Dielectric mismatch

Consider a point charge Q inside a semi-infinite medium of dielectric constant 6i at a distance d away from a planar interface that separates the first medium 

Therefore, an effective dielectric constant gg may be identified in describing the electric potential at distances far away from Q, by writing the above equation as 

If the dielectric constants of an aqueous medium and an oily cavity, such as the interior of a protein or a hydrophobic polymer, are taken to be 80 and 3, respectively, then the effective dielectric constant at large distances is roughly 40. Thus, the apparent dielectric constant can be different by a factor of two. The complementary problem of how a charge buried in an oily enclosure is subjected to an attractive force by its image charge, which is now present in the surrounding aqueous medium with higher dielectric constant, can be readily addressed from the above equations. 

Analogous calculations of the effects of dielectric mismatch due to curved interfaces become highly technical. Nevertheless, if one is interested in very accurate estimates of various forces near interfaces with dielectric mismatch. 

Charged interfaces in electrolyte solutions cause counterions to adsorb. A cloud with net opposite charge hovers around the interface, with a characteristic thickness comparable to the Debye length. At distances much larger than the Debye length, the effect of charged interfaces is essentially absent. 

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The presence in the cluster of a positively charged impurity has also been considered, analyzing, by first principles, the screening due to the Si-NCs [123,124]. A reduction of screening in Si nanostructures with respect to bulk Si has been already observed [52] and predicted [125]. This reduction is a fundamental process at the basis of the enhancement of both the electron-hole interaction and the impurity activation energies in nanosized objects, and is due to the fact that close to the surface there is a dielectric dead layer, with a finite-range reduction of the dielectric constant due to the dielectric mismatch at the nanocrystal-environment interface. 

We consider the cylindrical nanowire geometry shown in Fig. 17.1, with an incident plane wave normal to the cylinder axis and with an amplitude Eg. This is the simplest case to solve analytically and the one most often treated in experimental spectroscopic investigations of single nanowires. Possible orientations of linearly polarized incident light with respect to the wire axis are bounded by two cases. The first is the transverse magnetic (TM) polarization where the electric field is polarized parallel to the wire axis, and the second is the transverse electric (TE) polarization where the electric field is polarized perpendicularly to the wire axis. In TM polarization, the condition of continuity of the tangential electric field is expected to maximize the internal field, while in TE polarization, the dielectric mismatch should suppress the internal field. The incident plane wave may be expanded in cylindrical functions as ... 

The role of the dielectric mismatch between solvent and pore can be investigated via a simple model DNA, consisting of a rigid charged DNA fragment, where the translocation fi ee energy barrier can be easily computed see Fig. 17 for a schematic plot of the situation. 

Upon pore condensation with, say, ethanol, the effective dielectric constant of the medium surrounding the nanostructured Si changes and increases from 1 (vacuum) to 25 (ethanol). A simple model based on dielectric mismatch is discussed by Timoshenko et al. (2001) shows that in such conditions the activation energy of a B acceptor decreases from 105 to 30 eV. It should be kept in mind that B impurities are present in large concentrations in porous Si samples. 

Cultrera A, Boarino L, Amato G, Bordiga S, Mercuri F, Cartoixa X, Rurali R (2013) Molecular doping and gas sensing in Si nanowires from charge injection to reduced dielectric mismatch. JAppl Phys 114 204302... 

Density-functional theory (DFT) 2 Dielectric mismatch effect 3 Electron spin resonance (ESR) 1 Gas and liquid doping 1 Molecular doping 1... 

The conduction model is thought to be only valid for ER. suspensions in reaction with dc or low frequency ac fields. For high frequency ac fields, the polarization model is dominant [55,56]. As shown in Eq. (25) and (26), once the Wagncr-Maxwcll polarization is taken into account, the parameter P is detennined by the conductivity mismatch in dc or low frequency ac fields, and by the dielectric mismatch in high frequency fields (the low or high frequency is relative to the relaxation time of the Wagner-Maxwell polarization). The parameter p in the conduction model is ... 

Fig. 11 HOMO and LUMO energies for the rubreneA so interface system for different locations relative to the interface. The dielectric mismatch at the interface causes the rubrene HOMO and LUMO levels to pinch together and the C o HOMO and LUMO levels to pull apart. Reprinted with permission from [138]. Copyright 2013 American Chemical Society...

At the pore mouth, force fields may be generated by chemical decoration of the inside surface of the pore. More importantly, steep electric potential gradients may occur at the pore mouth due to the dielectric mismatch between the layer in which the pore is embedded and the rest of the system, in the presence of an applied voltage gradient. Furthermore, strong flow currents may arise at the... 

In realistic situations, the dielectric constants of the media across the interfaces are different. Consideration of the dielectric mismatch is quite important in assessing the quantitative aspects of forces among ions in the neighborhood of interfaces. A full treatment of dielectric heterogeneity (Frohlich 1958, Verwey and Overbeek 1999) in electrolyte solutions is beyond the scope of this book. Nevertheless, we shall consider one example in Section 3.2.6 to illustrate the effects due to dielectric mismatch across interfaces. Our primary... 

Figure 3.18 Effects of dielectric mismatch. The image charge is O. The potential due to Q at a distance r depends on whether r is in medium of ej or medium of 2-...

It must be cautioned that even the notion of the dielectric constant at such nanoscopic length scales is not accurate and it is necessary to compute the polarization forces. However, such calculations are yet to be performed for the ill-structured heterogeneous suspensions of polyelectrolyte molecules. Therefore, is taken to be different and smaller than the bulk value, in recognizing the existence of dielectric heterogeneity in these solutions. Since we do not know the value of q and the ion-pair distance d inside the worm, we combine these two quantities and define a parameter called the dielectric mismatch parameter b as... 

Energy of adsorbed ions. The gain in energy due to an adsorbed ion at the chain backbone depends on the microscopic details such as the ionic radii and the local dielectric constant, as discussed in Section 4.1.5. Using the phenomenological dielectric mismatch parameter defined in Equation 4.10, the free energy associated with the formation of ion pairs follows from Equation 4.11 as... 

This closed-form result for a readily gives its dependence on b (i.e., T (T)), the salt concentration Cs, the monomer concentration p, and the dielectric mismatch parameter 5. The value of a given by Equation 4.65 is substituted into Equation 4.55 to calculate the expansion factor li. A comparison of a and ii thus calculated with the separation approximation and with the numerical computation with full coupling reveals that the separation approximation is very good as long as the chain is not collapsed below the Gaussian chain size. 

The total free energy f = Fi -I- F2 -I- F3 -I- F4 -I- F5 is given in terms of the fraction of adsorbed counterions and coions (a s), the size of the polymer (h), temperature and the bulk dielectric constant (1b), the degree of polymerization (N), the monomer density (g), monovalent and divalent salt concentrations (Cj), and local dielectric mismatch parameters (6 and 62). The goal is to selfconsistently determine the fractions of the adsorbed ions (ai,a2, and af) and the size (Rg = (Nlli/6)) that minimize the free energy. Note that, for the salt-free case or if only the monovalent salt is present, there are only two variables, Qj and Rg, which require selfconsistent determination. Further, 62 does not play any role for monovalent salts. Therefore, it is a simultaneous minimization with respect to two variables... 

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