Surfaces diffusion

Diffusion is an activated process and is observed to obey an Arrhenius relationship of the form (where only one diffusion mechanism is involved — single site hopping, for example) 

This relationship is used in experiments and m is determined by measuring D at different temperatures. 

In terms of absolute rate theory, provided the transmission coefficient at the activation barrier is unity, the diffusivity D0 is given by the expression 

D is usually found (particularly in FEM experiments) using the approximation [246] for diffusion in a particular direction 

Nearest neighbour lateral interactions have been introduced into this programme by appropriately weighting the values of T depending on the value of z. For repulsive interactions, T is higher the higher z is, whereas the reverse is true for attractive interactions. The effects of such interactions on the surface potential energy barrier to diffusion are shown schematically in Fig. 45. The effect of n.n. interactions is considered to be simply additive in this method, that is 

The diffusion constant can be computed from transition-state theory. The pre-exponent depends on the square of the hopping length of a molecule. 

The surface sites, at a distance a, are separated by an energy barrier and diffusion is described by the hopping of adsorbates, grown one site to another, over the barriers. Therefore a is also called the hopping length. 

TABLE 4.10. Interval of Pre-exponential Factors for Some Elementary Surface 

The diffusion constant for CO diffusion on a metal surface can be estimated with the understanding that during diffusion, the carbon bond remains bonded to the surface. Diffusion then results in loss of only one degree of freedom in the transition state, namely, its frustrated translational motion that has a frequency Vj of 30 cm Since this implies that hv kT, becomes 

One important difference between diffusion and recombination processes is their activation energy. Their pre-exponents, however, appear to be related. The rate of associative desorption is proportional to the average surface area, whereas the diffusion constant relates to the square of the hopping length. 

The flux of material along the surface S is represented by a surface vector field j that is everywhere tangent to the surface S. If r is any unit vector that is tangent to the surface at a point, then the inner product j -t is the volume flow rate of material in the direction r per unit distance in 

Conservation of mass at each point on the surface requires that the net rate of material accumulation at that point due to surface flux is the negative of the divergence of the surface flux at that point. The net rate of material accumulation determines the surface speed Vn, which is the normal speed of the free surface relative to the material points currently on that surface, all referred to the reference configuration. Thus, in the present context. 

When coupled with (9.7), the surface normal speed is related to the surface chemical potential through 

This result provides the normal velocity of the surface in terms of local elastic strain energy density, the local mean curvature, and the surface energy density. 

The transport equation (9.7) is known by several names, with Nernst-Einstein equation and Pick s law being among them. The former identifier commonly refers to charge transport through an electrode and the latter refers to concentration driven transport. The first application in the study 

Migration of adsorbed molecules on the surface may contribute to transport of the adsorbates into the particle. This effect is very much dependent on the mobility oh the adsorbed, species, which is determined by the relative magnitude of the heat of adsorption and the activation energy of migration. 

As shown in Fig. 4.4, when the energy barrier, , existing between neighboring sites, is smaller than the heat of adsorption, Q , then it is easier to hop to the next site than to desorb into the bulk phase. 

The effective surface diffusion coefficient is defined by taking the gradient of the amount adsorbed as the driving force of diffusion. 

Fig 4 4 Cross sectional view of potential energy distribution of adsorption on solid surface 

Surface difl usion can be related to random walk in the direction of diffusion. When unit step is defined by length. Ax, and time. At, as shown in Fig. 4.S, then after n steps (after ndr), variation of the position X = Ax Ax Ax in steps) is given as 

The diffusion of adsorbates on the surface is a process which is important for reactions of the Langmuir-Hinshelwood type, where the molecules in question travel some distance on the surface before they react. The surface diffusion process is influenced by the structure and corrugation of the surface. Therefore the information which can be obtained by studying the diffusion process is of importance for the understanding of the surface structure and its influence on a number of processes, as adsorption/desorption, catalytic properties, and epitaxial growth. 

