Linear behavior

Solid-Fluid Equilibria The phase diagrams of binai y mixtures in which the heavier component (tne solute) is normally a solid at the critical temperature of the light component (the solvent) include solid-liquid-vapor (SLV) cui ves which may or may not intersect the LV critical cui ve. The solubility of the solid is vei y sensitive to pressure and temperature in compressible regions where the solvent s density and solubility parameter are highly variable. In contrast, plots of the log of the solubility versus density at constant temperature exhibit fairly simple linear behavior. 

This function together with the linear behavior of the resistance of the protection system is shown in Fig. 6-1. 

Several experiments will now be described from which the foregoing basic stiffness and strength information can be obtained. For many, but not all, composite materials, the stress-strain behavior is linear from zero load to the ultimate or fracture load. Such linear behavior is typical for glass-epoxy composite materials and is quite reasonable for boron-epoxy and graphite-epoxy composite materials except for the shear behavior that is very nonlinear to fracture. 

On the other hand, for aircraft and spacecraft structures, real laminate behavior is pretty typically linear. Laminate behavior is reasonably linear even with some 45° layers which you would expect to contribute their nonlinear shear deformation characteristic to the overall laminate and degrade its relative performance. If you go beyond the behavior of a laminate and look at a large structure, typically the load-response characteristics are linear. Even around a cutout, linear behavior exists. Beyond that apparent linear performance of many laminates, you might not like to operate in some kind of a nonlinear response regime. Certainly not when in a fatigue environment and probably not in a creep environment either would you like to operate in a nonlinear behavior range. 

For intermediate drift rates (4 < BN < 8), when chain conformations are already distorted, deviates from linear behavior and goes through a maximum at some critical value Bf. of the field, confirming earlier findings by Pandey et al. [103,104]. This critical bias B at which the velocity starts to decrease depends rather weakly on the density Cobs, turns out to be reciprocal to chain length A, implying that only when the total force, /c = B,N 9, acting upon the whole driven molecule, exceeds a certain threshold, which does not depend on the size of the macromolecule, the chains start to get stuck in the medium. 

Assuming a linear behavior, the conditions that have to be fulfilled by the different flowrates can be shown to be ... 

Calculation of TMB flowrates To calculate TMB flowrates, linear behavior of the adsorption isotherms for a feed concentration of 1 g is assessed. To check this point, we will use the criterion given in Equation (10). 

A linear regression was performed on the data, giving a slope of 1.08, an intercept of 1.922, and = 0.94. The fit of the data to the linear relationship is surprisingly good when one considers the wide variety of ionic liquids and the unloiown errors in the literature data. This linear behavior in the Walden Plot clearly indicates that the number of mobile charge carriers in an ionic liquid and its viscosity are strongly coupled. 

Linear viscoelasticity Linear viscoelastic theory and its application to static stress analysis is now developed. According to this theory, material is linearly viscoelastic if, when it is stressed below some limiting stress (about half the short-time yield stress), small strains are at any time almost linearly proportional to the imposed stresses. Portions of the creep data typify such behavior and furnish the basis for fairly accurate predictions concerning the deformation of plastics when subjected to loads over long periods of time. It should be noted that linear behavior, as defined, does not always persist throughout the time span over which the data are acquired i.e., the theory is not valid in nonlinear regions and other prediction methods must be used in such cases. 

Figure 8 shows an example of the most common behavior of AEam/0 as a function of adsorbate coverage. Linear behavior, if ever observed, is seen at the air/solution interface.93 At metal/solution interfaces, if chemical interactions with the metal can be ruled out, electrostatic interactions cannot be avoided, and these are responsible for the downward curvature.91 Upward curvatures are often observed at air/solution interfaces as a consequence of lateral interactions.95... 

According to Eq. (6.44) such a behavior may be analyzed using parameter Hi. In the range of Hi = 0.00791—0.0260 linear behavior of the bubble radius was observed, when Hi > 0.0260 exponential bubble growth took place (Fig. 6.24). 

Dimensional analysis shows that the behavior of the bubble radius with time depends on the parameter TI = qf (plUCpLATs). In the range of 17 = 0.0079-0.026, linear behavior was observed, and when 17 > 0.026 exponential bubble growth took place. 

