Lattice model adsorption

Various functional forms for / have been proposed either as a result of empirical observation or in terms of specific models. A particularly important example of the latter is that known as the Langmuir adsorption equation [2]. By analogy with the derivation for gas adsorption (see Section XVII-3), the Langmuir model assumes the surface to consist of adsorption sites, each having an area a. All adsorbed species interact only with a site and not with each other, and adsorption is thus limited to a monolayer. Related lattice models reduce to the Langmuir model under these assumptions [3,4]. In the case of adsorption from solution, however, it seems more plausible to consider an alternative phrasing of the model. Adsorption is still limited to a monolayer, but this layer is now regarded as an ideal two-dimensional solution of equal-size solute and solvent molecules of area a. Thus lateral interactions, absent in the site picture, cancel out in the ideal solution however, in the first version is a properly of the solid lattice, while in the second it is a properly of the adsorbed species. Both models attribute differences in adsorption behavior entirely to differences in adsorbate-solid interactions. Both present adsorption as a competition between solute and solvent. 

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-... 

Lattice models have been studied in mean field approximation, by transfer matrix methods and Monte Carlo simulations. Much interest has focused on the occurrence of a microemulsion. Its location in the phase diagram between the oil-rich and the water-rich phases, its structure and its wetting properties have been explored [76]. Lattice models reproduce the reduction of the surface tension upon adsorption of the amphiphiles and the progression of phase equilibria upon increasmg the amphiphile concentration. Spatially periodic (lamellar) phases are also describable by lattice models. Flowever, the structure of the lattice can interfere with the properties of the periodic structures. 

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 ... 

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. 

Tijssen, R., Schoenmakers, R.J., Bohmer, M.R., Koopal, L.K., and Billiet, H.A.H., Lattice models for the description of partitioning adsorption and retention in reversed-phase liquid-chromatography, including surface and shape effects, J. Chromatogr. A, 656, 135, 1993. 

Silberberg47) used a quasi-crystalline lattice model for the adsorption of flexible macromolecules. If it is assumed that an adsorbed polymer chain with P segments consists of ma trains of length i and mBi loops of length i, the total number of configurations of the chains is given by... 

In this review, we introduce another approach to study the multiscale structures of polymer materials based on a lattice model. We first show the development of a Helmholtz energy model of mixing for polymers based on close-packed lattice model by combining molecular simulation with statistical mechanics. Then, holes are introduced to account for the effect of pressure. Combined with WDA, this model of Helmholtz energy is further applied to develop a new lattice DFT to calculate the adsorption of polymers at solid-liquid interface. Finally, we develop a framework based on the strong segregation limit (SSL) theory to predict the morphologies of micro-phase separation of diblock copolymers confined in curved surfaces. 

Now possibilities of the MC simulation allow to consider complex surface processes that include various stages with adsorption and desorption, surface reaction and diffusion, surface reconstruction, and new phase formation, etc. Such investigations become today as natural analysis of the experimental studying. The following papers [282-285] can be referred to as corresponding examples. Authors consider the application of the lattice models to the analysis of oscillatory and autowave processes in the reaction of carbon monoxide oxidation over platinum and palladium surfaces, the turbulent and stripes wave patterns caused by limited COads diffusion during CO oxidation over Pd(110) surface, catalytic processes over supported nanoparticles as well as crystallization during catalytic processes. 

This approach was later extended to off-lattice models and a more detailed description of the transfer energy of the different amino acid residues [77]. Magainin, melit-tin, and several other amphipathic peptides were simulated. In these simulations, differences in the interaction of the peptides with the lipid phase were observed. For example, magainin only showed adsorption onto the lipid and no crossing of the lipid occurred, whereas melittin crossed the lipid and formed a stable transmembrane helix. These results are in full agreement with later studies reported by other research groups presented below, involving more elaborate simulation protocols and representations of the peptides and the lipid. These examples show the potential of computer simulations even when some simplifications have to be made to make the system computationally tractable. 

