Two-Component Models

S.M. Mithieux, S.G. Wise, M.J. Raftery, B. Starcher, A.S. Weiss, A model two-component system for studying the architecture of elastin assembly in vitro, J. Struct. Biol. 149 (2005) 282—289. 

Binary phase diagrams are very important for ceramics. The two most important cases for ceramics are the combination of a metal plus oxygen and the combination of two oxides. A model two-component system is shown in Figure 8.9 where we are now using the third dimension to display the data. 

Fig. 5.3. Coeiistence curve and crhical point for the model two-component lattice gas in the Pa tb-fdane. The lines are tielines, connecting coexisting phases, and the dot is the critical point.

Tuiri, S., Alborghetti, L., and Levi, M. 2008. Formulation and properties of a model two-component nanocomposite coating from organophUic nanoclays. Journal of Polymers Research 15 365—372. 

Even in the case of toluene-polystyrene solutions, xi2 is in the range 0.3-0.4. Thus a positive enthalpy of mixing tends to oppose the polymer dissolution in a solvent (AG > 0). In the Flory-Huggins model, two components can mix with each other only if the positive enthalpy term is compensated by the entropy term (-TA.S), which is always negative. 

A binary alloy of two components A and B with nearest-neighbour interactions respectively, is also isomorphic with the Ising model. This is easily seen on associating spin up with atom A and spin down with atom B. There are no vacant sites, and the occupation numbers of the site are defined by... 

If the scalar order parameter of the Ising model is replaced by a two-component vector n = 2), the XY model results. All important example that satisfies this model is the 3-transition in helium, from superfiuid helium-II... 

Though the solution procedure sounds straightforward, if tedious, practice difficulty is encountered immediately because of the implicit nature of the available flux models. As we saw in Chapter 5 even the si lest of these, the dusty gas model, has solutions which are too cumbersc to be written down for more than three components, while the ternary sol tion itself is already very complicated. It is only for binary mixtures therefore, that the explicit formulation and solution of equations (11. Is practicable. In systems with more than two components, we rely on... 

The Lennard-Jones potential is characterised by an attractive part that varies as r ° and a repulsive part that varies as These two components are drawn in Figure 4.35. The r ° variation is of course the same power-law relationship foimd for the leading term in theoretical treatments of the dispersion energy such as the Drude model. There are no... 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

In practice, such a fractionation experiment could be carried out by either lowering the temperature or adding a poor solvent. In either case good temperature control during the experiment is important. Note that the addition of a poor solvent converts the system to one containing three components, so it is apparent that the two-component Flory-Huggins model is at best only qualitatively descriptive of the situation. A more accurate description would require a... 

Markham andBenton. This model (34) is known as the extended Langmuir isotherm equation for two components, i and j. 

A method of resolution that makes a very few a priori assumptions is based on principal components analysis. The various forms of this approach are based on the self-modeling curve resolution developed in 1971 (55). The method requites a data matrix comprised of spectroscopic scans obtained from a two-component system in which the concentrations of the components are varying over the sample set. Such a data matrix could be obtained, for example, from a chromatographic analysis where spectroscopic scans are obtained at several points in time as an overlapped peak elutes from the column. 

Based on Hquid—Hquid equiHbrium principles, a general model of octanol—water partitioning is possible if accurate activity coefficients can be determined. First, phase equiHbrium relationships based on activity coefficients permit Hquid—Hquid equiHbrium calculations for the biaary octanol—water system. Because the two components are almost immiscible ia each other, two phases form an octanol-rich phase containing dissolved water, and a water-rich phase containing dissolved octanol. 

To illustrate the development of a physical model, a simplified treatment of the reactor, shown in Fig. 8-2 is used. It is assumed that the reac tor is operating isothermaUy and that the inlet and exit volumetric flows and densities are the same. There are two components, A and B, in the reactor, and a single first order reaction of A B takes place. The inlet concentration of A, which we shall call Cj, varies with time. A dynamic mass balance for the concentration of A (c ) can be written as follows ... 

For field-oriented controls, a mathematical model of the machine is developed in terms of rotating field to represent its operating parameters such as /V 4, 7, and 0 and all parameters that can inlluence the performance of the machine. The actual operating quantities arc then computed in terms of rotating field and corrected to the required level through open- or closed-loop control schemes to achieve very precise speed control. To make the model similar to that lor a d.c. machine, equation (6.2) is further resolved into two components, one direct axis and the other quadrature axis, as di.sciis.sed later. Now it is possible to monitor and vary these components individually, as with a d.c. machine. With this phasor control we can now achieve a high dynamic performance and accuracy of speed control in an a.c. machine, similar to a separately excited d.c. machine. A d.c. machine provides extremely accurate speed control due to the independent controls of its field and armature currents. 

To illustrate this theory, we consider a one-component fluid with the interaction between the same species given by Eq. (36). Obviously, the model differs from that described in Sec. (II Bl). In particular, the geometrical constraints, which determine the type of association products in the case of a two-component model, are no longer valid. If we restrict ourselves to the case L < cr/2, only dimers and -mers built up of rigid, regular polygons are possible. 

The model in question may serve as a benchmark, or as a reference system, for several extensions. In particular, the adsorption of simple fluids in cross-hnked and branched-chain molecules may be studied as the next logical step. Adsorption of a two-component fluid mixture in a matrix of chain molecules made of two types of monomer with different fluid-matrix affinity may exhibit interesting features. 

The electrolyte solution is modelled as a two-component, electroneutral system of point ions with charges ez, = ezL = ez. The density of the fluid is (p+ = pL = p /2). The fluid-fluid and fluid-matrix Coulomb interactions are... 

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