Lattice quasi-crystalline

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... 

Plastic crystals The long-range orientational order is lost, the long-range positional order is preserved, i.e. a plastic quasi-crystalline lattice is still present. 

Before passing on to the theoretical interpretation of these results we shall first indicate the geometrical parameters to be employed. Once again we shall assume that the solution can be described in terms of a quasi-crystalline lattice (c/. 

This idea has been realized in the excellent study [39]. The quasi- crystalline monolayer of the protein from the thermophilic bacterium Sulfolobus acidocaldarius was prepared on the flat support. Then the deposition of metal (Ta/W) on the obtained layer of adsorbed protein was followed by the removal of both the protein and the metal deposited on the protein using the ion bombardment. As a result the regular network of metal islands with the lattice parameter 22 nm equal to the distance between the centres of the adsorbed proteins was obtained. The islands of metal have a diameter of 15 nm and a thickness of Inm. 

Few studies have been made on transport processes involving concentrated solutions. In the concentrated solutions, in the range of dehydrated melt formation, incompletely hydrated melts and anhydrous salt melts, various structural models are described to define their properties, i.e. the free-volume model, the lattice-model and the quasi-crystalline model. Measured and calculated transport phenomena do not always represent simple ion migration of individual particles, but instead we sometimes find them to be complicated cooperative effects (27). 

A second mechanism for increasing disorder on melting which cannot be conveniently represented by a quasi-crystalline model for the melt involves the formation of association complexes. Quite generally, these can be defined as clusters of the units of structure (e.g., molecules or ions) in the crystal which have approximately the same distance between nearest neighbours as in the crystal lattice, but which need not have the full regularity of crystal packing. As already stated, only one particular form of cluster, the crystal nucleus can normally be extended indefinitely... 

Roe [1975] developed a quasi-crystalline lattice model for conditions where %cr (where... 

Figure 2.5a shows a snapshot from a Ru(0 001) surface with a small coverage of adsorbed O atoms at 300K. The O atoms are randomly distributed and move around like in a Brownian motion with a mean residence time (at 300K) of 60 ms at a certain adsorption site. However, due to the weak attraction between two adatoms with a minimum at a distance of 2flo ( o = lattice constant of the substrate), at higher coverages a separation into two phases, namely, a quasi-gaseous and a quasi-crystalline phase, takes place (Fig. 2.5b) [9]. Under present conditions, the two phases are in equilibrium with each other, a situation that is rationalized by the phase diagram depicted in Fig. 2.6a. In our case, the horizontal scale (composition) denotes the concentration of occupied sites (i.e., overall coverage 0). As long as 0 is small, we...

The calculation of the diffraction pattern for a periodic system revolves around the construction of the reciprocal lattice and subsequent placement of the first Brillouin zone however, in this case the aperiodicity of the pentagonal array requires a different approach due to the lack of translational symmetry. The reciprocal lattice of such an array is densely filled with reciprocal lattice vectors, with the consequence that the wave vector of a transmitted/reflected light beam encounters many diffraction paths. The resultant replay fields can be accurately calculated by taking the FT of the holograms. To perform the 2D fast Fourier transform (FFT) of the quasi-crystalline nanotube array, a normal scanning electron micrograph was taken, as shown in Fig. 1.13. 

The quasi-lattice model was developed by Roe (13) and Scheutjens and Fleer (14) (SF theory) The basic analysis considered all chain conformations as step-weighted random walks on a quasi-crystalline lattice that extends in parallel layers from the surface. This is illustrated in Figure 16.2 which shows a possible conformation of a polymer molecule at a flat surface. The partition function was written in terms of the number of chain configurations that were treated as connected sequences of segments. In each layer parallel to the surface, random mixing between the segments and solvent molecules was assumed, i.e. by using... 

Several approaches have been followed to understand the transport phenomena occurring in perfluoropolymeric membranes, and these include irreversible thermodynamics, use of the Nemst-Planck transport equation (Eq.l3), and various descriptions (e.g., free-volume model, lattice model, quasi-crystalline model) coupling the morphological and chemical properties of the membranes. The reader is referred to the references and citations in the reviews [81-87] for more details related to these models. 

