Adsorption Statistical mechanics treatment

The adsorption of soluble polymers at solid-liquid interfaces is a highly complex phenomenon with vast numbers of possible configurations of the molecules at the surface. Previous analyses of polymer adsorption have ranged in sophistication from very simple applications of "standard" models derived for small molecules, to detailed statistical mechanical treatments of the process. 

The integral molar quantities are of importance for modelling adsorption systems or in the statistical mechanical treatment of physisorption. For example, they are required for comparing the properties of the adsorbed phase with those of the bulk... 

Previous works dealing with disordered surfaces have been dedicated mainly to random, or correlated topographies. In the latter case, the combination of heterogeneity and ad-ad interactions effects produce complex behaviors on the equilibrium properties. An exact statistical mechanical treatment is unfortunately not yet available and, therefore, the theoretical description of adsorption has relied on simplified models. One way of circumventing this complication is the Monte Cado (MC) method, which has demonstrated to be a valuable tool to study surface processes [3,4],... 

The proposed thermodynamic treatment has two main advantages. It keeps the modelistic assumptions to a minimum and it is relatively simple. The first advantage increases the applicability and reliability of the treatment, since there is no need for questionable structural assumptions and / or drastical approximations, like those involved in statistical mechanical treatments. The second advantage allows many interesting adsorption phenomena, like co-adsorption and re-orientation processes as well as the effect of the specific adsorption of ions and the heterogeneity of the adsorbing surface on the equilibrium properties of adsorbed layers, to be readily taken into account. 

The adsorption process is assumed to exist in a reversible or quasi-reversible condition and a thermodynamic or statistical mechanical treatment is applicable. 

P-lO - Exact statistical mechanical treatment of a lattice model of hydrocarbon adsorption on zeolites... 

Statistical mechanics treatment, adsorption 16-P-lO Steam dealumination II-P-08 13-P-07 28-P-08 Steam reforming 24-P-23... 

Up to now the model has been applied with monomeric, dimeric and trimeric solute molecules. Although the study of these cases is not complete, possibly due to computational difficulties, it seems that some of the adsorption features are satisfactorily predicted only in the case of non-polar monomeric and polar dimeric solute molecules, provided that the latter exhibit certain orientations on the electrode surface. " In the case of polar monomeric and dimeric molecules that may adsorb either vertically or flat, the model does not give satisfactory predictions. This is shown in Figure 3 where the solid lines represent adsorption isotherms predicted by the model and the dotted lines represent the best Frumkin s isotherms that describe them. In the case of the trimeric solutes, the model predicts the existence of a surface phase transition. However, the transition properties, due to the use of an inappropriate statistical mechanical treatment, contradict thermodynamic and experimental data. Thus, despite its novelty the three-dimensional lattice approach has not given the expected results yet. 

Statistical-mechanical treatments of polymer adsorption at a planar surface have been pursued extensively using both analytical and Monte Carlo techniques. These procedures place polymer chain configurations in a one-to-one correspondence with random walk configurations on a lattice. While the analytical methods are limited to massless segments, and the Monte Carlo techniques are restricted to relatively short chains because of computational limitations, both provide results capable of experimental verification. The restriction to dilute solutions and non-interacting adsorbed molecules has been circumvented in recent theoretical treatments of concentrated pol3nner solutions. 

As was indicated in the previous subsection, Steele [16-18] has developed an analytical model for the gas-solid interaction in which the periodic nature of the adsorbate-adsorbent interaction is taken into account when the molecules of the former move parallel to the surface of the latter. Subsequently, Steele [19] developed a statistical mechanics treatment which, at least in the case of monolayer adsorption, allows the three-dimensional problem to be reduced to a two-dimensional one, and then simple models for the 2D fluid, such as the Lennard-Jones one, permits one to find theoretical values for the properties of interest in physisorption. 

Virial treatment provides a general method of analysing the low-coverage region of an adsorption isotherm and its application is not restricted to particular mechanisms or systems. If the structure of the adsorbent surface is well defined, virial treatment also provides a sound basis for the statistical mechanical interpretation of the adsorption data (Pierotti and Thomas, 1971 Steele, 1974). As indicated above, Kl in Equation (4.5) is directly related to kH and therefore, under favourable conditions, to the gas-solid interaction. 