The diffusion process itself is an activated process, i.e., the molecule has to jump from one site to another site separated from it by a barrier. In principle there is no problem in jumping over several sites, thereby increasing the effective diffusion rate [32]. In any case, the diffusion constant D T) is temperature dependent and a simple Arrhenius expression is often used to model the temperature dependence, i.e., 

Thus measured values of D T) as a function of temperature are plotted against 1/r in order to extract Dq and activation energy Eq [33]. 

A simple expression for the diffusion constant can be obtained using the transition state approach (see Appendix A). This theory gives the following expression for the rate constant for diffusion in the x-direction 

If the rate constants for jumping from site i to j, kij is known, they can be inserted in a rate equation for the probability distribution P, (0 at a given site as a function of time. Thus we have 

Direct measurement of surface diffusion is not feasible since the flux due to diffusion through the gas phase is always present in parallel. In order to study surface diffusion it is therefore necessary to eliminate the gas phase contribution. The normal procedure, which is illustrated by the study of Schneider and Smith, involves making measurements over a wide range of temperatures. The flux through the gas phase is determined from the high temperature measurements since under these conditions the surface flux can be neglected. The flux through the gas phase at the lower temperature is then found by extrapolation and subtracted from the measured flux in order to estimate the 

Knudsen diffusion, and this simplifies considerably the extrapolation. 

Representative data for surface diffusion of propane in a silica gel adsor- 

Surface diffusion is an activated process which is in some ways analogous to micropore or intracrystalline diffusion. The temperature dependence may be correlated by an Eyring equation  

The contribution of surface diffusion to the overall effective diffusivity, however, depends on the product KD rather than on the surface diffusivity alone (Eq. (5.20)]. Since the diffusional activation energy ( ) is generally smaller than the heat of sorption, this product, and therefore the relative contribution of surface diffusion, normally decreases with increasing temperature. Such a trend is illustrated by the data given in Table 5.1. 

The transfer of the adspecies within macrodistances is described with the aid of diffusion equations. The concentrations of the adspecies therein is characterized by the local degrees of lattice coverage 6(x) with adspecies / in the vicinity of the point with the coordinates a. In the absence of external fields the diffusion equation for each component has the form [154,155]  

The temporal evolution of the concentration profiles of the adspecies with allowance for their interaction seems to have been studied for the first time by Bowker and King [158]. Initially, the distribution of the adspecies density has been given up in the form of a step (this technique is often applied to surface diffusion studies). Consideration has been given to the concentration profiles in the case of attraction and repulsion of the adspecies to conclude that they can be used to estimate the lateral interactions. The applicability of the model to the description of diffusion in the O/W (110) system [159] is discussed. 

The effect of the collective jumps of the adspecies on their surface transfer (QCA, CM) has been treated by Tarasenko and Chumak [163]. The motion of the dimers and trimers on lattices with z — 4 and 6 was discussed. The dimer motion consists of the following steps (1) a jump of one of the adspecies with the associated change in the dimer orientation (2) a parallel transfer of the dimer relative to the line connecting its adspecies in the initial position, and (3) a longitudinal displacement of the dimer along the line connecting its atoms (1). On a lattice with z — 6 the first type of motion breaks into two cases (1) the distance between the dimer adspecies in the terminal state remains the same as that in the initial one, or (2) this distance increases. If the frequencies of the different jumps are similar to one another, then even a minor attraction between the adspecies results in a 

In the general case, for a multicomponent system at an arbitrary radius of interaction between the adspecies in a one-step approximation in time a similar treatment gives the following expression for the diffusion coefficients in the transfer equations  

The presence of the cofactor xi serves as a hint of a need for the presence of a correlation cofactor fc which is bound to appear during a more detailed kinetic derivation of the expression for the diffusion coefficient due to changes in the functional relation between the local density and the probabilities of multibody configurations, the relation being controlled by the adspecies concentration gradient. This factor has been considered in the case of binary alloys [155,165] (see Subsection 5.2)  

