Langmuir 2008, 24, 8719-8725
8719
Kinetic Adsorption Energy Distributions of Rough Surfaces: A Computational Study T. Panczyk,*,† T. P. Warzocha,‡ P. Szabelski,‡ and W. Rudzinski†,‡ Institute of Catalysis and Surface Chemistry, Cracow, Poland, and Department of Theoretical Chemistry, Faculty of Chemistry, Maria Curie-Skłodowska UniVersity, Lublin, Poland ReceiVed March 6, 2008. ReVised Manuscript ReceiVed May 5, 2008 The adsorption energy distribution usually refers to localized monolayers of adsorbate at thermodynamic equilibrium. Many papers have been published that analyze its influence on adsorption isotherms, heats of adsorption, and adsorption kinetics. However, the adsorption energy distribution, in its classical thermodynamic equilibrium sense, may be not as useful as expected. This is because many important processes involving adsorption have dynamic character and reactant particles have a finite time for penetration of the adsorbent. The above suggests that some adsorption centers located in less accessible fragments of the surface can be invisible in a dynamic process. However, under conditions allowing the thermodynamic equilibrium such adsorption centers could noticeably contribute to the adsorption energy distribution. The aim of this work is to measure the adsorption energy distributions of special rough surfaces using a dynamic method. This method is based on the molecular dynamics simulation of an ideal gas flowing over a sample surface. The ideal gas particles penetrate the surface, and at the moment of collision of a gas particle with the surface the Lennard-Jones potential energy is calculated. This energy can be identified with the adsorption energy at a given point on the surface. The surfaces used in the calculations have been created using two surface growth models (i.e., random deposition and ballistic deposition). The application of these highly disordered surfaces enables us to draw some general conclusions about the properties of real surfaces that are usually far from any deterministic geometry.
Introduction Energetic nonequivalence of adsorption centers is a common feature of actual solid surfaces.1–3 In catalytic systems, one can often observe various chemisorption energies of a given reactant corresponding to adsorption on various active centers.4–8 Physisorbed systems reveal even more complex behavior because the spectrum of possible adsorption energies is often so dense that it can be approximated as a continuous function.1–3,9,10 This function has been called the adsorption energy distribution, and it physically describes the probability χ of finding on the surface an adsorption site with a given adsorption energy ε. This function must normalize to unity, as was pointed out by Hill11 because the surface is of finite extent; that is ∫ χ(ε) dε ) 1 over the range of energies considered significant. At equilibrium the adsorption on an energetically heterogeneous surface is described by the integral equation * Corresponding author. Present address: Department of Chemistry, University of Cambridge, Cambridge CB2 1EW, U.K. E-mail: panczyk@ vega.umcs.lublin.pl. † Institute of Catalysis and Surface Chemistry. ‡ Maria Curie-Skłodowska University. (1) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: New York, 1992. (2) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: New York, 1988. (3) Ross, S.; Olivier, J. P. On Physical Adsorption; Interscience: New York, 1964, p. 123. (4) Brown, W. A.; Kose, R.; King, D. A. Chem. ReV. 1998, 98, 797. (5) Clark, A. The Theory of Adsorption and Catalysis; Academic Press: New York, 1970. (6) Kanervo, J. M.; Reinikainen, K. M.; Krause, A. O. I. Appl. Catal., A 2004, 258, 135. (7) Panczyk, T.; Gac, W.; Panczyk, M.; Dominko, A.; Borowiecki, T.; Rudzinski, W. Langmuir 2005, 21, 7311. (8) Panczyk, W.; Gac, W.; Panczyk, M.; Borowiecki, T.; Rudzinski, W. Langmuir 2006, 22, 6613. (9) Ross, S.; Olivier, J. P. J. Phys. Chem. 1961, 65, 608. (10) Kowalczyk, P.; Kaneko, K.; Terzyk, A. P.; Tanaka, H.; Kanoh, H.; Gauden, P. A. Carbon 2004, 42, 1813. (11) Hill, T. L. J. Chem. Phys. 1949, 17, 762.
