Electric polarization time-dependent case

An alternative approach that avoids many of the problems associated with electric polarization in dielectrics is TD-DFT. In the time-dependent case, the time change of the polarization induces a current, which may be considered an ultra-nonlocal functional of the charge density, and has been successfully used as an alternative additional variable for the description of dielectric properties of both solids and molecular systems. 

Relaxation processes are probably the most important of the interactions between electric fields and matter. Debye [6] extended the Langevin theory of dipole orientation in a constant field to the case of a varying field. He showed that the Boltzmann factor of the Langevin theory becomes a time-dependent weighting factor. When a steady electric field is applied to a dielectric the distortion polarization, PDisior, will be established very quickly - we can say instantaneously compared with time intervals of interest. But the remaining dipolar part of the polarization (orientation polarization, Porient) takes time to reach its equilibrium value. When the polarization becomes complex, the permittivity must also become complex, as shown by Eq. (5) ... 

Suppose now that two different two-step fields are applied (1) at t0 the electric field 2EX is turned on and at tx the field suddenly changes to Ex and (2) the field at tx is the same as in (1) but between t0 and tx it is Ex/2. It is clear from Fig. 9.14 that the polarization in these two cases is different, even for t > tx when the applied field is the same that is, the polarization at time t depends on the history of the applied field and not merely its instantaneous value. This is a specific example of a general conclusion that we made about the response of a linear medium to a time-dependent electric field (see Section 2.3). The polarization at all times greater than tx can be obtained by following reasoning similar to that which led to (9.37) we write P(t) in the form (9.36) and require that lim,A, P(f) = Pd(tx) + e0x0vEx the result is... 

The measurement of the polarization properties of light can be automated and improved by introducing a modulation of the polarization. Here a regular, time-dependent variation is introduced onto the optical properties of certain devices within either (or both) the PSG or PSA sections of the instrument. The modulation can be one of two types rotation of an optical element with fixed optical properties, or the modulation of the optical properties (retardation, for example) of an element with a fixed orientation. These are referred to as rotary modulators or field effect modulators, respectively. The latter name reflects the use of external fields (stress, electric or magnetic) to impart the modulation in these devices. In any case, a periodic oscillation is introduced into the signals that are measured that can effectively isolate specific optical properties in the sample. 

However the precise sequence of coordinate participation in the reaction path is solvent dependent. For the case just discussed, the water solvent is rapid, largely because of the small moment of inertia involved in the water molecule reorientations underlying the change of the electrical polarization. Dimethyl formamide (DMF) solvent is less rapid, and the resulting coordinate sequence on the way to the TS [3] is again in the order of decreasing slowness, but now the solvent coordinate is the slowest of the three, followed by the bend angle and finally the C - Cl bond stretch. The reaction path depends on the solvent time scale. 

The objective of this monograph is to describe and interpret the time dependence of the electrical response of dielectrics. Interpretation is difficult because the observable relationship between polarization and field is simple in the cases relevant for dielectric relaxation and because the measurements have relatively little information content. The response of the dielectric can be described by a set of linear differential equations and many models can be described which correspond to the same differential equations. When the dielectric relaxation of a given material has been measured the investigator is in the position of a man presented with a black box which has two terminals. He may apply alternating fields of various kinds and he may heat the box but he is not allowed to look inside. And he finds that the box behaves as if it contained a combination of capacitors and resistors. 

We now analyze a case where we have an instantaneous increase or a reduction of the electric field, E. This will lead to a polarization or depolarization process, which will follow with some delay or retardation due to the increase or reduction of the electric field, respectively. Consequently, in relation with a time-dependent variation of the electric field, E = E(t), the dielectric properties of the materials become dynamic events. In this regard, the time dependency of P = P(t) will not be the same as that of E = E (t), since the different polarization processes have different time delays, with respect to the appearance of the electric field. This delay is obviously related to the time-dependent behavior of the susceptibility % = %(t). 

In the macroscopic case, we will instead consider the polarization P (z) of the medium by a time-dependent electric field, and we may expand the polarization in orders of the applied external field as... 

The second approach to the approximate description of the dynamic solvation effects is based on the semiempirical account for the time-dependent electrical polarization of the medium in the field of the solute molecule. In this case, the statistical averaging over the solute-solvent intennolecular distances and configurations is presumed before the solution of the SchrOdinger equation for the solute and correspondingly, the solvent is described as a polarizable dielectric continuum. The respective electrostatic solvation energy of a solute molecule is given by the following equation[13]... 

