Constitutive macroscopic

As discussed in Section 1, the absolute permeability is a constitutive macroscopic property which arises in the local volume-averaged momentum balance, which is Darcy s law (Eq. (2)). It is interesting that, while the porosity can be defined independently of the equation of continuity, the permeability is... 

Generally speaking, intermolecular forces act over a short range. Were this not the case, the specific energy of a portion of matter would depend on its size quantities such as molar enthalpies of formation would be extensive variables On the other hand, the cumulative effects of these forces between macroscopic bodies extend over a rather long range and the discussion of such situations constitutes the chief subject of this chapter. 

A constitutive equation is a relation between the extra stress (t) and the rate of deformation that a fluid experiences as it flows. Therefore, theoretically, the constitutive equation of a fluid characterises its macroscopic deformation behaviour under different flow conditions. It is reasonable to assume that the macroscopic behaviour of a fluid mainly depends on its microscopic structure. However, it is extremely difficult, if not impossible, to establish exact quantitative... 

The concept of corresponding states was based on kinetic molecular theory, which describes molecules as discrete, rapidly moving particles that together constitute a fluid or soHd. Therefore, the theory of corresponding states was a macroscopic concept based on empirical observations. In 1939, the theory of corresponding states was derived from an inverse sixth power molecular potential model (74). Four basic assumptions were made (/) classical statistical mechanics apply, (2) the molecules must be spherical either by actual shape or by virtue of rapid and free rotation, (3) the intramolecular vibrations are considered identical for molecules in either the gas or Hquid phases, and (4) the potential energy of a coUection of molecules is a function of only the various intermolecular distances. 

The multi-shell fullerenes constitute the transition from fullerenes to macroscopic graphite. They present both the closed graphitic surface of fullerenes and the stacked layers interacting by van der Waals forces, as in graphite. 

Network properties and microscopic structures of various epoxy resins cross-linked by phenolic novolacs were investigated by Suzuki et al.97 Positron annihilation spectroscopy (PAS) was utilized to characterize intermolecular spacing of networks and the results were compared to bulk polymer properties. The lifetimes (t3) and intensities (/3) of the active species (positronium ions) correspond to volume and number of holes which constitute the free volume in the network. Networks cured with flexible epoxies had more holes throughout the temperature range, and the space increased with temperature increases. Glass transition temperatures and thermal expansion coefficients (a) were calculated from plots of t3 versus temperature. The Tgs and thermal expansion coefficients obtained from PAS were lower titan those obtained from thermomechanical analysis. These differences were attributed to micro-Brownian motions determined by PAS versus macroscopic polymer properties determined by thermomechanical analysis. 

In chemical education, the micro-world is a key perspective in viewing the macroscopic world around us. Among other perspectives, such as a personal, a socio-economical perspective, a political perspective, this key chemical perspective can offer students a view on the materials and substances they use, that they are made of and that constitute their material world. A micro-world perspective enables the understanding of mat r phenomena and offers the possibihty to improve the material world within social circumstances. 

A model is one of the main outcomes of ary scientific enquiry and hence is a major contributor to philosophy of science. A model may be defined as a simplified representation of a phenomenon (an object, system, event, process) or idea produced for the specific purpose of providing an explanation of that entity, the most important outcomes of which are the production of successful predictions of how it will behave under a range of circumstances (Gilbert, Boulter, Elmer, 2000). Entities can be modelled at the three levels at the macroscopic, by representing some of the aspects of the entity that can be seen at the sub-microscopic, by representing the ideas produced to explain the constitution and behaviour of the particles that constitute the entity and at the symbolic, by representing the symbols created to simplify the reference to such particles (as, for instance, chemical formulae and chemical equations). 

Nanotechnology is the branch of engineering that deals with the manipulation of individual atoms, molecules, and systems smaller than 100 nanometers. Two different methods are envisioned for nanotechnology to buUd nanostructured systems, components, and materials. One method is the top-down approach and the other method is called the bottom-up approach. In the top-down approach the idea is to miniaturize the macroscopic structures, components, and systems toward a nanoscale of the same. In the bottom-up approach the atoms and molecules constituting the building blocks are the starting point to build the desired nanostmcture [96-98]. 

The building blocks of all materials in any phase are atoms and molecules. Their arrangements and how they interact with one another define many properties of the material. The nanotechnology MBBs, because of their sizes of a few nanometers, impart to the nanostructures created from them new and possibly preferred properties and characteristics heretofore unavailable in conventional materials and devices. These nanosize building blocks are intermediate in size, lying between atoms and microscopic and macroscopic systems. These building blocks contain a hmited and countable number of atoms. They constitute the basis of our entry into new realms of bottom-up nanotechnology [97, 98]. 