θt(p) )
∫ θ(p, ε) χ(ε) dε
(1)
where θt is the total amount adsorbed on the energetically heterogeneous surface, whereas θ(p, ε) is called the kernel function or local isotherm. The functional form of the local isotherm depends on the assumed model of adsorption, for example, the Langmuir model for localized adsorption or the Volmer model for a mobile adsorbed film.12 Adsorption is the result of the net attractive force between a solid surface and an adsorbed particle. For physical adsorption, these forces arise from London-type dispersion forces (van der Waals forces) resulting from induced dipole-dipole interactions. This type of force is well approximated by the Lennard-Jones potential for pairwise interaction, and any irregularity in the local chemical composition, density, or geometry of the surface will cause a variation in the adsorptive potential at a given point on the surface. The existence of an adsorption energy distribution is therefore related to the surface irregularity (i.e., chemical, geometrical, etc.). Several theories have been proposed in the literature linking the adsorptive properties of surfaces to their state. Cerofolini13 assumed that the actual solid surfaces are never smooth at finite temperatures because of the displacement of surface atoms into higher energy states. He assumed that the lowest energy u0 corresponds to a fully coordinated site on the crystal face with the densest packing. Atoms on a rough surface have an excess energy, uex, relative to u0. A consideration of the relationship between the roughness of the surface and its adsorptive properties led Cerofolini to postulate that the number of atoms Ni with excess energy uiex is related to the number of the atoms N0 in the ground level u0 by the Boltzman distribution,1,13 (12) Cerofolini, G. F.; Rudzinski, W. In Equilibria and Dynamics of Gas Adsorption on Heterogenous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Studies in Surface Science and Catalysis; Elsevier: Amsterdam, 1997; Vol. 104. (13) Cerofolini, G. F. Surf. Sci. 1975, 51, 333.
10.1021/la800707v CCC: $40.75 2008 American Chemical Society Published on Web 07/01/2008
8720 Langmuir, Vol. 24, No. 16, 2008
( )
uiex Ni ) exp N0 kTf
Panczyk et al.
(2)
where Tf is a certain characteristic temperature not far from the melting temperature. Next, Cerofolini expressed the relation between uiex and adsorption energy using a formal Taylor expansion. He also postulated that the higher the excess energy of a given surface atom, the higher the adsorption energy. However, such an assumption ignores the effects of long-range van der Waals interactions. Another theoretical approach, proposed by Rudzinski and Everett1 was based on the assumption that the distribution of excess energy of surface atoms is described by Fermi-Dirac statistics. Their description does not require a monotonic increase in adsorption energy with increasing excess energy of surface atoms. Although it still does not account directly for the effects of long-range interactions. Both of these approaches were able to explain the existence of the well-known empirical adsorption isotherms. Moreover, they explained quite an intriguing observation that almost all experimental adsorption systems can be quite accurately described using only a few empirical isotherm equations.14 Another theoretical approach to relate the surface energetic heterogeneity to its geometric structure was based on the concept of pore size distribution of fractal surfaces.15,16 It has been shown that the pore size distribution function F(r) on the fractal surface scales with the fractal dimension
F(r) ∼ (3 - D)r2-D
(3)
where D is the fractal dimension. Next, assuming a certain relationship15 between pore size r and adsorption energy ε in a pore of size r one can derive the same functional expressions describing the adsorption energy distributions as those given by Cerofolini,13 and Rudzinski and Everett.1 However, the meaning of the parameters is different. The above presented literature results suggest that there is some general relationship linking the adsorption energy distribution to either the energetic state of surface atoms or the pore size distribution. The functional form of such a relationship is still unknown, but it can be anticipated that this form is the same for all adsorption systems whereas the individual properties of a given systems are determined by the parameters of this function.15 It should be also pointed out that the above conclusions concern the adsorption systems being at thermodynamic equilibrium. In dynamic processes, the concept of adsorption energy distribution (whatever its origin) must be reviewed. This is because the adsorbate particles have only a finite time to penetrate the adsorbent structure. This means that some adsorption centers located in less accessible regions of the surface can be invisible in a dynamic process although they are strongly energetically favorable. However, under conditions allowing thermodynamic equilibrium such adsorption centers could noticeably contribute to the adsorption energy distribution. The transport processes in geometrically irregular/fractal media have been extensively studied in the literature,17–21 mainly in terms of diffusion-limited reaction (DLR) models. However, (14) Hobson, J. P.; Armstrong, R. A. J. Phys. Chem. 1963, 67, 2000. (15) Rudzinski, W.; Lee, S. L.; Yan, C.; C; Panczyk, T. J. Phys. Chem. B 2001, 105, 10847. (16) Rudzinski, W.; Lee, S. L.; Panczyk, T.; Yan, C. C. J. Phys. Chem. B 2001, 105, 10857. (17) Coppens, M.-O. Catal. Today 1999, 53, 225. (18) Sheintuch, M. Catal. ReV. 2001, 43, 233. (19) Seri-Levy, A.; Avnir, D. Surf. Sci. 1991, 248, 258. (20) Chaudhari, A.; Yan, C. C.; Lee, S. L. Chem. Phys. Lett. 2002, 351, 341. (21) Chaudhari, A.; Yan, C. C.; Lee, S. L. Chem. Phys. 2005, 309, 103.
