Self-Consistent Fourier-Tschebyshev Representations of the First Normal Stress Difference in Large Amplitude Oscillatory Shear

1. Problem Statement

Large Amplitude Oscillatory Shear (LAOS) is a standard technique for characterizing the nonlinear viscoelasticity of soft materials. However, the vast majority of LAOS research focuses exclusively on the shear stress response. While normal stress differences (specifically the first normal stress difference, $N_1$) are known to become significant in highly elastic materials (e.g., polymer melts) at large strains, they remain underutilized in rheological analysis due to:

• Lack of Standardization: There is no unified nomenclature or framework for decomposing the higher-order spectral content of $N_1$ in LAOS.
• Measurement Complexity: $N_1$ is difficult to measure accurately, often requiring specialized geometries to avoid edge fracture, and the signals are often dominated by noise at low strains.
• Missing Self-Consistency: While theoretical links exist between shear stress and $N_1$ in the quasilinear limit (Small Amplitude Oscillatory Shear, SAOS), these relationships have not been systematically verified across the full range of LAOS strain amplitudes for complex materials.
• Data Efficiency: Traditional methods of mapping material functions across strain amplitudes and frequencies are time-consuming, which is problematic for time-sensitive or degrading materials.

2. Methodology

The authors propose a comprehensive framework to quantify $N_1$ in LAOS, combining theoretical development, numerical modeling, and advanced experimental techniques.

A. Theoretical Framework: Fourier-Tschebyshev Decomposition

The authors extend the established Fourier-Tschebyshev representation (previously used for shear stress) to the first normal stress difference.

• Notation: They propose a unified notation where $N_1$ is expanded in terms of even harmonics (due to symmetry) using Chebyshev polynomials ($T_k$).
  $$N_1 = \gamma_0^2 \left[ n_1^d + \sum_{k=2, \text{even}}^{\infty} \left( n_{1,k}' \cos(k\omega t) + n_{1,k}'' \sin(k\omega t) \right) \right]$$
  Here, $n_1^d$ is the DC (zeroth) term, and $n_{1,k}'$ and $n_{1,k}''$ represent the elastic and viscous coefficients for the $k$-th harmonic.
• Self-Consistency Relations: The paper derives asymptotic connections between the shear stress material functions ($G', G''$) and the $N_1$ coefficients in the quasilinear limit ($\gamma_0 \to 0$). For example:
  $$n_1^d(\omega) = G'(\omega)$$
  $$n_{1,2}'(\omega) = -G'(\omega) + \frac{1}{2}G'(2\omega)$$
  These relations serve as a rigorous check for instrument calibration and data validity.

B. Numerical Modeling

• Upper Convected Maxwell (UCM) Model: Used as a baseline quasilinear model to demonstrate that $N_1$ contains only zeroth and second harmonics in the linear/quasilinear regime.
• Giesekus Model: A fully nonlinear model used to simulate higher-order harmonics (4th, 6th, etc.) and strain-dependent behavior. The authors derived analytical solutions for Medium Amplitude Oscillatory Shear (MAOS) up to fourth order to validate their numerical extraction methods.

C. Experimental Technique: Gaborheometry

To overcome the time-constraints of traditional discrete strain sweeps, the authors employ Gaborheometry strain sweeps (GaborSS).

• Protocol: A single experiment involves a linearly ramped strain amplitude ($\gamma_0(t) = \gamma_i + rt$) applied at a fixed frequency.
• Analysis: The resulting time-varying signal is decomposed using a Discrete Gabor Transform (DGT). This localizes the signal in time (and thus strain amplitude), allowing the extraction of material functions ($n_1^d, n_{1,k}', n_{1,k}''$) as continuous functions of $\gamma_0$ from a single sweep.
• Geometry:
  • PDMS (Silicone): Measured using a standard Cone-and-Plate (CP) geometry to obtain true $N_1$.
  • TPU (Thermoplastic Polyurethane): Measured using a Cone-and-Partitioned Plate (CPP) geometry to prevent edge fracture. The authors analyze the "apparent" $N_1$ ($N_{1,app}$) and use the offset from theoretical predictions to estimate the ratio of second-to-first normal stress differences ($N_2/N_1$).

