Effective terms density

It is possible to calculate a number of different kinds of "effective" crosslink densities. Bauer et al have used a quantity they termed the "elastically effective crosslink density " (Cel) correlate cure with solvent resistance and other physical properties of coatings (7-10). The correlation was basically empirical. Formally, the is a calculation of the number of functional groups attached to the infinite network for which there are at least two other paths out to the network on the given polymer or crosslinker. Thus, chains with only one or two paths to the infinite network are excluded. The following expression can be written for... 

On the other hand, if the first and third terms on the right-hand side of Eq. (55) are small (which is true for highly viscous liquids), the density still appears in the equation and its increase results in a decrease in the bubble volume. This situation is applicable to the experiments of Davidson and Schuler (D8). Since no data are reported which pertain to the effect of density alone, a comparison is not possible. 

Stine et al. proposed a simple analytic model for patterned feature removal incorporating an effective density determination step [54]. Figure 10 defines terms used in the model, which reformulates Preston s equation [Eq. (2)] as a function of blanket rate K and effective pattern density... 

The paper of Gordon describes a model for diffusion-controlled reaction based on the "hole concept in liquids of Jost (Ref 1, p 459). in which the activation energy for diffusion is equated simply to pV. The marked effect of density, therefore, results from the strong dependence of pressure on density (p varying about as the density cubed) and the appearance of this factor in an exponential term. On this basis, Gordon derived an approximate expression for dependence of detonation velocity D on explosive density pQ. This equation is given on pp 833 and 836 of Gordon s paper. From this expression the critical diameter dc for composite explosives is related to an exponential function of density by ... 

As long as the Mach number is small—meaning the velocities are small compared to the sound speed—it is reasonable to assume that the incompressible continuity equation is a good approximation for isothermal, single-species flow. That is, velocity variations have little effect on density variations. As a result the simplifications associated with V-V 0 can be enjoyed. In practical terms, most consider that flows with Ma < 0.3 can be assumed to be gas-dynamically incompressible. 

Drijvers and Goethals 52) have reported that excess sulphide functions (monomer and polymer) and diethyl ether have no detectable effect on the dissociation of two sulphonium tetrafluoroborate salts in methylene chloride and nitrobenzene, when present in similar proportions to those in corresponding polymerisation reactions. In contrast to this, however, Jones and Plesch 51) have shown that the dissociation constant of triethyloxonium hexafluorophos-phate in methylene chloride at 0°C increases by a factor of - 2 when small quantities of tetrahydrofuran are added. The latter molecule has a lower dielectric constant than methylene chloride and might therefore be expected to reduce dissociation. These workers have interpreted their results in terms of specific solvation of the cation by ether molecules, with subsequent reduction in the effective charge density of the positive ion and hence in the coulombic force favouring ion pairing, e.g. 

In reality, however, the synergistic effects of density, temperature, and composition on retention (In k ) will necessitate the inclusion of one or more cross terms in equation 5, at least for some sets of experimental conditions. Since the precise relationship depends on the parent fluid, the modifier, and the location of the experimental conditions relative to the critical point, it is often difficult to make intuitive predictions about which cross-terms, if any, should be included. Such predictions may become easier as the knowledge base of SFC continues to grow. 

Catalyst manufacturers also define an effective solid density, determined by displacement of a given liquid (wster, ethanol). Values obtained with different liquids may vary substantially, depending on the extent to which the molecules of the liquid are able to penetrate into the pores of the catalyst. The term piece density is sometimes used when the measurement is performed with a liquid which does not substantially penetrate into the pores. 

The differential pressure drop for the completely fluidized bed can be written in terms of the effective solids density (p, — p ) and voidage e ... 

Eq. 94 can be used to analyze the effect of errors in the approximate quantities (functionals and potentials) on the interaction energy in weakly interacting complexes. The first six terms represent the zeroth-order contribution to the error i.e. do not account for the complexation induced change of the electron densities of the isolated monomers. Such terms can be expected to be dominant for weak complexes in the absence of strong electrostatic and polarization effects. Terms from the seventh to ninth represent the first-order contribution. Their relative significance increases... 

Hydrogen bond cooperativity is treated in the work of Sear and Jackson, by including in the association energy a three-body term which either increases or decreases the strength of the bonds in chains of three or more molecules. Their model (in essence, an association model) is able to account for the effect of density on the degree of association and, thus, it can describe vapor-liquid equilibria. 

Long-term effects of sublethal effects Often the sublethal effects of a PPP are described well on the level of individuals, whereas the likely impact on population level is less well known. In some cases, the effects of density dependence, for example, will make the effects much less pronounced on the population level than on the individual level. In other cases, the sublethal effects may have pronounced effects on population density and structure. Ecological models can be used to put the effects observed on the individual level into the more ecologically relevant population level. 

We now discuss the density of states term in Eq. (2). Neglecting again collective effects, the density of states contribution to f can be written as ... 

The effect of density of WPC on its durability in terms of oxidative depolymerization of its plastic matrix is described in detail in Chapter 15. Briefly, porosity in WPC, which is directly related to the decrease of density (specific gravity) of the material, provides a chemically reactive area for oxygen. Oxygen flows into pores and attacks WPC from inside, particularly at elevated temperatures. An increase in temperature by every 10°C accelerates the oxidative destruction of WPC by about... 

Figure 4- - Transverse average of the total effective potential of the Kohn-Sham equation Weff and its metal electrostatic and solvent effective terms Vee and Vsoiv versus the -coordinate of the supercell (part a). Transverse average of the metal valence electron density pe (part b). The metal slab in contact with waiter (solid lines) and in vacuum (dashed lines).

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