Site Energy Patches

Alternatively, an integral distribution function F may be defined as giving the fraction of surface for which the adsorption energy is greater than or equal to a given Q, 

Equation XVII-127 connects the functions 0(F, T), d(Q,P, T) and f Q) and, in principle, if any two are known or can be assumed, the remaining one can be calculated. As may be imagined, many choices of such pairs of functions have been examined, often designed so that Eq. XVII-127 can be handled analytically alternatively, various schemes of successive approximations may be used. The field has become somewhat of a happy hunting ground for physical chemists and there are numerous reviews of the now-extensive literature (see Refs. 144-147 the last is a personalized account). For this reason only some generic approaches will be discussed here. 

One may choose 6(Q,P,T) such that the integral equation can be inverted to give f Q) from the observed isotherm. Hobson [150] chose a local isotherm function that was essentially a stylized van der Waals form with a linear low-pressure region followed by a vertical step tod = 1. Sips [151] showed that Eq. XVII-127 could be converted to a standard transform if the Langmuir adsorption model was used. One writes 

The analysis is thus relatively exact for heterogeneous surfaces and is especially valuable for analyzing changes in an adsorbent following one or another treatment. An example is shown in Fig. XVII-24 [160]. This type of application has also been made to carbon blacks and silica-alumina catalysts [106a]. House and Jaycock [161] compared the Ross-Olivier [55] and Adamson-Ling 

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C. Point versus Patch Site Energy Distributions... 

The question is now whether Ag j ul ) may be determined from appropriate (adsorption) experiments and analysis. Unfortunately there is no unambiguous way to achieve this. The simplest way is to assume some distribution, then derive an isotherm equation and compare that with experiments to find the distribution parameter. However, such approaches are not generally unique in that different models may predict the same isotherm in addition, little is known a priori about the distributions. The opposite process of finding from experimental isotherms also requires some assumptions on the way in which the sites or patches are distributed and on the way molecules adsorb on spots with identical energy. In addition, great experimental accuracy is demanded. In conclusion, there is no unique solution, although a variety of procedures of limited applicability have been proposed to account for heterogeneity. 

The volume V, is that adsorbed on sites or patches having an adsorption energy Ayjj u. The subscript i denotes local in heterogeneity studies the isotherm, describing the adsorption on such homogeneous parts, is often called the local isotherm for which, as already stated, any isotherm equation of app. 1 can be substituted. Volumes pertaining to different local isotherms have to be added in... 

This latter situation requires that the increase of E due to induced heterogeneity on the lowest energy patch is larger than E q — Eoo, the range of E in the site heterogeneity. Substitution of Eq. (48) into (49) gives... 

It is possible to extend the analysis for the surface nonidealities. For example, a surface can be composed of patches, each of which has a uniform site energy distribution such that the Langmuir adsorption isotherm... 

The lateral interactions in the adsorbate can enhance or diminish the interaction energy in the surface. If the adsorption sites at the boundary between reconstructed and unreconstructed areas of surface are further distinguished from those inside these patches, we can introduce more interactions such as... 

The polydispersity of polymers results in competing adsoiption of the thermodynamically favored larger molecules for surface sites filled initially by smaller molecules. Different segments of a block copolymer may exhibit quite different adsoiption characteristics, complicating the rearrangement process farther. This is an effect of considerable interest in protein adsoiption, and is referred to as the rearrangement of a protein layer to maximize hydrophobic interaction of "oily" patches with low energy surfaces such as medical implant polymers. 

For some multimeric proteins, hydrophobic patches on the surface of subunits serve as interaction sites that favor polymerization in aqueous solutions. To allow polymerization to occur, the organized clusters ( clathrates ) of water around the hydrophobic sites must be removed. This process requires an input of thermal energy to melt the clathrates. Thus, the... 

The secondary site of eco binds to the protease over 20 A away from the active site and forms up to 30 van der Waals interactions and up to five additional hydrogen bonds. An additional important source of binding energy and association, the secondary site is composed of the 60 s and lOO s loops of eco and a hydrophobic patch near the protease residues 91 to 94 and the C-terminal a helix amino acids 236 to 242. This patch and helix separated from the 80 s loop accounts for the fold specificity of eco. Each inhibitor molecule forms an interaction with both proteases of the tetramer in a clamp configuration that can be adapted to fit most serine proteases. 

For O.25<0aparticles adsorbed on border shallow sites would mcrease the energy of the system due to the A-B interactions). The adsorption process follows until the deep patches become completely covered (0a=O.S) in this limit, the A isotherm exhibits the well known plateau due to the surface heterogeneity, which induces a plateau in the B isotherm. 

In the case of repulsive interactions, neither for Q->0 (random heterogeneous substrate) nor (2-> I (strongly correlated patches of sites) the energy correlation influences importantly the thermodynamic factor (and accordingly the collective diffusion). 

We assume that the substrate is represented by a two-dimensional square lattice of M=LxL adsorption sites, with periodic boundary conditions. Weak and strong sites, with adsorptive energies > and S2 (Sjspatially distributed either in a deterministic alternate way (chessboard topography) or in a non overlapping random way. 

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