Langmuir lattice

Pore diameters are in the molecular size range, where the distinction between whether the molecules are gas or liquid ceases to exist. The molecules aie under the effect of the interaction potential from the pore wall and their transport is according to the hopping diffusion like a solid-state vacancy mechanism. This mechanism can be applied well for amorphous stmctures and molecules that fit tightly in the pore. Membiane pore system is then considered as a Langmuir lattice of the sites that can be either vacant or occupied by the molecules. In such a case the flux of species /j can be written by... 

For simplicity the first mechanism is modelled by the partition function of Langmuir lattice... 

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. 

For structures with a high curvature (e.g., small micelles) or situations where orientational interactions become important (e.g., the gel phase of a membrane) lattice-based models might be inappropriate. Off-lattice models for amphiphiles, which are quite similar to their counterparts in polymeric systems, have been used to study the self-assembly into micelles [ ], or to explore the phase behaviour of Langmuir monolayers [ ] and bilayers. In those systems, various phases with a nematic ordering of the hydrophobic tails occur. 

A particularly simple lattice model has been utilized by Harris and Rice [129] and subsequently by Stettin et al. [130] to simulate Langmuir mono-layers at the air/water interface chains on a cubic lattice which are confined to a plane at one end. Haas et al. have used the bond-fluctuation model, a more sophisticated chain model which is common in polymer simulations, to study the same system [131]. Amphiphiles are modeled as short chains of monomers which occupy a cube of eight sites on a cubic lattice and are connected by bonds of variable length [132], At high surface coverage, Haas et al. report various lattice artefacts. They conclude that the study... 

The curves were determined from Eqs. 24"—26" in order to apply these in a numerical calculation one first has to know the values of the following functions at — 3°C Afz, the difference in chemical potential between the "empty Structure II lattice and ice Cpg. the Langmuir constant for propane in the larger cavities of Structure II (Cpi = 0 for geometrical reasons) Cmi> Cm2> the Langmuir constants for methane in the two types of cavities of Structure II. 

Langmuir, constants, 48 isotherm, 39 theorem, 15 Lattice model, 123... 

Projections, linearly independent, 293 Propagation, of polymerization, 158 Propane, hydrate, 10, 33, 43, 46, 47 hydrate thermodynamic data and lattice constants, 8 + iodoform system, 99 Langmuir constant, 47 water-hydrogen sulfide ternary system, 53... 

M.T.M. Koper, A.P.J. Jansen, and J.J. Lukkien, Lattice-gas modeling of electrochemical Langmuir-Hinshelwood surface reactions, Electrochim. Acta 45, 645-651 (1999). 

Most important for many applications of S-layer lattices in molecular nanotechnology, biotechnology, and biomimetics was the observation that S-layer proteins are capable of reassembling into large coherent monolayers on solid supports (e.g., silicon wafers, polymers, metals) at the air/water interface and on Langmuir lipid films (Fig. 6) (see Sections V and VIII). 

FIG. 16 Fomation of a Langmuir lipid monolayer at the air/subphase interface and the subsequent crystallization of S-layer protein, (a) Amphiphilic lipid molecules are placed on the air/subphase interface between two barriers. Upon compression between the barriers, increase in surface pressure can be determined by a Wilhelmy plate system, (b) Depending on the final area, a liquid-expanded or liquid-condensed lipid monolayer is formed, (c) S-layer subunits injected in the subphase crystallized into a coherent S-layer lattice beneath the spread lipid monolayer and the adjacent air/subphase interface. 

Hu and Ruckenstein s results (130) showed that on the reduced nickel-containing catalyst, the reaction took place by a Langmuir-Hinshelwood mechanism involving adsorbed CH4 and oxygen species. Furthermore, they indicated that a slow dynamic redox process consisting of lattice oxygen formation and its reduction by carbon species was at least partly responsible for the CO formation. 

Levine, Y. K., Kolinski, A. and Skolnick, J. (1993). A lattice dynamics study of a Langmuir monolayer of monounsaturated fatty acids, J. Chem. Phys., 98, 7581-7587. 

Studies of the order within surfactant monolayers have been reported for many decades. Multilayer assemblies have been studied by electron as well as infrared absorption. Motivated by an older model proposed for the orientation of molecules (Langmuir, 1933 Epstein, 1950), and by recent theoretical calculations, these two potential models for tilt disorder in the monolayer have been examined. Both models arise because the monolayer structure tries to compensate for the difference between the equilibrium head-head and chain-chain distances that each piece of the molecule would want to attain if it were independent. In one model, the magnitude of the tilt is fixed, but the tilt direction wanders slowly through the lattice. In the second... 

The foregoing expression represents the Langmuir adsorption isotherms, in which the Me2+ ion occupies two adsorption sites, the number of the lattice sites being dependent on the subphase pH. The interactions in the monolayer and the subphase are neglected. The magnitude of X (R2Me) can be estimated from the experimental known ratio (Me/RCOOH). 

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