Patchwise

Most calculations of f(Q) for a heterogeneous surface, using an adsorption isotherm assume a patchwise distribution of sites. Explain for what kind of local isotherm functions,/((2,P, T) this assumption is not necessary, and for which it is necessary. Give examples. 

The effects due to the finite size of crystallites (in both lateral directions) and the resulting effects due to boundary fields have been studied by Patrykiejew [57], with help of Monte Carlo simulation. A solid surface has been modeled as a collection of finite, two-dimensional, homogeneous regions and each region has been assumed to be a square lattice of the size Lx L (measured in lattice constants). Patches of different size contribute to the total surface with different weights described by a certain size distribution function C L). Following the basic assumption of the patchwise model of surface heterogeneity [6], the patches have been assumed to be independent one of another. 

Fig. 6a-d. Schematic view of adsorption from solution onto smooth, planar surfaces where the surface sites are considered to have the same area as the projected area of the solute of interest, a. Top, the ideal (Langmuir) case b. clustering of adsorbed solute due to attractive lateral interactions or positive cooperativity c. heterogeneous surface, i.e., two sets of binding sites d. patchwise heterogeneity or surface domains of different adsorptive properties... 

Most solid surface are also chemically inhomogeneous. Cassie considered a smooth but chemically patchwise heterogeneous surface. If there are two different kinds of region with... 

Thorn, L.H. et al., Polymer adsorption on a patchwise heterogeneous surface, Progr. 

As a trend, adsorbates tend to be mobile at high temperature and localized at low temperature. Between these limits there will be transitional states that are difficult to handle because the issue of surface heterogeneity also has to be considered. Whether or not an adsorbed molecule can move laterally depends on the activation energy of such displacements, a. The fraction that can move is proportional to exp(-a/RT). It is likely that there will be a certain distribution of a over the surface. Because of the exponential dependence, and because the distribution of the parts with low a (patchwise or homogeneous at the site level) also plays an important role, predictions with some general applicability are virtually impossible, as work by Jaronlec et al. has indicated. 

Figure 1.18. Patchwise partially mobile adsorbates. and are the fraction of the surface on which the adsorption Is localized and the fraction of the adsorbate that Is localized, respectively. Figure (a) localized fraction, figure (b) Isotherms. Parameters in the moleculeir partition functions correspond to nitrogen atoms, ignoring rotation.

A first and basic question is what is the nature of the distribution is patchwise or random "Patchwise" means that the surface consists of a set of small areas, say crystal faces, each of which is homogeneous and characterized... 

Let us. before giving illustrations, discuss some aspects of the statistical foundations. Model assumptions regarding the mode of adsorption (patchwise or random mobile or localized mono- or multilayer with or without lateral interaction ) are reflected in the natures of the local partition functions and in the way they combine to Q[N,N, T). Again, no general solution can be given models of different degrees of sophistication can be developed. 

Consider, by way of example, the simple case of localized monolayer adsorption without lateral interaction. Let there be /V j sites of type j. Tliese sites are distinguishable in the present situation we do not yet have to discriminate between patchwise and random distribution of these sites, but this becomes important as soon as lateral interaction has to be accounted for. We have Ns = Sj sj If Is the number of molecules adsorbed on sites j in... 

Let us assume the patchwise topography for active sites and take into account the lateral interactions between the neighbouring chemisorbed species in the mean-field approximation [32]. Then, the interaction potential between the first-order neighbouring... 

Only repulsive interactions between the chemisorbed species as their exponential dependence on the distance between them was taken into consideration. Assuming the Gamma distribution for p(k) and Gamma-, uniform or normal distribution for p(r) functions, approximate solutions of Eq. (56) for the reaction on patchwise and random topography have been obtained by its sequential integration as a long algebraic series [105]. 

Figure 15. (A) Diagram of patchwise type surface. (B) Diagram of random type surface. Two types of adsorption centres are denoted by crosses and circles.

In general the desorption kinetics from patchwise heterogeneous surfaces are given by ... 

The work carried out in [5,62 66] allows the direct extension of the adsorption isotherms developed in the previous section to heterogeneous surfaces. Here, for simplicity, we restrict our study to extend only the adsorption isotherm (57) with n = 1 and za = 0 to random and patchwise surfaces. 

Here, K denotes the r.h.s. of Eq. (79) and A = Uo/kT is an heterogeneity factor which increases with increasing heterogeneity of the adsorbing surface. Homogeneous surfaces correspond to A = 0. At patchwise surfaces Eq. (80) should be solved numerically with respect to 0, and the total adsorption isotherm is determined again numerically from Eqs. (81) and (82). 

Figure 10. Adsorption isotherms on random (—) and patchwise (---) heterogeneous surfaces...

The present treatment is restricted to non interacting patchwise heterogeneity and random heterogeneity. The first part of this section is concerned with the theory of monocomponent ion adsorption on heterogeneous surfaces, the second part with multicomponent ion adsorption. 

The overall proton adsorption from an indifferent electrolyte solution on a heterogeneous surface can simply be described as the sum of the local adsorption contributions. The effect of lateral interactions should be taken into account in the local isotherm. For a patchwise surface with a discrete distribution of intrinsic affinity constants the total... 

The distinction between patchwise and random heterogeneity can be expressed by the equation for Hs. For random heterogeneity where the local site density matches the... 

For patchwise heterogeneity with non-interacting patches, each patch has its own, smeared out, patch potential and Hg becomes ... 

Prediction of patchwise heterogeneous surfaces is possible when detailed information on the double layer parameters is available. Such calculations are very useful to obtain theoretical insight in he behaviour of patchwise heterogeneous surfaces. Some examples of this type of calculations can be found in refs. [32,101,102]. 

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