Sigmoidal dielectric function

Plotting ixbase VS. pH gives a sigmoidal curve, whose inflection point reflects the apparent base-pAi, which may be corrected for ionic strength, I, using Equation 6.11 in order to obtain the thermodynamic pATa value in the respective solvent composition. Parameters A and B are Debye-Hiickel parameters, which are functions of temperature (T) and dielectric constant (e) of the solvent medium. For the buffers used, z = 1 for all ions ao expresses the distance of closest approach of the ions, that is, the sum of their effective radii in solution (solvated radii). Examples of the plots are shown in Figure 6.12. 

Figures 3-5 that the dielectric relaxation again reveals only a single a relaxation for the mixtures. These are, however, noticeably broader than the a relaxation of the pure polymers. The temperatures of the loss maxima, when plotted (Figure 7) as a function of wu the weight fraction of PPO in the mixtures, do not display the smooth monotonic increase in T0 vs. Wi that was shown by both the Vibron and the DSC results. Instead, there is a pronounced increase in Tg above = 0.5 to give a sigmoid curve for this relation. Some reservations should be attached to this observation inasmuch as data for only three polyblend compositions are available nevertheless a qualitatively similar phenomenon is observed in the analysis of the intensity of the y peak (below). Further, if only the stronger maxima in the dynamical mechanical data are considered— i.e.y if the secondary peaks and shoulders which led to the identification of two phases are omitted—then a similar sigmoid curve is found. The significance of this observation is discussed later.

This model has been, and still is, widely used especially for some specific applications. An older use is in the description of dielectric saturation effects around ions. The origin is the Debye model, not completely satisfying and thus subjected over the years to many variants. The spherical symmetry of the problem suggests the use of a distance dependent function s(r). The functions belonging to this family are often called sigmoidal functions because their spatial profile starts from a low value and increases monotonically to reach... 

The simple distance-dependent dielectric has no physical basis and so it is not generally recommended, except when no alternative is possible. More sophisticated distance-dependent functions can also be employed. Many of these have an approximately sigmoidal shape in which the relative permittivity is low at short distances and then rises towards the bulk value at long distances. One example of such a function is [Smith and Pettit 1994] ... 

Jayaram et al. performed a systematic study of the effects of electrostatic interactions on the counterion condensation around DNA. They used a 20-mer of electrically neutral sodium-DNA, with the DNA fixed in its canonical B form. The mobile counterions were placed randomly in a 50 A radius cylinder around the DNA, and the solvent was modeled as a dielectric continuum. Four dielectric treatments, ranging from Coulombic interactions with constant dielectric to a dielectric saturation model with a modified Coulombic potential introducing dielectric discontinuity, were studied. The dielectric saturation model used a modified Hingerty sigmoidal function... 

---