We have seen that the earlier methods of micropore analysis were either essentially empirical or based on questionable assumptions. In contrast, molecular simulation and DFT offer the prospect of a more rigorous treatment since they are based on the fundamental principles of statistical mechanics. However, it must be kept in mind that to solve the statistical mechanical Hamiltonian, it is necessary to know the exact position of the force centres in the solid structure and also the potential functions which govern the solid-fluid and fluid-fluid interactions. In view of the complexity of most porous adsorbents, it is not surprising that so far most attention has been given to the adsorption of small, spherical molecules in pores of uniform geometry -particularly cylindrical or slit-shaped pores (Steele and Bojan, 1997). 

D. Nicholson, N.G. Parsonage, Computer Simulation and the Statistical Mechanics of Adsorption, Academic Press (1982). (Physical approach, more specialized and more advanced than our treatment.)... 

The Langmuir adsorption isotherm was developed by Irving Langmuir in 1916 from kinetic considerations to describe the dependence of the surface fractional coverage of an adsorbed gas on the pressure of the same gas above the adsorbent surface at a constant temperature. The Langmuir isotherm expression was re-derived thermodynamically by Volmer and statistically mechanically by Fowler. In his original treatment, Langmuir made several assumptions for his model ... 

Physisorption arises from the van der Waals forces, and these forces also condense gas molecules into their liquid state. Thus, in principle, there is no reason to stop upon completion of a monolayer during physisorption. Indeed, the formation of multi-layers, which are basically liquid in nature, is very common in physisorption experiments. Brunauer, Emmett and Teller developed a theory in 1938 to describe physisorption, where the adsorbate thickness exceeds a monolayer, and this isotherm equation is known by the initials of the authors (B.E.T.). The original derivation of the B.E.T. equation is an extension of Langmuir s treatment of monolayer adsorption from kinetic arguments. Later, in 1946, Hill derived this equation from statistical mechanics. In the B.E.T. isotherm, it is assumed that ... 

Local Isotherm Equations.—To be able to achieve a tractable solution to the integral equation (15), approximate models for local isotherm equations have to be postulated. The previous discussion drew attention to the role of intrinsic heterogeneity although recognizing that a rigorous treatment is difficult for most real systems. Except for helium, hydrogen, and possible neon, classical statistical mechanics ordinarily give an accurate representation of the adsorption system. 

The second section of the book addresses phenomena associated with the gas—oxide (and vapor—oxide) interface. The question of adsorption at the solid— gas interface has been explored using innumerable theoretical methods. In Chapters 10 and 11, two of the more successful methods based on statistical mechanics and thermodynamics are examined. These treatments represent the current state of gas— solid adsorption theory in the field of gas—solid adsorption, a field that has been actively evolving for many decades. 

The determination of more comprehensive coking mechanisms and rate equations requires simultaneous treatment of all experimental data to enable all the relevant parameters related to coking to be considered. After analysing the experimental data, numerical values of the rate and adsorption equilibrium constants were determined by statistical tests, and models were rejected if a negative constant was estimated at more than one temperature. It was found that the hyperbolic type of decay, as described in Equation (1), gives the best fit from the 9 models tested because it gives the least error from the sum of squares analysis [8],... 

In this chapter, we review the basic mechanisms underlying adsorption of long-chain molecules on solid surfaces such as oxides. We concentrate on the physical aspects of adsorption and summarize the main theories which have been proposed. This chapter should be viewed as a general introduction to the problem of polymer adsorption at thermodynamical equilibrium. For a selection of previous review articles see Refs 1—4, while more detailed treatments are presented in two books on this subject [5,6]. We do not attempt to explain any specific polymer/oxide system and do not emphasize experimental results and techniques. Rather, we detail how concepts taken from statistical thermodynamics and interfacial science can explain general and universal feamres of polymer adsorption. The present chapter deals with equilibrium properties whereas Chapter 3 by Cohen Stuart and de Keizer is about kinetics. 

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