Lattice models play a central role in the description of polymer solutions as well as adsorbed polymer layers. All of the adsorption models reviewed so far assume a one-to-one correspondence between lattice random-walks and polymer configurations. In particular, the general scheme was to postulate the train-loop or train-loop—tail architecture, formulate the partition function, and then calculate the equilibrium statistics, e.g., bound fraction, average loop... 

The formulation of a proper surface boundary condition is a delicate matter, as noted by DiMarzio (1965) and de Gennes (1969). Lattice models simply require that P(i, s) = 0 for layers i < 0, a form proven correct by DiMarzio (1965). In continuum models, chains intersecting the surface undergo both reflection and adsorption, the relative amount of each depending on the energy of contact at the surface. The result is a mixed boundary condition expressed by de Gennes (1969) as... 

This situation Is typical for solid-liquid interfaces where, during adsorption, the area A remains constant. In lattice models this is Interpreted as the constancy of the total number of sites. With adsorption from solution at liquid-liquid Interfaces this restriction Is often absent then one can distinguish between adsorption at given A and that at constant Interfaclal tension y or. for that matter, at constant x. 

With realizations of the Vycor glass as shown in Fig. 1, mean field theory applied to the lattice model (1) provides a simple and sufficiently realistic method to examine fluid adsorption behavior on a coarse-grained level. In particular, the local density on each site pi = (riitj) is self-consistently determined by... 

To examine the dynamical aspect of the hysteretic behavior, we consider the system geometry shown in Fig. 4. The porous material of length L in the z-direction is bounded by the gas reservoirs at z = 0 and z = L. Periodic boundary conditions are imposed on the X and y-directions. In typical experimental situations, starting from a (quasi)-equilibrium state, the external vapor pressure of the gas reservoir is instantaneously changed by a small amount, which induces gradual relaxations of the system into a new state. This geometry was used in recent work on dynamics of off-lattice models of adsorption (Sarkisov and Monson,... 

Model materials similar to the M41 family can be modeled using lattice MC simulation under the assumption of non silica polymerization conditions. The structures observed are in agreement with experimental evidence with respect to the surfactant/silica ratio, temperature and surfactant architecture. Even when the resulting structures have a strong influence of the lattice constraints, adsorption properties on such materials are in good agreement with experimental observations. Adsorption properties of modeled materials are similar to experimental observations on MCM-41 type materials and show heterogeneity at low pressures. Such behavior is not observed in smooth cylinders. 

With regards to the repulsive case, the model adsorption isotherm reproduces the MC data fairly well for w/keT. The discrepancy in the case w/kal links to the order-disorder phase transitions that dimers develop on a square lattice at w/kBTe w3, 6c =l/2 and w/kaTc 5, 0c =2/3 (see ref. [5]). These transitions can not be reproduced by the approximation (mean field) used in eq. (8). As it is clear from Figure 2, the simulated isotherm for w/keT=4 display a plateau (characteristic of an ordered phase) at 6=>l/2 since w/kBTc <4. However, only a knee around 0=2/3 is visible at this temperature because w/ksTc 4. A deeper discussion of the referred phase transitions can be found in ref. [5]. 

Imbihl et al. was adapted to include modifications of oxygen adsorption on the 1 X 1 phase caused by defects or irregularities in the CO adlayer 317). Also the dynamics of the surface structure transition were descried differently, using a Ginzberg-Landau equation and a lattice model to calculate energies of the different states of the surface. These two modifications resulted in a better fit of the experimental data. 

It may be good to note here that various molecular cross-sections have now been considered. In the treatment of adsorption on solid surfaces was introduced. Interpreting this area in terms of lattice models is not a property of the adsorptive molecule but of the adsorbent. It is possible to imagine a situation where greatly exceeds the real molecular cross-section. On the other hand, for mobile monolayers on homogeneous surfaces is the real molecular cross-section or, for that matter, it is the excluded area per molecule. To avoid an undue abundance of symbols we have used the same symbol for both situations, for instance in table 3.3 in sec. 3.4e. It is to be expected that a and a, obtained by compression of monolayers, are more similar to the a s for adsorbed mobile monolayers on homogeneous substrates than to those for localized monolayers. 

---