2670
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Tracer Diffusion on Correlated Heterogeneous Surfaces K. Sapag,? F. Bulnes,t J. L. Riccardo,+J V. Pereyra,tpg and G. Zgrablich'9tJ Instituto de Investigaciones en Tecnologfa Qutmica (INTEQUI), Universidad Nacional de San Luis-CONICET, C.C. 290,5700 Sun Luis, Argentina, and Centro Regional de Estudios Avanzados (CREA), Gobierno de la Provincia de Sun Luis, C.C. 256,San Luis, Argentina Received October 20,1992. I n Final Form: March 19,199P Tracer diffusion on heterogeneoussurfaces is studied by Monte Carlosimulations. The energetic surface topography is described by a dual site-bond model where site and saddle points (bond) energies are incorporated in a general way. The case of uniform energetic distributions is studied and the behavior of the diffusion coefficient is analyzed for several topographies: random traps, random barriers, random sites-bonds, correlated traps, correlated barriers, and correlated sites-bonds, for different correlation degrees. The conditions under which the temperature dependenceof the tracer diffusioncoefficient shows a non-Arrhenian behavior are discussed. The jump correlation factor is also analyzed since the jump mechanism plays a fundamental role in this treatment. Surface diffusion emerges as an important tool for the characterization of the energetic topography on heterogeneous surfaces. Introduction Diffusion on heterogeneous media is a complex and open problem that has received the attention of researchers in the last years.13 Analytical methods and simulation have been employed to analyze this phenomenon, but there are few exact results to account for the diffusion mechanism of particles on heterogeneous substrates.3 The description of different surface processes on heterogeneous substrates, like adsorption and diffusion of adsorbed gases, requires an adequate description of the surface energetic heterogeneity. We do not consider here the effect of the geometrical heterogeneity which has been dealt with elsewhere.ll3 The problem of diffusion on heterogeneous substrates involves not only the different energies of the adsorptive sites but also the distribution of saddle point energies between sites (bond energies) in order to evaluate the activation energy for the migration. This problem is formulated in the context of the "dual site-bond model": which gives a suitable description of the energetic surface topography. Two well-known models have been employed to describe the heterogeneous medium: the random trap model (RTM) and the random barrier model (RBM). But analytical solutions for the diffusion problem have been obtained only for particular cases.3 In this work, the dual sitebond model formulation is employed to obtain the RTM, the RBM, the random s i t e bond model (RSBM), the correlated trap model (CTM), the correlated barrier model (CBM), and the correlated site-bond model (CSBM). For convenience, we assume that the surface is represented by a square lattice of adsorptive sites, which are interconnected by bonds representing the saddle point between the nearest neighbor sites. Both sites and bonds have a uniform distribution of energies. We study here INTEQUI t CREA i Present address: Institiit fUr Theorestische Physik, Johannes Gutemberg Univereitit Mainz, Germany. e Abstract published in Advance ACS Abstracts, August 15,1993. (1) Havli, S.; Ben-Avraham, D. Adu. Phys. 1987,36,695. (2) Bouchaud, J. P.; Georgee, A. Phys. Rep. 1990, 195, 127. (3) Haus, J. W.; Kehr, K. W. Phys. Rep. 1987,150,263. (4) V. MayagoitiaandF.RojasInFundamentalsofAdsorption;Lmpia, A. I., Ed.; New York, 1987.