An important class of cycles with non-linear behavior is represented by situations when coupling occurs between cycles of different elements. The behavior of coupled systems of this type has been studied in detail by Prigogine (1967) and others. In these systems, multiple equilibria are sometimes possible and oscillatory behavior can occur. There have been suggestions that atmospheric systems of chemical species, coupled by chemical reactions, could exhibit multiple equilibria under realistic ranges of concentration (Fox et ai, 1982 White, 1984). However, no such situations have been confirmed by measurements. 

The electrode shows linear behavior in the immediate vicinity of the working point on the calibration curve EMF = EO + 5 logio(C). 

Experiments show that the equilibrium concentration of dissolved gas increases linearly with the partial pressure of the gas. The equation describing this linear behavior is Henry s law [g3s(c2 <7)]eq =. h CPgas)gq The... 

A thorough insight into the comparative photoelectrochemical-photocorrosion behavior of CdX crystals has been motivated by the study of an unusual phenomenon consisting of oscillation of photocurrent with a period of about 1 Hz, which was observed at an n-type CdTe semiconductor electrode in a cesium sulfide solution [83], The oscillating behavior lasted for about 2 h and could be explained by the existence of a Te layer of variable width. The dependence of the oscillation features on potential, temperature, and light intensity was reported. Most striking was the non-linear behavior of the system as a function of light intensity. A comparison of CdTe to other related systems (CdS, CdSe) and solution compositions was performed. 

For PVA and PCP the dependences on V of log(MyJ)v and log (My)y (Figures 2a, 2b and 4) show similar trends as for SRM 706. Noticeable downward curvature of log(M,y)y at low molecular weight occurs in each case, in contrast to relatively linear behavior of log(MjJ)y. Anomalously large values of g result (Figures 3 and 5). In the highest molecular weight region, a sharp downturn in the log (My)y dependence on V is apparent which is accompanied by very Tow g values. 

Here N is the distance between points p and q measured along the perpendicular from the point q to the geoid. Fig. 2.9a. The linear behavior of the normal potential implies that the field y is constant between the geoid and the reference ellipsoid. The change of sign in Equation (2.260) is related to the fact that the field has a direction, which is opposite to the direction of differentiation. As follows from the first equation of the set (2.258 and 2.260) we have... 

The I U) characteristic of the arrays showed a linear behavior over a broad voltage range. If each cluster is assumed to have six nearest neighbors and a cluster-to-cluster capacitance of 2 x 10 F is implied, the total dot capacitance will be 1.2 x 10 F. A corresponding charging energy can thus be approximated to 11 meV, which is only about half of the characteristic thermal energy at room temperature. This excludes a development of a Coulomb gap at room temperature. 

NN can be used to select descriptors and to produce a QSPR model. Since NN models can take into account nonlinearity, these models tend to perform better for log S prediction than those refined using MLR and PLS. However, to train nonlinear behavior requires significantly more training data that to train linear behavior. Another disadvantage is their black-box character, i.e. that they provide no insight into how each descriptor contributes to the solubility. 

Since the capacitor without electrolyte shows a linear behavior of the electrostatic potential between the electrodes, (x), at the hypothetical moment when the electrolyte is added to the system (t = 0 s), the electrochemical potentials of the ions. 

Figure 12.8 CO-saturated electrolyte in the thin cell of Fig. 12.2. (a) CO oxidation the first and second scans are shown, (b) Comparison with the CO stretch frequency shift. (Filled circles denote linear Stark tuning behavior while open circles correspond to deviations from linear behavior during oxidation.)...

The linear distance dependence seen for AQ-DNA(3) is not observed to be universally independent of specific DNA base sequence. This is clearly revealed by examination of AQ-DNA(4) and AQ-DNA(5). Plots of the distance dependence of strand cleavage at the GG steps in these oligomers are shown in Fig. 11. Both show stepped rather than linear behavior, and the size of... 

From Figure 4 (a) and (b) it is seen that especially the initial rate of adsorption (adsorption within the first 5 min of the experiment) as well as the adsorption isotherm at pH=6 have a near linear behavior. The situation is less clear at pH=3, where the initial rate increases much slower with initial gold concentration and the adsorption isotherm shows non-linear behavior. This shows furthermore, that pH=6 is more favorable for the adsorption of more gold from solution. 

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