Ideal two-phase model

Figure 5.8 Scattering length density profile in an ideal two-phase model.

The most important theoretical result for the ideal two-phase model is the prediction that I(q) should decrease as q A for large q, and moreover that the proportionality constant should be related to the total area S of the boundaries between the two phases in the scattering volume. In other words, as q - oo,... 

A real material never completely fulfills the idealization stipulated in the ideal two-phase model. Therefore certain modifications have to be introduced either to the theoretical expressions derived or to the experimentally observed intensity data before the two can be compared. We discuss separately the effects arising from the presence of heterogeneities within each phase and from the diffuseness of the interface boundaries. 

The main theoretical results derived from the ideal two-phase model, i.e., Equation (5.70) relating the invariant Q to the phase volumes and the Porod law (5.71), are no longer valid and need be modified when in the two-phase system the phase boundaries are diffuse. To see the modifications necessary to these theoretical results, we represent by p r) the scattering length density distribution in a two-phase material with diffuse boundaries and by p d(r) the density distribution in the (hypothetical) system in which all the diffuse boundaries in the above have been replaced by sharp boundaries. The two are then related to each other25 by a convolution product... 

Structure with Variable Lamella Thickness The sharp peaks predicted by Equation (5.135) are broadened and reduced in height if various types of imperfections degrade the structure from the ideal two-phase model envisioned above. Most of such imperfections find their counterparts in three-dimensional crystals, and the methods of analysis to account for such imperfections, developed in Section 3.4, can be applied to the small-angle scattering equally well and need not be elaborated here again. There is, however, one type of imperfection... 

Determination of Crystallinity WAXD has been routinely used to determine the degree of crystallinity in semicrystalline polymers. The method usually involves one assumption, that is, the system is considered as an ideal two-phase model containing only crystalUne... 

Volume and mass-based expressions for the degree of crystallinity are easily derived from the experimentally measured density (p) of a semi-crystalline polymer. The method is based on an ideal crystalline and liquid-like two-phase model and assumes additivity of the volume corresponding to each phase... 

Figure 8.21 shows functions of the distorted topologies that are not pointed at the origin, and j (0) < 1. The reason is that the presented model is not an ideal two-phase system, because it considers smooth transitions of the electron density between the crystalline and the amorphous layers. 

If the structural entities are lamellae, Eq. (8.80) describes an ensemble of perfectly oriented but uncorrelated layers. Inversion of the Lorentz correction yields the scattering curve of the isotropic material I (5) = I (s) / (2ns2). On the other hand, a scattering pattern of highly oriented lamellae or cylinders is readily converted into the ID scattering intensity /, (53) by ID projection onto the fiber direction (p. 136, Eq. (8.56)). The model for the ID intensity, Eq. (8.80), has three parameters Ap, dc, and <7C. For the nonlinear regression it is important to transform to a parameter set with little parameter-parameter correlation Ap, dc, and oc/dc. When applied to raw scattering data, additionally the deviation of the real from the ideal two-phase system must be considered in an extended model function (cf. p. 124). 

Ruland and Smarsly [84] study silica/organic nanocomposite films and elucidate their lamellar nanostructure. Figure 8.47 demonstrates the model fit and the components of the model. The parameters hi and az (inside H ) account for deviations from the ideal two-phase system. Asr is the absorption factor for the experiment carried out in SRSAXS geometry. In the raw data an upturn at. s o is clearly visible. This is no structural feature. Instead, the absorption factor is changing from full to partial illumination of the sample. For materials with much stronger lattice distortions one would mainly observe the Porod law, instead - and observe a sharp bend - which are no structural feature, either. 

Analytical solutions for x and y as functions of the bed-length, z, and time, t, are available [45,52], The expressions are a useful extension of two-phase model applied to plug-flow. These two models are appropriate in describing the extraction of crushed or broken seeds to recover the seed oil, either in shallow beds or in plug flow. As shown by Sovova [52], applying the plug-flow model requires corrections for non-ideal residence-time distribution (non-plug flow) of the fluid in contact with the solid. 