the behavior of a single adsorbed particle (tracer particle) moving on the surface, which guarantees that the effects on the particle motion are due entirely to the substrate characteristics. The tracer particle is initially in a random selected adsorptive site and then walks on the surface by means of thermally activated jumps over the bonds to nearest neighbors sites. This treatment is based on the random walk theory and Monte Carlo simulation, which is similar to that employed by Limoge and B o c q ~ e t .The ~ mean square displacement versus time and the temperature behavior for different energetic topographies are analyzed. The particle behavior differs for the different surface models, making the analysis of the correlation degree between jumps important. The jump correlation factor provides a measure of the randomless degree between jumps in a sequence. Dual Site-Bond Description In dealing with surface diffusion, an adequate description of the heterogeneous surface requires consideration of not only the adsorptive site energies but also the bond energies.8t7 The respective energetic distribution functions cannot be, in general, completely independent. Statistically, site energies and bond energies are described by their probability density functions Fs(Es)and FB(EB) and by the distribution functions S(Es)and&&), respectively.6
S(E,) = s,EaFs(E)dE;
&E,) = s,EBF,(E)dE
(1)
A fundamental principle must be satisfied. It is the construction principle, which states: "For a given adsorptive site, ita energy must always be greater than, or, a t least, equal to, the energy of any of ita bonds" (the energies are referred to in absolute value). To observe this principle, it is necessary to state two self-consistent laws. The first law states: B(E) 1 S(E) for every E This law establishes the relation between the distribution (5) Limoge, Y.; Bocquet, J. L. Acta Metall. 1988,36,1717. (6) Mayagoitia, V.; Rojas, F.; Pereyra, V.; Zgrablich,G.Surf. Sci. 1989, 221, 394. (7) Mayagoitia,V.; Rojas,F.;Ricardo, J. L.; Pereyra, V. D.; Zgrablich, G . Phys. Reu. B 1990,41,7150.
0743-7463/93/2409-2670$04.00/00 1993 American Chemical Society
Tracer Diffusion on Heterogeneous Surfaces
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functions ensuring that the amount of bonds with energy lower than E is enough to link sites of energy lower than
E. Since overlap between the frequency curves is allowed, there can exist some bonds with energy greater than that of some sites not connected to them. To ensure appropriate assignments of bonds to sites, a second law is required. This can be formulated in terms of the joint probability density function of finding a site with energy Ese (Es,Es + a s ) and a connected bond with energy EBE (EB,EB+ UB), defined as:
F(Es,EB)
= Fs(Es) FB(EB)~(Es$B)a s
(2) where ~ ( E ~ , Eisa B correlation ) function depending on the overlap I between the energetic distributions (0 S I 5 l), which is defined as the common area under the functions Fs and FB. Thus, a second law is stated as b(ES,EB) = 0 for E B > E, (3) This second law, and the definition (2) leads to the following properties
This means that there is certainty to find (i) a bond with energy Es or smaller, for a site of a given energy Es, eq 4, and (ii) a site with energy EB or larger for a bond of a given energy EB,eq 5. For general distributions, it is possible to obtain an expression for 4 by means of the following procedure: Sites with increasing energy are chosen, starting with those with the lowest energy, and a set of bonds is assigned to them, each one having an energy determined with the maximum randomness allowed by the Construction Principle and by the availability of bonds. The procedure leads to the determination of the correlation function in the form'
The function 4 determines completely the statistical description of the heterogeneoussurface through the joint site-bond probability density given by eq 2 where the site and bond frequency functions Fs(Es) and FB(EB) can be completely arbitrary as long as they fulfill the first law. In what follows, we shall use uniform distributions for the site and bond frequency functions, i.e.
Fo
forb1 5 EB 5 b2
It can be shown that for uniform distributions, the correlation function becomes6
d~(&&) = exP(-dEs$B)I/(1- n } / ( l - I) where I = Fo(b, -SI) and
(8)
Natural energetic correlations arise when bonds and sites energy density functions overlap. The overlapping degree characterizes correlations from an energetic point of view. However, it has also been stressed the importance of describing energy correlations through a topological parameter: the correlation length r0.l4 The meaning of ro is the following: the radius of a region centered in a given site within which all sites have energies highly correlated. A relation between the overlapping degree I and the correlation length ro, for uniform distributions, has already been proposed as ro = I/(1 -I) (10) It is clear from eq 10 that I = 0 implies ro = 0 (random bonds and traps) and for I 1,ro 03 (large homotatic patches). For intermediate I we observe the more interesting and realistic cases corresponding to finite correlation lengths. In Figure 1we show different adsorptive energy topographies which can be generated by different choices of site and bond distributions and different overlapping degrees. Monte Carlo simulations of such surfaces can be generated by the method described in refs 8 and 9. Once the desidered surface is generated, the tracer diffusion coefficient can be studied, which is the purpose of this work.
- -
Diffusion on Heterogeneous Surfaces In the conventional theory of random walks on homogeneous media, the tracer diffusion coefficient is given by (11)
where D H ,=~ 1/2d (d is the Euclidean space dimension), t is the time that is spent in the walk, and ( R 2 ( t ) )is the mean quadratic displacement of the tracer particle at time t. For the simulated random walk, the tracer particle jumps from a site to another one by means of thermal activated jumps. Then, the diffusion coefficient depends on the temperature T and D His ~a constant corresponding to
T=
03.