If we assume that the rates of adsorption and desorption are both large compared with the surface migration rate, the surface and bulk concentrations of each species will be almost in equlibriura, and hence will be 

The form of the functions f depends on Che particular isotherm used for example the Langmuir isotherm gives the familiar relation 

In general, therefore, the surface flux of each substance is linearly related Co all the concentration gradients in the adjacent bulk phase. The coefficients in this linear relation depend on the bulk phase concentrations, 

A rather simpler situation arises when the bulk concentrations are sufficiently small that the adsorption isotherms approach linearity. Then (7,4), for example, shows that 

The form of this flux relation is the same as equation (2.5) for 

If the dominant mechanism of deposition involves the formation of adatoms followed by surface diffusion to steps, the relation between current and electrode potential becomes complicated. The essential 

The incorporation of the adatoms at the steps should be fast because no charge transfer is involved hence the adatom concentration should attain its equilibrium value  

Under stationary conditions dc /dt = 0, and an ordinary differential equation results with Eq. (10.5) as boundary conditions, which can be solved explicitly by standard techniques. The resulting expression for the current density is  

Ao has the meaning of a penetration length of surface diffusion. We can distinguish two limiting cases  

Ao L the two terms involving L/Xo cancel, surface diffusion is fast, the deposition of adatoms is rate determining, and Eq. (10.6) reduces to the Butler-Volmer equation. 

Furfher caufion is warranted given that not all of the elementary mechanisms comprising surface diffusion are fully understood. Traditional hops to nearest neighbors may be accompanied by many other processes including site exchange whereby the adsorbate and a surface atom exchange positions resulting in a net 

Because growth mechanisms for thin films are so strongly determined by how atoms are transported across the surface, it is appropriate to pause to consider diffusion of atoms at the atomic scale in more detail. 

The consequence of these predictions is that single adatoms wUl tend to stay away from this type of surface step and growth will not occur there until the dimer bound to the lower step edge is broken. The repulsion of adatoms by the step would also enhance nucleation of new adatom islands in the surrounding region because they have two directions to move on the open terrace but only one near the step. This 

Surfaces are heterogeneous on the atomic scale. Atoms appear in flat terraces, at steps, and at kinks. There are also surface point defects, vacancies, and adatoms. These various surface sites achieve their equilibrium surface concentrations through an atom-transport process along the surface that we call surface diffusion. Adsorbed atoms and molecules reach their equilibrium distribution on the surface in the same way. This view of surface diffusion as a site-to-site hopping process leads to the random-walk picture, in which the mean-square displacement of the adsorbed particle along the. r-component of the coordinate is given by 

The frequency v with which an atom with a vibrational frequency uq will escape from a site depends on the height AE of the potential-energy barrier it has to climb in order to escape  

Surface diffusion has so far been discussed in terms of a single surface atom. However, on a real surface many atoms diffuse simultaneously and in most diffusion experiments the measured diffusion distance after a given diffusion time is an average of the diffusion lengths of a large, statistical number of surface atoms. A statistical thermodynamic treatment in terms of macroscopic parameters leads to the 

Experimentally, the diffusion coefficient D is obtained by using a relationship between the diffusion rate and concentration gradient, namely. Pick s second law of diffusion in one dimension  

Diffusion experiments at surfaces are designed to measure self-diffusion or the diffusion of adsorbates. The techniques used [49-55) may provide atomic-scale diffusion data or macroscopic diffusion parameters. The techniques that provide atomic-level information include (a) field ion microscopy, which can be used to observe the surface migration of isolated adatoms or clusters of atoms, (b) field electron microscopy, and (c) scanning tunneling microscopy (for descriptions of the techniques, see references [56-68]. Macroscopic mass transport along the surface can be monitored by the use of radiotracers or by techniques that monitor the restructuring of surfaces as a function of time. 

This value is 70% greater than the experimental result—evidence that the random-pore model is not very suitable for Vycor. 

