H-function representations for stretched exponential relaxation and non-Debye susceptibilities in glassy systems

Area of Science:

• Condensed matter physics
• Dielectric spectroscopy
• Materials science

Background:

• Non-Debye relaxation processes are crucial for understanding material behavior.
• Existing models often rely on approximations or lack general analytical solutions.
• Accurate characterization of relaxation dynamics is essential across various scientific disciplines.

Purpose of the Study:

• To derive general analytical expressions for non-Debye relaxation processes in both time and frequency domains.
• To provide a unified framework for analyzing stretched exponential and other common relaxation functions.
• To investigate the validity of established relationships between different relaxation parameters.

Main Methods:

• Derivation of complex frequency-dependent susceptibility functions using H-functions for stretched exponential relaxation.
• Transformation of frequency-domain susceptibility functions (Cole-Cole, Cole-Davidson, Havriliak-Negami) into time-domain relaxation functions.
• Mathematical analysis to compare and contrast different relaxation models.

Main Results:

• Analytical expressions for the complex frequency-dependent susceptibility of the stretched exponential relaxation function are presented for all stretching exponent values.
• Time-domain relaxation functions corresponding to Cole-Cole, Cole-Davidson, and Havriliak-Negami susceptibilities are derived using H-functions.
• A commonly assumed correspondence between the stretching exponent of Kohlrausch functions and the stretching parameters of Havriliak-Negami susceptibilities is shown to be generally invalid.

Conclusions:

• The derived analytical expressions offer a more comprehensive tool for analyzing non-Debye relaxation phenomena.
• The findings necessitate a re-evaluation of established parameter correlations in dielectric relaxation studies.
• This work provides a rigorous mathematical foundation for understanding complex relaxation dynamics in materials.