Lattice Field Model

A number of theoretical models have been proposed to describe the phase behavior of polymer—supercritical fluid systems, eg, the SAET and LEHB equations of state, and mean-field lattice gas models (67—69). Many examples of polymer—supercritical fluid systems are discussed ia the Hterature (1,3). 

To illustrate the complexity of the phase behavior in a more compact way it is instructive to employ a mean-field lattice-gas model. The relative simplicity of the grand potential... 

The behavior of an adsorbate on a single patch of size L has been represented by the familiar two-dimensional lattice gas model Hamiltonian with the added term resulting from the presence of a boundary field ... 

The answer that Zuse offers to the first question is to use irregularly organized lattices (recall T.D. Lee s random lattice field theory ([tdlee85a] section 12.5.3) and Ilachinski s speculative reasons for developing his SDCA model ([ilach87] see section 8.8) ... 

The simplest treatment of the lattice-gas model is through the mean-field or randommixing approximation, which is treated in a number of textbooks (see, e.g.. Refs. 1 and 4). We give a short summary of its application to liquid-liquid interfaces, since it nicely illustrates under what conditions the phases separate. 

The lattice gas model of Bell et al. [33] neither gave any detailed mechanism of the orientational ordering nor separated the contributions of the headgroup and the acyl chain. Lavis et al. [34] discussed Ref. 33 critically and concluded that the sharp kink point in the isotherm at transition was an artifact of the mean field approximation used. An improved correspondence to experimental data was claimed by the use of the real-space renormalization group method [35]. The same authors returned to the problem [35] and concluded that in addition to the orientation of the molecules, chain melting had to be included in a model which could interpret the phase transitions. 

Figure 3a shows the mean-field predictions for the polymer phase diagram for a range of values for Ep/Ec and B/Ec. The corresponding simulation results are shown in Fig. 3b. As can be seen from the figure, the mean-field theory captures the essential features of the polymer phase diagram and provides even fair quantitative agreement with the numerical results. A qualitative flaw of the mean-field model is that it fails to reproduce the crossing of the melting curves at 0 = 0.73. It is likely that this discrepancy is due to the neglect of the concentration dependence of XeS Improved estimates for Xeff at high densities can be obtained from series expansions based on the lattice-cluster theory [68,69].

Leermakers, F. A. M. and Cohen Stuart, M. A. (1996). Self-consistent-field lattice gas model for the surface ordering transition of n-hexadecane, Phys. Rev. Lett., 76, 82-85. 

Although the simple mean-field expression (Eqn (7.10)) for a lattice-gas model has been used to understand intercalation systems qualitatively... 

Classical Free-Electron Theory, Classical free-electron theory assumes the valence electrons to be virtually free everywhere in the metal. The periodic lattice field of the positively charged ions is evened out into a uniform potential inside the metal. The major assumptions of this model are that (1) an electron can pass from one atom to another, and (2) in the absence of an electric field, electrons move randomly in all directions and their movements obey the laws of classical mechanics and the kinetic theory of gases. In an electric field, electrons drift toward the positive direction of the field, producing an electric current in the metal. The two main successes of classical free-electron theory are that (1) it provides an explanation of the high electronic and thermal conductivities of metals in terms of the ease with which the free electrons could move, and (2) it provides an explanation of the Wiedemann-Franz law, which states that at a given temperature T, the ratio of the electrical (cr) to the thermal (k) conductivities should be the same for all metals, in near agreement with experiment ... 

Adsorbed ions, specifically, 886 Adsorbent, 969 Adsorption, 971 contact, 959 electrical field, 929 and equation of state, 931 ionic, summary, 964 irreversible, 969, 970 lattice gas models of, 965 nonlocalized, 928, 958 organic and inorganic, 972 of intermediates, 1192... 

What is next Several examples were given of modem experimental electrochemical techniques used to characterize electrode-electrolyte interactions. However, we did not mention theoretical methods used for the same purpose. Computer simulations of the dynamic processes occurring in the double layer are found abundantly in the literature of electrochemistry. Examples of topics explored in this area are investigation of lateral adsorbate-adsorbate interactions by the formulation of lattice-gas models and their solution by analytical and numerical techniques (Monte Carlo simulations) [Fig. 6.107(a)] determination of potential-energy curves for metal-ion and lateral-lateral interaction by quantum-chemical studies [Fig. 6.107(b)] and calculation of the electrostatic field and potential drop across an electric double layer by molecular dynamic simulations [Fig. 6.107(c)]. 

In many production routes, and also during processing, polymer systems have to undergo pressure. Changes in the volume of a system by compression or expansion, however, cannot be dealt with in rigid-lattice-type models. Thus, non-combinatorial free volume ( equation of state ) contributions to AG have been advanced [23 - 29]. Detailed interaction functions have been suggested (but all of them are based on adjustable parameters, for blends, e.g., Mean-field lattice gas [30], SAFT [31], specific interactions [32]), and have been succesfully applied, for example, by Kennis et al. [33]. 

According to a two-sublattice mean field model, the EMD of inter-metallic compounds is determined by the competition between the EMDs of the rare earth sublattice and the transition metal sublattice. In the Th2Zn]7-type crystal lattice, the Sm sublattice prefers c-axis anisotropy to others because of the positive Stevens factor (aj) of Sm, and the Fe sublattice contributes to the c-plane anisotropy. Both Co and carbon in Sm2(Fe1 xCox)17C>. help to enhance the contribution of the Sm sublattice dominant to the EMD, but the effect of carbon is larger than that of Co. 

The Classical Free-Electron Theory. The classical free-electron theory considers that the valence electrons are virtually free everywhere in the metal. The periodic lattice field of the positively charged ions is evened out into a uniform potential inside the metal. The major assumptions of this model are (1) an electron can pass from one atom to another and (2) in the absence of an electric field elec-rons move randomly in all directions, and their movements obey the laws of classical mechanics and the kinetic theory of gases. In an electric field electrons... 

In considering the vibronic side-bands to be expected in the optical spectra when we augment the static crystal field model by including the electron-phonon interaction, we must know the frequencies and symmetries of the lattice phonons at various critical points in the phonon density of states. We shall be particularly interested in those critical points which occur at the symmetry points T, A and at the A line in the Brillouin zone. Using the method of factor group for crystals we have ... 

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