Macroscopic inhomogeneities

Ab-initio studies of surface segregation in alloys are based on the Ising-type Hamiltonian, whose parameters are the effective cluster interactions (ECI). The ECIs for alloy surfaces can be determined by various methods, e.g., by the Connolly-Williams inversion scheme , or by the generalized perturbation method (GPM) . The GPM relies on the force theorem , according to which only the band term is mapped onto the Ising Hamiltonian in the bulk case. The case of macroscopically inhomogeneous systems, like disordered surfaces is more complex. The ECIs can be determined on two levels of sophistication ... 

In the past two decades, 129Xe NMR has been employed as a useful technique for the characterization of the internal void space of nanoporous materials. In particular, the xenon chemical shift has been demonstrated to be very sensitive to the local environment of the nuclei and to depend strongly on the pore size and also on the pressure [4—6], Assuming a macroscopic inhomogeneity resulting from a distribution of adsorption site concentrations, 129Xe NMR spectra of xenon in zeolites have been calculated, and properties such as line widths, shapes as well as their dependence on xenon pressure can be reproduced qualitatively. A fully quantitative analysis, however, remains difficult due to the different contributions to the xenon line shift. (See Chapter 5.3 for a more detailed description of Xe spectroscopy for the characterization of porous media.)... 

Boss, et al., fitted Gq. (17) to their data vs. vdi enabling them to determine fp and D . At solvent concentration approaching vdiI = 0.95, the data revealed an enhancement above the value predicted by Eq. (17) as fitted to the lower-concentration data. The authors argued that under these circumstances macroscopic inhomogeneities in concentration (and hence in the free-volume distribution) should exist and enhance the diffusivity. Above v > 0.99 the polymer coils no longer overlapped substantially, depriving the solvent molecules of a set of obstacles fixed with respect to the laboratory, and solvent diffusion became related principally to intrinsic viscosity. 

Consider a macroscopically inhomogeneous system of linear geometry. If the number of particles z between coordinate and ( +d( ) at time t = 0 is z(t,0), what is the number of particles z(, t) at a predetermined coordinate between and ( + d ) after time r has elapsed In order to answer this question, we define a function fT ( . (f) as the probability of finding the particle at. a distance ( - ) after... 

It is important to point out that the observed two-component EPR spectra are an intrinsic property of the lightly doped LSCO and are not due to conventional chemical phase separation. We examined our samples using x-ray diffraction, and detected no impurity phases. Moreover, the temperature dependence of the relative intensities of the two EPR signals rules out macroscopic inhomogeneities and points towards a microscopic electronic phase separation. The qualitatively different behavior of the broad and narrow EPR signals indicates that they belong to distinct regions in the sample. First we notice that the broad line vanishes at low temperatures. This... 

These three models illustrate the different effects that can influence the conductivity near E, . The results are summarized as follows, with the additional comment that the conclusions apply only to material without macroscopic inhomogeneities. 

The second problem facing the researcher, and the one which is the subject of this paper, is to determine which, if any, effective modulus theory accurately predicts the acoustic wave velocity and attenuation in a microscopically or macroscopically inhomogeneous material. 

Open questions also exist in the case of macroscopically inhomogeneous deformation, as it occurs for instance in the presence of aggressive environments Even less well known are those inhomogeneous deformation mechanisms which are induced by certain morphological features crack-like defects in the spherulitic morphology, second phases, void or crack nucleating particles, or certain micro-structural elements behaving differently such as spherulites and spherulite boundaries. 

The same experiment was carried out several times with different pieces of Nafion 390 cut from a large sheet all data showed similar trends in current efficiency and electro-osmotic water with increasing current density. However, results from each membrane were nonidentical, presumably because of macroscopic inhomogeneities. 

To observe morphology in polymer blends, one should start with a careful observation by the naked eye and then by OM. This will provide information on a length scale of 10 /on, thus about rather macroscopic inhomogeneities, often created by inadequate control of extrusion, solution casting, etc. The macroscopic inhomogeneities can be critical for the materials performance. 

On visual inspection, blood vessels appear to be fairly homogeneous and distinct from surrounding connective tissue. The inhomogeneity of the vascular wall is realized when one examines the tissue under a low-power microscope, where one can easily identify two distinct structures the media and adventitia. For this reason the assumption of vessel wall homogeneity is applied cautiously. Such an assumption may be valid only within distinct macroscopic structures. However, few investigators have incorporated macroscopic inhomogeneity into studies of vascular mechanics [ 1 ]. 

Alternatively, the level of stmctural complexity may be affected in a totally different manner employing co-assembly of chemically unlike molecules instead of self-assembly of chemically identical molecules [12-20]. We refer to the resultant micelles as mixed micelles or co-micelles to indicate that this type of micelle consists of more than one type of molecule, whereas classical micelles consist of identical molecules (polydispersity effects not taken into account). Consider two chemically distinguishable amphiphilic molecules A-B and C-D. Self-assembly into A/C or B/D micelles consisting of a core-shell structure, with a core solely consisting of A or C units and a shell solely consisting of B or D units, will only occur if the A or C units are solvophobic and the B or D units are solvophilic. However, if all units (A and B, or C and D) are solvophobic, phase separation will occur on a macroscopic level and result in a macroscopically inhomogeneous two-phase system. Conversely, if all units (A and B, or C and D) are solvophilic, phase separation... 

Figure 4.30 Anisotropy Created by Macroscopic Inhomogeneity, (a) Uniform biomaterial, (b) with one spheroid of low conductivity, or (c) with high conductivity.

Macroscopic alignment, 3-3, 3-12 Macroscopic inhomogeneities, 16-2 Magnesium metal, 9-2 Magnetic susceptibility... 

A number of approaches to connect multiple-scale simulation in finite-element techniques have been published [31-34], They are able to describe macroscopically inhomogeneous strain (e.g., cracks)—even dynamic simulations have been performed [35]—but invariably require artificial constraints on the atomistic scale [36], Recently, an approach has been introduced that considers a system comprising an inclusion of arbitrary shape embedded in a continuous medium [20], The inclusion behavior is described in an atomistically detailed manner [37], whereas the continuum is modeled by a displacement-based Finite-Element method [38,39], The atomistic model provides state- and configuration-dependent material properties, inaccessible to continuum models, and the inclusion in the atomistic-continuum model acts as a magnifying glass into the molecular level of the material. 

The microscopic and macroscopic inhomogeneity of all solids causes an unequal distribution of the masses. If the solid accelerates, then different forces affect it depending on the mass distribution. If the body is to follow a certain movement, then resulting forces deviating by the direction of motion must be compensated by reactive forces. 

The harder thin layer then exhibits the patterned structures, whose size can be reduced while minimizing deformations. Another possibility is the use of a metal lamella covered with a thin structured PDMS ]17]. Because of the stiffness of the metal, roof collapse between the structures is reduced still, the metal is flexible enough to compensate macroscopic inhomogenities and to pattern curved surfaces ]11]. 

In actual fact, macroscopic inhomogeneities are always present. In the externally driven stirring process, perfect mixing is a mathematical limit rather than a reality. This is even more true in real world systems, where mixing is achieved by the system s own hydrodynamic flow. In short, contrary to what is usually assumed, a typical reaction takes place in a non-uniform environment. The question that arises naturally is, therefore, how the chemical system responds to this ubiquitous constraint. 

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