One component approximation

Similar calculations were also performed in the strong segregation limit. In this case, the two-phase and disordered homogeneous regions were found to be smaller, and the phase boundaries were more vertical (see Fig. 6.49) (Shi and Noolandi 1995). This phase diagram was interpreted on the basis of interfacial curvature. If the diblocks are completely segregated, the phase boundaries are determined by the total composition / (=

phase boundaries are parallel to this line (dashed line sloping to the left in Fig. 6.49) ( one-component approximation ). This explains the approximately parallel... 

It is instructive to break the derivation procedure into several steps. As a first step, we adopt a scalar, one component approximation and write the UPPE in the following form ... 

Although the full four-component treatment with the Dirac Hamiltonian is ideal, the computation of four-component wave functions is expensive. Thus, since small components have little importance in most chemically interesting problems, various two- or one-component approximations to the Dirac Hamiltonian have been proposed. From Eq. 10.32, the Schrodinger-Pauli equation composed of only the large component is obtained as... 

The value of s.s.d. is five times more in case of one component approximation than in case of two. 

If (pi were a real-valued function, we would have to have two different radial parts for the two components of the 2-spinor (pi in order to obtain a nodeless radial density distribution as in the four-component case. In a one-component approximation, which neglects all spin-dependent terms and which therefore does not resemble the four-component density, additional nodes may occur as in the Schrbdinger case. 

The evaluation of DKH property operators up to any predefined order in the external potential requires a computational scheme, which cannot rely on a direct noniterative numerical method since odd operators have no representation in a truly two-component formalism (which includes the one-component approximation). Only the product of two odd operators may be evaluated. 

Figure 5.17 SCF phase diagrams of binary (AB) /(AB)p copolymer blends based on the one-component approximation of Matsen and Bates [199]. In (a), the thermodynamic incompatibility (xN) is provided as a function of the volume fraction of the p copolymer (< ) for copolymers with f = -fi = 0.25 (the subscripts 1 and 2 refer to copolymers a and P, respectively). In (b), the effect of copolymer compositions (J and fi) on phase stability at xN = 20 and = 0.5 is shown. (Compiled from Matsen, M. W. and Bates, F. S. Macromolecules 28, 7298, 1995, and reprinted with permission. Copyright (1995) American Chemical Society.)...

When, for a one-component system, one of the two phases in equilibrium is a sufficiently dilute gas, i.e. is at a pressure well below 1 atm, one can obtain a very usefiil approximate equation from equation (A2.1.52). The molar volume of the gas is at least two orders of magnitude larger than that of the liquid or solid, and is very nearly an ideal gas. Then one can write... 

We now turn to a mean-field description of these models, which in the language of the binary alloy is the Bragg-Williams approximation and is equivalent to the Ciirie-Weiss approxunation for the Ising model. Botli these approximations are closely related to the van der Waals description of a one-component fluid, and lead to the same classical critical exponents a = 0, (3 = 1/2, 8 = 3 and y = 1. 

Although the previous paragraphs hint at the serious failure of the van der Waals equation to fit the shape of the coexistence curve or the heat capacity, failures to be discussed explicitly in later sections, it is important to recognize that many of tlie other predictions of analytic theories are reasonably accurate. For example, analytic equations of state, even ones as approximate as that of van der Waals, yield reasonable values (or at least ball park estmiates ) of the critical constants p, T, and V. Moreover, in two-component systems... 

Clearly, Eq. (E.12) shows that to a first approximation the elecbonic energy varies linearly with displacements in p, increasing for one component state while decreasing for the other. Thus, the potential minimum cannot be at p = 0. This is the statement of the Jahn-Teller theorem for a X3 molecule belonging to the D3 , point gioup. 

The density functional approach of Refs. 91, 92 introduces a correction to the wall-particle direct correlation function resulting from the HNCl approximation (see Eqs. (32)-(34)). A correction to Eq. (34) reads (we drop the species label because the model is one-component)... 

Applying equation (6.59) presents problems in that B 2 is often not known. When components 1 and 2 are very similar in their molecular properties one can approximate Bi2 as the mean of Bu and B22 That is. 

The explicit form of the functional Fh is of course unknown and in practical applications has to be approximated. In order to facilitate the aeation of these approximations one decomposes Fh into a sum of other functionals that focuses all the unknowns into one component, the exchange-correlation functional, Fxo... 

The Chapman-Enskog theory of flow In a one-component fluid yields the following approximation to the momentum balance equation (Jil). 

The results of the simple DHH theory outlined here are shown compared with DH results and corresponding Monte Carlo results in Figs. 10-12. Clearly, the major error of the DH theory has been accounted for. The OCP model is greatly idealized but the same hole correction method can be applied to more realistic electrolyte models. In a series of articles the DHH theory has been applied to a one-component plasma composed of charged hard spheres [23], to local correlation correction of the screening of macroions by counterions [24], and to the generation of correlated free energy density functionals for electrolyte solutions [25,26]. The extensive results obtained bear out the hopeful view of the DHH approximation provided by the OCP results shown here. It is noteworthy that in... 

The maximum outlet concentrations from each mixer are given in Table 6.8. The maximum outlet concentrations given in the table stem from approximate process data. It is important to notice that there is maximum outlet concentration defined for only one component from each mixer. This is since each mixer will at any point only contain residue of a specific product. The inlet and outlet concentrations of the other products will be equal. The outlet concentration from each mixer varies due to the varying nature of the products. 

The acceptance criteria for the Gibbs ensemble were originally derived from fluctuation theory [17]. An approximation was implicitly made in the derivation that resulted in a difference in the acceptance criterion for particle transfers proportional to 1/N relative to the exact expressions given subsequently [18]. A full development of the statistical mechanics of the ensemble was given by Smit et al. [19] and Smit and Frenkel [20], which we follow here. A one-component system at constant temperature T, total volume V, and total number of particles N is divided into two regions, with volumes Vj and Vu = V - V, and number of particles Aq and Nu = N - N. The partition function, Q NVt is... 

P-type delayed fluorescence is so called because it was first observed in pyrene. The fluorescence emission from a number of aromatic hydrocarbons shows two components with identical emission spectra. One component decays at the rate of normal fluorescence and the other has a lifetime approximately half that of phosphorescence. The implication of triplet species in the mechanism is given by the fact that the delayed emission can be induced by triplet sensitisers. The accepted mechanism is ... 

In case (2), isotope filtering will (ideally ) remove all protons bound to the chosen hetero-nuclear isotope(s). In the case of 13C and/or 15N filtering (requiring 13C and/or 15N labeling of one component), there will be approximately the following selectivity for the unlabeled component ... 

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