In contrast, the tortuosity predicted by the random-pore model would be 1/e = 3.2., 

A final word should be said about the variety of porous materials. Porous catalysts cover a rather narrow range of possibilities. Perhaps the largest variation is between monodisperse and bidisperse pellets, but even these differences are small in comparison with materials such as freeze-dried beef, which is like an assembly of solid fibers, and freeze-dried fruit, which appears to have a structure like an assembly of ping-pong balls with holes in the surface to permit a continuous void phase.  

Surface migration is pertinent to a study of intrapellet mass transfer if its contribution is significant with respect to diffusion in the pore space. When multimolecular-layer adsorption occurs, surface diffusion has been explained as a flow of the outer layers as a condensed phase. However, surface transport of interest in relation to reaction occurs in the monomolecular layer. It is more appropriate to consider, as proposed by deBoer, that such transport is an activated process, dependent on surface characteristics as well as those of the adsorbed molecules. Imagine that a molecule in the gas phase strikes the pore wall and is adsorbed. Then two alternatives are possible desorption into the gas (Knudsen diffusion) or movement to an adjacent active site on the pore wall (surface diffusion). If desorption occurs, 

deBoer, in Advances in Catalysis, vol. VIII, p. 18, Academic Press, Inc., New York, 1959. 

The model of localized adsorption essentially implies that each adsorption center is isolated that is, completely surrounded by a potential barrier with the height equal to the desorption energy. Then the adsorbed entity can visit some other sites only through repeated desorption-adsorption steps. In Sect. 4.3 it was shown that even the shortest displacements between the two events are of the mean molecular free flight. At STP, it means mostly hundreds of distances between adjacent adsorption sites, not just a few as it is highly schematically shown in Fig. 5.21. 

On the other hand, if the thermal energies are close to or exceed the desorption energy, like in the bottom of Fig. 5.21, its undulation has smaller influence on the lateral motion of the adsorbate. The migration becomes less restricted and the adatoms move rather freely, being unconfined to specific sites. The situation can be better described as two-dimensional Brownian motion [88], in which a coefficient of viscous friction simulates dumping by the substrate [90], The surface diffusion 

The potential energy schematic shown in the middle of Fig. 5.21 seems to outline the most common relations between the thermal energies and the characteristics of desorption and lateral diffusion which take place in IC and TC experiments. The schematic does not illustrate the geometry of the lateral movement. Indeed, even if an adsorbed entity approaches the level of the desorption energy, it continues to stay within the monolayer thickness. This follows from the fact that the attractive Lennard-Jones potential is inversely proportional to a high power of the distance. The one-dimensional graphs in Fig. 5.21 are also oversimplified in the sense that the adsorption potential is a function of both lateral dimensions. Nevertheless, such sketches allow useful qualitative conclusions. 

The specific regularities in adsorption of metallic adatoms on metal surfaces, like the small diffusion barriers compared with the sublimation energies, are due to the peculiar nature of metallic bonds. They cannot be extended to the case of 

Activated adsorption is primarily found with dissociative adsorption as can be rationalized on the basis of Fig. 1.4. Adsorption in the molecular state, A2,ad (he., trapping), is sometimes denoted as intrinsic precursor from where the activation barrier for dissociation has to be surmounted. Usually at least two neighboring free adsorption sites are required for this process, so for random distribution of the adsorbates in the Langmuir picture the sticking coefficient is expected to vary with coverage as s =Sq(1 —5). Again, as with nondissociative adsorption, such a situation is found only in exceptional cases since usually various complications (such as the influence of defects or the need for more than two adjacent vacant sites, etc.) come into play. 

The periodic arrangement of atoms in a single-crystal surface causes a periodic sequence of potential wells separated by an energy barrier, the activation energy for surface diffusion E (Fig. 1.9). In fact, this barrier depends on the direction of motion, but the adsorbed particle will always jump to a neighboring site along the path of minimum activation energy, which is then 

FIGURE 1.9. Potential of a chemisorbed particle along a certain direction on a single-crystal surface (a) lifetime against desorption t urf determined by the adsorption energy (b) surface residence time for motion Xsite determined by the activation energy for diffusion E. (See color insert.) 

