Reciprocal conductance relaxation

Figure 2. Reciprocal conductance relaxation times as a function of the square root of TBAP concentration in media and concentration range where conductance indicates a preponderance of the simple ions. TBAP in diphenyl ether (D— 3,55) at 318 ( ). TBAP in benzene-chlorobenzene (60 40 v/v D = 3, 55) at 298 (A). TBAP in benzene-chlorobenzene (60 40 v/v D = 3, 64) at 298 and 350 atm (2). TBAP in benzene-chlorobenzene (60 40 v/v D = 3, 36) at 323 (0).

Figure 3. Dependence of the reciprocal conductance relaxation time on TBA— salt concentration in media and concentration range where conductance indicates an important fraction of triple ions. TBA-bromide in benzene-nitrobenzene (3, 22 vol % D = 2, 90) at 298 ( ). TBA-picrate in benzene-chlorobenzene (16 vol % D = 2, 78) at 298 (0).

As a second example we turn to the analysis of the conductance relaxation observed in tetrabutylammonlun plcrate in benzene-chlorobenzene over the concentration range 0.1—2 x 10 M as shown in Figure 3. A qualitative, but important difference with the previous example is the linear relation between reciprocal relaxation time and total TBAP-concentratlon. 

Since the uncertainty of the numerical values in K3 makes the test of eq. 4b somewhat hazardous, the temperature, and pressure, dependence of the conductance relaxation was investigated. Temperature, and pressure, dependence of intercept and slope of the experimental reciprocal relaxation time vs. concentration is given in Table III. If the reciprocal relaxation time indeed is functionally described by eq. 14 b then we are able to calculate these temperature, and pressure, dependences from previously obtained experimental data. 

Debye and Falkenhagen predicted that the ionic atmosphere would not be able to adopt an asymmetric configuration corresponding to a moving central ion if the ion were oscillating in response to an applied electrical field and if the frequency of the applied field were comparable to the reciprocal of the relaxation time of the ionic atmosphere. This was found to be the case at frequencies over 5 MHz where the molar conductivity approaches a value somewhat higher than A0. This increase of conductivity is caused by the disappearance of the time-of-relaxation effect, while the electrophoretic effect remains in full force. 

Considering these different limiting forms of the recombination term an Important tentative conclusion emerges the concentration dependence of the reciprocal relaxation time is a direct measure of the main ionic recombination process and yields therefore information on the ionic species present in solution. A linear dependence on total ion-pair concentration would therefore indicate unilateral triple ion formation or, if both kinds of triple ions are present as indicated by conductance, a sufficient difference in their stability. At this point it should be noted that the usual method of Fuoss and Draus... 

Hydrated Zeolites. Figure 3 gives a typical plot of the conductivity vs. the reciprocal temperature for hydrated NaF86.5. The other samples behave qualitatively in the same way. Conduction and dielectric absorption are superposed. The position of the maximum of dielectric absorption is frequency dependent it shifts to higher temperatures with increasing frequency. In some favorable cases a second conduction phenomenon is observed on the low temperature side of the relaxation phenomenon (Figure 3). Because of a lack of reproducibility we cannot interpret it. 

In Section 4, we have examined, from a fundamental point of view, how temperature and cure affect the dielectric properties of thermosetting resins. The principal conclusions of that study were (1) that conductivity (or its reciprocal, resistivity) is perhaps the most useful overall probe of cure state, (2) that dipolar relaxations are associated with the glass transition (i.e., with vitrification), (3) that correlations between viscosity and both resistivity and dipole relaxation time are expected early in cure, but will disappear as gelation is approached, and (4) that the relaxed permittivity follows chemical changes during cure but is cumbersome to use quantitatively. 

Eventually there is a critical frequency above lO s at which the ionic cloud cannot adjust anymore to the ion s movements in the right way because there is too much inertia to execute the rapid changes required by the oscillating applied field. The reciprocal of this critical frequency is called the relaxation time of the asymmetry of the ionic cloud. As a consequence, an increase in conductivity occurs at this frequency because there is no longer more charge behind the ion than in front. This increase in conductance at the critical frequency is called the Debye effect. It is part of the evidence that shows that the ionic atmosphere is indeed present and functioning according to the way first calculated by Debye. 

Fig. 13. Hydration dependence of protonic conduction. The dielectric relaxation time, Ts, is shown versus hydration, h, for lysozyme powders. The relaxation time is proportional to the reciprocal of the conductivity. (A) H20-hydrated samples solid curve, lysozyme without substrate , lysozyme with equimolar (GlcNAc)< at pH 7.0 , with 3x molar (G1cNAc)4 at pH 6.5. The relaxation time is nearly constant between pH 5.0 and 7.0. (B) HjO-hydrated samples solid curve, lysozyme without substrate 9, lysozyme with equimolar (GlcNAcb at pH 7.0. From Careri etal. (1985). 

We would expect intuitively that tan 0 emd the Deborah number De are related, since both refer to the ratio between the rates of an imposed process and that (or those) of the system. The exact shape of this relationship depends on the number and nature(s) of the releixation process(es). So let us anticipate [3.6.4 la] for the loss tangent of a monolayer in oscillatory motion, which describes a special case of [3.6,12], namely -tan0 = t]°(o/K°. Here, (o is the imposed frequency, equal to the reciprocal time of observation, t(obs) =< . The quotient K° /t]° also has the dimensions of a time in fact it is the surface rheological equivalent of the Maxwell-Wagner relaxation time in electricity, (Recall from sec. 1.6c that for the electrostatic case relaxation is exponential ith T = e/K where e e is the dielectric permittivity and K the conductivity of the relaxing system. In other words, T is the quotient between the storage and the dissipative part.) For the surface rheological case T therefore becomes The exponential decay that is required for such a... 

