Quadratic law

This is referred to as quadratic law of mixture shown in curve 4. The parameter K involves an interaction between components A and B and provides an expression for the interfacial effect. 

Using this estimate behind, it is not difficult to establish that for the convergence of iterations in accordance with a quadratic law, it suffices to choose the initial approximation so as to satisfy the condition... 

We must stress, however, that the Black-Halperin analysis has been conducted for only a single substance, namely, amorphous silica, and systematic studies on other materials should be done. The discovered numerical inconsistency may well turn out to be within the deviations of the heat capacity and conductivity from the strict linear and quadratic laws, repsectively. Finally, a controllable kinetic treatment of a time-dependent experiment would be necessary. 

We have carrried out an analysis of the multilevel structure of the tunneling centers that goes beyond a semiclassical picture of the formation of those centers at the glass transition, which was primarily employed in this chapter. These effects exhibit themselves in a deviation of the heat capacity and conductivity from the nearly linear and quadratic laws, respectively, that are predicted by the semiclassical theory. 

We examine next the cyclic voltammetric responses expected with nonlinear activation-driving force laws, such as the quasi-quadratic law deriving from the MHL model, and address the following issues (1) under which conditions linearization can lead to an acceptable approximation, and (2) how the cyclic voltammograms can be analyzed so as to derive the activation-driving force law and to evidence its nonlinear character, with no a priori assumptions about the form of the law. 

If the kinetics of electron transfer does not obey the Butler-Volmer law, as when it follows a quadratic or quasi-quadratic law of the MHL type, convolution (Sections 1.3.2 and 1.4.3) offers the most convenient treatment of the kinetic data. A potential-dependent apparent rate constant, kap(E), may indeed be obtained derived from a dimensioned version of equation (2.10) ... 

This favorable situation may not be encountered in every case. With radical reductions endowed with high intrinsic barriers, the half-wave potential reflects a combination between radical dimerization and forward electron transfer kinetics, from which the half-wave potential cannot be extracted. One may, however, have recourse to the same strategy as with the direct electrochemical approach (Section 2.6.1), deriving the standard potential from the half-wave potential location and the value of the transfer coefficient (itself obtained from the shape of the polarogram) under the assumption that Marcus-Hush quadratic law is applicable. 

The initial rise section in Figure 7.7, which follows the quadratic law (i-f) given by the preexponential term in Eq. (7.13), corresponds to formation of the first monolayer. 

Higher order blending laws have been proposed (215). The quadratic law,... 

This polydispersity dependence is probably still too weak, perhaps because of the unjustified assumption about the form of H12. The quadratic law leads directly to the experimentally observed behavior Je° oc c 2 if one of the components is solvent the linear law gives J°oc c. ... 

The initial rise section in Figure 7.7, which follows the quadratic law (/—r2) given by the pre-exponential term in Eq. (7.13), corresponds to the formation of the first mono-layer. Figure 7.7 also shows the theoretical i-t transients for the formation of successive layers under conditions of progressive nucleation. The theoretical current-time transient for the three-dimensional nucleation is shown in Figure 7.8. The difference between the 2D and 3D nucleation (Fig. 7.7 and 7.8) is in the absence of damped oscillations in the latter case. A comparison between the theoretical and the experimental transients for the 2D polynuclear multilayer growth is shown in Figure 7.9. 

Equation (8) undoubtedly is identical to corresponding Semenov s equation which describes kinetics of N active sites in branched chain reaction with quadratic law of chain termination and zero order of initiation [5], However the essence of processes is different. 

A dependence close to a linear law is observed down to 100 K. At low temperature, both the thermal expansion and the pressure coefficient are small. Therefore, the constant-volume temperature dependence of the resistivity does not deviate from the quadratic law observed under constant pressure. At this stage it is interesting to stress that the theory of the resistivity in a half-filled band conductor [63], including the strength of the coulombic repulsions as derived from NMR data (Section III.B), should lead to a more localized behavior than that observed experimentally in Fig. 14. 

Since the early work of Kerr in 1875, the results of almost all authors and particularly those of the Le Fevre centre point to a value of Aji independent of the electric fidd strength in gases and liquids, and charact istic of the substance under investigation (Kerr constant). O Konski and co-workers have shown that in solutions of macromolecules Kerr s quadratic law does not hold, and electric saturation of optical birefringence is generally observed. 

As seen from Fig. 5.25, in the initial monolayer region, calculated and simulated transients follow the quadratic law given by the pre-exponential term of eq. (5.15). In the further course of the transients, the theoretical curves go slightly astray. This is obviously connected with the different approximations used in the calculation of the contribution of the (n + l)th layer on top of the th layer. The most reliable results seem to be obtained by nucleation and disk growth Monte Carlo simulations (circles in Fig. 5.25) [5.43, 5.44]. 

Quantitative agreement, of course, is not to be expected, as this critical intensity is not susceptible of exact definition at all. The comparison shows that the intensities for which the quadratic law is transformed into the linear law are of the order of magnitude anticipated, and also that the behaviour at different temperatures becomes intelligible. 

As a result of these considerations, then, we may say that the fact that the quadratic law ceases to be valid at a definite field intensity, and the actual magnitude of this field intensity, are alike intelligible. The essential explanation of the effect lies not in the forces acting on the electronic spin, but in the Lorentz force which affects the motion of the electron in its path. An effect of the correct magnitude, however, is not obtained unless the forces to which the electron is subjected in the metal are also taken into account. 

We emphasize again that the porous inserts are of infinite length and interpreted as an area z [0,/t] u [2H - h,2H] with distributed local mass force f = -AU U a l, for which a = 1 expresses the linear force law, and a = 2 is hold for the quadratic law [219], This kind of a flow generalizes the flow problems for smooth or rough... 

The problem considered here differs from the canonical problem by the presence of a source term (i.e. the force) on the right-hand side of the complete Navier—Stokes equations (3.29). This force vanishes outside the EPR, for z (h, 1 - h), is opposite to the local flow direction, and is proportional to some power of its velocity (here, we consider the linear or quadratic law). The boundary condition at the entrance x = 0 is evident, U = 1, V = 0 (homogeneous velocity distribution). There are non-slip conditions on the walls z = 0 and z = 1. The further formulation of the problem is somewhat different for linear and quadratic EPRs. 

The velocity distribution inside the EPR, z [0,h, can be obtained by means of a solution of the second-order ordinary differential equation (3.133). As the equation uses the quadratic law for the distributed drag force, it allows a numerical solution only. The following algorithm can be suggested ... 

The molecule contains Si-0 bonds. FTIR suggested that the Si-0 bonds move when the side groups move. Hence, ferroelectric liquid crystalline polymers have higher rotational viscosities than small molecular mass ferroelectric liquid crystals. In Figure 6.44 the relation of rotational viscosity r/ and molecular weight Mw at 600 °C is plotted, rj increases as Mw increases and the quadratic law is observed. 

Troitskij claims rediscovery of a quadratic law, first identified by Lundmark (1925) and later confirmed by Segal (1976). This is not strictly correct. From the available data Lundmark established a clear non-linear relationship of the... 

There is no Curie-Weiss behaviour immediately above but only at a much higher temperature, the Bums temperature, T. Between and the relationship between the relative permittivity and the temperature is usually better described by a quadratic law ... 

---