Physical origin of dielectric loss The foregoing conclusions correspond to a static description or cases for which the polarization can perfectly follow the oscillation of the electric field. Indeed, the electric field orientation depends on time with a frequency equal to 2.45 GHz (the electric field vector switches its orientation approximately every 10 s). The torque exercised by the electric field induces rotation of polar molecules, but they cannot always orient at this rate. The motion of the particles will not be sufficiently rapid to build up a time-dependent polarization P(t) that is in equilibrium with the electric field at any moment. This delay between electromagnetic stimulation and molecular response is the physical origin of the dielectric loss. 

Equation (12) is very widely applicable. As an action A(r), mechanical, electrical, or magnetic force fields may be considered. Even the response of the polymer to a temperature jump can be treated this way. As a response R t), the mechanical compliance or modulus ouy be used. In tbe dielectric case, the external electric field in the classical meaning may be used as tbe action, that is, A(r) s (i) in V/m. As a response, the dielectric polarization field P l), expressed in terms of tbe pennittivity by Eq. (9X or the displacement >(r) > c c(0 E(t) may be used. Substituting these functions into Eq. (12). integrating by parts and considering the limiting values, one obtains for the time-dependent dielectric permittivity ... 

The polarization of the electric field is determined by the motion of the elementary charge producing the time-dependent dipole moment. If the electron is performing linear oscillations, equations (2.69) and (2.70) show that the radiation fields are also linearly polarized. In the more general case of elliptical or circular motion, the... 

In addition to internal charge transport in transient cases the external current can also be compensated by dielectric effects (displacement current by local polarization). This transient charging current is characterized by electrical capacitances. They are considered in more detail in Section 7.3.3. Capacitive effects (charge storage) are generally responsible for time dependences. Apart from dielectric effects, storage phenomena can also occur if the stoichiometry changes by virtue of the current flow. Such chemical capacitances (see Section 6.7.4) will be treated in more detail in Section 7.3.4 (cf. also Section 7.4). 

Below we first offer some brief remarks on SDFT, and then turn to the more complex problems posed by orbital currents, described by CDFT. Both, SDFT and CDFT also have a relativistic version, on which we make some brief comments in Sect. V11.2. Next, we briefly consider the simpler (but still not trivial) case of external electric fields. Finally, we describe the peculiar problems arising in the calculation of the electric polarization and the orbital magnetization of extended systems. We do not address, in this chapter, time-dependent electromagnetic fields, for which we refer the reader to reviews of the large and flourishing field of TD-DFT. ... 

If the polarization of a given point in space and time (r, t) depends only on the driving electric field at the same coordmates, we may write tire polarization as P = P(E). In this case, we may develop the polarization m power series as P = = P - + P - + P - +, where the linear temi is = X] Jf/ Pyand the... 

In the first case, that is with dipoles integral with the main chain, in the absence of an electric field the dipoles will be randomly disposed but will be fixed by the disposition of the main chain atoms. On application of an electric field complete dipole orientation is not possible because of spatial requirements imposed by the chain structure. Furthermore in the polymeric system the different molecules are coiled in different ways and the time for orientation will be dependent on the particular disposition. Thus whereas simple polar molecules have a sharply defined power loss maxima the power loss-frequency curve of polar polymers is broad, due to the dispersion of orientation times. 

Fig. 51—Dependence of electric current on time for different EEF polarities. Liquid glycerin. Load 4 N. Line (a) and (b) represent the positive and negative cases, respectively. Positive EEF intensities of 518.6 kV/cm during the initial period of 60s then...

As the existence of MChA can be deduced by very general symmetry arguments and the effect does not depend on the presence of a particular polarization, one may wonder if something like MChA can also exist outside optical phenomena, e.g. in electrical conduction or molecular diffusion. Time-reversal symmetry arguments cannot be applied directly to the case of diffusive transport, as diffusion inherently breaks this symmetry. Instead, one has to use the Onsager relation. (For a discussion see, e.g., Refs. 34 and 35.) For any generalized transport coefficient Gy (e.g., the electrical conductivity or molecular diffusion tensor) close to thermodynamic equilibrium, Onsager has shown that one can write... 

---