Adsorption lattice theories

Of particular interest has been the study of the polymer configurations at the solid-liquid interface. Beginning with lattice theories, early models of polymer adsorption captured most of the features of adsorption such as the loop, train, and tail structures and the influence of the surface interaction parameter (see Refs. 57, 58, 62 for reviews of older theories). These lattice models have been expanded on in recent years using modem computational methods [63,64] and have allowed the calculation of equilibrium partitioning between a poly-... 

Clearly Fig. 7 must actually have a maximum at high asymmetry since this corresponds to negligible anchor block size and therefore to no adsorption (ct = 0). The lattice theory of Evers et al. predicts this quantitatively [78] and is, on preliminary examination, also able to explain some aspects of these data. From these data, the deviation from power law behavior occurs at a number density of chains where the number of segments in the PVP blocks are insufficient to cover the surface completely, making the idea of a continuous wetting anchor layer untenable. Discontinuous adsorbed layers and surface micelles have been studied theoretically but to date have not been directly observed experimentally [79]. 

A lattice theory of solutions has been proposed (.k) to describe the adsorption-desorption phenomena in zeolites. There are several reasons for this choice (a) forming a solid solution by two substances is analogous to the forming of an adsorbed phase in the cavities of a zeolite, (b) the theory of solutions is well understood and its mathematical techniques powerful, and (c) since the state-of-the-art in description of adsorption phenomena in... 

The "solid-solution lattice theory" model of adsorption on zeolites has been shown to describe the experimental results in the literature with accuracy comparable to all existing theories, even though these theories are in many instances semi-empirical. Since the theory is related to actual physical phenomena, systematic studies can be made of the effect on adsorption of changing the zeolite. 

It can be concluded that the "solid-solution lattice theory" model shows great promise in describing adsorption on zeolites. 

In chapter 1.3 a number of examples of elaborations have already been given, mostly using lattice statistics. All of them Involve a "divide and rule" strategy, in that the system (i.e. the adsorbate) is subdivided into subsystems for which subsystem-partition functions can be formulated on the basis of an elementary physical model. For instance, in lattice theories of adsorption one adsorbed atom or molecule on a lattice site on the surface may be such a subsystem. In the simplest case the energy levels, occurring in the subsystem-partition function consist of a potential energy of attraction and a vibrational contribution, the latter of which can be directly obtained quantum mechanically. Having... 

Other justifications are (i) the fact that the Langmuir adsorption isotherm equation can also be derived klnetically, without making the restriction that there are localized sites (see sec. 11.1.5a), (ii) that thermodynamic derivations without this restriction can also be given (see above), and (iii) the experience that lattice theories can often represent properties of liquids satisfactorily. 

In the second paper of the series [150] the authors extend their treatment with a mean field lattice theory. They found that the adsorbed amount in the pores is smaller and the step in the isotherm shifts to lower chemical potential than in a flat surface in the same conditions. They also established that the influence of the curvature on the phase transtition increases with the length of the headgroup. The shift of the phase transition increases with the adsorption energy. 

The SCLF method was developed by Koopal and coworkers [46-50] to describe adsorption of surfactant molecules at the solution-solid interface. The method derives from two earlier statistical thermodynamic lattice theories (1) the Flory and Fluggins [51 ] model describing properties of polymers in solution, and (2) the methods of Scheutjens and coworkers [52-55] developed to describe the properties of polymer molecules adsorbed at the solution-solid interface and in associated mesomorphic solution structures such as micelles and vesicles. 

Most microscopic theories of adsorption and desorption are based on the lattice gas model. One assumes that the surface of a sohd can be divided into two-dimensional cells, labelled i, for which one introduces microscopic variables Hi = 1 or 0, depending on whether cell i is occupied by an adsorbed gas particle or not. (The connection with magnetic systems is made by a transformation to spin variables cr, = 2n, — 1.) In its simplest form a lattice gas model is restricted to the submonolayer regime and to gas-solid systems in which the surface structure and the adsorption sites do not change as a function of coverage. To introduce the dynamics of the system one writes down a model Hamiltonian which, for the simplest system of a one-component adsorbate with one adsorption site per unit cell, is... 

