Quasi-one-dimensional approach

The Quasi-One-Dimensional Approach to the Normal Phase of the Bechgaard Salts... 

Abstract This chapter deals with capillary instability of straight free liquid jets moving in air. It begins with linear stability theory for small perturbations of Newtonian liquid jets and discusses the unstable modes, characteristic growth rates, temporal and spatial instabilities and their underlying physical mechanisms. The linear theory also provides an estimate of the main droplet size emerging from capillary breakup. Formation of satellite modes is treated in the framework of either asymptotic methods or direct numerical simulations. Then, such additional effects like thermocapiUarity, or swirl are taken into account. In addition, quasi-one-dimensional approach for description of capillary breakup is introduced and illustrated in detail for Newtonian and rheologically complex liquid jets (pseudoplastic, dilatant, and viscoelastic polymeric liquids). 

The linear stability theory is exceptional in the sense that it can be fuUy based on the three-dimensional equations of fluid dynamics. All the additional effects lead to either direct numerical simulations or the asymptotic approximations. One of the most natural ways of the asymptotic description of the dynamics of jets is the quasi-one-dimensional approach. In the quasi-one-dimensional approximation. 

Summarizing, in the linear stability theory of capillary breakup of thin free liquid jets, the quasi-one-dimensional approach allows for a simple and straightforward derivation of the results almost exactly coinciding with those obtained in the framework of a rather tedious analysis of the three-dimensional equations of fluid mechanics. This serves as an important argument for further applications of the quasi-one-dimensional equations to more complex problems, which do not allow or almost do not allow exact solutions, in particular, to the nonlinear stages of the capillary breakup of straight thin liquid jets in air (considered below in this chapter). 

In the quasi one-dimensional approach, when the meniscus length is smaller or about equal to the channel width w, the original... 

Fig. 2 Schematic representation of the quasi-one-dimensional approach for solving the pulse voltage problem...

The quasi-one-dimensional model of flow in a heated micro-channel makes it possible to describe the fundamental features of two-phase capillary flow due to the heating and evaporation of the liquid. The approach developed allows one to estimate the effects of capillary, inertia, frictional and gravity forces on the shape of the interface surface, as well as the on velocity and temperature distributions. The results of the numerical solution of the system of one-dimensional mass, momentum, and energy conservation equations, and a detailed analysis of the hydrodynamic and thermal characteristic of the flow in heated capillary with evaporative interface surface have been carried out. 

The most reliable technique to find the global optimum by means of common methods is the transition from the quasi-two-dimensional approach (Fig. 5.3b,c) to a complete two-dimensional one. It consists of a certain number of experiments as shown in Fig. 5.4. 

In this and the next sections we discuss two groups of molecule-based conducting magnets at which the %-d interaction works effectively. The first approach is the use of quasi one-dimensional electronic systems as the re-electron layers, and the other strategy is to increase the magnitude of the %-d interaction by the introduction of intermolecular halogen-halogen contacts. 

We use the same approach to classify the different nanostructures for Titania. The term one-dimensional (ID) nanostructures indicate nanocrystals in which elongation only in one direction is above this threshold (about 10 nm). This class of ID nanostructures comprises different types of nano-ordered materials, such as nanorods, -wires, -coils, -fibers, -pillars (or -columns) and -tubes. We prefer to use the term quasi one-dimensional nanostructures, because often the dimensions are larger than the indicated threshold, although elongation along one main axis still exists. When the diameter of the nanorod, nanowire or nanotube becomes smaller, there is often a significant change in the properties with respect to crystalline solids or even two-dimensional systems. A bismuth nanowire is an excellent example, which transforms into a semiconductor, as the wire diameter becomes smaller.145... 

An adequate quantitative description of such a situation requires a two- or even three-dimensional approach. Today, a great variety of numerical models are available that allow us to solve such models almost routinely. However, from a didactic point of view numerical models are less suitable as illustrative examples than equations that can still be solved analytically. Therefore, an alternative approach is chosen. In order to keep the flow field quasi-one-dimensional, the single well is replaced by a dense array of wells located along the river at a fixed distance xw (Fig. 25.2c). Ultimately, the set of wells can be looked at as a line sink. This is certainly not the usual method to exploit aquifers Nonetheless, from a qualitative point of view a single well has properties very similar to the line sink. 

The statistical thermodynamics analysis of -mers adsorption in a one-dimensional lattice provides an intuitive approach to linear molecules confined in quasi-one-dimensional nanotubes. More elaborated analytical solutions that incorporate nearest and next-nearest-neighbors between fc-mer s ends can be obtained by applying the mapping proposed in the present work. 

