Site-number density

Let us analyze these results one step further and ask about a quantitative measure of the Kirkendall effect. This effect had been detected by placing inert markers in the interdiffusion zone. Thus, the lattice shift was believed to be observable for an external observer. If we assume that Vm does not depend on concentration and local defect equilibrium is established, the lattice site number density remains constant during interdiffusion. Let us designate rv as the production (annihilation) rate of the vacancies. We can derive from cA+cB+cv = l/Vm and jA +/ B +./v = 0 that... 

In the previous p per, we have given just a formalism of STCF for the site number density representation, which has been outlin above. In what follows, a very preliminary numerical results of STCF for a Cl—>G process is presented based on the theories described in the previous sections. The calculation has beoi carried out for a variety of polar liquid as solvent including methyl chloride (MeCl), acetonitrile (MeCN), methanol (MeOH) and wato-. Methyl chloride and acetcHiitrile represent a class of simple rqnotic dipolar liquids while water does those liquids which feature the extensive hydrogen-bond network. The alcohol shows characteristics in between those two classes of liquids. Ihe calculation is performed at the room temperature (298 K) for all solvents except MeCl. For MeO, its liquid temperature (249 K 1 atm) is chosen. The Edward-McDonald (EM), SPC and TII models are os i fcr MeCN, water and MeOH, respectively, while the parametos detramined by Jorgensen et al. is employed for MeCl. Those models use the same functional form for the intermolecular site-site interaction, namely... 

Consider first linear polymers composed of identical spherical sites that interact intermolecularly via a pair decomposable site-site hard-core potential of diameter d. The dimensionless reduced fluid density is where p , = Np is the site number density. 

Figure 6. Predicted interchain radial distribution function for a hard-core polyethylene melt described by three single-chain models atomistic RIS at 430 K, overlapping (lid = 0.5) SFC model with appropriately chosen aspect ratio and site number density (see text), and the Gaussian thread model (shifted horizontally to align the hard core diameter with the value of rld = l).

To illustrate how the effect of the adsorption on the modulus of the filled gel may be modelled we consider the interaction of the same HEUR polymer as described above but in this case filled with poly(ethylmetha-crylate) latex particles. In this case the particle surface is not so hydrophobic but adsorption of the poly (ethylene oxide) backbone is possible. Note that if a terminal hydrophobe of a chain is detached from a micellar cluster and is adsorbed onto the surface, there is no net change in the number of network links and hence the only change in modulus would be due to the volume fraction of the filler. It is only if the backbone is adsorbed that an increase in the number density of network links is produced. As the particles are relatively large compared to the chain dimensions, each adsorption site leads to one additional link. The situation is shown schematically in Figure 2.13. If the number density of additional network links is JVL, we may now write the relative modulus Gr — G/Gf as... 

The principal effect of larger L is to introduce into the regions of high density (traffic jams) an oscillation in the density rij as a function of lattice site number j. As would be expected, these solutions are progressively damped in the direction of decreasing j. 

Let us consider the atomic processes that take place during chemical diffusion on three neighboring lattice planes. As in Figure 4-10, we label these planes p-, p, and p+ 1. n (= np(i)) designates the number density of species i on plane p such that = n°. The total number of nearest neighbor sites of ip on plane p— 1 (p+1) is... 

With respect to the rate of nonrandom nucleation, the essence of the rate equation (6.7) is unchanged. However, we may have a spectrum of preferential nucleation sites p, with number density g (p), Therefore, Rn becomes... 

Fig. 10. Typical adsorbent surfaces can be considered to have very high binding site ( ligand ) densities, resulting in multivalent interactions with adsorbed protein. If the multivalent interactions are of sufficient number and energy, the adsorptive interactions is irreversible ...

The macroscopic polarization of the phase is given by equations 1 and 2, where Di is the number density of the ith conformation, jlj is the component of the molecular dipole normal to the tilt plane when the ith conformation of the molecule is oriented in the rotational minimum in the binding site, ROFj is the "rotational orientation factor", a number from zero to one reflecting the degree of rotational order for the ith conformation, and e is a complex and unmeasured dielectric constant of the medium (local field correction). 

Fig. 12 Nucleation density of foamed PPE/SAN blends vs number of theoretically available nucleation sites (=particle density of the PPE phase). The dotted lines represent the theoretical nucleation density at different phase sizes of PPE (reprinted from [47])...

The protein is now represented as, B, for bivalent. The units of Kai are in M-1, while Ka2 has units of dm2/moles (1/number density).85 The total surface site density [S]s can be expressed as ... 

Figure 9.4 Schematic illustration of site-selective deposition of CoTMPyP on gold NP surfaces and tuning the quantities of the as-anchored electiocatalysts by number density of gold NPs.65 (Reprinted with permission from W. Cheng et al., J. Phys. Chem. B 2004,108, 19146-19154. Copyright 2004 American Chemical Society.)...

Initially, the elasticity of concentrated polymer systems was ascribed to the existence of a network in the system formed by long macromolecules with junction sites (Ferry 1980). The sites were assumed to exist for an appreciable time, so that, for observable times which are less than the lifetime of the site, the entangled system appears to be elastic. Equation (1.44) was used to estimate the number density of sites in the system. The number of entanglements for a single macromolecule Z = M/Me can be calculated according to the modified formula... 

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