Coordination lattice

Since A Aw is the change in interaction energy per 1,2 pair, it can be expressed as some multiple of kT per pair or of RT per mole of pairs. It is also conventional to consolidate the lattice coordination number and x into a single parameter x, since z and x are not measured separately. With this change of notation Az Aw is replaced by its equivalent xRT, and Eq. (8.41) becomes... 

If the more accurate expression (5 ) for/t given in the footnote on p. 500 is used, additional terms containing the lattice coordination number z appear in the entropy of mixing expression. 

In practice, the scheme as explained above is not implemented. The consecutive generation of all possible chain conformations is a very expensive step. The reason for this is that there are of the order of ZN number of conformations, where Z is the lattice coordination number. A clever trick is to generate a subset of all possible conformations and to use this set in the SCF scheme. This approach is known in the literature as the single-chain mean-field theory, and has found many applications in surfactant and polymeric systems [96]. The important property of these calculations is that intramolecular excluded-volume correlations are rather accurately accounted for. The intermolecular excluded-volume correlations are of course treated on the mean-field level. The CPU time scales with the size of the set of conformations used. One of the obvious problems of this method is that one should make sure that the relevant conformations are included in the set. Typically, the set of conformations is very large, and, as a consequence, the method remains extremely CPU intensive. 

Host lattice Coordination Eu2+ ion Absorption bands Crystal field splitting Centre 54 level Ref. 

There have been many attempts to describe the process of mixing and solubility of polymer molecules in thermodynamic terms. By assuming that the sizes of polymer segments are similar to those of solvent molecules, Flory and Huggins derived an expression for the partial molar Gibbs free energy of dilution that included the dimensionless Flory Higgins interaction parameter X = ZAH/RT, where Z is the lattice coordination number. It is now... 

Both the components of E and the elements of the electric field gradient as given by Eqs. (8.30) and (8.32) are with respect to the reciprocal-lattice coordinate system. A transformation is required if the values in the direct-space coordinate systems are needed. To obtain the elements of the traceless V tensor, the quantity — (47t/3)pe(r) = — (47r/3K) F(H) exp ( — 27tiHT) must be subtracted from each of the diagonal elements VEU. 

The symbol z is used in the literature for the ratio i / c and should not be confused with the lattice coordination number a. 

Before discussing the dynamics of bond making/breaking at surfaces, it is helpful to consider generic PES topologies encountered in the various experimental systems and neglect the coupling to the lattice coordinates. 

As a result of the repulsion, the distance between the atoms in a single molecule AB will be larger than it is in the lattice. If the crystal belongs to the NaGl type, then for n we have to take a value of 10 the second term, 12/25 = 0-4, is small compared with the first. As a first approximation, B can be replaced by bcjrn where c is the lattice coordination number. 

These comparatively lipophilic ligands have been conceived as ion carriers for systems such as ion-selective electrodes, 29 therefore they do not have high stability constant values, but require fast complexation kinetics in order to achieve rapid equilibration. Nevertheless, it has been possible to recover crystalline species which often have present additional water molecules to help stabilize the crystal lattice. Coordinative participation of the carbonyl oxygen... 

Fig. 103. a. Principle of the Weissenborg moving-film goniometer, arranged for recording equatorial layer by normal-beam method, b. Determination of reciprocal lattice coordinates for. pots on equatorial layer. 

YV eissenberg photograph of the substance (Et3As)2(Hg012)2 (triclinic rotation axis, r zero level), with chart for reading off reciprocal lattice coordinates. (R. C. Evans and H. 8. Peiser)... 

We have also performed calculations for higher diffusion rates (D = 100) and for the triangular lattice (coordination number z — 6). The qualitative behaviour is in complete agreement with the calculation presented here. For the case S = 1 the increase of the diffusion rate or change of the lattice structure leads to a very small shift of the phase transition point y to higher values of j/n- This trend is clear because the reactivity of the H atoms is increased by the larger mobility. For S < 1 nearly no effect is observed which means that the system s behaviour is mainly dominated by the number of activated sites. The correlation of the adsorbed particles are rather small as expected for S < 1. 

For miscible blend phases, these parameters need to be described as a function of the blend composition. In a first approach to describe the behavior of the present PPE/PS and SAN/PMMA phases, these phases will be regarded as ideal, homogeneously mixed blends. It appears reasonable to assume that the heat capacity, the molar mass of the repeat unit, as well as the weight content of carbon dioxide scale linearly with the weight content of the respective blend phase. Moreover, a constant value of the lattice coordination number for PPE/PS and for SAN/PMMA can be anticipated. Thus, the glass transition temperature of the gas-saturated PPE/SAN/SBM blend can be predicted as a function of the blend composition (Fig. 17). Obviously, both the compatibilization by SBM triblock terpolymers and the plasticizing effect of the absorbed carbon dioxide help to reduce the difference in glass transition temperature between PPE and SAN. 

For the PPE/PS phase, the previously described Chow equation can be combined with the Couchman equation to estimate the Tg as a function of the blend composition. The results are highlighted in Fig. 25. For the prediction, the heat capacity and the molar mass of the repeat unit of the PPE/PS blends is regarded to scale linearly with the mass content of the blend partners, and a constant lattice coordination number of z = 2 is used [75]. While the addition of PS to PPE allows one to reduce continuously the Tg in presence of carbon dioxide, the plasticization effect is less pronounced, mainly driven by the decreasing solubility via addition of PS. 

Lattice coordination numbers (2) and the cell volumes (vjj) for both the pure components and mixture lattices are assumed to have the same value. The partition function for this ensemble can be formulated following Equation 2. It is assumed now that the partition function, far from the binary critical point can be approximated by its largest term. Since molecule segments and holes can distribute themselves non-randomly, the partition function must incorporate terms to account for this effect. The nonrandomness correction rjj... 

The vector that defines the crystallographic direction should be situated in such a way that it passes through the origin of the lattice coordinate system. 

Constants A l7 A0, and Ai are determined by the choice of the lattice in our calculations, A i =Ai =0.25 and A0 = 0.5. This choice corresponds to the simple cubic lattice, which is commonly used in the application of lattice SCFT to polymer-clay composites. To relate the lattice coordinate to "real" dimensions, we must also specify the value of the lattice size a here, we set a = 0.4 nm. (Later, in Section 3.5, when describing our results, we will automatically convert all distances from lattice units to nanometers.)... 

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