Double-valued behavior

The presence of the pores adds two parameters - the pore volume fraction and the pore radius. The predicted Rp Increases as the pore radius decreases suggesting a preference tor small pore packings. However, for a small pore radius of 1.0 pm a single value of the separation factor corresponds to two values of the particle diameter (13). Such double-valued behavior Is of course undesirable In an analytic technique. 

Monodisperse melts appear to exhibit a plateau region in the stress vs shear rate flow curve [51,62,65]. The capillary flow behavior actually closely resembles the oscillatory shear behavior in the sense that the flow curve essentially overlaps on the absolute value of complex modulus G vs the oscillation frequency (0 [62]. Thus.it appears that the transition-like capillary flow behavior of highly entangled monodisperse melts reflects constitutive bulk properties of the melts and is not interfacial in origin. It remains to be explored whether this plateau indeed manifests a real constitutive instability, i.e., whether it is double-valued. 

Simulations—isoergic and isothermal, by molecular dynamics and Monte Carlo—as well as analytic theory have been used to study this process. The diagnostics that have been used include study of mean nearest interparticle distances, kinetic energy distributions, pair distribution functions, angular distribution functions, mean square displacements and diffusion coefficients, velocity autocorrelation functions and their Fourier transforms, caloric curves, and snapshots. From the simulations it seems that some clusters, such as Ar, 3 and Ar, 9, exhibit the double-valued equation of state and bimodal kinetic energy distributions characteristic of the phase change just described, but others do not. Another kind of behavior seems to occur with Arss, which exhibits a heterogeneous equilibrium, with part of the cluster liquid and part solid. 

Figure 3 again shows the nonuniformity among the diflFerent types of species considered toward the heavy-molecule terminus of the triple-point loci. In addition, the double-valued thermal conductivity behavior is evident from the isotherms which lie above the diatomic molecule triple-point locus. The crossover is seen to occur at reduced temperatures so that tritium should be the heaviest molecule exhibiting a thermal conductivity maximum along its vapor pressure curve. 

Based on the above discussion, there are a number of factors that affect the double-layer behavior and the corresponding EDL capacitance, such as the concentration and size of ions, the ion-specific adsorption, the ion-solvent interaction, and the solvent in the electrolytes. The thickness of the EDL is typically on the order of several angstroms in aqueous solution. Since the distance separating the charges in an EDL is extremely small, the specific capacitance (capacitance normalized by the effective surface area) of EDL can reach a very high value in the aqueous electrolyte. In contrast, the specific capacitance of a typical parallel-plate capacitor is quite small... 

H H2, we know that it is synunetric across the symmetry hne which extends radially outward from the origin at = 0 (i.e. to the right of the conical intersection). It is also symmetric across the two symmetry lines which extend radially outward from the origin at = 120°. The geometric phase alters the symmetry of the real electronic wave function for H3, so that it is also antisymmetric across the symmetry line which extends radially outward from the origin at = 7t (i.e. to the left of the conical intersection). By symmetry, it is also antisymmetric across the two symmetry lines which extend radially outward from the origin at = 60°. The antisynunetric behavior is a direct consequence of the wave function s double-valuedness. In order to satisfy Fermi statistics for aU nuclear geometries, the product of the nuclear motion wave function and nuclear spin wave function must also be double-valued and be antisynunetric across the synunetry hnes at (f) = 0, (f> = 120°, and = —120°. The product must also be synunetric across the symmetry lines at = tt, = 60°, and 4> = —60°. 

Since the expression (41) is deduced for a sphere whose radius is large compared with the molecules of the liquid, it is not known to what extent the behavior of atomic and small molecular ions should be in accordance with (41). It is clear that, if (41) were applicable, the value of the mobility should vary inversely with the viscosity. If for any ion the K on the left-hand side of (41) is set equal to the constant force acting on the ion in a field of unit intensity, the v on the right-hand side of (41) becomes equal to the mobility u. Since K is independent of temperature the product of u and ij should be independent of temperature. From Table 42 it will be seen that at 25°C the viscosity of water is almost exactly half the viscosity at 0°C thus, according to (41) the mobility u of each ion should be double. 

Mishuk et a/.675,676 have applied the modified amplitude demodulation method to electrochemically polished pc-Bi in aqueous NaF solution. The curves of the real component of the nonlinear impedance Z" as a function of the electrode potential, unlike pc-Cd and pc-Pb, intersect for various cNaF at E - -0.62 V (SCE),674 i.e., at Ea=0 for pc-Bi, as obtained by impedance.666-672 The different behavior of pc-Bi from pc-Cd and pc-Pb at a > 0 has been explained by the semimetallic nature of pc-Bi electrodes. A comparison of inner-layer nonlinear parameter values for Hg, Cd, and Bi electrodes at a < 0 shows that the electrical double-layer structure at negative charges is independent of the metal.675,676... 

With regard to the composition of the electrical effect, examination of the p values reported in Table XVII shows that in six of the sets which gave significant correlation, the localized effect is predominant (in these sets, either Pr < 50 or / is not significant). Thus it would appear that in so far as substituent effects are concerned, there are two major classes of electrophilic addition to the carbon-carbon double bond predominance of the localized effect or predominance of the delocalized effect. This behavior may well be accounted for in terms of the reaction mechanism. The rate-determining step in the electrophilic addition reaction is believed to be the formation of an intermediate which may be either bridged or a free carbonium ion. 

FIGURE 21.8 The behavior of (a) chain stiffness, kz and (b) viscosity of the chain, tj2 against the extension. The values of 2 and -172 were calculated by a double-Voigt model. 

---