For heterogeneous media the time dependence of the mean quadratic displacement satisfies' ( R 2 ( t ) )a t2Idw where dw 1 2 (12) The exponent 2/dw suggests the diffusion type. The case with dw = 2 corresponds to normal diffusion. Anomalous diffusion is presented when dw > 2 and this behavior is characteristic when the process occurs on a highly amorphous medium, such as geometrically fractal medium with heterogeneous site and saddle point energies. However, in random walks on heterogeneous media of a dimension greater than, or equal to 2, with short range correlation, the normal diffusion behavior is again obtained for long times.2 (8) Sapag, K.; Ricardo, J. L.;Pereyra,V.; 'Zgrablich, G. Submitted for publication. (9) Sapag,K. Tesis de Licenciature,Univenridad Nacionalde San Luis, Argentina, 1991.
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2672 Langmuir, Vol. 9, No. 10,1993
In the problem of tracer diffusion, the particle diffuses on a medium that consists of N-adsorbent sites separated by energetic barriers. We only consider transitions to nearest neighbors. The thermally activated jump of the particle from site i to sitej is governed by the corresponding activation energy (Ea&. Then the transition probability between sites i and j is given by wij = u exp(-P E a 2 ]
(18)
Here
Figure 1. One-dimensionalrepresentations of adsorptive energy
along a direction x on the surface (right-handside) for site-bond distributions with different degrees of overlapping I (left-hand side).
For heterogeneous media, the following relation, analogous to eq 11 is proposed
-
where DoT is the tracer diffusion coefficient for T a. If the tracer particle is in the position &t) (from the origin) after N jumps, we can write N
R(t) =
E?,
Eadij = (E," - E:) (19) where E Bis~the ~ bond energy between site i and site j, and E,' is the adsorptive energy of site i. The parameter u is an arbitrary jump frequency; /3 = l/kT, where k is the Boltzmann constant and T is the absolute temperature. At low temperature, the blind ant model is inefficient, since unsuccessful jumps must be attempted before giving a step. Limoge and Bocquet6 propose an alternative method for saving computational time. This new method consists in the performance of a random walk on a given network, simultaneously evaluating the total time of walk, considering: (a) the jump direction is randomly selected in agreement with the z jump frequencies W i j , where z is the coordination number; (b) the residence time at site i corresponding to the lth visit, til, is also randomly selected from a Poisson distribution with mean ( ~ i ) . The total jump frequency for site i is z
191
wi = c w i j
where ii denotes the displacement of the particle corresponding to the ith step. Then, the mean quadratic displacement can be expressed by
and the mean residence time ( r i ) = l / W i . The total time of walk is given by
where a is the step length and f the jump correlation factor.10
i is the visited site and 1the number of visits corresponding to this site. A more efficient way, would be to take
11 '
(22) but it will be correct only if we take a great number of jumps and if only mean values are evaluated. For the calculation, we create bidimensional square lattices of 100 X 100 sites, with periodic boundary conditions. Both site and bond energies are assigned through the dual model. The Monte Carlo step number (N), which actually represents the jump number, is varied, from 2 to 2 X 104, averaging over n walks for each value of N. We have taken til
(16) with Pk = a?h. The jump correlation factor can be calculated in the limit for large N through
For diffusion on homogeneous surfaces, each jump is completely independent from the preceding one, obtaining f = 1 for every temperature. The Simulation Technique The traditional scheme for the treatment of diffusion on heterogeneous media can be viewed as the blind ant model on the percolation cluster,ll Le., any jump is attempted. Whether the jump is successful or not depends on the transition probability. (10)Bakker, H. In Diffusion in crystalline solids; Much, G.;Nowick, A.; Eda.; Academic Preee: New York, 1984. (11) Stauffer,D. In Introduction to Percolation Theory; Taylor and Francis: London, 1985.