3. Key Contributions

1. Unified Framework: Established a rigorous Fourier-Tschebyshev decomposition for $N_1$ in LAOS, providing a standardized nomenclature for material coefficients ($n_1^d, n_{1,k}', n_{1,k}''$).
2. Self-Consistency Verification: Demonstrated experimentally and theoretically that the asymptotic relations between shear stress and $N_1$ hold true for real fluids, providing a new method for validating rheometer calibration.
3. High-Throughput Data Acquisition: Successfully adapted Gaborheometry to $N_1$ measurements, enabling the rapid generation of "rheological fingerprints" (Pipkin diagrams) spanning multiple strain amplitudes in a single experiment.
4. Novel Material Insights:
   • Revealed that for the Giesekus model, the 4th harmonic intensity peaks at a critical strain amplitude dependent on the anisotropy parameter $\alpha$.
   • Showed that for complex TPU melts, the $N_1$ response is significantly more complex than predicted by single-mode models, with distinct phase shifts in higher harmonics.
   • Provided a method to estimate $N_2/N_1$ ratios using the offset between measured $N_{1,app}$ and theoretical $G'$ in CPP geometries.

4. Results

• Theoretical Validation: Analytical solutions for the UCM and Giesekus models confirmed that the proposed Fourier-Tschebyshev coefficients satisfy the derived self-consistency relations at low strains. The Giesekus model successfully predicted the emergence of 4th harmonics at higher strains.
• PDMS (Weakly Elastic):
  • Lissajous curves of $N_1$ vs. strain were "butterfly-shaped" with a non-zero mean.
  • Experimental data at low-to-moderate strains ($\gamma_0 < 5$) matched the self-consistency predictions ($n_1^d \approx G'$).
  • Higher harmonics were not detectable due to the low elasticity of PDMS and instrument noise floors.
• TPU (Strongly Elastic):
  • Gaborheometry successfully captured the evolution of $N_1$ from linear to highly nonlinear regimes.
  • Pipkin Diagrams: Contour plots of $n_1^d$, $|n_{1,2}^*|$, and $|n_{1,4}^*|$ revealed complex dependencies on strain amplitude and frequency that deviated from simple Giesekus predictions.
  • $N_2$ Estimation: The offset between the measured $n_{1,app}^d$ and the linear $G'$ allowed the estimation of $N_2/N_1 \approx -0.54$, consistent with liquid-crystal-like behavior in the hard blocks of the TPU.
  • Higher Harmonics: The 4th harmonic was clearly resolved above the noise floor, providing a sensitive probe of the material's nonlinear microstructure.

5. Significance

• Enhanced Rheological Fingerprinting: By incorporating $N_1$ data, researchers can now generate more complete "digital twins" of complex fluids. This is crucial for distinguishing between materials that may have similar shear stress responses but different normal stress behaviors.
• Constitutive Model Discovery: The rich datasets generated by this framework (including higher harmonics and self-consistency checks) are ideal for training machine learning models and discovering new constitutive equations (e.g., via Rheological Universal Differential Equations).
• Industrial Application: The ability to rapidly characterize nonlinear normal stresses in thermoplastic melts (like TPU) using Gaborheometry is highly relevant for industrial processes such as film stretching and fiber spinning, where normal stresses dominate flow stability.
• Instrumentation Validation: The self-consistency checks provide a robust, physics-based method for calibrating normal force transducers, ensuring data reliability in high-strain experiments.

In summary, this paper bridges a critical gap in rheology by providing a systematic, self-consistent, and efficient method to analyze the first normal stress difference in LAOS, transforming it from a difficult-to-measure artifact into a primary source of material characterization data.