For adsorbates out of local equilibrium, an analytic approach to the kinetic lattice gas model is a powerful theoretical tool by which, in addition to numerical results, explicit formulas can be obtained to elucidate the underlying physics. This allows one to extract simplified pictures of and approximations to complicated processes, as shown above with precursor-mediated adsorption as an example. This task of theory is increasingly overlooked with the trend to using cheaper computer power for numerical simulations. Unfortunately, many of the simulations of adsorbate kinetics are based on unnecessarily oversimplified assumptions (for example, constant sticking coefficients, constant prefactors etc.) which rarely are spelled out because the physics has been introduced in terms of a set of computational instructions rather than formulating the theory rigorously, e.g., based on a master equation. 

The simplest model of polymers comprises random and self-avoiding walks on lattices [11,45,46]. These models are used in analytical studies [2,4], in particular in the numerical implementation of the self-consistent field theory [4] and in studies of adsorption of polymers [35,47-50] and melts confined between walls [24,51,52]. 

The Monte Carlo method as described so far is useful to evaluate equilibrium properties but says nothing about the time evolution of the system. However, it is in some cases possible to construct a Monte Carlo algorithm that allows the simulated system to evolve like a physical system. This is the case when the dynamics can be described as thermally activated processes, such as adsorption, desorption, and diffusion. Since these processes are particularly well defined in the case of lattice models, these are particularly well suited for this approach. The foundations of dynamical Monte Carlo (DMC) or kinetic Monte Carlo (KMC) simulations have been discussed by Eichthom and Weinberg (1991) in terms of the theory of Poisson processes. The main idea is that the rate of each process that may eventually occur on the surface can be described by an equation of the Arrhenius type ... 

At present it is impossible to formulate an exact theory of the structure of the electrical double layer, even in the simple case where no specific adsorption occurs. This is partly because of the lack of experimental data (e.g. on the permittivity in electric fields of up to 109 V m"1) and partly because even the largest computers are incapable of carrying out such a task. The analysis of a system where an electrically charged metal in which the positions of the ions in the lattice are known (the situation is more complicated with liquid metals) is in contact with an electrolyte solution should include the effect of the electrical field on the permittivity of the solvent, its structure and electrolyte ion concentrations in the vicinity of the interface, and, at the same time, the effect of varying ion concentrations on the structure and the permittivity of the solvent. Because of the unsolved difficulties in the solution of this problem, simplifying models must be employed the electrical double layer is divided into three regions that interact only electrostatically, i.e. the electrode itself, the compact layer and the diffuse layer. 

This chapter is concerned with the application of liquid state methods to the behavior of polymers at surfaces. The focus is on computer simulation and liquid state theories for the structure of continuous-space or off-lattice models of polymers near surfaces. The first computer simulations of off-lattice models of polymers at surfaces appeared in the late 1980s, and the first theory was reported in 1991. Since then there have been many theoretical and simulation studies on a number of polymer models using a variety of techniques. This chapter does not address or discuss the considerable body of literature on the adsorption of a single chain to a surface, the scaling behavior of polymers confined to narrow spaces, or self-consistent field theories and simulations of lattice models of polymers. The interested reader is instead guided to review articles [9-11] and books [12-15] that cover these topics. 

From the fit one obtains values of and a. Note how the electronic polarizability of the adsorbed molecules gives the absorptance a nonlinear coverage dependence. However, there exist several systems that do not follow Eqs. (2) and (3). This can be caused either by a coverage dependent change in the electronic structure, that is an additional chemical shift, or because the system exhibits clustering or the molecules occupy more the one adsorption site, since the theory assumes a random filling of the adsorbate lattice. 

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