=
(Ti)
n = 10'1~ (23) During the simulation process we compute ( R z ( t ) )and the time t corresponding to each walk of N steps. From eq 13, the following relation for a very long time is obtained
To evaluate the interesting quantities in the process, we plot, for every case, In ( R 2 ( t ) versus ) In(t). So, for a given T (at a very long time) the slope of the function provides valuable information about the diffusion exponent dw and
Tracer Diffusion on Heterogeneous Surface8 "
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Langmuir, Vol. 9, No. 10, 1993 2673 10
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Figure 2. Logarithm of normalized tracer diffusion coefficient versus @E,for different correlation lengths,ro,in the trap model. Dashed lime represents the theoretical result, the solid line repreaenta the homogeneous case, and the symbols represent the Monte Carlo simulation result. EB= +2 kcal/mol,Es = +3 kcal/ mol, and A& = 2 k c a m o l (uniform distribution of Ea).
the Y-axisintersectiongives the value of ln(DT/DoT),which is necessary to analyze the Arrhenian behavior. The estimation of the correlation factor f is realized by means of eq 17.
Ob
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retically obtained,l3 with an expression given by
with Results and Discussion Trap Model. A trap surface has the same energy for all bonds and an energy distribution for the adsorptive sites. The most important characteristic of this model is that the transition probability from site i to a nearest-neighbor site j, only depends on the energy of site i, independently of the jump direction. In order to study several cases of trap topographies, lattices with random traps (ro = 0) and correlated traps (ro > 0) were generated. For constructing a correlated trap lattice, we first construct a lattice of traps and bonds by assigning the energy to each element in a Markovian sequence by using the dual site-bond distribution of eq 2. Then we set all bond energies equal to the minimum trap energy (in absolute value). Since we can initially choose the overlap degree between traps and bond energies, the correlation length between trap energies is determined through eq 10. Energetic correlation between sites means how similar are the energies corresponding to neighbor sites. Then, for high overlap (high 0, we should find a great similarity between energies of neighbor sites. For the RTM problem it has been demonstrated that, if the initial condition corresponds to a stationary state, then the diffusion process is normal.12 Analyzing the plot of l n ( R 2 ( t ) vs ) ln(t),obtained from Monte Carlo simulation data, we find that the tracer diffusion is normal (dw = 21, independent of temperature correlation degree and time. In Figure 2 we plot MOT/ DOT)vs @ad in order to compare the different cases of traps and the corresponding homogeneous case. A clear non-Arrhenian behavior is observed, which does not depend on the energetic correlation. Besides, the results agree with the tracer diffusion coefficient the0
This behavior can be explained as follows: At high temperatures, the particle has enough energy to leave (somewhat easily) any site of the surface. Moreover, the jump probability does not depend on the jump direction, then the particle sees the surface like a homogeneous surface. At low temperatures, the particle is trapped for a long time in the deeper sites, which causes a decreasing on the diffusion coefficient. The correlation factor depends neither on the overlap nor on the temperature, like in the homogeneous case. We find f = 1 (Figure 3) for all the considered cases. This result is the expected one, since the transition probability only depends on the occupied site. Barrier Model. Here, all sites have the same energy and the barriers have again a uniform distribution of energies. A surface with random barriers (RBM) can be easily generated in the usual way. The procedure followed to simulatea surface with correlatedbarriers (CBM)is similar to that used for constructing a surface of correlated traps (CTM). We first construct a surface of traps and barriers with uniform distributions having a given overlapping degree I. Then all trap energies are equaled to the maximum barrier energy (in absolute value). Finally the correlation length ro of barrier energies is calculated. Analyzing the long-time behavior of l n ( R 2 ( t ) vs ) ln(t) for RBM and CBM, we fiid that, dw = 2 (normaldiffusion) for different temperatures. In Figure 4 we show an Arrhenian-type plot for different correlation lengths. The dashed line corresponds to a = Ea - EB. homogeneous surface, with When the temperature increaees,all the cases approach the homogeneous behavior as should be expected. At high temperatures, the particle has enough energy to jump any barrier on the surface, so the energetic differencebetween
(12)Haw, J. W.;Kehr, K.W.;Lyklema, J. W.Phys. Rev. E 1982,15, 2905.
(13)Bulnw, F.;Riccardo, J. L.;Zgrablich, G.;Pereyra, V. Surf. Sci. 1992,260,304. Bulnw, F.Twin de Licenciatura, Universidad Nacional de san Luis, Argentina, 1991.
