With surface diffusion

Adsorption kinetics are especially interesting when compared with surface diffusion rates of the adsorbate. This is because of the theoretical possibility that nonspecific and reversible adsorption of a ligand (say, a hormone), followed by two-dimensional diffusion on the membrane, may enhance the reaction rate with a specific binding patch (say, a hormone receptor).(1161I7) A similar effect might enhance the reaction rates between a surface-immobilized enzyme and bulk-dissolved substrate, thereby speeding some reactions in industrial chemistry. 

Clearly this is a very interesting problem and of great practical relevance, very well suited to Monte Carlo simulation. At the same time, simulations of such problems have just only begun. In the context of crystal growth kinetics, models where evaporation-condensation processes compete with surface diffusion processes have occasionally been considered before . But many related processes can be envisaged which have not yet been studied at all. 

The initial theoretical treatment of these mechanisms of deposition was given by Lorenz (31-34). The initial experimental studies on surface diffusion were published by Mehl and Bockris (35, 38). Conway and Bockris (36, 40) calculated activation energies for the ion-transfer process at various surface sites. The simulation of crystal growth with surface diffusion was discussed by Gilmer and Bennema (43). 

Binh et al. (104, 105) performed experimental investigations of surface diffusion under ultra-high vacuum and found that free evaporation must also be considered for temperatures higher than about 0.65 Tm (the melting point) for clean surfaces. They presented a thermal grooving model (106-108) in which free evaporation was considered simultaneously with surface diffusion. In this... 

Using the computer programs discussed above, it is possible to extract from these breakthrough curves the effective local mass transfer coefficients as a function of CO2 concentration within the stable portion of the wave. These mass transfer coefficients are shown in Figure 15, along with the predicted values with and without the inclusion of the surface diffusion model. It is seen that without the surface diffusion model, very little change in the local mass transfer coefficient is predicted, whereas with surface diffusion effects included, a more than six-fold increase in diffusion rates is predicted over the concentrations measured and the predictions correspond very closely to those actually encountered in the breakthrough runs. Further, the experimentally derived results indicate that, for these runs, the assumption that micropore (intracrystalline) resistances are small relative to overall mass transfer resistance is justified, since the effective mass transfer coefficients for the two (1/8" and 1/4" pellets) runs scale approximately to the inverse of the square of the particle diameter, as would be expected when diffusive resistances in the particle macropores predominate. 

So how can one handle situations where the steps are far apart One answer lies in the pioneering work of Burton et al. (1951), also known as the BCF model. The BCF model is a continuum PDE that describes adsorption of atoms to and desorption from terraces along with surface diffusion on terraces [see Eq. (2) below for a simplified version of the BCF model]. When the concentration of adatoms is relatively large, nucleation between distant steps is most likely to occur, because the probability of a diffusing adatom to reach steps before encountering another adatom is low. Under these conditions, the BCF model is inadequate since it does not account for nucleation. Furthermore, the boundary conditions in the BCF model ignore the discrete nature of steps and treat them... 

Adsorption Entropy on Heterogeneous Surfaces with Surface Diffusion... 

One question that arises with such an approach is how well the model parameters associated with surface diffusion and the chemical and electrochemical reactions can be extracted from the current, potential and ex situ surface morphology data, given the complex nature of the interactions of the additives with the surface (e.g. see Table 4.3). A key point is that current and potential curves and the surface morphology are very sensitive to changes in the experimental inputs (shown in Table 4.2), indicating that... 

Separation with Surface Diffusion and Capillary Condensation... 

Selective surface adsorption with surface diffusion... 

More generally, adsorption is controlled by a combination of transport mechanisms in macropores or micropores, depending on the pore size distribution, the sorbate concentration, the isotherm, and other conditions. The combining bulk diffusion with surface diffusion gives the effective macropore diffusivity ... 

Ponec and co-workersl l have recently demonstrated a relationship between the Tamman temperature and the Ri and R2 exchange rate for some first row transition metal oxides as shown in Fig. 6. In this study, the Tamman temperature plotted is the ratio of the exchange reaction temperature to the Tamman temperature of the oxide. This was taken to be indicative of the importance of diffusion, with surface diffusion occurring in the 0.2-0.5 temperature range and a bulk diffusion beyond 0.5. 

Fig. 13 Current transients i(t) for Au (111), miscut < 0.5°, in 0.05 M H2SO4 obtained after a singie potentiai step from 1 = 0.75 V (region II) to various final potentials in region iii. The experimentai traces are given as individual data points, the solid lines represent theoretical curves calculated with the parameters of the numerical fit to a model combining (a) an adsorption process (Eq. 7) and (b) one-step nucleation according to an exponential law with surface diffusion-controlled growth (Eq. 34), (reprinted from Ref. [299]. Copyright 1997 by VCH Verlagsgesellschaft mbH Weinheim).

Fig. 23 Current transients ofthe dissolution of a Cu UPD MLon Pt(lll) in 1 mM Cu + +0.1 M H2SO4, obtained when stepping the potential from 1 = 0.50 V to various final potentials as indicated in the figure. The transients with final potentials lower than 0.67 V could be modeled by assuming two successive hole nucleation processes according to an exponential law coupled with surface diffusion-controlled growth (cf Eq. (30) and Eq. (33)). The inset shows, as an example, the fit (----) for the experimental transient (-----) 1 = 0.50 V —> 2 = 0.65 V [407].

Single and multiple potential step experiments demonstrated that the macrokinetics of the formation of the phy-sisorbed uracil film represents a first-order phase transition and follows the exponential law of nucleation (cf. Eq. (34)) in combination with surface diffusion-controlled growth [183]. In situ STM [20, 478, 479] and time-resolved SEIRAS studies [475] suggest that these processes are strongly related to the formation/breaking of uracil-water and water-water hydrogen bonds within the Helmholtz region. 

The Avrami-exponent m = 1.48 points to nucleation according to an exponential law coupled with surface diffusion-controlled growth (Eq. (34) and Eq. (35), solid line). The experimental results in panels (b) and (c) indicate that the spectroscopic and the electrochemical transients probe different interfacial properties of the dissolution process and illustrate the complementary information of both approaches (reproduced from Ref. [475]). 

To deal with surface diffusion, it is usually assumed that (Cunningham and Williams, 1980)... 

A similar realistic model for surface growth during c.b. processing is random deposition with surface diffusion ... 

FIGURE 31.9. Random deposition modei with surface diffusion, (a) Schematic view of the modei in a trianguiar iattice. (b) Simuiation resuit for a square iattice. Reprinted from [16] with kind permission of Springer Science + Business Media. 

---