Three macroscopic dimensions

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 ... 

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. 

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—... 

ABSTRACT According to relevant documents and payment in the form of questionnaires, 224 cases of the domestic excavation accident in past two decades are collected in this paper. Based on processing and analysis of accident cases, the common risk events and factors are determined in the construction of deep excavation. Meanwhile the relativity between risk of quality and risk of safety and risk of investment is analyzed according to causes and losses of the accident, and the graph of macroscopic logical relation established between the risk of three objective dimension, which provides a reference for comprehensive risk management excavation construction process and basehne data for the theoretical analysis of risk. 

Quality risk accidents, safety risk accidents and investment risk accidents of deep foundation pit are not independent from each other, when consider one kind of risk accidents and its factors separately we may leave out some inner relation of them and then get a distorted risk recognition result. Via combing accident cases of deep foundation pit, the article carried out an analysis on the relationship among quality risk, security risk and investment risk and then established a macroscopic logic diagram of the influence rule of the three objective dimensions, reached a tentative conclusion as follows ... 

Chain units that are involved in forming intermolecular cross-links require special attention as far as crystallization is concerned. When a sufficient number of intermolecular cross-links are imposed on a collection of linear polymer chains, a three-dimensional network structure reaching macroscopic dimensions is developed. Such structures are termed infinite networks. According to theory (1,2) the initial formation of a network occurs when the fraction of cross-linked units p exceeds a critical value pc that is expressed as... 

A qualitative change in properties is achieved by coupling all the polymer chains of a sample together, thereby building up a three-dimensional network. This is the basic structure of rubbers. In fact, a rubber represents one huge macromolecule of macroscopic dimensions, with properties depending on the cross-link density and the functionality of the junction units. 

When the average valence is higher than 2, this implies that all monomeric units of a sample are connected to each other by covalent bonds and that a unique polymer chain of macroscopic dimension and same size as that of the sample is eventually obtained. One can then regard its molecular size as infinitely large, and the system is known as three-dimensional (see Figure 1.3). 

Fjualiy if cross-linking increases between the chains, we have three-y / mersiona aggregates w hich may extend over macroscopic dimensions. Ihf- best exiimple is water with its tetrahedral open structure... 

The study of chemistry deals essentially with three main aspects the macroscopic approach, the representational/symbolic dimension and the submicroscopic level of thinking (Johnstone, 1991, 2007). This chapter considers the role of the macroscopic component, taking also into account the other two levels. 

Before deriving the rate equations, we first need to think about the dimensions of the rates. As heterogeneous catalysis involves reactants and products in the three-dimensional space of gases or liquids, but with intermediates on a two-dimensional surface we cannot simply use concentrations as in the case of uncatalyzed reactions. Our choice throughout this book will be to express the macroscopic rate of a catalytic reaction in moles per unit of time. In addition, we will use the microscopic concept of turnover frequency, defined as the number of molecules converted per active site and per unit of time. The macroscopic rate can be seen as a characteristic activity per weight or per volume unit of catalyst in all its complexity with regard to shape, composition, etc., whereas the turnover frequency is a measure of the intrinsic activity of a catalytic site. 

Velocity maps of simple or complex liquids, emulsions, suspensions and other mixtures in various geometries provide valuable information about macroscopic and molecular properties of materials in motion. Two- and three-dimensional spin echo velocity imaging methods are used, where one or two dimensions contain spatial information and the remaining dimension or the image intensity contains the information of the displacement of the spins during an observation time. This information is used to calculate the velocity vectors and the dispersion at each position in the spatially resolved dimensions with the help of post-processing software. The range of observable velocities depends mainly on the time the spins... 

Several kinds of intermediate states exist between the state of highest order in a crystal having translational symmetry in three dimensions and the disordered distribution of particles in a liquid. Liquid crystals are closest to the liquid state. They behave macroscopically like liquids, their molecules are in constant motion, but to a certain degree there exists a crystal-like order. 

Common ideas about dimensionality include that three-dimensional objects have all three-dimensional parameters (length, width, depth) in the same macroscopic size range. Two-dimensional objects have one length which is considerably smaller than the other two (consider a line such that the length and thickness of the line may be determined but the depth is so much smaller that it may essentially be considered as a two-dimensional object). A one-dimensional object has two dimensions much smaller than the third. An example of this is the carbon nanotube, which has a length much longer than the tube diameter. Dimensionality essentially adds... 

Minerals have definite chemical compositions when analyzed at the macroscopic scale, but as exemplified by solid solutions, the local chemical composition may vary within fixed limits. As crystalline phases, whether chemical end members or solid solutions, minerals have internal atomic structures said to possess long-range order on average in three dimensions. 

Minerals are generally regarded as crystalline phases formed as a result of geological processes. As a (bulk) crystalline phase in the classical sense, a mineral must satisfy the conditions of long-range structural order in three dimensions, and homogeneity with respect to its macroscopic physical and chemical properties. 

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