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Langmuir 1989,5, 1044-1047
The Effect of Adsorbate-Induced Surface Reconstruction on Diffusion of Adsorbed Particles V . P . Zhdanov Institute of Catalysis, Novosibirsk 630090, USSR Received October 12, 1988. In Final Form: April 7, 1989
The effect of adsorbate-induced surface reconstruction on diffusion of adsorbed particles is considered in the framework of the lattice-gas model. The reconstruction is assumed to be the continuous phase transition that is described by using the phenomenological Landau theory. At low coverages (e < 0.5), the reconstruction is shown to result in reduction of the diffusion coefficient in comparison with the value derived for the unreconstructed surface. At high coverages (0 > 0.51, diffusion is strongly anisotropic. It is facilitated in one direction and suppressed in another. The contribution of reconstruction to the apparent preexponential factor for diffusion obeys the compensation effect and is of the order of 10-2-104. 1. Introduction Surface diffusion is of considerable intrinsic interest and is also important for understanding the mechanism of surface reactions, ranging from the simplest, like recombination of adsorbed particles on surfaces, to the complex processes encountered in heterogeneous catalysis.lp2 Its intrinsic interest arises from the dynamical and statistical features of particles in adsorbed overlayers. In particular, diffusion at a finite coverage representa an example of a nontrivial phenomenon in a many-body dynamical system. To simulate diffusion at finite coverages, the lattice-gas models are customarily used.2 In the framework of these models, the surface lattice is usually assumed to possess well-defined two-dimensional periodicity closely resembling the atomic ordering in the bulk. This assumption is correct for many systems. In some instances, however, the surface lattice reconstructs into a phase with new This can occur spontaneously with temperature, or it can be induced by adsorbed particles. The purpose of the present paper is to consider the effect of the adsorbateinduced reconstruction on diffusion of adsorbed particles. The reconstruction is regarded as a continuous phase transition. The detailed description of diffusion depends on whether the continuous phase transition is displacive or order-disorder. If the transition is displacive, the lattice atoms vibrate above T, in well-defined vibrational modes about the ideal positions, and below T, the atoms vibrate about positions which are gradually displaced from the ideal arrangement. On the other hand, in an order-disorder phase transition the lattice atoms are randomly displaced above T,, with small vibrations about the displaced positions, and below T, the displacementsare partly or completely ~ r d e r e d .We ~ use the model (section 2) that is more applicable to the displacive phase transitions, because the model does not take into account the shortrange order in arrangements of the lattice atoms. However, our final conclusions are justified both for displacive and order-disorder phase transitions. The description of diffusion depends also on whether reconstruction is fast or slow compared to diffusion. We (1)Ehrlich, G.; Stolt, K. Annu. Reu. Phys. Chem. 1980, 31, 603. Gomer, R. Vacuum 1983,33,537.Naumovets, A.G.;Vedula, Yu. S. Surf. Sci. 1985,4,365.Doll, J. D.; Voter, A. F. Annu. Rev. Phys. Chem. 1987, 38,413. (2)Zhdanov, V. P.;Zamaraev, K. I. Usp. Fiz. Nauk 1986,149,635 (English translation: Sou. Phys. Usp. 1986,29,755). (3)Estrup, P. J. In Chemistry and Physics of Solid Surfaces. V; Vanselow, R., Howe, R., Eds.; Springer: Berlin, 1984;p 205. Willis, R. In Dynamical Phenomena at Surfaces, Interfaces and Superlattices; Nizzoli, F., Rieder, K.-H., Willis, R. F., Eds.; Springer: Berlin, 1985;p 126. Ying, S. C. Ibid; p 148. (4)Cowley, R. A. Adu. Phys. 1980,29,1.
