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Better Resolved Low Frequency Dispersions by the Apt Use of Kramers-Kronig Relations, Differential Operators, and All-In-1 Modeling.

van Turnhout J - Front Chem (2016)

Bottom Line: It proved also useful to go around the KK conversion altogether.The all-in-1 minimization turned out to be also highly useful for the dielectric modeling of a suspension with the complex dipolar coefficient.It guarantees a secure correction for the electrode polarization, so that the modeling with the help of the differences ε' and ε″ can zoom in on the genuine colloidal relaxations.

View Article: PubMed Central - PubMed

Affiliation: Department of Chemical Engineering, Sect. Organic Materials and Interfaces, Delft University of Technology Delft, Netherlands.

ABSTRACT
The dielectric spectra of colloidal systems often contain a typical low frequency dispersion, which usually remains unnoticed, because of the presence of strong conduction losses. The KK relations offer a means for converting ε' into ε″ data. This allows us to calculate conduction free ε″ spectra in which the l.f. dispersion will show up undisturbed. This interconversion can be done on line with a moving frame of logarithmically spaced ε' data. The coefficients of the conversion frames were obtained by kernel matching and by using symbolic differential operators. Logarithmic derivatives and differences of ε' and ε″ provide another option for conduction free data analysis. These difference-based functions actually derived from approximations to the distribution function, have the additional advantage of improving the resolution power of dielectric studies. A high resolution is important because of the rich relaxation structure of colloidal suspensions. The development of all-in-1 modeling facilitates the conduction free and high resolution data analysis. This mathematical tool allows the apart-together fitting of multiple data and multiple model functions. It proved also useful to go around the KK conversion altogether. This was achieved by the combined approximating ε' and ε″ data with a complex rational fractional power function. The all-in-1 minimization turned out to be also highly useful for the dielectric modeling of a suspension with the complex dipolar coefficient. It guarantees a secure correction for the electrode polarization, so that the modeling with the help of the differences ε' and ε″ can zoom in on the genuine colloidal relaxations.

No MeSH data available.


Related in: MedlinePlus

Inserting Dl = lnEl/ln2 in a symbolic operator provides a versatile tool to get an expansion in El, which in turn provides an easy-to-use conversion frame, in this case for ε″ to dε′∕dlnω.
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Figure 12: Inserting Dl = lnEl/ln2 in a symbolic operator provides a versatile tool to get an expansion in El, which in turn provides an easy-to-use conversion frame, in this case for ε″ to dε′∕dlnω.

Mentions: The use of these unusual symbolic operators looks at first sight not easy. The job was facilitated a lot by inserting for Dl = lnEl/lnh in the tan and cot operator. Let us illustrate this for Equation (4.10), see also Figure 12. If we take the logarithmic derivative of both sides, we get


Better Resolved Low Frequency Dispersions by the Apt Use of Kramers-Kronig Relations, Differential Operators, and All-In-1 Modeling.

van Turnhout J - Front Chem (2016)

Inserting Dl = lnEl/ln2 in a symbolic operator provides a versatile tool to get an expansion in El, which in turn provides an easy-to-use conversion frame, in this case for ε″ to dε′∕dlnω.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4862991&req=5

Figure 12: Inserting Dl = lnEl/ln2 in a symbolic operator provides a versatile tool to get an expansion in El, which in turn provides an easy-to-use conversion frame, in this case for ε″ to dε′∕dlnω.
Mentions: The use of these unusual symbolic operators looks at first sight not easy. The job was facilitated a lot by inserting for Dl = lnEl/lnh in the tan and cot operator. Let us illustrate this for Equation (4.10), see also Figure 12. If we take the logarithmic derivative of both sides, we get

Bottom Line: It proved also useful to go around the KK conversion altogether.The all-in-1 minimization turned out to be also highly useful for the dielectric modeling of a suspension with the complex dipolar coefficient.It guarantees a secure correction for the electrode polarization, so that the modeling with the help of the differences ε' and ε″ can zoom in on the genuine colloidal relaxations.

View Article: PubMed Central - PubMed

Affiliation: Department of Chemical Engineering, Sect. Organic Materials and Interfaces, Delft University of Technology Delft, Netherlands.

ABSTRACT
The dielectric spectra of colloidal systems often contain a typical low frequency dispersion, which usually remains unnoticed, because of the presence of strong conduction losses. The KK relations offer a means for converting ε' into ε″ data. This allows us to calculate conduction free ε″ spectra in which the l.f. dispersion will show up undisturbed. This interconversion can be done on line with a moving frame of logarithmically spaced ε' data. The coefficients of the conversion frames were obtained by kernel matching and by using symbolic differential operators. Logarithmic derivatives and differences of ε' and ε″ provide another option for conduction free data analysis. These difference-based functions actually derived from approximations to the distribution function, have the additional advantage of improving the resolution power of dielectric studies. A high resolution is important because of the rich relaxation structure of colloidal suspensions. The development of all-in-1 modeling facilitates the conduction free and high resolution data analysis. This mathematical tool allows the apart-together fitting of multiple data and multiple model functions. It proved also useful to go around the KK conversion altogether. This was achieved by the combined approximating ε' and ε″ data with a complex rational fractional power function. The all-in-1 minimization turned out to be also highly useful for the dielectric modeling of a suspension with the complex dipolar coefficient. It guarantees a secure correction for the electrode polarization, so that the modeling with the help of the differences ε' and ε″ can zoom in on the genuine colloidal relaxations.

No MeSH data available.


Related in: MedlinePlus