When an external electric field is imposed on an electrolyte solution by electrodes dipped into the solution, the electric current produced is proportional to the potential difference between the electrodes. The proportionality coefficient is the resistance of the solution, and its reciprocal, the conductivity, is readily measured accurately with an alternating potential at a rate of 1 kHz in a virtually open circuit (zero current), in order to avoid electrolysis at the electrodes. The conductivity depends on the concentration of the ions, the carriers of the current, and can be determined per unit concentration as the molar conductivity Ae. At finite concentrations ion-ion interactions cause the conductivities of electrolytes to decrease, not only if ion pairs are formed (see Sect. 2.6.2) but also due to indirect causes. The molar conductivity Ae can be extrapolated to infinite dilution to yield Ae" by an appropriate theoretical expression. The modern theory, e.g., that of Fernandez-Prini (1969), takes into account the electrophoretic and ionic atmosphere relaxation effects. The molar conductivity of a completely dissociated electrolyte is ... 

The root. i simply indicates that infinite distances are correlated with infinite time, S2 is the reciprocal of the Debye relaxation time, and 3 is the kinetic relaxation frequency of the system. Depending on the kinetic parameters of the chemical process, the kinetic relaxation frequency can be faster or slower than the Debye frequency of the system. If the kinetic relaxation frequency is much smaller than the Debye mode, it can be determined experimentally by conductance fluctuation analysis. 

The simplest model of relaxation using these reduced properties is shown in the Formal Graph in the case study abstract. Two projections from above in a two-dimensional horizontal plane are given in Graph 11.25, one with the elementary properties and one with the composed path formed by a combination of the reciprocal of the permittivity with the conductivity, assuming that the two properties are homothetic (i.e., proportional). 

Electrolyte solutions in low polar media are characterized by a very complex conductance behaviour mainly due to the long range of Coulombic interaction. However, at low electrolyte concentration, a simple ion-pair dissociation into free ions is the predominant process. This situation is experimentally come out by the reciprocal relation between equivalent conductance and the square root of electrolyte concentration. In relaxation experiments this simple behaviour is likewise characterized by a linear dependence of the reciprocal relaxation time of the ionisation equilibrium on the square root of total concentration. It is therefore relatively easy to delimit the experimental conditions within which the simple ion- pairing process is present and not obscured by the progressive emergence of higher ionic aggregates, e.g, triple ions, quadrupoles. .. 

The melt viscosity r] is closely associated with the dielectric relaxation time. The temperature dependence of the melt viscosity for each oligomer was evaluated and was compared to both the dielectric relaxation time and the dc conductivity. Figure 2 shows the melt viscosity of five oligomers as a function of the reciprocal Kelvin temperature. Figures 3 and 4 demonstrate the temperature dependences of three properties, rj, a, and t, for Epikote 834 and Epikote 1002, respectively. Log-log plots of melt viscosity versus dc conductivity for three oligomers, Epikote 828, 1001, and 1004, are shown in... 

It is common practice to define the dielectric transition frequency by the maxinu of the c (co) or tan 6(u>) curves, although these are somewhat shifted with respect lo each other. Neither of these methods is correct, because the relaxation time distribution is usually nonsymmetric Tbc transition frequency may be defined as that corresponding lo Ibe derivative of the real part c by log(frequency). According to Eq. (19). ibis corresponds lo the reciprocal of the maximum of the relaxation time distribution. The first derivative is usually sufficient, e is affected by the ohmic conductivity only when in-lerfocial polarization is dominant in the frequency range studied. 

The electric modulus is the reciprocal of the permittivity M =l/s. Generally, for a pure conduction process, a relaxation peak would be observed in the frequency spectra of the imaginary component M" and no peak would take place in the corresponding plot of s", such is our case in all the SlLLPs. 

An adsorbed atom or a molecule being in its excited state is characterized by a finite lifetime which is determined by the reciprocal of the decay rate of this state. The finiteness of the lifetime leads to a broadening of the lines in the optical spectra of the adsorbate. Besides spontaneous emission which occurs also for free atoms and molecules, adsorbed species have other specific channels of relaxation, conditioned by their proximity to the surface. Any relaxation process must obey the conservation law of energy and therefore it takes place only if there is a substrate excitation which can accept the energy that the excited adsorbate releases. Therefore, possible decay mechanisms are determined by the energy spectrum of the substrate and thus generally are different for metals, semiconductors and dielectrics. They can be broadly classified as being mediated by photons, phonons, electron-hole pairs and conduction electrons. 

In addition the material would, in general, have a dc conduction so the capacitor would pass a time independent leakage current I(t) equal to A F where A is the conductance (reciprocal of resistance) of the sample-filled capacitor. A is proportional to the specific conductivity (in units of Q" cm ) of the dielectric. Note that the time-dependent charge, which arises from relaxation processes or space-charge development, has a derivative which is a current but this is not to be associated with the steady leakage current of a dc conductivity. 

---