assume that reconstruction is fast, and consequently, the system is in local thermodynamic equilibrium. 2. Model of Reconstruction There exist various statistical models that have been used to construct phase diagrams of systems that undergo the adsorbate-induced surface r e c o n s t r u ~ t i o n . ~ , ~We ~”~ use the semiphenomenological model that has been proposed by Lau and Yingg (see also ref 2) to describe the effect of low-coverage adsorption of hydrogen on the reconstruction of the W(OO1) surface. In this case, the experimentally observed ~ ( 2 x 2 )structure of the reconstructed surface corresponds to zigzag displacementsof the lattice atoms in the directions K = ( r / a ) ( l , l ) or K‘ = (r/a)(l,-l), where a is the lattice spacing (Figure 2). If attention is confined to displacements in the direction K , the average displacement vector and the average occupation numbers of the adsorption sites can be represented in the form ( Ul) = a& cos (KR:) (1) (ni)= 9
+ m sin (KR,)
(2)
where Rlo is the coordinate of the lattice atom before reconstruction, Ri is the adsorbed particle coordinate, p and m are free parameters, K is the unit vector, and 9 is the coverage defined so that tJ= 1 corresponds to two adsorbed particles per one lattice atom on the surface. The total free energy of the system (per one adsorption site) is represented as (3) F = F, + Fad + Fint F, = r ( T - T , ) @ + up4 (4)
+ 2c2(g2 - m2)+ ( T / 2 )X (1 - 0 - m ) ] + (T/2)[(0 - m) In (9 - m ) + (1 - 9 + m ) In (1 - 9 + m ) ]
Fad = 2t1g2
[(e + m) In (e + m ) + ( 1 - 9 - m) In
(5) Fint= -Amp (6) where F, is the phenomenological Landau expression for the surface free energy (T,is the temperature of the phase transition of the clean surface and r and v are constants which, to the first approximation, are temperature independent), Fadis the overlayer free energy calculated in the mean-field approximation (el and cz are the nearest- and (5)Lau, K.H.; Ying, S. C. Phys. Reu. Lett. 1980,44,1222. (6)Ying, S.C.; Roelofs, L. D. Surf. Sci. 1983,125,218;1984,147,203. Roelofs, L.D.; Hu, G. Y.; Ying, S. C. Phys. Reu. 1983,B28, 6369. (7)Roelofs, L. D.; Wendelken, J. F. Phys. Reu. 1986, B34, 3319. Roelofs, L. D.Phys. Rev. 1986,B34, 3337; Surf. Sci. 1986,178,396. (8)Inaoka, T.; Yoshimori, A. Surf. Sci. 1982,115, 301. (9)Zhdanov, V. P.Surf. Sci. 1985,164,L807.
0143-1463/8912405-1044$01.5O/O 0 1989 American Chemical Society
Diffusion of Adsorbed Particles
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Figure 1. Arrangement of particles on the W(oO1) surface before (a) and after (b)reconstruction. The black circles denote tungsten atoms, and diamonds denote the adsorption sites for hydrogen atoms. In the latter case (after reconstruction), the average occupation numbers of the adsorption sites, indicated by + and -, are respectively 0 + m and 0 - m, where m is the order parameter.
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Figure 2. Phase diagram of the system. next-nearest-neighbor lateral interactions between adsorbed particles), and Fintis the additional interaction energy of adsorbed particles and surface atoms (X is an interaction parameter). The Boltzmann constant is set to unity. Minimization of the total free energy with respect to the parameters Q and m yields the following equations for determining these parameters: 2r(T - T J Q+ 4v$ - Xm = 0 (7) XQ
+ 4e2m (T/2) In [(O
+ m ) ( l - 0 + m)/(O- m ) ( l - 0 - m ) ] = 0 (8)
Using eq 3-8, we can construct a phase diagram of the system and calculate the coverage dependence of the chemical potential of adsorbed particles. The described semiphenomenological model is applicable both to the displacive and order-disorder transitions. However, the model is more adequate in the former case because it does not take into account the short-range order in the arrangement of the lattice atoms. In concluding this section, it should be emphasized that we use the described model to study general aspects of surface diffusion in systems that undergo reconstruction. We do not claim to a detailed description of the H/W(001) system because of two reasons. First, the phase transition in this system seems to be order-disorder' (it is of interest that 10 years ago the dominant opinion was that the transition is displacive).1° Second, the phase diagram of (10)Iglesfield, J. E.Vacuum 1981, 31, 663.
the H/W(001) system is more complex than that predicted in the framework of the described model: in addition to an H-induced 42x2) structure at low coverages,the phase diagram contains a prominent region where the surface is incommensurate with the bulk l a t t i ~ e . ~ Assuming that diffusion occurs via activated jumps of adsorbed particles between nearest-neighbor lattice sites, we have the following general expression for the diffusion coefficient (see eq 8 in ref 1 1 ) : 1 aI.1
D = DOexp(p/T)F ~ T P o o exp(-t,*/T) ,,
(9)
where Dois the coefficient of diffusion on the unreconstructed surface a t low coverages, 1.1 is the chemical potential of adsorbed particles; PW,, is the probability that a pair of the nearest-neighbor sites (in the direction of diffusion) is empty, with the environment being marked by the index i and E,* is the variation of the top energy of the activation barrier due to lateral interaction of the activated complex with the nearest adsorbed particles and displaced lattice atoms. The chemical potential is known to be related to the energy of particles. Using eq 9, we assume the energy of a particle to equal zero if the surface is unreconstructed and the sites neighboring this particle are empty. Equation 9 is derived" for diffusion in systems that do not undergo reconstruction. However, all the arguments presented in ref 11 are correct also for systems that undergo the continuous phase transition, provided that the topological structure of the arrangement of adsorption sites does not vary during reconstruction (e.g., in the case under consideration, adsorption sites form the same square lattice before and after reconstruction). The lateral interaction of the activated complex with its environment, E,*, is customarily assumed to be weak in comparison with other interactions. For this reason, we neglect the former interaction and rewrite eq 9 as
where POOis the average probability that a pair of the nearest-neighbor sites is empty. There is no difficulty in including the interaction ti* in the analysis. We have avoided doing so to keep the number of parameters in the model small. The analysis shows that the average jump rate is proportional to exp(p/T).l' Thus, this term of eq 10 makes the main contribution to the coverage dependence of the diffusion coefficient. The average probability Poodepends on whether diffusion occurs along or perpendicular to the zigzag rows. This term result in anisotropy of diffusion. The anisotropy certainly obeys the symmetry requirements imposed by the substrate. Finally, the term &/a0 leads to the correct description of thermodynamic equilibrium." Assuming that local thermodynamic equilibrium occurs during diffusion and using eq 3-8, we derive the following expression for the chemical potential: (0 + m)(O - m ) aF - = 4c10 + 4t20 + ( T / 2 ) In p ao ( 1 - 0 - m ) ( l - 8 + m)
If diffusion occurs perpendicular to zigzag rows, the average occupation numbers of the nearest-neighbor sites are 0 + m and 0 - m (see eq 2). Accordingly, the proba(11) Zhdanov, V. P. Surf. Sci. 1985, 149,L13.