these studies are carried out on the mesoscopic level where concentration gradients within fractal fjords can be defined.17 The analysis of the elementary interactions of gas particles with solid substrate atoms involves the microscopic description; therefore, to determine the local changes in adsorption energy on the fractally rough surface one has to apply such methods as Monte Carlo or molecular dynamics simulations. Therefore, the aim of this work is to study the energetic properties of rough surfaces in a dynamic processes by means of MD simulations. More precisely, this involves the construction of a map of gas-solid interaction energy and its analysis for various degrees of surface geometric irregularity. We call such a map the kinetic adsorption energy distribution (KAED) because the applied method of its determination is purely dynamic and thus such distributions can be more adequate in real adsorption kinetics or surface reactions. Adsorption energy distributions are usually determined using adsorption isotherms or a temperature-programmed desorption technique. Therefore, they contain an element of arbitrarily chosen model of elementary gas-solid interactions (e.g., a local adsorption isotherm or a model of an elementary desorption step). Usually both models are greatly simplified because the next step in the determination of the adsorption energy distribution is a very demanding task from a computational point of view. Therefore, it would be very difficult to directly compare our result (which represents a direct measure) with the experimentally determined adsorption energy distributions. Also, the dynamic nature of our KAEDs makes a direct comparison rather impossible. However, as will be discussed in the section devoted to strongly irregular surfaces, some kind of mapping between our results and the behavior of the adsorption energy distribution derived from the fractal approach can be found. It seems that the effects of a strong surface irregularity then become dominant and hence some similarities in the behavior of our KAEDs and those derived from the fractal approach can be observed.
Methods The methodology applied here consists of two steps, namely, the creation of rough surfaces and the determination of the adsorption energy map of gas particles interacting with those surfaces. The studied surfaces were created using two simple growth models, namely, random deposition and ballistic deposition. Both of these models are thoroughly discussed in the literature;22 however, a short description of their most important features will be convenient for further discussion. The random deposition can be simply outlined as follows: an atom on the regular L × L lattice is chosen at random. Then, an additional atom is placed atop, and the number of atoms in this node is increased by 1. This procedure is repeated until the total desired number of atoms is deposited. In this study, the starting surface structure was the regular lattice of atoms, and the deposition occurred in on-top positions. Using the random deposition method, one can obtain surfaces with fractal dimensions from 2 to 2.5 depending on the number of atoms deposited. The initial perfectly regular surface corresponds to D ) 2 (no atoms deposited), and the surface in the stationary state has a fractal dimension of 2.5.23,24 The ballistic deposition can be outlined as follows. A particle is released from a randomly chosen position above the surface, located at a distance larger than the maximum height of the interface. The particle follows a straight vertical trajectory until it reaches the surface, whereupon it sticks. The particle sticks to the first surface atom along its trajectory in the nearest-neighbor position. This procedure (22) Barabasi, A. L.; Stanley, H. E. Fractal Concepts in Surface Growth; Cambridge University Press: New York, 1995. (23) Panczyk, T. J. Comput. Chem. 2007, 28, 681. (24) Panczyk, T.; Warzocha, T. P.; Rudzinski, W. Appl. Surf. Sci. 2007, 253, 5846.
