Adsorption on heterogeneous surfaces

Surfaces are usually not perfectly homogeneous. Different crystal faces are exposed, defects and other deviations from the perfect lattice are present. Often there are different types of molecules as in steel (e.g. Fe, C, Ni, Co) or in glass (e.g. Si02, B, Na, K) and their concentrations on the surface might vary locally. 

For the adsorption isotherm on a well determined homogeneous part of the surface 0H(Q, P) the Langmuir equation is often used. 

A relatively well-known adsorption isotherm, the Freundlich isotherm [366] 

Often one would like to conclude something about the distribution of the binding energies using a measured adsorption isotherm. This is difficult. Usually certain assumptions concerning 0H have to be made [381], 

This Report is concerned with the surface heterogeneity of gas-solid and liquid-solid systems. It presents a historical development of the subject with particular emphasis placed on work published in the period 1970 to July 1981. 

Properties of the solid/gas and solid/liquid interfaces are dependent on the chemical structure and topology of the solid surface. The chemical potential of the adsorbate, whether vapour, gas, or liquid, is dependent upon the nature of the surface and the spatial variation of the adsorption potential energy. Generally the development of physical adsorption theory has tended to concentrate on homogeneous surfaces, although the subsequent applications have been made in a haphazard manner to a wide class of absorbents commonly termed heterogeneous since the details of surface heterogeneity are invariably unknown, the entire development and results derived therefrom are disputable. 

It is interesting to reflect on the situation described over thirty years ago by Roginskii and Todes The inhomogeneity of the surfaces of most solids is borne out by a number of independent facts. The achievement of electron microscopy justifies the hope that the time is not very far off when the peculiar structure of an active surface will probably be open to direct observation and measurement. At present however, only integral and indirect methods of 

Here A° is a constant related to the partition function for the adsorbate and usually assumed to be temperature dependent and independent of the adsorption energy this approximation will be discussed in Section 3. 

Langmuir himself was the first to generalize his equation Let us assume that the surface contains several different kinds of elementary spaces representing the fractions 3i, 182, p3, etc., of the surface so that Pi + 2 + p3 + -.. = 1 . This approach led to an equation describing the total adsorption, t, on a heterogeneous surface  

The adsorption-desorption reaction in Eq. 4.3 has been applied to soils in an average sense in a spirit very similar to that of the complexation reactions for humic substances, discussed in Section 2.3.11 Although no assumption of uniformity is made, the use of Eq. 4.3 to describe adsorption or desorption processes in chemically heterogeneous porous media such as soils does entail the hypothesis that effective or average equilibrium (or rate) constants provide a useful representation of a system that in reality exhibits a broad spectrum of surface reactivity. This hypothesis will be an adequate approximation so long as this spectrum is unimodal and not too broad. If the spectrum of reactivity is instead multimodal, discrete sets of average equilibrium or rate constants—each connected with its own version of Eq. 4.3—must be invoked and if the spectrum is very broad, the sets of these parameters will blend into a continuum (cf. the affinity spectrum in Eq. 2.38). 

A simple model approach to the heterogeneity of surface reactivity in soils can be developed by assuming that Eq. 4.3 [without the dissociable species Q(aq)] applies to each member of a set of parallel adsorption-desorption react ions involving the same reactant aqueous species, but with differing ail sorbent and adsorbate species. A rate law for each of the parallel reactions can be postulated in the following form (cf. Eq. 1.34)  

The first term on the right side of Eq. 4.17 describes an adsorption reaction. Under experimental conditions such that [C] remains constant (e.g., flow-through reactions) and the second term can be ignored, the adsorption rate law has the exponential mathematical solution (cf. Eq. 1.56)  

By analogy with the affinity spectrum model in Eq. 2.38, the sum on the right side of Eq. 4.20 can be replaced with an integral over the probability, p(k) dk, that a rate coefficient k has a value between k and k + dk 12 

One possible model choice for p(k) that is of widespread use in statistical applications, because of its simplicity and flexibility, is the two-parameter gamma distribution 13 

---

Another school has also developed and attempted to understand the functional dependence of adsorption on heterogeneous surfaces on the vapor pressure and temperature. Various empirical or semiempirical equations were proposed [24-26] and used later to represent experimental data and to evaluate EADF by inverting Eq. (1), which belongs to the class of linear Fredholm integrals of the first kind [27]. 

M. Jaroniec, R. Madey. Physical Adsorption on Heterogeneous Surfaces. Amsterdam Elsevier, 1988. 

Van Riemsdijk.W.A. Bolt, G.H. Koopal, L.K. Blakemeer, J. (1986) Electrolyte adsorption on heterogeneous surfaces. Adsorption models. 

Various attempts have been made to modify the Langmuir model. One of the best known is that of Fowler and Guggenheim (1939), which allowed for adsorbate-adsorbate interactions in a localized monolayer on a uniform surface. However, on an empirical basis the Fowler-Guggenheim equation turns out to be no more successful than the original Langmuir isotherm. The highly complex problem of localized adsorption on heterogeneous surfaces has been discussed by Rudzinski and Everett (1992). 

A more practical analytical equaiton for the adsorption on heterogeneous surfaces is... 

Multilayer adsorption on heterogeneous surfaces has been analysed in some detail by Jaronlec et al. ). A statistical multilayer theory by Ash et al. ) also deserves special attention. Here the authors considered mixtures of monomers... 

W. Rudzinski and D. Everett, Adsorption of gases on heterogeneous surfaces, Acad. Press, N. York (1992) W. Rudzinsky, W. Steele and G. Zgrablich, Equilibria and dynamics of gas adsorption on heterogeneous surfaces . Elsevier,Amsterdam (1996)... 

Quinones, I. Guiochon, G. Extension of a Jovanovic-Freundlich isotherm model to multicomponent adsorption on heterogeneous surfaces. J. Chromatogr. A 1998, 796, 15-40. 

C.M. Lastoskie, N. Quirke and K.E. Gubbins, in "Equilibria and Dynamics of Gas Adsorption on Heterogeneous Surfaces Studies in Surface Science Catalysis. , W. Rudzinski, W.A. Steele G. Zgrablich (Eds.), 104 745 Elsevier, Amsterdam (1997). 

Another possible approach to the evaluation of the required effective quantities is to start with the effective entropy change. Hill [98,99] considered the statistical mechanics of the localized unimolecular adsorption on heterogeneous surfaces to propose formulae for the configurational entropy. 

---