The crystalline and noncrystalline phases in polyamide fibers do not appear to be governed by what may be defined as thermodynamie equilibria, nor is there evidenee for definite boundaries between a phase, characterized by a simple or complex state of order and an essentially amorphous phase. It is therefore quite obvious that the morphological structure of nylons cannot be described adequately in terms of a simple two-phase model according to which ideally ordered crystallites exist together with eompletely amorphous domains. This model constitutes merely one of the two limiting cases the other is that of a paracrystal according to which all deviations from the ideal crystal are ascribed to defects and distortions of the crystal lattice [275-277]. 

Fig. 5.27 Schematic diagrams of the idealized two-sphrae model showing microstiuctural characteristics of a solid-state sintering and b liquid-phase sintering. Reproduced with permission from [1]. Copyright 2003, CRC Press...

Taking into account the given pecularities one may suggest a two phase model for description of irreversible development of cavitation zone which occurs experimentally. The SW spreading over real liquid is calculated as an approximation of ideal compressed liquid due to the low initial concentration of gaseous phase. When in the RW the pressure of cavitation threshold Pj, [1] is achieved its instantaneous relaxation occurs up to the magnitude (the pressure of saturation vapor). 

Figure 10.1 Sketch of an idealized two-sphere model comparing the microstructural aspects of (a) solid-state sintering with (b) liquid-phase sintering.

The phenomenon of phase transitions in two dimensions is of great fundamental interest and has therefore drawn a considerable amount of attention.i 2 insoluble monomolecular layers at a water-air interface provide a quite ideal two-dimensional model system with an isotropic substrate and an easily controllable density of molecules. At low densities they often exhibit a two-dimensional gas behavior,3 whereas at higher densities transitions to liquid and solid states can be found. In many systems, the liquid phase is further divided into the so-called liquid-expanded (LE) and liquid-condensed (LC) phases.4 Though observed and intensively studied, the nature of the LE-LC phase transition is still controversial. 

Early bubble models were too much idealized and therefore in conflict with observed facts, especially for large scale reactors. Also the predictive power of these models is limited. In recent models empirical relations are introduced for the three main factors determining the exchange between bubbles and dense phase bubble size, rising velocity and mass transfer coefficients. As the empirical relations have a limited range of validity, these models can often not replace general two phase models applied in combination with experimentally observed pareuneters. 

Averaging the fuzzy interface into a ID density profile transforms the sharp discontinuous density step of the ideal two-phase system into a smooth continuous sigmoidal shap>e. It was shown by Ruland that this sigmoidal density transition can be modeled in real space by convoluting the sharp density step with a... 

Ideal adhesion simply means the adhesion expected under one or another model situation of uniform materials having intimate contact over a well-defined area. In these cases, the important quantity is the work of adhesion wab between two phases, which is given by... 

The mathematical model describing the two-phase dynamic system consists of modeling of the flow and description of its boundary conditions. The description of the flow is based on the conservation equations as well as constitutive laws. The latter define the properties of the system with a certain degree of idealization, simplification, or empiricism, such as equation of state, steam table, friction, and heat transfer correlations (see Sec. 3.4). A typical set of six conservation equations is discussed by Boure (1975), together with the number and nature of the necessary constitutive laws. With only a few general assumptions, these equations can be written, for a one-dimensional (z) flow of constant cross section, without injection or suction at the wall, as follows. 

The quality of the mean-field approximation can be tested in simulations of the same lattice model [13]. Ideally, direct free-energy calculations of the liquid and solid phases would allow us to locate the point where the two phases coexist. However, in the present studies we followed a less accurate, but simpler approach we observed the onset of freezing in a simulation where the system was slowly cooled. To diminish the effect of supercooling at the freezing point, we introduced a terraced substrate into the system to act as a crystallization seed [14]. We verified that this seed had little effect on the phase coexistence temperature. For details, see Sect. A.3. At freezing, we have... 

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