The Fickian diffusion constant is, on the other hand, related to the hopping of individual particles as sketched by the potential in Fig. 1.9 through the Einstein-Smoluchowski equation 

The latter technique was also applied for demonstrating the equality of Dx and D (in the absence of adsorbate-adsorbate interactions) with N atoms diffusing on a Ru(0 001) surface [40]. Exclusive dissociation of NO at 300K at monoatomic steps creates a quasi 6-function of Nad concentration at f = 0. Determination of their mean square displacement away from the step as a function of time leads to a straight line whose slope yields D = (3.4 0.4) x lO cm /s (Fig. 1.10). The solution of Pick s second law for the indicated initial concentration profile (where the N atoms do not cross the step) yields a Gaussian of the form n x,t) = NAx/y/nD t where n(x, t) is the number of 

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An alternative defining equation for surface diffusion coefficient Ds is that the surface flux Js is Js = - Ds dT/dx. Show what the dimensions of Js must be. 

It was commented that surface viscosities seem to correspond to anomalously high bulk liquid viscosities. Discuss whether the same comment applies to surface diffusion coefficients. 

Protein adsorption has been studied with a variety of techniques such as ellipsome-try [107,108], ESCA [109], surface forces measurements [102], total internal reflection fluorescence (TIRE) [103,110], electron microscopy [111], and electrokinetic measurement of latex particles [112,113] and capillaries [114], The TIRE technique has recently been adapted to observe surface diffusion [106] and orientation [IIS] in adsorbed layers. These experiments point toward the significant influence of the protein-surface interaction on the adsorption characteristics [105,108,110]. A very important interaction is due to the hydrophobic interaction between parts of the protein and polymeric surfaces [18], although often electrostatic interactions are also influential [ 116]. Protein desorption can be affected by altering the pH [117] or by the introduction of a complexing agent [118]. 

It is known that even condensed films must have surface diffusional mobility Rideal and Tadayon [64] found that stearic acid films transferred from one surface to another by a process that seemed to involve surface diffusion to the occasional points of contact between the solids. Such transfer, of course, is observed in actual friction experiments in that an uncoated rider quickly acquires a layer of boundary lubricant from the surface over which it is passed [46]. However, there is little quantitative information available about actual surface diffusion coefficients. One value that may be relevant is that of Ross and Good [65] for butane on Spheron 6, which, for a monolayer, was about 5 x 10 cm /sec. If the average junction is about 10 cm in size, this would also be about the average distance that a film molecule would have to migrate, and the time required would be about 10 sec. This rate of Junctions passing each other corresponds to a sliding speed of 100 cm/sec so that the usual speeds of 0.01 cm/sec should not be too fast for pressurized film formation. See Ref. 62 for a study of another mechanism for surface mobility, that of evaporative hopping. 

The state of an adsorbate is often described as mobile or localized, usually in connection with adsorption models and analyses of adsorption entropies (see Section XVII-3C). A more direct criterion is, in analogy to that of the fluidity of a bulk phase, the degree of mobility as reflected by the surface diffusion coefficient. This may be estimated from the dielectric relaxation time Resing [115] gives values of the diffusion coefficient for adsorbed water ranging from near bulk liquids values (lO cm /sec) to as low as 10 cm /sec. 

One might expect the frequency factor A for desorption to be around 10 sec (note Eq. XVII-2). Much smaller values are sometimes found, as in the case of the desorption of Cs from Ni surfaces [133], for which the adsorption lifetime obeyed the equation r = 1.7x 10 exp(3300// r) sec R in calories per mole per degree Kelvin). A suggested explanation was that surface diffusion must occur to desorption sites for desorption to occur. Conversely, A factors in the range of lO sec have been observed and can be accounted for in terms of strong surface orientational forces [134]. 

