Ideal two-dimensional models

The phenomenon of phase transitions in two dimensions is of great fundamental interest and has therefore drawn a considerable amount of attention.i 2 insoluble monomolecular layers at a water-air interface provide a quite ideal two-dimensional model system with an isotropic substrate and an easily controllable density of molecules. At low densities they often exhibit a two-dimensional gas behavior,3 whereas at higher densities transitions to liquid and solid states can be found. In many systems, the liquid phase is further divided into the so-called liquid-expanded (LE) and liquid-condensed (LC) phases.4 Though observed and intensively studied, the nature of the LE-LC phase transition is still controversial. 

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. 

This is the customary relation [3] for the collision frequency in an ideal two-dimensional gas moving freely in a constant potential field provided by the surface, except for the absence of a /2 factor. The kinetic theory derivation of the collision frequency involves the relative velocity, which is the origin of they/2 factor. In the present model, there are two types of collisions, firstly with the localised adatoms, when the y/2 factor would not be appropriate, and secondly with mobile adatoms, when it would. However, the degree of approximation in the treatment is such that the refinement of a /2 factor is unwarranted it is mentioned merely to clarify its presence in expressions found in the literature. 

The model developed above serves as a convenient starting point for carrying out a dynamical analysis of the nucleation problem from the perspective of the variational principle of section 2.3.3. A nice discussion of this analysis can be found in Suo (1997). As with the two-dimensional model considered in section 2.3.3, we idealize our analysis to the case of a single particle characterized by one degree of freedom. In the present setting, we restrict our attention to spherical particles of radius r. We recall that the function which presides over our variational statement of this problem can be written genetically as... 

The easiest model for a desorption process can be derived when the adsorbed amount is assumed to be located in an ideal two-dimensional layer Tj (Fig. 4.4) and the change in concentration in this layer with time over the initial concentration is T(t) -. ... 

Structure with Variable Lamella Thickness The sharp peaks predicted by Equation (5.135) are broadened and reduced in height if various types of imperfections degrade the structure from the ideal two-phase model envisioned above. Most of such imperfections find their counterparts in three-dimensional crystals, and the methods of analysis to account for such imperfections, developed in Section 3.4, can be applied to the small-angle scattering equally well and need not be elaborated here again. There is, however, one type of imperfection... 

Deng, D., Argon, A. S., and Yip, S. (1989a) A molecular dynamics model of melting and glass transition in an idealized two-dimensional material I, Phil. Trans. Roy. Soc., 329, 549-573. 

If (10.2.16) is not met, one has to include non-ideal effects in the two-dimensional phase description. It is worth pointing that in the model of ideal two-dimensional gas the lower point of phases co-existence for the phases 1 and 2 ( 2c) is not defined at all, while the upper one ( j ) is introduced only formally. In the models of nonideal gas these points can be determined from the coverage versus the chemical potenticJ curve. 

On the other hand, as applied to the submonolayer region, the same comment can be made as for the localized model. That is, the two-dimensional non-ideal-gas equation of state is a perfectly acceptable concept, but one that, in practice, is remarkably difficult to distinguish from the localized adsorption picture. If there can be even a small amount of surface heterogeneity the distinction becomes virtually impossible (see Section XVll-14). Even the cases of phase change are susceptible to explanation on either basis. 

While the smooth substrate considered in the preceding section is sufficiently reahstic for many applications, the crystallographic structure of the substrate needs to be taken into account for more realistic models. The essential complications due to lack of transverse symmetry can be dehneated by the following two-dimensional structured-wall model an ideal gas confined in a periodic square-well potential field (see Fig. 3). The two-dimensional lamella remains rectangular with variable dimensions Sy. and Sy and is therefore not subject to shear stresses. The boundaries of the lamella coinciding with the x and y axes are anchored. From Eqs. (2) and (10) one has... 

Two-dimensional compressible momentum and energy equations were solved by Asako and Toriyama (2005) to obtain the heat transfer characteristics of gaseous flows in parallel-plate micro-channels. The problem is modeled as a parallel-plate channel, as shown in Fig. 4.19, with a chamber at the stagnation temperature Tstg and the stagnation pressure T stg attached to its upstream section. The flow is assumed to be steady, two-dimensional, and laminar. The fluid is assumed to be an ideal gas. The computations were performed to obtain the adiabatic wall temperature and also to obtain the total temperature of channels with the isothermal walls. The governing equations can be expressed as... 

While in the ideal case S(g) according to Eq. (57) clearly reflects the singular behavior at T due to substrate inhomogeneity and/or limited resolution the actual behavior of scattering data is quite smooth see Fig. 13b for an example. A detailed analysis of finite resolution effects on the structure factor of two-dimensional lattice gas models has been presented by Bartelt et af. If the... 

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