon to take place, if a suitable value is given to h; for two-dimensional lattices, as we shall see later, and for higher temperatures, this value of F.;is still smaller. In Fig. 2, 0, vs hfor different values of Jis plotted. One can observe the similarity between these curves and the isotherms of avan der Waals gas and, therefore, the analogy between this phenomenon and first-order phase transitions. 6 An interesting parameter is the relaxation time (scaled by H) of the degree of adsorption far from astabie state. We have found that essentially two relaxation times exist, corresponding to initial and final stages, respectively. The final relaxation time, v, does not depend on the initial state but does characterize the steady state. The behavior of v depends on Jand h, and is shown in Fig. 3. For values of J less than Jc there is obviously only one relaxation time, which increases as hdecreases (and, consequently, 0, increases). As regards values of Jgreater than Jc, for h&h, (/t, denoting the value of /t corresponding to critical aline at agiven J), there is only one relaxation time, 7&, which increases smoothly as hincreases. When h=h„while ~~ follows the aforementioned behavior, arelaxation time ~2, corresponding to anew steady state, emerges. At the beginning, rz is very great (rz ~as hh, ), decreases abruptly, and then continues decreasing smoothly, even at h=hb (hb denoting the value of /t corresponding to critical bline at agiven J). At h=hq, r~ ~, and for h)hq only rz exists. As Jdecreases, the difference hb — h, decreases and at the critical point, ~here hb =h„both steady states coalesce and the relaxation time tends to become infinite. Finally, we extend the formulation of this problem to a two-dimensional lattice with coordination number z(z =3, 4, 6). Again, we follow the guideline of Ref. 2, extended to atwo-dimensional case, and use the Bragg-Williams e FIG. 1. Stability diagram in the (J,h) plane. 1and 3denote the number of steady states in each region of the plane. FIG. 2. Steady states of the system as afunction of h, Jbeing a parameter. BRIEF REPORTS 33 0.4 100 500 FIG. 3. Scaled relaxation time as afunction of h, Jbeing a parameter. The numbers in brackets denote the values of J. FIG. 4. Stability diagram in the (J,h) plane for various twodimensional lattices with coordination number zand comparison with the linear chain. approximation. We then obtain the kinetic equation d(wj =(1— 0)(JO+ 1)'— hO(J'0+ 1)'=P, +t(O) .(3) d(Hr) The right-hand side of (3) is apolynomial of the (z+ 1)th degree and we must solve the equation P,+t(O) =0 to find the steady states. One should expect that the stability diagram shown in Fig. 1drastically changes, because the equation P,+&(0)=0has, in general, z+1roots. However, this change does not take place. We have performed numerical calculations for Jgoing from — 1to very high values, and for hfrom very low positive values to very high ones, i.e., we have swept practically all the region of the plane J&— 1, h&0. We have always found one or three steady states (two of the states being stable in the latter case) and the stability diagrams in the J— hplane are similar to that obtained for the linear chain, with critical points Jc=1.000 and bc=8.000, for z=3; Jc=0.666 and h~=7.716, for z=4; and Jc =0.400 and Ac=7.529, for z=6 (Fig. 4). Thus, we can conclude that, for the two-dimensional case, the lattice coordination number and the degree of nonlinearity of the rate equation, which are correspondingly greater than those of the one-dimensional case, do not essentially influence the results of the problem. 'G. Nicolis and I. Prigogine, Self-Organization in Wonequilibrium Systems (Wiley-lnterscience, New York, 1977). 3. J, Luque and A. Cordoba, 3. Chem. Phys. 76, 6393 (1982). ~A. Cordoba and J. J, Luque, Phys. Rev. 826, 4028 (1982); 28, 7353 (1983). 4A. Cordoba and J. J. Luque, Phys. Rev. 831, 8111 (1985). 5R. J.Glauber, J. Math. Phys. 4, 294 (1963). ~A. Nitzan, P. Ortoleva, J. Deutch, and J. Ross, J. Chem. Phys. 61, 1056 (1974).