Mobility of this second kind is illustrated in Fig. XVIII-14, which shows NO molecules diffusing around on terraces with intervals of being trapped at steps. Surface diffusion can be seen in field emission microscopy (FEM) and can be measured by observing the growth rate of patches or fluctuations in emission from a small area [136,138] (see Section V111-2C), field ion microscopy [138], Auger and work function measurements, and laser-induced desorption... 

The sequence of events in a surface-catalyzed reaction comprises (1) diffusion of reactants to the surface (usually considered to be fast) (2) adsorption of the reactants on the surface (slow if activated) (3) surface diffusion of reactants to active sites (if the adsorption is mobile) (4) reaction of the adsorbed species (often rate-determining) (5) desorption of the reaction products (often slow) and (6) diffusion of the products away from the surface. Processes 1 and 6 may be rate-determining where one is dealing with a porous catalyst [197]. The situation is illustrated in Fig. XVIII-22 (see also Ref. 198 notice in the figure the variety of processes that may be present). 

Linderoth T R, Florsch S, Laesgaard E, Stensgaard i and Besenbacher F 1997 Surface diffusion of Pt on Pt(110) Arrhenius behavior of iong ]umps Phys. Rev. Lett. 78 4978... 

Dunphy J C, Sautet P, Ogietree D F, Dabbousi O and Saimeron M B 1993 Scanning-tunneiing-microscopy study of the surface diffusion of suifur on Re(OOOI) Phys. Rev. B 47 2320... 

George S M, DeSantoio A M and Haii R B 1985 Surface diffusion of hydrogen on Ni(IOO) studied using iaser-induced thermai desorption Surf. Sol. 159 L425... 

Schultz K A and Seebauer E G 1992 Surface diffusion of Sb on Ge(111) monitored quantitatively with optical second harmonic microscopy J. Chem. Phys. 97 6958-67... 

Zhu X D, Rasing T H and Shen Y R 1988 Surface diffusion of CO on Ni(111) studied by diffraction of optical second-harmonic generation off a monolayer grating Phys. Rev. Lett. 61 2883-5... 

Jaklevic R C and Elie L 1988 Scanning-tunnelling-microscope observation of surface diffusion on an atomic scale Au on Au(111) Rhys. Rev. Lett. 60 120... 

Figure C2.11.6. The classic two-particle sintering model illustrating material transport and neck growtli at tire particle contacts resulting in coarsening (left) and densification (right) during sintering. Surface diffusion (a), evaporation-condensation (b), and volume diffusion (c) contribute to coarsening, while volume diffusion (d), grain boundary diffusion (e), solution-precipitation (f), and dislocation motion (g) contribute to densification.

The incorporation of surface diffusion into a model of transport in a porous medium is quite straightforward, since the surface diffusion fluxes simply combine additively with the diffusion fluxes in the gaseous phase. 

It is therefore reasonable to go ahead with the construction of models for the gaseous phase diffusion without considering surface diffusion, though of course it must be incorporated before the models are used predictively. 

The literature of surface diffusion is now quite extensive. A review of the basic ideas, with reference to many of the earlier papers, is given by Dacey [44], and a good selection of references including more recent work can be found in Aris [45]... 

The flux N (a,w) is the Sum of contributions from a gaseous phase flux and a flux due to surface diffusion. The surface diffusion contribution is given by equation (7.7) or, more generally, by the corresponding relation which follows from equation (7.5). For the gaseous phase contribution Feng and Stewart assume flux relations of the dusty gas form, (5.1)- ... 

The disposable parameters in these equations are the permeability 0, the surface diffusion factor y, the tortuosity function . (a) (which also... 

Since the void fraction distribution is independently measurable, the only remaining adjustable parameters are the A, so when surface diffusion is negligible equations (8.23) provide a completely predictive flux model. Unfortunately the assumption that (a) is independent of a is unlikely to be realistic, since the proportion of dead end pores will usually increase rapidly with decreasing pore radius. 

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