A single-sheet model-discrimination protocol for structural memory in 5-interval thixotropy tests of soft-glassy inks

Technical Summary: A Single-Sheet Model-Discrimination Protocol for Structural Memory in 5ITT Tests

Problem Statement
Five-interval thixotropy tests (5ITT) are standard protocols for probing the breakdown and recovery of soft-glassy and thixotropic materials (e.g., inks and pastes). While a recovery ratio greater than unity or a significant contrast between low- and high-shear viscosities is often interpreted as evidence of "structural memory," the paper argues this interpretation is incomplete. A 5ITT curve can exhibit recovery even within a purely descriptive framework that lacks an internal state variable. The core rheological problem addressed is how to determine, using standard 5ITT and flow-sweep data, whether a material genuinely requires an internal structural-memory variable for its constitutive description, rather than being adequately described by a memoryless, regime-dependent baseline.

Methodology
The paper proposes a rigorous, falsifiable model-discrimination workflow applied to public 5ITT data (specifically the DoEofInksRheology_GLT dataset). The protocol consists of six key components:

1. Single-Sheet Analysis: Instead of averaging trajectories across a family, the method fits each experimental sheet (individual repeat) separately to preserve data variability.
2. Model Comparison:
   • Memoryless Baseline: A simple two-state model where viscosity depends only on the current shear regime (Low or High), defined by two parameters ($\eta_L, \eta_H$).
   • Structural-Memory Model: A minimal first-order discrete-time internal-variable model. It introduces a scalar structural coordinate ($a_i$) that relaxes toward regime-dependent equilibrium values ($a^{(L)}_{eq}, a^{(H)}_{eq}$) with specific relaxation factors ($\alpha_L, \alpha_H$). This model has four parameters.
3. Fit Metric: Both models are fitted to the full 5ITT viscosity trajectory in log-space to weight relative errors evenly across orders of magnitude. The goodness of fit is measured by the Root Mean Square Error (RMSE) of log-viscosity.
4. Parameter Penalization: The Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) are calculated to penalize the structural-memory model for its additional parameters. A negative $\Delta$AIC or $\Delta$BIC (structural minus memoryless) indicates the structural model provides a statistically significant improvement.
5. History-Order Negative Control: To verify that the model captures physical history dependence rather than merely fitting two viscosity levels, the 5ITT protocol order is scrambled ($S_1 \to S_2 \to S_4 \to S_3 \to S_5$). A valid structural-memory model should show a significant increase in error under this scrambled order, whereas a memoryless model would not.
6. Steady-Flow Checks: Flow-sweep data are used to confirm that the materials follow a yield-stress-like behavior (Herschel-Bulkley), ensuring the time-dependent models operate on a physically consistent steady-flow branch.

Key Results
The protocol was applied to two families of water-based soft-glassy ink formulations: the "55" family (3 repeats) and the "Ref" family (6 repeats).

• 55 Family (Primary Evidence):
  • The structural-memory model reduced the log-space viscosity RMSE by 76.6 ± 0.6% compared to the memoryless baseline.
  • The improvement remained strongly favored after penalization: $\Delta$AIC = −187.9 ± 3.4 and $\Delta$BIC = −183.5 ± 3.4.
  • Under the scrambled-order control, the structural model's error increased by 219 ± 6%, confirming the model relies on the physical sequence of shear history.
• Ref Family (Secondary Evidence):
  • The structural-memory model reduced RMSE by 47.3 ± 16.4%.
  • All six repeats showed negative $\Delta$AIC and $\Delta$BIC values, though the improvement was more variable than in the 55 family.
  • The scrambled-order error increase was 68 ± 51%, indicating a weaker but still present history dependence.

Significance and Claims
The paper claims to advance rheological methodology by transforming 5ITT analysis from a descriptive exercise (calculating recovery ratios) into a falsifiable model-discrimination test.

• Modest Scope: The authors explicitly state this is not a universal constitutive law nor a claim that a single variable suffices for all thixotropic materials. The model is discrete, lacks continuous stress overshoot or aging spectra, and does not resolve complex phenomena like shear banding or wall slip.
• Core Contribution: The work demonstrates that for the specific public benchmark used, a single internal memory coordinate materially improves full-trajectory fits even after strict parameter penalization.
• Validation: The methodology successfully distinguishes between a simple regime-switching description and a history-dependent structural model, particularly when the data quality is high (as seen in the 55 family). The failure of the structural model under scrambled-order conditions serves as a critical negative control, validating that the fitted parameters represent physical memory rather than statistical overfitting.

In conclusion, the paper provides a specific, testable workflow to determine if standard 5ITT data contains sufficient information to justify the inclusion of an internal structural-memory variable in constitutive modeling.