Kinetic Energy Distributions of Rough Surfaces
Langmuir, Vol. 24, No. 16, 2008 8721
Figure 2. Evolution of the fractal dimensions of surfaces obtained from random and ballistic depositions as a function of the number of monolayers deposited. Figure 1. Objects created using both surface growth models for the deposition of 1.25 monolayers on the initial (100) lattice of surface atoms.
is repeated until the total desired number of particles is deposited. In this study, the starting surface structure was the regular L × L lattice of atoms, and deposition occurred only in the nearest-neighbor positions. The ballistic deposition technique produces surfaces with fractal dimensions from 2 to 3 whose structures can represent those of foamlike silicas or carbons.25 However, in our case neither random deposition nor ballistic deposition creates fractal objects in a strict mathematical sense. The studied objects reveal lower (diameter of the surface-building atom) and upper (diameter of the widest pore) cutoffs of scaling laws. Therefore, the fractal dimension of these objects represents only a kind of measure of the surface roughness. However, we will sometimes refer to their fractal dimension when discussing their properties in the framework of the fractal approach to adsorption phenomena. An illustration of sample objects obtained from these two deposition techniques is given in Figure 1. Next, measurements of the fractal dimension of the obtained surfaces were performed. The applied method results from the basic property of fractal objects. Namely, the area S of the fractal object scales with its linear dimension, L26
S ≈ LD
(4)
where D is the fractal dimension. The areas of the objects were determined in the following way. Every atom of the object was replaced by a cube of the same diameter d, and the number of cube faces exposed outside the object was counted. In the case of ballistic deposition, the obtained object can be composed of a certain number of bubbles whose inner surface cannot contribute to the outer surface area. Thus, to obtain the surface fractal dimension of the object, the area of bubbles has been subtracted from the total area obtained from counting the number of cube faces creating the interface. The deposition of N atoms on an initially regular lattice of L2 atoms increases the object volume/mass by d3N, where d is the diameter of the atom. If the system size is rescaled so that L′ f RL, then the equivalent object is obtained after the deposition of the same mass/volume because rescaling occurs in all dimensions. Therefore, for a rescaled system the number of atoms of diameter d required to obtain the equivalent object mass/volume is R3N. To determine the fractal dimension of the object after the deposition of N atoms, a few rescaled equivalent systems of size RL (for example, R takes values of 2, 3, and 4) and the number of deposited atoms R3N must be generated. Next, from the tangent of the linear plot of ln S versus ln L the fractal dimension of the object can be obtained. Figure 2 shows how the fractal dimension of the objects changes (25) Panczyk, T.; Warzocha, T. P.; Rudzinski, W. Appl. Surf. Sci. 2008, 254, 2285. (26) Avnir, D. The Fractal Approach to Heterogeneous Chemistry; Wiley: New York, 1989.
with an increasing number of deposited atoms in both surface growth models. The determination of the kinetic adsorption energy distribution functions on the rough surfaces was done using the simplified molecular dynamics simulation technique. The most important assumptions of this technique concern the interactions. It was assumed that only hard core repulsions exist between gas particles. When a gas molecule makes contact with a solid atom, then its motion is not affected by any attractive interactions (i.e., also in this case only hard core repulsions were assumed). However, at the moment of closest contact of the gas particle with the solid the Lennard-Jones potential energy of the molecule was calculated and recorded. In that way, a map of potential energy of various places on the rough surface was determined. The simulations, with periodic boundary conditions in the x-y directions, were performed in a box of size L × L × Lz, where L ) 50d is the size of the bottom of the box whereas Lz is the height of the box. A slitlike pore was composed of two identical rough surfaces separated by a distance Lz ) 300d. The height of the largest object studied was 26d (ballistic deposition), thus the distance Lz ) 300d was sufficient to prevent any contribution to the energy from the opposite object. The choice of this distance was also dictated by earlier results on the basis of the analysis of collision frequency with similar surfaces.23–25 Such a distance ensures that the calculation results are representative of a single surface (i.e., without any pore effects). The gas molecules were placed inside the simulation box with randomly selected positions. The number of molecules in the box was not more than 100 because of the CPU time savings. Larger number of gas particles did not reveal any changes in the calculation results. A detailed description of the simulation scheme can be found elsewhere.23 It is generally based on a numerical solution of the Newton equations of motion. Because of only the assumed hard core repulsions it was not necessary to introduce any thermostatting technique or calculate forces. In the case of the collision of two gas particles or a gas particle with a surface atom, the velocity components of gas particles were recalculated using the well-known relations developed for the noncentral ideally elastic collision of two balls with a finite diameter. Thus, the gas particles within the system behave like an ideal gas; they do not experience any attractive forces induced by the surface atoms or other gas particles. This ensures the totally uncorrelated motion of gas particles within the complex interior of the solid and ensures the statistical independence of successive probes of interaction energy weighted only by the accessibility of a given region of the solid surface. The interaction energy of a gas particle in a given region of the solid surface was determined as follows. At the moment of collision of the gas particle with the surface atom (i.e., when the distance between them equals the sum of radii), the adsorptive potential of the gas particle in this place εi is calculated by summing the LennardJones contributions from the surrounding surface atoms
8722 Langmuir, Vol. 24, No. 16, 2008
εi )
[( ) ( ) ]
∑ uij ) ∑ 4ε0 j
j
Panczyk et al.
