Two macroscopic dimensions

Thin Films Structures with Two Macroscopic Dimensions, 387... 

However, the application potential can be fnlly exploited only if suitable methods to control the structure and to prepare ordered particle arrays over macroscopic dimensions are available. Various methods have been tried to organize mesoscale particles in two or three dimensions. Among them are methods already being used for the organization of molecules at interfaces, while other methods were developed especially for the organization of particles ... 

Even though this contribution is always negative, the total capacity must be positive - otherwise the capacitor would accumulate charge spontaneously. Thus Eq. (17.4) is only valid if f > rjm, so that there is no electronic overlap between the two plates. Similarly the use of a macroscopic dielectric constant in Eq. (17.5) presupposes a plate separation of macroscopic dimensions, and again the total capacity is positive. Only unphysical models or bad mathematical approximations can produce negative interfacial capacities, which enjoyed a brief spell of fame under the name of the Cooper-Harrison catastrophe [2]. 

Abstract In this work, we explain how small surfactant molecules with dimensions of one to two nanometers, when dissolved in water with a concentration of a few weight percent, organize themselves spontaneously into various structures with a long-range order over macroscopic dimensions of several centimeters, even though the molecules are all in the liquid state. It follows that two molecules that are more than a million times their main length apart stUl point, on average, in the same direction in three dimensional space. 

In the simplest molecular theory now available, there are two major assumptions. The first is that m/mnolccular interactions are unimportant, that is, they do not change with deformation. The second is that the deformation is affine, which means that the molecular deformation is the same as the macroscopic deformation. More precisely, the assumption is that the cross-link positions vary in a simple linear manner with the macroscopic dimensions. 

Israelachvili (1992) discusses the various approximate and analytic formalisms for A, with the conclusion that where two macroscopic isotropic phases 1 and 2 interact across an isotropic medium 3, a suitable approximation to the relevant Hamaker constant, A132, valid for Z greater than molecular dimensions is ... 

The structure of an usual crystal, or of the perfectly regular chain of a theoretical CP, is specified completely by a small number of parameters, whatever the size of the system. Defects in these perfect structures, such as a dislocation or a soliton, are well defined in the same way. Except for PDAs, which are not considered in this section, this is not true of a real CP sample. It is usually disordered at all scales in a variable way, up to macroscopic dimensions for instance, the morphologies of the two surfaces of a fibrillar PA film are different (see, e.g., Fig. 5 in Ref. 17), or that of an electrochemically synthesized CP film varies continuously along its thick-... 

A thermodynamic system is a part of the physical universe with a specified boundary for observation. A system contains a substance with a large amount of molecules or atoms, and is formed by a geometrical volume of macroscopic dimensions subjected to controlled experimental conditions. An ideal thermodynamic system is a model system with simplifications to represent a real system that can be described by the theoretical thermodynamics approach. A simple system is a single state system with no internal boundaries, and is not subject to external force fields or inertial forces. A composite system, however, has at least two simple systems separated by a barrier restrictive to one form of energy or matter. The boundary of the volume separates the system from its surroundings. A system may be taken through a complete cycle of states, in which its final state is the same as its original state. 

The theory of Kaluza and Klein [89, 90] is based on an observation that of two macroscopic forces of Nature only gravitation can be ascribed to geometric features of four-dimensional space-time. In order to incorporate another interaction the logical development would be to consider an additional dimension and to examine if extra degrees of freedom provided by 15 covariant components of the five-dimensional symmetric tensor needed to specify the line element... 

Another instructive exercise consists in examining the free expansion of a gas, depicted in Fig. 1.8.2. The gas is initially confined to a space of volume VA, volume VB being totally evacuated. A hole of macroscopic dimensions is now opened in the diaphragm separating the two volumes the gas ultimately occupies the total volume VA + VB. What is the work involved in this process The answer is not absolutely straightforward If the system is taken with boundaries encompassing only the volume VA, complications arise because... 

The favored structure for most phospholipids and glycolipids in aqueous media is a bimolecular sheet rather than a micelle. The reason is that the two fatty acyl chains of a phospholipid or a glycolipid are too bulky to fit into the interior of a micelle. In contrast, salts of fatty acids (such as sodium palmitate, a constituent of soap), which contain only one chain, readily form micelles. The formation of bilayers instead of micelles by phospholipids is of critical biological importance. A micelle is a limited structure, usually less than 20 nm (200 A) in diameter. In contrast, a bimolecular sheet can have macroscopic dimensions, such as a millimeter (10 nm, or 10 A). Phospholipids and related molecules are important membrane constituents because they readily form extensive bimolecular sheets (Figure 1211). 

The total volume of a cylindrical sample may be divided into two regions by defining a sphere round any given molecule within the sample large enough to be of macroscopic dimensions and small compared with the size of the sample. If the sample is placed in a uniform magnetic field Hq, the field experienced by the nucleus at the centre of the sphere is made up of three parts—... 

A rather large body of evidence has been accumulated over the past 10 to 15 years indicating that the chemistry of the interphase between the metal per se and the homogeneous bulk of the electrolyte may determine the kinetics of the corrosion process. This interphase must be visualized as being distinctly different in composition from either the metal or the electrolyte and extending from microscopic to perhaps even macroscopic dimensions. As a consequence the overall kinetics of the corrosion process are determined by a complex interplay between the reaction rates at the two interfaces made up by the boundaries of the interphase and transport phenomena in the interphase itself. 

In passing we note that the functions in the set g are completely delocalized over the region of sites defined by the localized particle-antiparticle basis h, while the f-basis contains all possible phase-shifted contributions from each site in accordance with Eqs. (56) and (57) above. Some interrelationships can be recognized here. The first connection concerns Coleman s so-called extreme state [18], cf. the theories of superconductivity and superfluidity based on ODLRO. The second observation relates to the identification of the present finite dimensional representation as a precursor for possible condensations, developing correlations and coherences that may extend over macroscopic dimensions. If h is a set of two-particle determinants and the iV-particle fermionic wave function is constructed from an AGP, antisymmetrized geminal power, based on i, see Eq. (57), then the reduced density matrix can be represented as... 

If H-bonding end groups interact to create dynamic cross-links with a functionality exceeding two, then a transient network spanning macroscopic dimensions can... 

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