Dispersion of Time Constants

In general, the distribution of the time constants t = can arise from 

The distribution of solution resistances is observed in the Hull cell [354] displayed in Fig. 8.12, where the distance between the working and the counter electrode changes between li and l2- 

A simulation of impedances for different values of is shown in Fig. 8.13. When li I2, dispersion of the impedances is observed displaying a CPE-like behavior but only in a limited frequency range. Adding a charge transfer resistance in parallel with the double-layer capacitance leads to a skewed semicircle with a radius lower than the assumed / ct- When the two electrodes are parallel, no dispersion is observed and a correct result is obtained. Simulations of impedances in the presence of the redox reaction represented by R t is displayed in Fig. 8.14, where distorted semicircles are obtained. 

It should be added that in the foregoing cases, the current lines are not parallel as the current is looking for a path of least resistance. 

Newman found that at disk electrodes current distribution is nonuniform in the radial direction (known as the primary [355] and secondary [356] current distributions), which leads to impedance dispersion [357]. Recently, Huang et al. [310,358, 359] continued these studies in more detail using a local impedance approach. Global admittance corresponds to the integration of the local admittances over the total disk area. Impedance can also be defined (and experimentally measured) locally as a function of the position on the electrode surface. In the case of the disk geometry, it changes radially from the disk center, r = 0, to the disk radius, r = ro- The authors distinguished two types of distribution of time constants  

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The oxygen vacancy defects appear to function as deep traps for holes and recombination centers for electrons and shallow traps for electrons [66] (the limited dispersion of time constants in transient current experiments has been attributed to the absence of deep electron traps). No optical absorbance is associated with occupied defects but instead a bleaching of the absorption... 

The dispersion of time constants for conduction and recombination lead to dispersive features in both time-resolved and frequency-resolved measurements. These... 

Another possible source of dispersion of time constants is the dispersion of capacitances arising from atomic-scale surface inhomogeneities such as grain boundaries, crystal faces on a polycrystalline electrode, or other variations in surface properties [313, 334, 350, 358]. However, experimental studies do not confirm this hypothesis. In fact, an increase in surface roughness, that is, of surface inhomogeneities, does not decrease the CPE parameter (p but increases it and such a surface becomes more similar to the ideal capacitance [313, 334, 350]. 

As was shown earlier, the presence of the CPE of fractal impedance produces a distribution of the time constants. In addition, other elements such as the Warburg (semi-infinite or finite-length) linear or nonlinear diffusion, porous electrodes, and others also produce a dispersion of time constants. Knowledge about the nature of such dispersion is important in the characterization of electrode processes and electrode materials. Such information can be obtained even without fitting the experimental impedances to the corresponding models, which might be still unknown. Several methods allow for the determination of the distribution of time constants [378, 379], and they will be briefly presented below. 

A breakthrough curve with the nonretained compound was carried out to estimate the axial dispersion in the SMB column. A Peclet number of Pe = 000 was found by comparing experimental and simulated results from a model which includes axial dispersion in the interparticle fluid phase, accumulation in both interparticle and intraparticle fluid phases, and assuming that the average pore concentration is equal to the bulk fluid concentration this assumption is justified by the fact that the ratio of time constant for pore diffusion and space time in the column is of the order of 10. ... 

The aimealing kinetics of the light-induced defects are shown in Fig. 6.29. Several hours at 130 °C are needed to anneal the defects completely, but only a few minutes at 200 C. The relaxation is nonexponential, and in the initial measurements of the decay the results were analyzed in terms of a distribution of time constants, Eq. (6.78) (Stutzmann, Jackson and Tsai 1986). The distribution is centered close to 1 eV with a width of about 0.2 eV. Subsequently it was found that the decay fits a stretched exponential, as is shown in Fig. 6.29. The parameters of the decay-the dispersion, p, and the temperature dependence of the decay time, t - are similar to those found for the thermal relaxation data and so are consistent with the same mechanism of hydrogen diffusion. The data are included in Fig. 6.23 which describes the general relation between x and D,. The annealing is therefore the process of relaxation to the equilibrium state with a low defect density. 