σ rij
12
-
σ rij
6
(5)
where ε0 is some energy unit, rij is the distance separating the gas particle and surface atom, and σ is the sum of radii of a gas particle and a surface atom divided by 21/6 (then the minimum in the LJ potential appears at the closest contact and equals -ε0). The summation is carried out within the sphere of diameter rcut ) 5d, ensuring a negligible contribution to εi from the surface atoms placed at distances larger than rcut. In other words, the LJ tail correction was completely negligible regardless of the diameter of a gas particle. Once the summation is performed, the gas particle reflects from the surface atom and continues wandering within the system until it collides with another surface atom and measures the adsorption potential at this new place. Note that the LJ energy determined in the above way does not modify the motion equations. It only shows what adsorption energy the gas particle would have when it appears at this place in an actual adsorption process. During the simulation, the calculated energies εi are collected, and the probability χi of the occurrence of a given value of εi is determined. After a sufficiently large number of probes, the spectrum of χi versus εi finally evolves to the invariant and continuous function χ(ε). This is just the kinetic adsorption energy distribution function (KAED). The term “kinetic” reflects the fully deterministic status of that function. It accounts for various degrees of access of various surface places; that is, the probability of occurrence of the states that are hardly accessible from the gas phase are proportionally decreased in that function. On the contrary, the equilibrium adsorption energy distribution function does not take into account the accessibility of a given adsorption center. This is of course justified because at equilibrium the adsorbate particles have sufficient time to penetrate all possible adsorption centers. The function χ(ε) originates mainly from surface geometric irregularities and van der Waals interactions. It does not account for any specific short-range interactions, that is, chemical or, as postulated by Cerofolini,1,13 induced by the unsaturated bonds between surface atoms. The possible contribution from the specific interactions could be accounted for by adding some additional energy terms δi to εi at a given surface atom; however, in the present state the values of δi cannot be reasonably predicted. It must be also noted that the interaction energy appearing in the function χ accounts only for submonolayer adsorption. The contribution from the particles at distances from the solid surface larger than the sum of a gas particle and a solid atom radius is neglected here. However, as shown by Knippenberg et. al.,27 in the case of multilayer adsorption the definition of the adsorption energy associated with single adsorption center or a single gas particle is not simple. In such cases, the heat of adsorption becomes a distance-dependent quantity.27
Results and Discussion Flat Surfaces. At the beginning, let us consider the behavior of the function χ(ε) in the case of a perfect regular (100) lattice of atoms. Such a lattice is a macroscopically perfectly flat surface; however, on the microscopic level it reveals a periodic density map (Figure 3). Therefore, the perfect (100) surface is not energetically uniform, and one can distinguish three specific adsorption sites: top, hollow, and bridge. Thus, the adsorption energy distribution for (100) plane should reveal three distinct peaks with a relative amplitude of 1:1:2. The calculations were carried out for three various ratios s of gas molecule diameter to the surface building atom diameter (i.e. s ) 0.5, 1, and 2). The summation of the LJ contributions from the subsurface atoms was accounted for up to the rcut distance. The energy unit ε0 in the LJ potential was the same for all diameters of gas molecules (i.e., the interaction energy of a single pair at (27) Knippenberg, M. T.; Stuart, S. J.; Cooper, A. C.; Pez, G. P.; Cheng, H. J. Phys. Chem. B 2006, 110, 22957.
