## 2 Materials and Methods
### Characterization of Samples  
#### 2.2.1 Rheological Characterization  
The viscoelastic properties of the samples were determined by low-amplitude dynamic viscoelastic tests (SAOS) using an AR 2000-controlled stress rheometer (TA Instruments, New Castle, DE, USA) together with a 40 mm diameter aluminum plate-plate sensor (PP40). First, the extension of the linear viscoelastic zone (LVR) was determined by means of stress sweeps at a fixed frequency of 1 Hz. Then, frequency sweep measurements were performed at a stress located within the LVR. The dependence of the storage modulus on frequency can be fitted to a potential law equation whose shape is as follows:  
\[\mathrm{G^{\prime}}=\mathrm{a}\cdot\mathrm{\omega^{b}} \tag{1}\]  
where a is related to the value of the storage modulus and b to the degree dependence of \(\mathrm{G^{\prime}}\) on frequency.  
To determine flow behavior, steady state measurements were performed using a [PERSON] 40-controlled stress rheometer (Haake, Karlsruhe, Germany), with a geometry of coaxial cylinders with a sand-blasted surface (CC25 sensor, Ri = 1.25 cm and Re/Ri = 1.085). A multistep stress control protocol was applied with 120 s per point. The experimental data were fitted to the Carreau model:  
\[\eta=\frac{\eta_{0}}{\left(1+\left(\dot{\gamma}/\dot{\gamma}_{c}\right)^{2} \right)^{(\frac{1-n}{2})}} \tag{2}\]  
where \(\eta\) is the apparent viscosity, \(\dot{\gamma}\) is the shear rate, and \(\eta_{0}\), \(\dot{\gamma}_{c}\) and n are the fitting parameters. \(\eta_{0}\) is the so-called zero-shear viscosity, n is the flow index, and \(\dot{\gamma}_{c}\) is the critical shear rate, related to the onset of shear thinning behavior. It should be noted that the Carreau model has been simplified by eliminating the apparent viscosity at a very high-shear rate (\(\eta_{\infty}\)), since this Newtonian region has not been attained.  
The intervals used in both sweeps, stress or frequency, and flow curves depended on the characteristics of the sample, encompassing values ranging from 0.01 to 100 Pa, from 10 to 0.01 Hz, and from 1 to 100 Pa, respectively.  
All rheological tests were carried out 24 h after sample preparation, with a fresh sample, in duplicate, and at 20 \({}^{\circ}\)C \(\pm\) 0.1 \({}^{\circ}\)C. After loading the sample into the sensor system and prior to rheological measurements, an equilibration time of 300 s was allowed.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Sample**} & **Rotor–Stator** & **Rotor–Stator** & **Ultrasound** & **Ultrasound Time** & **Energy/Mass** \\  & **Time** & **Rate** & **Amplitude** & **(\%)** & **(min)** & **(\%)** \\ \hline U-2 min-40\% & 0 & 0 & 40 & 2 & 6.64 \\ \hline U-2 min-60\% & 0 & 0 & 60 & 2 & 31.17 \\ \hline U-2 min-80\% & 0 & 0 & 80 & 2 & 44.15 \\ \hline U-2 min-100\% & 0 & 0 & 100 & 2 & 61.88 \\ \hline L-0 min & 2 & 4000 & 0 & 0 & 62.70 \\ \hline LU-10 min-100\% & 2 & 4000 & 100 & 10 & 408.75 \\ \hline LU-20 min-100\% & 2 & 4000 & 100 & 20 & 712.50 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Samples studied, processing variables, and energy per mass unit applied to the aqueous dispersions of flaxseed fiber.