The impedance models developed in Chapters 9,10,11, and 12 are based on the assumption that the electrode behaves as a uniformly active surface where each physical phenomenon or reaction has a single-valued time constant. The assumption of a uniformly active electrode is generally not valid. Time-constant dispersion can be observed due to variation along the electrode surface of reactivity or of current and potential. Such a variation is described in Section 13.1.1 as resulting in a 2-dimensional distribution. Time-constant dispersion can also be caused by a distribution of time constants that reflect a local property of the electrode, resulting in a 3-dimensional distribution. 

The presence of time-constant (or frequency) distribution is frequently modeled by use of a constant-phase-element (CPE), discussed in Section 13.1. As discussed in Section 13.1.3, use of a CPE eissumes a specific distribution of time constants that may apply only approximately to a given system. The objective of this chapter is to describe specific situations for which time-constant dispersion can be predicted based on fundamental phenomena such as are Eissociated with distributions of mass-transfer rates and Ohmic currents. 

While assumption that the time constant is distributed can be better than assuming that the time constant has a single value, the physical system may not follow the specific distribution implied in equation (13.7). The examples presented in the subsequent sections illustrate systems for which a time-constant dispersion results that resembles that of a CPE, but with different distributions of time constants. 

A monolithic system is comprised of a polymer membrane with dmg dissolved or dispersed ia it. The dmg diffuses toward the region of lower activity causiag the release of the dmg. It is difficult to achieve constant release from a system like this because the activity of the dmg ia the polymer is constantly decreasiag as the dmg is gradually released. The cumulative amount of dmg released is proportional to the square root of time (88). Thus, the rate of dmg release constantly decreases with time. Again, the rate of dmg release is governed by the physical properties of the polymer, the physical properties of the dmg, the geometry of the device (89), and the total dmg loaded iato the device. 

A 2-1. two-necked, round-bottomed flask equipped with a magnetic stirrer (Note 1) is fitted with a 250-ml. pressure-equalizing constant-rate dropping funnel and a condenser, the top of which is connected to a mercury trap to prevent the entrance of air during the reaction and for the detection of gas evolution. The dropping funnel is removed, and 35 g. (0.85 mole) of sodium hydride dispersed in mineral oil is added (Note 2). The mineral oil is removed by washing the dispersion four times with 100-ml. portions of benzene (Note 3). The benzene is removed with a pipet after the sodium hydride is allowed to settle (Note 4). 

A dispersion of the sample is placed on top of a liq of greater density. The rate of sedimentation is detd by measuring the sediment vol at fixed time intervals. The results are converted to a size distribution by Stoke s Law Nitrogen Adsorption. The amt of N adsorbed on a sample is detd by carefully measuring the press change of a known vol of N exposed to a known wt of dry mat at constant temp. The info is used to detn the surface area which is converted to a particle size distribution Turbidometric Methods. The absorption of a beam of light passing thru a suspended sample in a suitable liq is measured as a function of time. 

Equation (17) indicates that the entire distribution may be determined if one parameter, av, is known as a function of the physical properties of the system and the operating variables. It is constant for a particular system under constant operating conditions. This equation has been checked in a batch system of hydrosols coagulating in Brownian motion, where a changes with time due to coalescence and breakup of particles, and in a liquid-liquid dispersion, in which av is not a function of time (B4, G5). The agreement in both cases is good. The deviation in Fig. 2 probably results from the distortion of the bubbles from spherical shape and a departure from random collisions, coalescence, and breakup of bubbles. 

Figure 12.5. Ethylene oxidation on Pt finely dispersed on Au supported on YSZ.7 Effect of the current 1 on x 1, where x is the time constant measured during a galvanostatic transient experiment with I as the applied current x is obtained by fitting either r/r0=exp(-t/x) or l-exp(-t/x) to the experimental data depending on the sign of the current and whether the reaction is electrophilic or electrophobic, (a) Positive values of I for electrophilic (squares, T=371°C, pO2=18.0 kPa, Pc2H4=0-6 kPa) and electrophobic behavior (circle, T=421°C, p02=l 4.8 kPa, Pc2H4 CU kPa) (b) negative currents, electrophilic behavior (T=421°C, p02=14.8 kPa, pC2H4=0.1 kPa. Reprints with permission from Academic Press.

By the use of very small sensing cells and electronic systems with very small time constants, the fixed wavelength detector can be designed to give a very fast response at high sensitivity and very low dispersion and... 

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