Figure 3. Atomic density map of the (100) lattice of atoms. One can distinguish three types of adsorption sites: T, top; B, bridge; H, hollow. The ratio among them is 1:1:2 T/H/B.
Figure 4. Kinetic adsorption energy distributions (KAED) for the perfect (100) lattice of atoms and various ratios s of gas molecule diameter to the surface building atom diameter (ε ) -ε0).
the closest contact was the same (-ε0)). Figure 4 shows the resulting adsorption energy distributions for the perfect (100) lattice. An analysis of the results shown in Figure 4 leads to the conclusions that the behavior of the KAEDs is more complex than could be expected on the basis of the density map shown in Figure 3. The KAEDs do not reveal the distinct peaks corresponding to the interaction with the T, H, and B sites. The only distinguishable states are the interaction with the top site (which must correspond to the lowest observable adsorption energy) and an ideal location of gas particle in the hollow site (the highest energy with nonzero probability). The intermediate adsorption energy states are difficult to link with the particular gas particle-solid atom configurations. Also, the small lambdashaped peaks cannot be definitely attributed to any particular configurations. A very interesting observation is that the small gas molecules (s ) 0.5) can directly interact with top sites with the highest probability. Direct interaction with the hollow (most energetically favorable) sites is almost impossible without involving an additional acceleration of gas particles toward this site. Thus, if the adsorption energy on the top and hollow sites is similar, then the occupation of the hollow sites would be very small compared to the occupation of the top sites. One can also notice a strong shift of KAED with the increasing diameter of gas particle. This is understandable because large
Kinetic Energy Distributions of Rough Surfaces
Figure 5. KAEDs determined for small numbers of deposited surfacebuilding atoms on the initially perfect lattice. (A-C) Depositions of 0.0032, 0.0256, and 0.1024 monolayers, respectively. Solid, dashed, and dotted lines correspond to very small gas particles (s ) 0.5), with the same diameter as that of surface atoms (s ) 1) and very large gas particles (s ) 2), respectively.
particles, upon collision with the surface, have more surface atoms in close proximity than do the smaller ones. Therefore, their interaction energy is enhanced compared to that of small particles. Moderately Disordered Surfaces. The deposition of up to 0.1 ML (monolayer) leads to surfaces that exhibit weak or moderate disorder (fractality). Figure 2 shows that in that range of deposition both surface growth models lead to objects with the same or very similar fractal dimensions that are not higher than 2.4. The KAEDs determined for such surfaces are shown in Figure 5. Of course, they are the same irrespective of the surface growth model applied because in the limit of such small depositions the difference between these models is hardly visible. As can be seen in Figure 5, the increasing surface disorder leads to substantial changes in KAEDs mainly in the low-energy part. This is related to the existence of surface atoms stuck on top of the surface atoms forming the initial lattice and having no neighbors in close proximity. The contribution to the energy coming from the interaction with such surface atoms is smaller because of the lack of or a very weak contribution from their neighbors. Despite the small number of such atoms, their existence is strongly reflected in the behavior of KAEDs. The first low-energy peak rapidly increases with increasing deposition. For the smallest gas particles, it even becomes dominant for the deposition of 0.1 ML, which means that the probability of interaction of gas particles with such surface atoms is very high although their number is relatively low. Figure 5 gives also some insight into the role of surface defects in the energetic behavior of surfaces. Small depositions (i.e., up to 0.0256 ML) (Figure 5B) can be viewed as defects decorating the perfect (100) plane. The deposition of 0.0256 ML corresponds to ca. 2.5% coverage by the extra surface atoms. Looking at
Langmuir, Vol. 24, No. 16, 2008 8723