From this expression one finds that f(T) approaches 1 for T —> 0 (by integration) and /(T) 2(1 - T/TCMF) - Z 2 for T -> TCMF. TCMF denotes the mean-field transition temperature. For quasi one-dimensional systems p1 has an additional factor C-2 (the inverse area perpendicular to the chain). 

Amici and Thalmeier (1998) used the quasi one-dimensional model mentioned in Section 4.9.1. In their approach the presence of ferromagnetically ordered Flo layers with the magnetic moments oriented perpendicular to the tetragonal c-axis is adopted and the competition of the RKKY interaction along the c-axis with the crystalline electric field is analyzed in order to determine the transition between the commensurate antiferromagnetic structure and the incommensurate c spiral shown in Figure 39. 

None of the approaches described above is completely realistic for quasi-one-dimensional organic conductors since there is always some Coulomb repulsion between them, which does not fall in any of the limits considered above. The U and V terms are essentially molecular, and usually U > V > 0, with U > 1 eV, which is on the order of magnitude of the bandwidth,... 

Thus, inter-atomic distances and the atomic state in Tetracarbon are fundamentally different from all the known forms of carbon. The differences between clear and hard diamond on the one hand, and soft and black graphite on the other hand, illustrate the differences among Tetracarbon and other forms of carbon. The distance between the neighboring sp -carbon atoms within the Tetracarbon chain is about 1.3 A, whereas the distance between the carbon chains is 4.80-5.03 A. It is interesting to note that in some respects Tetracarbon is similar to tubulenes, as it can be considered as tubulene in the limit when the diameter of the tube approaches the diameter of carbon atom. Nevertheless, in Tetracarbon the hybridization state of carbon atoms changes from sp to sp. It is basically a new purely one-dimensional sp -carbon modification with one-dimensional electron band structure, whereas tubulene is a quasi-one-dimensional material in which the number of one-dimensional electron bands increases with increasing tubulene diameter. Tetracarbon and tubulene are also similar in that the carbon chains in Tetracarbon are oriented normally to the surface of the film, similar to the orientation in tubulene. 

Since many-body optical transitions in zero-dimensional objects was demonstrated experimentally, it is important to assess this phenomenon from the perspective of the well established field of many-body luminescence. This is accomplished in the present chapter. Below we review the many-body luminescence in various systems studied to date experimentally and theoretically. We then demonstrate that many-body luminescence from highly excited zero-dimensional objects has unique features due to large number of discrete lines. This discreteness unravels the many-body correlations that are otherwise masked in the continuous spectrum of luminescence from infinite systems. We describe in detail the emergence of such correlations for a particular nanostructure geometry - semiconductor nanorings - using the Luttinger liquid approach for quasi-one-dimensional finite-size systems. 

A further development of the displayed approach to the electron-phonon interaction and relaxation processes in crystals activated by lanthanide ions came from the high-resolution study of Pr CsCdBr3 [7]. These crystals have recently attracted a considerable interest being a promising material for up-conversion lasers due to the property of their quasi-one-dimensional lattice to incorporate Ln ions in pairs. 

The typical space-size characterizing a plasma is the Debye radius, which is a linear measure of electroneutrality and shielding of external electric fields. The typical plasma time scale and typical time of plasma response to the external fields is determined by the plasma frequency illustrated in Fig. 3-19. Assume in a one-dimensional approach that all electrons at X > 0 are initially shifted to the right on the distance xq, whereas heavy ions are not perturbed and remain at rest. This results in an electric field, which pushes the electrons back. If = 0 at X < 0, this electric field acting to restore the plasma quasi-neutrality can be found at x > xq from the one-dimensional Poisson equation as... 

A different approach to correlate porous silicon conductivity with material porosity was described in Ref Aroutiounian and Ghulinyan (2003). In this work, the conductivity was shown to be mainly crystalline for porosities much lower than the percolation threshold at 57 %, while a fractal behavior was observed at porosities near percolation threshold. For higher values of porosities, the conductivity was described as a quasi-one-dimensional hopping. The report concluded that in PS with increasing porosity, at lower temperatures, the dimension of the chaimels of electrical current flow decrease fi"om 3 to 1, as described by the Mott law for amorphous semiconductors. However, the model results described in this work show some deviation fi"om the experimental results. 

Tsironis, G. P. and Pnevmatikos, S. 1989. Proton conductivity in quasi-one-dimensional hydrogen-bonded systems—nonlinear approach. Phvs. Rev. B. 39(10), 7161-7173. Tuckerman, M. E., Laasonen, L., Sprik, M. and Parrinello, M. 1994. Ab initio simulations of water and water ions. 6, A93-A100. 

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