Square-root time-temperature dependence

MDR WITH Square-Root Time-Temperature Dependence [26]... 

The increase in fiber-matrix interfacial shear strength can be predicted from a purely mechanistic viewpoint. Rosen [20], Cox [21], and Whitney and Drzal [22] have shown that the square root of the shear modulus of the matrix appears explicitly in any model of the interfacial shear strength. It has been demonstrated experimentally [23, 24] that the fiber-matrix interfacial shear strength has a dependence on both the product of the strain-to-failure of the matrix times the square root of the shear modulus and on the difference between the test temperature and Tg when the interfacial chemistry is held constant. 

Murabayashi coworkers (Nakajima et al., 2001, 2002) investigated the PL properties of rutile and anatase powders at room temperature in air in the absence or presence of reactants such as methanol or ethanol, with the goal of understanding the mechanisms of gas-phase photocatalytic reactions. In experiments with rutile, they observed that the PL intensity in the presence of methanol or ethanol increased linearly with the square root of UV-irradiation time, as shown in Figure 15. They also found a linear time dependence of the integrated amount of photodesorbed O2 (Nakajima et al., 2002). The time dependence of the PL intensity and the effect of O2 photodesorption were explained (as mentioned above) in terms of band bending of the powder. These results, however, were not observed for anatase powders. The authors explained that such a difference in PL behavior was related to the difference in photocatalytic activities of these alcohols. 

If D only depends on temperature (and thus not on concentration or time), the diffusion process is called Fickian. Simple gases show Fickian diffusion and so do many dilute solutions (even in polymers).The diffusivity can be determined directly either from sorption or from permeation experiments. In the first case the reduced sorption, c(t) / (cfX, cG), is plotted versus the square root of the sorption time and D is calculated from the equation ... 

Retention time and the peak efficiency depend on the carrier gas flow rate retention time is also directly proportional to column length, while resolution is proportional to the square root of the column length. For packed columns, the carrier gas flow rate is usually expressed in milliliters per minute at atmospheric pressure and room temperature. It is measured at the detector outlet with a soap film flow meter while the column is at operating temperature. Unless otherwise specified in the individual monograph, flow rates for packed columns are 60 to 75 mL/min for 4-mm id columns and 30 mL/min for 2-mm id columns. 

Potts and Rollefson studied the photochemical reaction between H2 and CI2 over the temperature range 140-300 °K. At the lower end of this range solid or liquid chlorine was always present in the system so that the reaction rate could be calculated from measurements of the change in pressure with time using a sulfuric acid manometer. At the higher temperatures where the CI2 pressure decreased as the reaction proceeded, the rates were calculated from measurement of hydrogen pressures before and after the reaction. At temperatures below 172 °K, Potts and Rollefson found the rate of HCl formation to depend upon the square root of the light intensity whereas above 200 °K the rate was proportional to the first power of the intensity. The rates were proportional to [H2] at all... 

We now turn to the discussion of isotope effects in hydrogen diffusivity. The classical diffusion theory predicts that the pre-exponential factor Dq in Eq. (26.17) is inversely proportional to the square root of the mass of a diffusing particle, while the activation energy does not depend on this mass. According to these predictions, the diffusion coefficient of D atoms should be s/2 times lower than that of H atoms over the entire temperature range. However, the isotope dependence of hydrogen diffusivity in all metal-hydrogen systems studied so far shows deviations from the predictions of the classical theory. In particular, the measured effec-... 

Weller (81) studied the pressure dependence of the synthesis over a cobalt-thoria-kieselguhr catalyst, at about 1 atmosphere pressure using very short contact times so that the concentration of synthesis products was very low. Under these conditions the synthesis rate increased approximately as the square root of the synthesis gas pressure. The temperature coefficient in Weller s tests corresponded to an activation energy of 26 kcal./ mole. Storch and his coworkers (55) obtained a value of 25 kcal./mole at long contact times corresponding to the Ruhrchemie process. 

Steinemann (140) found that, at a given temperature, the dielectric relaxation time was inversely proportional to the HF concentration if the concentration was high, and inversely proportional to the square root of the concentration if it was lower. (The relations are listed in Table III.) Thus, the concentration dependence of the dielectric relaxation time does not follow a simple law. The formulation depends on the assumptions one makes about the nature of the relaxation process or processes (ion translation or molecular rotation), carrier concentration and dissociation, and the energies involved. 

Simulation methods construct the wavefunction (or at positive temperature the /V-body density matrix) by sampling it and therefore do not need its value everywhere. The complexity then usually has a power-law dependence on the number of particles, T< N, where the exponent typically ranges from 1 s 5 4, depending on the algorithm and the property. The price to be paid is that there is a statistical error which decays only as the square root of the computer time, so that T e. ... 

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