Figure 5B, one can notice a non-negligible contribution to the KAEDs from the surface defects. Moreover, the ratio of interactions with defects is higher than could be expected from the number of defects. This ratio can be calculated as the ratio of the area belonging to the first peak to the area of the total distribution. It takes values of 6.4, 10.8, and 18.4% for s ) 0.5, 1, and 2, respectively. Clearly, the probability of interactions with defect sites is enhanced compared to their presence on the surface (i.e., 2.5%). Most probably, these excess surface atoms, sticking out of the plane of regular surface atoms, generate a higher kinetic cross section for collision than do the regular ones. High-energy parts of KAEDs are almost unaffected by the increasing surface disorder; that is, the absolute minima of the potential energy do not change significantly at the considered depositions. Some changes in KAED are visible only for large gas particles. This is due to the existence of such surface structures where the large gas particles can fit simultaneously to a larger number of atoms than they could on the flat surface. Generally, the results shown in Figure 5 suggest that the mean adsorption energy will be shifted to the left on the energy scale when the surface roughness increases. This could imply that the increasing surface disorder induces the occurrence of adsorption energies lower than those observed for the initial perfect lattice of atoms. This is understandable while taking into account the fact that the initial lattice corresponds to the highest possible density of surface atoms, hence the strong LJ contributions to the adsorption energy. The successive deposited layers of surface atoms reveal much a lower density, thus their contributions to the LJ interaction energy are smaller. Strongly Disordered Surfaces. Depositions above 0.1 ML lead to substantial differences between the fractal dimensions of surfaces obtained using the discussed growth models. Similarly, the morphologies of those surfaces as well as their energetic properties become dependent on the surface growth model. Therefore, the KAEDs for surfaces corresponding to higher depositions are presented separately on the two panels in Figure 6. The left panel corresponds to the surfaces obtained using the random deposition mechanism whereas the right panel corresponds to the ballistic deposition process. In the case of random deposition, the distributions evolve toward roughly exponentially decreasing functions, especially for small gas particles. The most probable adsorption energies correspond to the low-energy limit of KAED. Obviously in that limit the gas particles interact mainly with the topmost and wellisolated surface atoms. As the size of gas particles increases, the distributions become fuzzier; however, one can still notice a few distinct peaks in the low-energy limit. These peaks can be attributed to the interaction of gas particle with micropores composed of various numbers of walls. The behavior of surfaces obtained using ballistic deposition is significantly different than those from random deposition. Here one can observe that the most probable adsorption energy is shifted to the right from the lowest adsorption energy. This means that the gas particle interacts mainly with surface clusters, contrary to the case of random deposition where the most probable interaction was with the isolated topmost surface atoms. The shapes of KAEDs are also qualitatively different; they can be roughly approximated by the right-hand widened quasi-Gaussian functions. For large gas particles and the highest deposition studied, the shape of KAED can be even approximated by a uniform distribution. One can also notice that the spectrum of the possible adsorption energies is denser in the case of ballistic deposition: the KAEDs are smoother than previously.
8724 Langmuir, Vol. 24, No. 16, 2008
Panczyk et al.
Figure 6. KAEDs for surfaces corresponding to higher depositions shown in the legends. (A-C) Random deposition and (D-F) ballistic deposition. Solid, dashed, and dotted lines stand for various diameters of gas particles (i.e., various values of the parameter s, which is the ratio of gas particle diameter to solid atom diameter).