The molecular models of rubber elasticity relate chain statistics and chain deformation to the deformation of the macroscopic material. The thermodynamic changes, including stress are derived from chain deformation. In this sense, the measurement of geometric changes is fundamental to the theory, constitutes a direct check of the model, and is an unambiguous measure of the mutual consistency of theory and experiment. 

Bone is an extremely dense connective tissue that, in various shapes, constitutes the skeleton. Although it is one of the hardest structures in the body, bone maintains a degree of elasticity owing to its structure and composition. It possesses a hierarchical structure and, as most of the tissues, is nanostructured in fact, it is a nanoscaled composite of collagen (organic extracellular matrix) and hydroxycarbonate apatite, (HCA, bone mineral). This nanostructure is in intimate contact with the bone cells (several microns in size), which result (at the macroscopic level) in the bone tissue. Figure 12.2 shows the bone hierarchical ordering from the bone to the crystalline structure of HCA. 

Molecular calculations provide approaches to supramolecular structure and to the dynamics of self-assembly by extending atomic-molecular physics. Alternatively, the tools of finite element analysis can be used to approach the simulation of self-assembled film properties. The voxel4 size in finite element analysis needs be small compared to significant variation in structure-property relationships for self-assembled structures, this implies use of voxels of nanometer dimensions. However, the continuum constitutive relationships utilized for macroscopic-system calculations will be difficult to extend at this scale because nanostructure properties are expected to differ from microstructural properties. In addition, in structures with a high density of boundaries (such as thin multilayer films), poorly understood boundary conditions may contribute to inaccuracies. 

Alloys are classified broadly in two categories, single-phase alloys and multiple-phase alloys. A phase is characterized by having a homogeneous composition on a macroscopic scale, a uniform structure, and a distinct interface with any other phase present. The coexistence of ice, liquid water, and water vapor meets the criteria of composition and structure, but distinct boundaries exist between the states, so there are three phases present. When liquid metals are combined, there is usually some limit to the solubility of one metal in another. An exception to this is the liquid mixture of copper and nickel, which forms a solution of any composition between pure copper and pure nickel. The molten metals are completely miscible. When the mixture is cooled, a solid results that has a random distribution of both types of atoms in an fee structure. This single solid phase thus constitutes a solid solution of the two metals, so it meets the criteria for a single-phase alloy. 

Practically nothing is known about the chemical behavior of [5]radialenes. An exception is the electrochemistry of systems 139-141. These potent electron-donating molecules show only one pair of reversible waves involving a net transfer of four electrons. In fact, these systems constitute the first examples of a single-wave four-electron transfer with only one macroscopic redox site in an organic redox system87,89. The corresponding tetracationic salts could be isolated. 

The previous argument is valid for all observables, each represented by a characteristic operator X with experimental uncertainty AX. The problem is to identify an elementary cell within the energy shell, to be consistent with the macroscopic operators. This cell would constitute a linear sub-space over the Hilbert space in which all operators commute with the Hamiltonian. In principle each operator may be diagonalized by unitary transformation and only those elements within a narrow range along the diagonal that represents the minimum uncertainties would differ perceptibly from zero. 

Reaction rates are macroscopic averages of the number of microscopical molecules that pass from the reactant to the product valley in the potential hypersurface. An estimation of this rate can be obtained from the energy of the highest point in the reaction path, the transition state. This approach will however fail when the reaction proceeds without an enthalpic barrier or when there are many low frequency modes. The study of these cases will require the analysis of the trajectory of the molecule on the potential hypersurface. This idea constitutes the basis of molecular dynamics (MD) [96]. Molecular dynamics were traditionally too computationally demanding for transition metal complexes, but things seem now to be changing with the use of the Car-Parrinello (CP) method [97]. This approach has in fact been already succesfully applied to the study of the catalyzed polymerization of olefins [98]. 

Electrochemistry at electrodes with microscopic dimensions (e.g., a disk of 10 j,m diameter) and nanoscopic dimensions (e.g., a disk of <100 nm diameter) constitutes one of the most important frontiers in modern electrochemical science [25]. Such micro- and nanoscopic electrodes allow for electrochemical experiments that are impossible at electrodes of macroscopic dimensions (e.g., disks of mm diameter we call such electrodes macroelectrodes ). Examples of unique opportunities afforded by micro- and nanoscopic electrodes include the possibility of doing electrochemistry in highly resistive media and the possibility of investigating the kinetics of redox processes that are too fast to study at electrodes of conventional dimensions (both are discussed in detail below). In addition, microscopic electrodes have proven extremely useful for in vivo electrochemistry [62]. 

Recent times have seen much discussion of the choice of hydrodynamic boundary conditions that can be employed in a description of the solid-liquid interface. For some time, the no-slip approximation was deemed acceptable and has constituted something of a dogma in many fields concerned with fluid mechanics. This assumption is based on observations made at a macroscopic level, where the mean free path of the hquid being considered is much smaller... 

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