Sapag et al.
2674 Langmuir, Vol. 9, No. 10,1993 0
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Figure 4. Logarithm of normalized tracer diffusion coefficient versus @E, for different correlation lengths ro in the barrier model. Dashed l i e represents the homogeneous case and the symbols the Monte Carlo simulation. Es = +2 kcal/mol; EB= +1 kcal/mol; AEB = 2 kcal/mol (uniform distribution of DB). 10
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Figure 6. Logarithm of normalized tracer diffusion coefficient versus @E&for different correlation lengths ro in the site-bond model. Dashed line represents the homogeneous case and the symbols, the Monte Carlo simulation. Es = +3 kcal/mol; AEs = 2 kcal/mol; AEB= 2 kcal/mol; E B = +1,+2, +2.4, +2.06 kcal/ mol for Z = 0 (ro = 0),0.5jr07 1),0.7 (ro = 21, and 0.93 (ro 5* 101, respectively. Both distributions are uniform. In general, if the temperature decreases there will be a high degree of jump correlation cf 0). This can be understood as follows: At low temperature, in the simple case of random barriers, the particle spends much time in performing backward jumps on the very low energetic barriers. This phenomenon has been called by other the flip-flop effect. When there is some correlation between neighboring barriers (ro > 11, the particle jump onto a collection of low energetic barriers and then goes back to the same or similar path and is reflected sometimes by the higher barriers at the patch boundary. Site-Bond Model. We employ uniform distributions of the same width (A& = A& = 2 kcal/mol) and the following mean energies: (i) RSBM (ro = O), Es = 3 kcal/ mol, EB = 1kcal/mol; (ii) CSBM (ro = l),Es = 3 kcal/mol, EB = 2 kcal/mol; (iii) CSBM (ro = 2), Es = 3 kcal/mol, EB = 2.4 kcal/mol; (iv) CSBM (ro = lo), Es = 3 kcal/mol, EB = 2.86 kcal/mol. From the analysis of simulation data we find that the long-time behavior of the mean quadratic displacement is lineal with time (dw = 2) and depends neither on the temperature nor on the correlations (ro). From these data and eq 24, we find the values of ln(DT/D0T>for different temperatures. In Figure 6, we plot ln(DT/D0T,versus @ h a n d compare the different cases with that of the homogeneous surface. A clear non-Arrhenian behavior for the site-bond models is observed like that for the trap model. When roincreasea, the curves clearly separate from the corresponding to the homogeneous case. Limoge and Bocquetls predicted this behavior for ro = 0. For understanding this behavior, we must observe that the site-bond surface can be viewed as a mixed case of traps and barriers (Figures 2 and 4). For I = 0, the RTM presents a greater deflection (but in the opposite sense) than the RBM, as regards the homogeneous problem in the Arrhenius plot. It seems reasonable that for the “mixed case”, the energetictraps contributionto the behavior of the diffueion coefficientbe more important. The deflectionof the curves also increases with ro, since the trap contribution does not vary with the overlapping, but the barrier contribution mimetizes the trap case (see Figures 2 and 4). For high correlation, the surface becomes as a set of large quasi-homogeneous patches separated by large steps where each patch plays the role of a large energetic trap.
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Figure 5. Jump correlation factor versus 100/Tfor different correlation lengths in the barrier model. The symbols are the results from Monte Carlo simulation. The energetic values are the same aa those in Figure 4. different barriers becomes irrelevant. Then the particle sees the substrate like a homogeneous surface. At low temperatures, the dependence on the energetic correlation is very important. When the correlation length EO increases, the behavior is like that of the homogeneous case, and for great values of FO the system behaves like a trap model. For the random barriersmodel and at low temperatures, the particle can walk on a low energetic barriers pathway. This implies an increasing in the diffusion coefficient, which comes to be greater than the one corresponding to the homogeneous case. For ro # 0, neighbor barriers have similar energies, and when ro increases, the barriers tend to form homotatic patches.14 Each cluster (of similar barriers) behaves like a homogeneous surface. At low temperatures, the particle is trapped in some patch of low energy barriers and the observed behavior is like that corresponding to trap surface. We could say that the particle behavior for the barrier models in the long time limit is quasi-Arrhenian. In Figure 5 we plot the correlation factor f , v8 100/Tfor diffeEent correlation degrees. At high temperature, all the curves behave like the homogeneous case cf = 1).At low temperature, if the energetic correlation length increases, then the jump correlation decreases. (14)Riccardo, J. L.Ph.D. Theeie, Univeraidad Nacional de Sau Luis, Argentina, 1991. Riccardo, J. L.;Chade, M.; Pereyra, V.; Zgrablich, G. Feature Article, Longmuir 1992,8, 1518.
