Isotropic, definition

In Chapter III, surface free energy and surface stress were treated as equivalent, and both were discussed in terms of the energy to form unit additional surface. It is now desirable to consider an independent, more mechanical definition of surface stress. If a surface is cut by a plane normal to it, then, in order that the atoms on either side of the cut remain in equilibrium, it will be necessary to apply some external force to them. The total such force per unit length is the surface stress, and half the sum of the two surface stresses along mutually perpendicular cuts is equal to the surface tension. (Similarly, one-third of the sum of the three principal stresses in the body of a liquid is equal to its hydrostatic pressure.) In the case of a liquid or isotropic solid the two surface stresses are equal, but for a nonisotropic solid or crystal, this will not be true. In such a case the partial surface stresses or stretching tensions may be denoted as Ti and T2-... 

The traditional view of emulsion stability (1,2) was concerned with systems of two isotropic, Newtonian Hquids of which one is dispersed in the other in the form of spherical droplets. The stabilization of such a system was achieved by adsorbed amphiphiles, which modify interfacial properties and to some extent the colloidal forces across a thin Hquid film, after the hydrodynamic conditions of the latter had been taken into consideration. However, a large number of emulsions, in fact, contain more than two phases. The importance of the third phase was recognized early (3) and the lUPAC definition of an emulsion included a third phase (4). With this relation in mind, this article deals with two-phase emulsions as an introduction. These systems are useful in discussing the details of formation and destabilization, because of their relative simplicity. The subsequent treatment focuses on three-phase emulsions, outlining three special cases. The presence of the third phase is shown in order to monitor the properties of the emulsion in a significant manner. 

The term quasi-isotropic iaminate is used to describe laminates that have isotropic extensionai stiffnesses (the same in all directions in the plane of the laminate). As background to the definition, recall that the term isotropy is a material property whereas laminate stiffnesses are a function of both material properties and geometry. Note also that the prefix quasi means in a sense or manner. Thus, a quasi-isotropic laminate must mean a laminate that, in some sense, appears isotropic, but is not actually isotropic in all senses. In this case, a quasi-isotropic... 

If we now transfer our two interacting particles from the vacuum (whose dielectric constant is unity by definition) to a hypothetical continuous isotropic medium of dielectric constant e > 1, the electrostatic attractive forces will be attenuated because of the medium s capability of separating charge. Quantitative theories of this effect tend to be approximate, in part because the medium is not a structureless continuum and also because the bulk dielectric constant may be an inappropriate measure on the molecular scale. Eurther discussion of the influence of dielectric constant is given in Section 8.3. 

Thus, if we wish to compare the eigenvectors to one another, we can divide each one by equation [57] to normalize them. Malinowski named these normalized eigenvectors reduced eigenvectors, or REV". Figure 52 also contains a plot of the REV" for this isotropic data. We can see that they are all roughly equal to one another. If there had been actual information present along with the noise, the information content could not, itself, be isotropically distributed. (If the information were isotropically distributed, it would be, by definition, noise.) Thus, the information would be preferentially captured by the earliest... 

Given these differences between rigid and flexible conduit, let us examine the differences between steel and RTR pipe, both of which are, of course, flexible conduits. First, steel pipe is by definition constructed from a material, steel, that for our purposes is a homogeneous isotropic substance. Therefore, steel pipe can be considered to have the same material properties in all directions that is, it is equally strong in both the hoop and longitudinal directions [Fig. 4-2(b)]. 

The physical meaning of and f L.., is obvious they govern the relaxation of rotational energy and angular momentum, respectively. The former is also an operator of the spectral exchange between the components of the isotropic Raman Q-branch. So, equality (7.94a) holds, as the probability conservation law. In contrast, the second one, Eq. (7.94b), is wrong, because, after substitution into the definition of the angular momentum correlation time... 