An analysis of Figure 6 leads to the conclusion that either qualitative or quantitative properties of KAEDs strongly depend on the surface morphology. Noticeable differences between surfaces obtained using the two growth mechanisms appear above the deposition of ca. 0.1 ML, and then the fractal dimensions of these surfaces are different for the same deposition. Accordingly, the KAEDs reveal behavior that is representative of only a given surface growth model, and the fractal dimension of a given surface cannot definitely modulate its energetic properties. However, let us quickly recall Cerofolini’s theory of the origin of surface energetic heterogeneity.1,13 According to Cerofolini, the general form of the adsorption energy distribution function has the same functional form for all gas-solid systems and individual properties of those systems are coded in a few parameters of that function. Cerofolini’s approach is based on purely energetic arguments, and it does not directly account for geometric disorder. Rudzinski et.al.15,16 derived the same form of the general adsorption energy distribution relying only on geometric arguments. This function has the following form:
(
χ(ε) ≈ (3 - D)(a + b(ε - ε0)) exp -a(3 - D)(ε - ε0) b (3 - D)(ε - ε0)2 (6) 2
)
where D is the surface fractal dimension and a, b, and ε0 are constants that are characteristic for a given gas-solid system. Although the function (eq 6) is derived by assuming thermodynamic equilibrium in the gas-solid system, it surprisingly well predicts the qualitative behavior of the KAEDs analyzed in this work (especially those for high deposition in Figure 6). Let us note that when the value of the parameter b in eq 6 is close to zero then the function (eq 6) becomes exponentially decreasing, just as in the case of the random deposition shown in Figure 6. However, when the value of parameter a is close to zero then
the function (eq 6) becomes a right-hand-widened quasi-Gaussian function, which is the behavior observed for ballistic deposition in Figure 6. Thus, we can draw two important conclusions; namely, the results of the direct determination of adsorption energy distributions (although not representative of thermodynamic equilibrium) support the theory developed by Cerofolini and Rudzinski et.al.1,13,15,16 However, our KAEDs seem to follow the general rules developed for equilibrium adsorption energy distributions on the geometrically disordered surfaces. However, the above conclusions are valid for strongly disordered surfaces only. The behavior of KAEDs in the range of deposition of up to 0.1 ML (i.e., for weakly disordered surfaces) cannot be approximated by the function (eq 6). This is not surprising because the origin of the function (eq 6) is related to the fractal concept. Therefore, the assumptions that lie behind the development of eq 6 become less and less justified when the surface is approaching planar geometry.
Summary and Conclusions This work is devoted to the analysis of energetic properties of geometrically irregular surfaces created using two simple, well-known surface growth models.22 Although these models are idealized, they are helpful for qualitative studies of the behavior of real systems under conditions of strong surface geometric irregularity. The kinetic adsorption energy distributions (KAED) determined in this work carry important information concerning the surface energetic heterogeneity in dynamic processes (adsorption kinetics and surface reactions). They account for the accessibility constraints of various surface places, thus the probability corresponding to a given molecule-surface interaction energy also contains the probability of collision of the gas particle with a given surface atom.
Kinetic Energy Distributions of Rough Surfaces
The adsorption energies were determined by taking into account only van der Waals interactions, thus any specific interactions coming from unsaturated bonds of surface atoms were not included. The KAEDs were determined for various sizes of gas particles and various degrees of surface irregularity controlled by the number of deposited monolayers on the initially perfect lattice. Additionally, the fractal dimensions of the obtained structures were calculated. The evolution of KAEDs with increasing surface disorder is shown in Figures 4-6 for various diameters of the probe gas particles. In the range of weakly disordered surfaces 2 < D < 2.4, the KAEDs contain sharp, low-energy peaks corresponding to the interaction with the topmost surface atoms. In this range of D, both surface growth models lead to the same KAEDs. Assuming that small deposition (Figure 5B) mimics the presence of surface defects, it can be concluded that the ratio of the interactions with the surface defects is higher than could be expected by relying only on their relative amount. This is especially seen for large gas particles where the first low-energy peak fills 18% of the total area below the distribution function although the proportion of defect atoms is only 2.5%. The KAEDs strongly depend on the diameter of the probe gas particle. The larger the gas particle, the stronger its interaction
Langmuir, Vol. 24, No. 16, 2008 8725
with the surface. This is related to a higher number of surface atoms that large particle can approach very closely. Therefore, even a small surface disorder leads to the development of small adsorption energies corresponding to the interaction with the topmost layer of surface atoms that, in turn, reduces the interaction with the denser lower layers of atoms. High deposition, generating strong surface disorder, leads to distributions depending on the surface growth model. These distributions can be roughly approximated by an exponential function (random deposition) or a right-hand-widened quasiGaussian function (ballistic deposition). Such behavior was predicted by the theory1,13,15,16 of surface energetic heterogeneity based on an equilibrium assumption. Acknowledgment. T.P. expresses his thanks to the Foundation for Polish Science for the financial support of the KOLUMB programme. T.P.W. expresses his thanks to the British Council for the financial support of the Young Scientists Programme. Supporting Information Available: Visualizations of the MD trajectories for s ) 2 with the object obtained using the ballistic deposition of 3.3 ML and for s ) 1 with the object obtained using the random deposition of 3.3 ML. This material is available free of charge via the Internet at http://pubs.acs.org. LA800707V
