Slow polarization

Inclusion of part of the electronic polarization into the inertial polarization is due to strong interaction between the nuclei and electrons of the medium. The (slow) change in the nuclei positions inevitably produces polarization of the electron shells of the solvent molecules. Therefore, the latter also contribute to the slow polarization. 

When one places an electron into the donor molecule, the equilibrium fast polarization, which is purely electronic forms first. Being independent of the electron position, it is unimportant for the dynamics of electron transfer. Afterward the average slow polarization Pg, arises that corresponds to the initial (0 charge distribution (the electron in the donor). The interaction of the electron with this polarization stabilizes the electron state in the donor (with respect to that in the isolated donor molecule) (i.e., its energy level is lowered) (Fig. 34.1). At the same time, a given configuration of slow, inertial polarization destabilizes the electron state (vacant) in the acceptor (Fig. 34.1). Therefore, even for identical reactants, the electron energy levels in the donor and acceptor are different at the initial equilibrium value of slow polarization. 

The reorganization energy of the slow polarization is, roughly speaking, almost one-half of that for the bulk solution. 

The free-energy surfaces of the initial and final states, Uf ,r) and Uf(P,r), then involve two contributions the parabolic free energy as a function of the slow polarization, (7/(P) and (7/(P), and nonparabolic molecular potential f/ (r) and... 

The reorganization energy of the slow polarization for the reactions at metal electrodes can be calculated with the use of Eqs. (34.11). For a spherical model of the reacting ion, it is equal approximately to... 

For simple outer-sphere electron transfer reactions, the effective frequency co is determined by the properties of the slow polarization of the medium. For a liquid like water, where the temporal relaxation of the slow polarization as a response to the external field is single exponential, tfie effective frequency is equal to... 

In a static field both components of the polarization contribute, and the static value es of the dielectric constant must be used in Eq. (6.25). The slow polarization is obtained by subtracting Pf, which gives ... 

The reorganization of the solvent molecules can be expressed through the change in the slow polarization. Consider a small volume element AC of the solvent in the vicinity of the reactant it has a dipole moment m = Ps AC caused by the slow polarization, and its energy of interaction with the external field Eex caused by the reacting ion is —Ps Eex AC = —Ps D AC/eo, since Eex = D/eo- We take the polarization Ps as the relevant outer-sphere coordinate, and require an expression for the contribution AU of the volume element to the potential energy of the system. In the harmonic approximation this must be a second-order polynomial in Ps, and the linear term is the interaction with the external field, so that the equilibrium values of Ps in the absence of a field vanishes ... 

If the material exhibits slow polarization currents during switching, then the bridge technique fails, unless one uses an identical sample for the nuUing capacitance and employs a resistance ratio bridge. 

It should be noted, however, that in spite of qualitative agreement with experiment, the numerical results obtained should be considered with caution because the data on solvation energies are available only in the limit of slow polarization dynamics. Further investigations are necessary for going beyond this limit. 

The solvent polarization can be formally decomposed into different contributions each related to the various degrees of freedom of the solvent molecules. In common practice such contributions are grouped into two terms only [41,52] one term accounts for all the motions which are slower than those involved in the physical phenomenon under examination (the slow polarization), the other includes the faster contributions (the fast polarization). The next assumption usually exploited is that only the slow motions are instantaneously equilibrated to the momentary molecule charge distribution whereas the fast cannot readjust, giving rise to a nonequilibrium solvent-solute system. 

In the case of vibrations of solvated molecules the same two-term partition can be assumed, but in this case the slow term will account for the contributions arising from the motions of the solvent molecules as a whole (translations and rotations), whereas the fast term will take into account the internal molecular motions (electronic and vibrational) [42], After a shift from a previously reached equilibrium solute-solvent system, the fast polarization is still in equilibrium with the new solute charge distribution but the slow polarization remains fixed to the value corresponding to the solute charge distribution of the initial state. 

Of course, prior to any realization of a laser setup, the formation of polarized muonic hydrogen needs to be verified in a first experimental phase. Therefore a slow polarized muon beam would be required. These steps could start already at low intensity muon sources such as PSI or RAL where depending on the achievable polarization and fluxes also a first experiment (presumably with low... 

The transition state theory assumes that as the reacting species proceeds over the energy barrier, the medium adjusts rapidly enough to stay in equihbrium. Classical electron-transfer theory takes the first step away from this idea by distingnishing rapid and slow polarization dne to electronic and atomic motion, respectively. More recently, as faster reactions have been studied, interest has turned to the effects of rates of motion of solvent molecules. It is stiff possible to retain the notion of the solvent as a continnum, by introdncing polarizations that respond at... 

When a polymer dielectric is used, there is an additional complicating factor that slow polarization of the dielectric causes an instability in a direction opposite to the bias-stress instability and the hysteresis in organic semiconductors thus, there are two competing mechanisms, with a possible crossover between them after a certain stress period [15,16]. Slow polarization in a polymer dielectric is often due to residual polar solvent in the dielectric or water absorption from the air. It is natural to characterize this type of dielectric behavior by analyzing the frequency-dependent capacitance C(f) at sufficiently low frequencies. 

Applying of a ramp voltage on MIM structure creates the polarization current, which is a result of the slow polarization phenomena. A relaxation time of this polarization depends on the applied electric field and it is strongly related to the appearance of the partial discharges (Macur Domansky 1991). 

In practical sitiratiorrs, electrochemictJ systems are often more complex than the simple model assumed by the polarization resistance method. The presence of biofilms on the metal sirrface may introduce a capsrcitance as well as resisttmce to the interface. Moreover, the biofilm may introduce additional electrochemical reactions and adsorptive processes, which can lead to nonlinear polarization behavior. Even so, a polarization resistance value can be found as long as a sirfficiendy slow polarization scan rate (determined by the rate of the slowest reaction present) is used to maintain steady state conditions and a correction can be made for solution smd biofilm resistances. 

Polarization processes are extremely important in HR lluicls. Generally, there are four kinds of polarizations in a non-aqueous system containing no electrolytes or ions. They are electronic, atomic, Debye and the interfacial polarizations (the Wagner-Maxwell polarization). If the particulate material is an ionic solid, ionic displacement polarization should also be considered. The Debye and the intcrfacial polarizations arc rather slow processes as compared with electronic and the atomic polarizations. Usually, the former two polarizations arc called the slow polarizations, appearing at low frequency fields, whereas the last two are termed fast polarizations, appearing at high frequencies. 

For a heterogeneous system, slow polarizations should always occur once an electric field is applied. Those slow polarizations include the Debye polarization, the interfacial polarization, and the electrode polarization if there is electrolyte in the system. Those slow polarizations will result in an unstable dc current dc current decays with time and finally become stable, as shown in Figure 28. Once the applied electric field is turned off, the dc current goes down to zero quickly and continuously drops to a negative current and then gradually decays to zero. The charge and discharge curves... 

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