Neal and Nader [260] considered diffusion in homogeneous isotropic medium composed of randomly placed impermeable spherical particles. They solved steady-state diffusion problems in a unit cell consisting of a spherical particle placed in a concentric shell and the exterior of the unit cell modeled as a homogeneous media characterized by one parameter, the porosity. By equating the fluxes in the unit cell and at the exterior and applying the definition of porosity, they obtained... 

Crystal data and parameters of the data collection (at -173°, 50 < 20 < 450) are shown in Table I. A data set collected on a parallelopiped of dimensions 0.09 x 0.18 x 0.55 mm yielded the molecular structure with little difficulty using direct methods and Fourier techniques. Full matrix refinement using isotropic thermal parameters converged to R = 0.I7. Attempts to use anisotropic thermal parameters, both with and without an absorption correction, yielded non-positive-definite thermal parameters for over half of the atoms and the residual remained at ca. 0.15. 

Its first invariant A] is equal to zero by definition. The second and third invariants of this tensor arc A — atJaJt and A3 — atjajkakt, respectively. The range of physically allowed values of A2 and A3 is bounded and represented by the so-called Lumley triangle in the (A3, A2) plane (Lumley, 1978). The distanced = (Ay + Af) from the isotropic state, i.e., from the origin (A2 — 0, A3 — 0), is a measure of the degree of anisotropy. See also Escudie and Line (2006) for a more extensive discussion as to how to quantify and visualize how different from isotropic turbulence a stirred vessel is. 

The purpose of this chapter is to introduce the effect of surfaces and interfaces on the thermodynamics of materials. While interface is a general term used for solid-solid, solid-liquid, liquid-liquid, solid-gas and liquid-gas boundaries, surface is the term normally used for the two latter types of phase boundary. The thermodynamic theory of interfaces between isotropic phases were first formulated by Gibbs [1], The treatment of such systems is based on the definition of an isotropic surface tension, cr, which is an excess surface stress per unit surface area. The Gibbs surface model for fluid surfaces is presented in Section 6.1 along with the derivation of the equilibrium conditions for curved interfaces, the Laplace equation. 

This is the definition of the surface tension according to the Gibbs surface model [1], According to this definition, the surface tension is related to an interface, which behaves mechanically as a membrane stretched uniformly and isotropically by a force which is the same at all points and in all directions. The surface tension is given in J m-2. It should be noted that the volumes of both phases involved are defined by the Gibbs dividing surface X that is located at the position which makes the contribution from the curvatures negligible. 

The definition of ji-j has been extended to include terms that are quadratic in btj (Wouters et al. 1996). Strictly speaking, die first equality is only valid at high Reynolds numbers where the scalar- and viscous-dissipation terms are uncorrelated (i.e., locally isotropic). 

In order to extract some more information from the csa contribution to relaxation times, the next step is to switch to a molecular frame (x,y,z) where the shielding tensor is diagonal (x, y, z is called the Principal Axis System i.e., PAS). Owing to the properties reported in (44), the relevant calculations include the transformation of gzz into g x, yy, and g z involving, for the calculation of spectral densities, the correlation function of squares of trigonometric functions such as cos20(t)cos20(O) (see the previous section and more importantly Eq. (29) for the definition of the normalized spectral density J((d)). They yield for an isotropic reorientation (the molecule is supposed to behave as a sphere)... 

Navard and Haudin studied the thermal behavior of HPC mesophases (87.88) as did Werbowyj and Gray (2), Seurin et al. (Sp and, as noted above, Conio et al. (43). In summary, HjPC in H2O exhibits a unique phase behavior characterized by reversible transitions at constant temperatures above 40 C and at constant compositions when the HPC concentration is above ca. 40%. A definitive paper has been recently published by Fortin and Charlet ( who studied the phase-separation temperatures for aqueous solutions of HPC using carefully fractionated HPC samples. They showed the polymer-solvent interaction differs in tiie cholesteric phase (ordered molecular arrangement) from that in the isotropic phase (random molecular arrangement). 

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