Dielectric permittivity particle-size dependence

Fig. 2.6 The particle size dependence of dielectric permittivity of BaTi03 powders [22]...

Electrophoresis — Movement of charged particles (e.g., ions, colloidal particles, dispersions of suspended solid particles, emulsions of suspended immiscible liquid droplets) in an electric field. The speed depends on the size of the particle, as well as the -> viscosity, -> dielectric permittivity, and the -> ionic strength of the solution, and it is directly proportional to the applied electric field. In analytical as well as in synthetic chemistry electrophoresis has been employed to separate species based on different speeds attained in an experimental setup. In a typical setup the sample is put onto a mobile phase (dilute electrolyte solution) filled, e.g., into a capillary or soaked into a paper strip. At the ends of the strip connectors to an electrical power supply (providing voltages up to several hundred volts) are placed. Depending on their polarity and mobility the charged particles move to one of the electrodes, according to the attained speed they are sorted and separated. (See also - Tiselius, - electrophoretic effect, - zetapotential). 

The permittivity of these ceramic dielectrics varies between 1000 and 12 000 depending upon the particle size of the BaTi03-powder and its chemical composition. These also affect the temperature dependence of its permittivity. 

The extended simple point charge (SPC/E) model [59] is used. This model is known to give reasonably accurate values of static dielectric permittivity of liquid water at ambient conditions [60]. The MD simulations were performed for both H2O and D2O with the system size of 1024 particles at 220 K, 240 K, 267 K, 273 K, 300 K, and 355 K. The parallel molecular dynamics code for arbitrary molecular mixtures (DynaMix) is implemented by Lyubartsev and Laaksonen [61]. The simulations have been carried out on a Linux cluster built on the Tyan/Opteron 64 platform, which enables calculations of relatively long trajectories for a system of 1024 water molecules. The simulation run lengths depend on temperature and are in the range between 1 ns and 4 ns for the warmest and coldest simulation, respectively. As the initial condition was a cubic lattice, the equilibration time was chosen to be temperature dependent in the range from 200 ps at 355 Ktol ns at 200K. 

It is generally believed, that the critical parameters (temperature and radius of size-driven phase transition) of nanogranular ceramics can be extracted, e.g., from the position of dielectric permittivity maximum observed either at varying temperature and fixed mean radius or at varying radius and fixed temperature. However, the particles size distribution in real samples leads to the transition temperature distribution. This, in turn, is dependent on the parameters of size distribution function (see Eq. 3.89) and hence on the sample quality. The above dependences generate the essential scattering of critical parameters, obtained by different authors so that measured Td and Rcr depend rather on sample quality than correspond to actual physical values. 

T, R) is the temperature and size dependent dielectric permittivity of incipient ferroelectric nanoparticles of radius R, (x is the fermion effective mass, is the effective permittivity of the particle environment, 8q is the dielectric permittivity of vacuum (in SI units). Due to the high values of e(r, R) the radius (T, R) > 5 nm is much higher than the lattice constant a = 0.4 nm, proving the validity of the effective mass approximation as well as the self-consistent background for the introduction of dielectric permittivity in the continuous medium approach [60]. 

Contrary to binary oxides the exchange integral depends on the particle radius and temperature via the size and temperature dependence of the dielectric permittivity e TJt). In particular, one has to substitute s T,R) for 82 in Eq. (4.6) for exchange integral, where r is average nanoparticle radius. The calculations of dielectric permittivity dependence on temperature and particles size have been performed analogously to those described in Sects. 3.2.2.2 and 3.2.2.3 and lead to Barrett-type formula, which could be found in Ref. [48]. In Fig. 4.18 two roots (rji fixed temperature and particle... 

The dielectric permittivity (e) of solids often changes with the particle size. The dependence is a result of a complicated interplay of several factors, it can vary widely in magnitude and even have different sense. Let us consider the influence of different factors on e under transition from bulk solids to nanophases. 

For time-dependent electrical perturbation, the typical assumption is that the metal nanoparticle behaves as a dielectric, characterized by a frequency-dependent permittivity ( >). Permittivities experimentally determined on bulk sample are almost invariably used. They need to be corrected with terms depending on the particle size. In fact, when the size of the metal particle has the same order of magnitude of the mean free path of conduction electrons in the bulk of the solid (tens of nanometers), it is necessary to take into account the scattering of the electrons at the metal particle surface. This is one aspect of a more general class of phenomena, known as quantum size effects. They are tightly related to the confinement of electrons in the metal particle and hence to the loss of the band structures typical of a bulk metal. Since this phenomenon regards mainly the valence... 

The permittivity discussed so far depends only on frequencies and, through the relaxation times, on the particle size. It is well-known that the dielectric response of materials, in particular metal, is non-local, i.e. the polarization vector induced at a certain point depends on the values of the electric field in all other points. In the reciprocal space language, we can say that e(plane-waves in which the probing electric field can be decomposed. The permittivity of metals such as Ag, Au and Cu at optical frequencies mainly depends on the behavior of both the valence electrons, which is close to that of a free-electron gas, and the core of the metal. As we did before, the total dielectric constant of the metal ([Pg.239]

The amount of charge at the interface depends on the field strength and the dielectric properties (conductivity and permittivity) of the particle and the electrolyte. However, there is a slight asymmetry in the charge density on the particle which gives rise to an effective or induced dipole across the particle. Note that if the field is removed the dipole disappears, it is induced . The magnitude of the dipole moment depends on the amount of charge and the size of the particle. For a spherical particle in an electrolyte subject to a uniform applied electric field, three cases can be considered ... 

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