1046 Langmuir, Vol. 5, No. 4, 1989
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Figure 4. Order parameters as a function of coverage. bilities that sites are empty are 1 - 8 -m and 1 - 0 + m. Thus the average probability that a pair of the nearestneighbor sites is empty is Poo= (1- e - m ) ( l - e
+ m) = (1- e)2 - m2
(12)
If diffusion occurs along zigzag rows, the average occupation numbers of sites locate in zigzag rows and near zigzag rows are 9 + m and 0 - m, respectively. Accordingly, the probabilities that pairs of the nearest sites are empty are (1- 0 - m)2and (1- 0 m)2.The average probability that a pair of the nearest-neighbor sites is empty is Poo = [(I - 0 - mI2 + (1 - 0 + m ) 2 ] / 2= (1- el2 + m2 (13) Equations 7, 8, and 10-13 enable one to describe the effect of reconstruction on diffusion. For the sake of comparison, we have analyzed also the diffusion coefficient for the unreconstructed surace by using eq 10-13 and putting X = m = 0.
+
4. Results and Discussion To calculate the phase diagram (Figure 2)) the order parameters (Figures 3 and 41, and the diffusion coefficient (Figures 5-7), we have used the set of parameters ( T , = 300 K, r = 400, u = lo6 K, el = e2 = 200 K, and X = lo4 K) that is about the same as the set in ref 5. A t low coverages (e < 0.5) and low temperatures ( T < T,), the reconstruction is seen (Figure 5) to result in reduction of the diffusion coefficiefit in comparison with the
value derived for the unreconstructed surface. This is explained by preferential occupation of adsorption sites located in zigzag rows (see Figure 1). The energy of these sites is lower than the energy of sites on the unreconstructed surface. As a result, diffusion is suppressed. This effect is somewhat smeared at T > T, (Figure 6). At high coverages (0 > 0.5) and low temperatures, the adsorption sites located in zigzag rows are completely occupied and the sites located near zigzag rows are partly occupied. The energy of the latter sites is higher than the energy of sites on the unreconstructed surface. As a result, diffusion along zigzag rows is facilitated (Figure 5 and 6). On the other hand, diffusion perpendicular to zigzag rows is again suppressed by strongly bound particles that occupy sites located in zigzag rows. Thus, diffusion is strongly anisotropic at 0 > 0.5 and T < T,. The Arrhenius plot (Figure 7) shows that the apparent preexponential factor for diffusion is dependent on coverage. The contribution of reconstruction to the apparent preexponentialfactor is of the order of 10-2-104. Moreover, the compensation effect is seen to occur. I t is of interest that reconstruction can also lead to the compensation effect in the coverage dependence of the apparent Arrhenius parameters for desorption.12 Finally, it is necessary to point out that diffusion of adsorbed particles in systems that undergo the induced (12) Zhdanov, V. P. Surf. Sci. 1989,209,523. Zhdanov, V. Chem., in press.
P. J.Phys.
Langmuir, Vol. 5, No. 4 , 1989 1047
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Figure 5. Diffusion coefficient as a function of coverage at T = 250 and 300 K. The solid and broken lines correspond respectively to diffusion along and perpendicular to zigzag rows presented in Figure lb. The circles correspond to diffusion on the unreconstructed surface.
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Figure 7. Coefficient of diffusion along (a) and perpendicular
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reconstruction represents an example of a nontrivial phenomena. For this reason, the detailed description of diffusion in these systems is difficult, and we have used in our analysis rather rough approximations (the meanfield approximation for adsorbed particles and the phenomenological Landau approximation for free energy of the surface that, in fact, is equivalent to the mean-field
approximation). These approximations are not correct a t T = Tc(8),and some our particular results (e.g., the stepwise dependence of the diffusion coefficient on coverage and temperature a t T = Tc(B))seem to be artificial. However, the other general conclusions are believed to be correct because they are supported by clear physical arguments.
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