Cole-Cole diagram

Cole-Cole diagram of the dielectric function of water at 0 ° C

Current data on the permittivity of water (20 ° C), here as a Bode diagram

The Cole-Cole diagram (also: Cole-Cole circle or Cole-Cole plot ) represents complex material parameters, such as impedances or the permittivity of dielectric materials, as a locus in the Gaussian plane as a function of frequency (see also: phasor diagram ). A Cole-Cole diagram in the narrower sense shows the permittivity, while a Nyquist diagram in the narrower sense shows the impedance. In many cases, the terms Cole-Cole diagram and Nyquist diagram are used synonymously and generally for the representation of these frequency-dependent quantities in the Gaussian plane. The Cole-Cole diagram is also similar to the Smith diagram that was designed a few years later as an aid to RF technology .

Historical

The name of the Cole-Cole diagram comes from the two brothers Kenneth S. Cole and Robert H. Cole , who from 1931 together carried out experimental investigations into the impedance of biological tissue. In 1941 they published a pioneering discussion of the frequency dependence of permittivity , which they supplemented in another joint work in 1942. The biophysicist Kenneth S. Cole had used the diagram as early as 1928.

Properties and meaning

A typical Cole-Cole diagram describes a semicircle with the center point on the real axis (see picture). The real part of the relative permittivity (dielectric constant ) is read on the abscissa of the Cole-Cole diagram and its negative imaginary part (dielectric losses) is read on the ordinate . ${\displaystyle \varepsilon '}$ ${\displaystyle \varepsilon =\varepsilon '-j\varepsilon ''}$ ${\displaystyle \varepsilon ''}$

The relative permittivity of substances depends on temperature and frequency. The frequency dependence can be represented as a Cole-Cole diagram according to the following relationship, where the angular frequency and i is the imaginary unit . ${\displaystyle \varepsilon }$ ${\displaystyle \omega }$

${\displaystyle \varepsilon =\varepsilon _{\infty }+{\frac {\varepsilon _{s}-\varepsilon _{\infty }}{1+(i\omega \tau )^{1-\alpha }}}}$

The locus is approximately a semicircle, the position and size of which depends on four parameters which, for the example of water at room temperature or the dielectrically very similar muscle tissue, have the following values:

• The static dielectric constant , i.e. the relative permittivity of the dielectric at a frequency of 0 Hz ${\displaystyle \varepsilon _{s}}$ ${\displaystyle \varepsilon _{s}\approx 80}$
• The relative permittivity at very high frequencies ${\displaystyle \varepsilon _{\infty }\approx 6}$
• The relaxation time constant ${\displaystyle \tau \approx 10~\mathrm {ps} }$
• The Cole exponent, it is for muscle tissue and for water ${\displaystyle 1-\alpha \approx 0{,}8}$ ${\displaystyle 1-\alpha \approx 1}$

Some important characteristic parameters of the dielectric under investigation can be derived from the Cole-Cole diagram. The Cole exponent , the relaxation time or its reciprocal value are used for this . The Cole-Cole circle has two real intersections with the abscissa. The locus has its maximum at the resonance frequency . The figure (above) shows the locus curve of the relative permittivity of water for a temperature of 0 ° C. At this temperature is and . ${\displaystyle \alpha }$ ${\displaystyle \tau }$ ${\displaystyle \omega _{c}=1/\tau }$ ${\displaystyle \omega _{c}}$ ${\displaystyle \varepsilon _{s}\approx 88}$ ${\displaystyle \varepsilon _{\infty }\approx 4}$

See also

• Debye equation

literature

• Kenneth S. Cole: Electric Phase Angle of Cell Membranes ( online , PDF, 9 pages)
• Application note from Agilent : Basics of Measuring the Dielectric Properties of Materials. Pp. 13-19. ( online , PDF; 0.5 MB)
• The Complex Dielectric Constant of Pure and Sea Water from Microwave Satellite Observations (PDF; 491 kB)

Web links

Wikibooks: Collection of tables chemistry / material data water  - learning and teaching materials

Individual evidence

1. ↑ Kenneth Stewart Cole: Citation Classic - Dispersion and Absorption in Dielectrics .1. Alternating-Current Characteristics . In: Current Contents / Physical Chemical & Earth Sciences . tape 3 , January 21, 1980, p. 61 ( Citation Classic Commentaries on the Eugene Garfield site [PDF; accessed June 27, 2015]).
2. ↑ Kenneth Stewart Cole, Robert H. Cole: Dispersion and Absorption in Dielectrics I. Alternating Current Characteristics . In: American Institute of Physics (Ed.): The Journal of Chemical Physics . tape 9 , no. 4 . AIP Publishing, doi : 10.1063 / 1.1750906 .
3. ↑ Kenneth Stewart Cole, Robert H. Cole: Dispersion and Absorption in Dielectrics - II Direct Current Characteristics . In: American Institute of Physics (Ed.): The Journal of Chemical Physics . tape 10 , no. 2 . AIP Publishing, S. 98-105 , doi : 10.1063 / 1.1723677 .
4. ↑ Kenneth Stewart Cole: Electric impedance of suspensions of spheres . In: The Rockefeller Institute for Medical Research (Ed.): The Journal of General Physiology . tape 12 , no. 1 . The Rockefeller University Press, September 20, 1928, pp. 29-36 , PMID 19872446 , PMC 2323685 (free full text).