FIGURE 14.4 Schematic representation of a polymer molecule adsorbing on a flat surface. A quasi-crystalline lattice is depicted with segments filling layers that are parallel to the surface (random mixing of segments and solvent molecules in each layer is assumed). 

The outcome, though more or less strongly disordered can still be referred to the parent crystal lattice, i.e. the liquid is quasi crystalline. However, if the molecules can form short range hydrogen bonds, based on atoms of low atomic number such as N, O, F, S the quasi crystalline fluid generally rearranges its molecules so as to decrease the enthalpy by... 

For this purpose it was assumed that the molecules are located on a quasi-crystalline lattice and can be moved about on this lattice without affecting the spacing of the energy levels of the system. This will be discussed again in Chapter 14 in connexion with regular solutions and the equations above will also be applied to the problem of adsorption. 

In 8 1 it was shown that equations (14 4)-(14 6) lead to a statistical deduction of the laws of ideal solutions. For this purpose it was supposed that a solid or liquid solution can be approximated by a quasi-crystalline lattice, and also that the A and B molecules are of a sufficiently equal shape and size for them to be interchangeable between the lattice sit without change of lattice structure and without change in the lattice vibrations or the internal states of the molecules. Before mixing there is only one geometrical arrangement and after mixing there are... 

The following is based on a derivation given by Everett,f with some simplifications of the notation. The necessary assumptions are (a) a gas molecule can only be adsorbed at a finite number of positions, called the sites , on the surface of the solid (b) the quantum states of adsorbed gas molecules are the same for all sites and independent of the presence of neighbouring molecules. The first of these is analogous to the assumption of a quasi-crystalline lattice in the case of a solution and its purpose is to give a countable number of configurations. Its justification depends, of course, on the fact that the surface is atomic in structure and may be expected to have potential energy wells , where adsorption takes place most readily. The assumption... 

Roe [207] used a quasi-crystalline lattice model to determine the properties of the interface between two coexisting liquid phases, where one or both of the components are of polymeric nature. For y much larger than the critical value Xc. the composition transition at the interface is expected to be sharp. In this limit. Roe predicted that ... 

The adsorption of polymers from solutions strongly depends on the thermodynamic quality of solvent and the interaction energy between polymer and surface. All theories of adsorption include the thermodynamic parameter of interaction of the Flory-Huggins theory %i2. The thermodynamic interaction between polymer and solvent determines the conformation of macromolecules in solutions and thus the conditions of its interaction with the surface. The interaction between polymer and surface is characterized by the parameter of thermodynamic interaction, which was introduced by Silberberg, using the model of quasi-crystalline lattice of the surface layer, describing the properties of polymer solutions. This parameter may be determined as follows ... 

The present chapter deals with the thermodynamic properties of pure f-mers while Ch. XVII is devoted to mixtures. The only type of systems containing r-mers which is tractable at present in a quantitative way is that in which a quasi-crystalline lattice is preserved. If this condition is not fulfilled only methods valid for very dilute solutions are available (cf. Ch. V) but these will not be considered here. We diall briefly discuss the criterion for the existence of a quasi-lattice in 2. The restriction imposed by this criterion forces us to consider situations in which the thermal expansion remains small. For this reason both Ch. XVI and XVII are restricted to systems far from the critical point. It is possible to improve the situation by considering "holes or other fluctuations in density (cf. Ch. VII) but we shall not go into details (Bellemans [1953]). 

The differences observed between the quasi-racetnic mixture and the mixture of enantiomers with the same configuration are analogous to the differences between the racemic and the optically active forms of the pure compounds. Thus, a kind of two-dimensional isomorphy exists between the molecular systems in question. The different properties of optically active and racemic forms in solid surface phases have been taken as evidence of an orderly arrangement of the molecules in some kind of two-dimensional crystalline lattice. 

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