(15) Gomer, R.Rep. B o g . Phys. 19SO,53, 917.
(16)Limoge,Y.; Bocquet, J. L.Phye. Reo. Lett. 1990, 65, 1.
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From the obtained results, it is clear that the disagreement with the expected short time results arises due to the averaging of the dependent variable time of walk t, which presents great fluctuations under these conditions. However, both methods provide similar results when a greater number of steps or a higher temperature is considered. The short time regime duration decreases when the temperature Tincreases and increases when the correlation length ro increases, as it was to be expected. A more exhaustive treatment of the tracer diffusion on heterogeneous substrates at short-time regime is now being worked out.
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CORRELATED BARRIER MODEL ( r o = l )
Figure 8. Logarithm of mean square displacement versus logarithm of t for correlation length t o = 1 and for different temperatures,in the barriermodel. Squaresand points represent resulta from simulationmethods,taking as independentvariable N and t , respectively.
In Figure 7 we plot f versus 100/T for different overlapping degrees. Evidently, although in the Arrhenius plot the behavior is like that for traps, the jump process is quite different. Qualitatively, the behavior of the correlation factor (Figure 7) is as in the barrier problem, due to the asymmetry in the jump process. The explanation for this behavior is the same as that for the barrier case. Short-Time Behavior. The simulation method used provides incorrect results in studying the diffusive surface phenomena,at short time an_datlowtemperature. Indeed, at relative temperature 0 = E,JkT = 5 or greater, and for a short time walk (20 steps or less), a behavior corresponding to dw < 2 was found, specially in the barrier and site-bond models. To analyze the short time behavior, a modified simulation method was developed, where the independent variable is now the time of walk, instead of the number of steps. As an example, in Figure 8,ln(R2)vs ln t for the barrier model with correlation length ro = 2 and for different temperatures is plotted. Squares and points represent the Monte Carlo results corresponding to the LimogeBocquet method and the modified method, respectively.
Conclusions We have studied the tracer diffusion on energetically heterogeneous surfaces by performing Monte Carlo simulations. The different energetic models (RTM, RBM, RSBM, CTM, CBM, and CSBM) were constructed by means of the dual description. This model gives a more realistic description of the heterogeneous surface. The analysis of the diffusion process was based on the random walk theory. The quantities that were obtained by simulation are the mean quadratic displacement, the time of walk, and the number of steps. We considered the long time behavior and we found normal diffusion for every case. This result depends neither on energetic correlation nor on temperature. The behavior of the diffusion coefficient and of the correlation factor was studied, in all cases, for different temperatures. It arises from the Arrhenius plot of the diffusion coefficient that the results can be separated in two regions: (i) the barrier region, between x-axis and the bisectrix; (ii) the trap region, between the bisectrix and y-axis. For the case of traps, a very important result was found both the tracer diffusion coefficient and the jump correlation factor do not depend on the energetic correlation. Besides, the results agree with the theoretical prediction. In the barrier model, we have obtained a quasi-Arrhenian behavior. The curve goes from barrier region to trap region as the overlap increases. But the jump correlation factor is in this case very different from the one corresponding to the trap model. The study of this factor may be very useful to obtain some information about the energetic topography of the surface. The site-bond model can be considered as a mixed case between trap model and b-arrier model. The behavior of the curve In (DT/DoT)vs OE,& is similar to that of the trap model, and the correlation factor is like that of the barrier model. The short-time behavior of tracer diffusion deserves a deeper analysis since it appears to be very sensitive to energy correlations. The tracer diffusion analysis on real heterogeneous surfaces could be a very important tool for the energetic characterization of the surface. The jump correlation factor plays a fundamental role in the determination of the surface type. A complementary use of other surface phenomenalike adsorption and collective surface diffusion should be helpful for such a characterization.
