Kramers-Kronig Relations for Nonlinear Rheology: 1. General Expression and Implications

Abstract

The principle of causality leads to linear Kramers-Kronig relations (KKR) that relate the real and imaginary parts of the complex modulus $G^{\ast }$ through integral transforms. Using the multiple integral generalization of the Boltzmann superposition principle for nonlinear rheology, and the principle of causality, we derived nonlinear KKR, which relate the real and imaginary parts of the $n^{\text{th}}$ order complex modulus $G_n^{\ast }$. For $n$=3, we obtained nonlinear KKR for medium amplitude parallel superposition (MAPS) rheology. A special case of MAPS is medium amplitude oscillatory shear (MAOS); we obtained MAOS KKR for the third-harmonic MAOS modulus $G_{33}^{\ast }$; however, no such KKR exists for the first harmonic MAOS modulus $G_{31}^{\ast }$. We verified MAPS and MAOS KKR for the single mode Giesekus model. We also probed the sensitivity of MAOS KKR when the domain of integration is truncated to a finite frequency window. We found that that (i) inferring $G_{33}^{\prime \prime }$ from $G_{33}^{\prime }$ is more reliable than vice-versa, (ii) predictions over a particular frequency range require approximately an excess of one decade of data beyond the frequency range of prediction, and (iii) $G_{33}^{\prime }$ is particularly susceptible to errors at large frequencies.

22 pages, 4 figures

References in corpus (1)

• Medium Amplitude Parallel Superposition (MAPS) Rheology of a Wormlike Micellar Solution

Cited by in corpus (4)

• Large Amplitude Oscillatory Shear Study of a Colloidal Gel at the Critical State
• The Method of Harmonic Balance for the Giesekus Model under Oscillatory Shear
• Harmonic Balance for Differential Constitutive Models under Oscillatory Shear
• When Spectroscopies Speak the Same Language: Unifying Rheology, Electrochemical Impedance, and Dielectrics