Dielectric Theories

Fit of dielectric spectroscopy data with a set of \(N\) discrete equidistant Debye modes. The number of modes can be selected by pressing the Up/Down arrows in the theory window. The frequencies of the Debye modes are equally distributed in logarithmic scale between a minimum frequency, \(\omega_\text{min}\), and a maximum frequency, \(\omega_\text{max}\), which can be fixed or set free by ticking the corresponding checkboxes. The position of \(\omega_\text{min}\) and \(\omega_\text{max}\) can be changed by dragging the leftmost and rightmost modes with the mouse. The vertical position of the modes can be changed by dragging the yellow points.

Each mode contributes to the dielectric relaxation spectrum through the following formulas:

\[\begin{split}\epsilon_i'(\omega) &= \Delta\epsilon_i \frac{1}{1+ (\omega \tau_i)^2}\\ \epsilon_i''(\omega) &= \Delta\epsilon_i \frac{\omega \tau_i}{1+ (\omega \tau_i)^2}\end{split}\]

with \(\Delta\epsilon_i\) the amplitude and \(\tau_i\) the characteristic relaxation time of the mode \(i\) (inverse of the frequency).

The parameters of the theory are the number of modes (which is fixed by the user and is not minimized), \(\omega_\text{min}\), \(\omega_\text{max}\), and a value of \(\Delta\epsilon_i\) for each mode. \(\Delta\epsilon_i\) is calculated in logarithmic scale.

Warning

The theory can only be applied to one file per data set. If more than one file is active in the current data set, the theory will be applied to the first one in the list of active files.

Havriliak-Negami modes 

Summary 

Description 

Fit dielectric spectroscopy data with a set of \(N\) discrete Havriliak-Negami modes. Each mode contributes a relaxation strength \(\Delta\epsilon_i\), a characteristic time \(\tau_i\), and the common shape parameters \(\alpha\) and \(\gamma\).

The mode frequencies are equally distributed on a logarithmic scale between a minimum frequency, \(\omega_\text{min}\), and a maximum frequency, \(\omega_\text{max}\). RepTate stores these bounds as logwmin and logwmax and uses \(\tau_i=1/\omega_i\).

For a frequency \(\omega\), the implemented complex response is

\[\epsilon^*(\omega) = \epsilon_\infty + \sum_i \frac{\Delta\epsilon_i} {\left[1+\left(i\omega\tau_i\right)^\alpha\right]^\gamma}\]

The calculated \(\epsilon'\) column is the real part of this response and the calculated \(\epsilon''\) column is minus the imaginary part, matching the sign convention used in the implementation. The theory output keeps the same frequency points as the input file.

The adjustable parameters exposed in the theory table are:

einf: unrelaxed permittivity, \(\epsilon_\infty\).
alpha: asymmetry parameter.
gamma: broadness parameter.
logwmin and logwmax: base-10 logarithms of the frequency range bounds, expressed in rad/s.
logDe00, logDe01, …: base-10 logarithms of the mode strengths \(\Delta\epsilon_i\).

The number of modes, nmodes, is controlled from the theory toolbar and is not fitted directly. When the theory is created, the initial number of modes is estimated from the frequency span of the data. The initial mode strengths are interpolated from the first file in the data set. The View modes button shows or hides the yellow mode markers; dragging the markers changes the frequency range and mode strengths in the current view coordinates.

The Havriliak-Negami modes theory is available in the Dielectric application as Havriliak-Negami modes and is intended for .dls dielectric spectroscopy files with columns \(\omega\), \(\epsilon'\), and \(\epsilon''\).

Warning

The theory can only be applied to one file per data set. If more than one file is active in the current data set, the theory will be applied to the first one in the list of active files.

Kohlrausch-Williams-Watts (KWW) modes 

Summary 

Description 

Fit dielectric spectroscopy data with a set of \(N\) discrete Kohlrausch-Williams-Watts modes. Each mode is a Laplace-Fourier transform of a stretched-exponential relaxation contribution with a relaxation strength \(\Delta\epsilon_i\), a characteristic time \(\tau_i\), and a common stretching exponent \(\beta\).

The mode frequencies are equally distributed on a logarithmic scale between a minimum frequency, \(\omega_\text{min}\), and a maximum frequency, \(\omega_\text{max}\). RepTate stores these bounds as logwmin and logwmax and uses \(\tau_i=1/\omega_i\).

For a frequency \(\omega\), the implemented frequency-domain response is calculated as

\[\begin{split}\epsilon'(\omega) &= \epsilon_\infty + \sum_i \Delta\epsilon_i\,K_c(\omega\tau_i,\beta)\\ \epsilon''(\omega) &= \sum_i \Delta\epsilon_i\,K_s(\omega\tau_i,\beta)\end{split}\]

where \(K_c\) and \(K_s\) are the cosine and sine KWW transform functions evaluated by the bundled libkww library. The theory output keeps the same frequency points as the input file and fills the calculated \(\epsilon'\) and \(\epsilon''\) columns.

The adjustable parameters exposed in the theory table are:

einf: unrelaxed permittivity, \(\epsilon_\infty\).
beta: stretched-exponential parameter, limited in the source to the range 0.1 to 2.0.
logwmin and logwmax: base-10 logarithms of the frequency range bounds, expressed in rad/s.
logDe00, logDe01, …: base-10 logarithms of the mode strengths \(\Delta\epsilon_i\).

The number of modes, nmodes, is controlled from the theory toolbar and is not fitted directly. When the theory is created, the initial number of modes is estimated from the frequency span of the data. The initial mode strengths are interpolated from the first file in the data set. The View modes button shows or hides the yellow mode markers; dragging the markers changes the frequency range and mode strengths in the current view coordinates.

The KWW modes theory is available in the Dielectric application as KWW modes and is intended for .dls dielectric spectroscopy files with columns \(\omega\), \(\epsilon'\), and \(\epsilon''\).

Warning

The theory can only be applied to one file per data set. If more than one file is active in the current data set, the theory will be applied to the first one in the list of active files.