## 3 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sample** & **\(\mathbf{\tau_{c}}\) (Pa)** & **a \(\pm\) SD\({}_{\mathbf{a}}\)** & **b \(\pm\) SD\({}_{\mathbf{b}}\)** & **R\({}^{2}\)** \\ \hline U-2 min-40\% & 12.6 & 24.51 \(\pm\) 0.560 & 0.14 \(\pm\) & \(<\)0.01 & 0.999 \\ \hline U-2 min-60\% & 12.6 & 22.01 \(\pm\) 0.020 & 0.16 \(\pm\) & \(<\)0.01 & 0.999 \\ \hline U-2 min-80\% & 12.6 & 21.59 \(\pm\) 0.01 & 0.17 \(\pm\) & \(<\)0.01 & 0.999 \\ \hline U-2 min-100\% & 12.6 & 21.21 \(\pm\) 0.17 & 0.16 \(\pm\) & \(<\)0.01 & 0.998 \\ \hline L-0 min & 12.6 & 30.86 \(\pm\) 0.05 & 0.13 \(\pm\) & \(<\)0.01 & 0.999 \\ \hline LU-10 min-100\% & 3.9 & 8.95 \(\pm\) 0.12 & 0.33 \(\pm\) & 0.01 & 0.995 \\ \hline LU-20 min-100\% & 0.4 & 0.20 \(\pm\) 0.00 & 0.65 \(\pm\) & \(<\)0.01 & 0.998 \\ \hline \end{tabular}
\end{table} 
Table 2: Critical stress to exit of the linear viscoelastic region and fitting parameters of the power law equation for the storage modulus dependence of frequency of the aqueous dispersions of flaxseed fiber under different treatments of ultrasounds. Temperature: 20 \({}^{\circ}\)C.  
Figure 1: Stress sweeps for aqueous dispersions of flaxseed fiber as a function of different treatments of ultrasounds. The tests were carried out at 20 \({}^{\circ}\)C, 24 h after preparation. dependence of both on frequency indicated that these samples showed a weak gel-like behavior in agreement with the results obtained in the literature with other dispersions of polysaccharides [43; 49]. An increase in amplitude percentage (A%) from 40% to 60% caused a slight decreasing trend in both viscoelastic moduli, G\({}^{\prime}\) and G\({}^{\prime\prime}\). A further rise of A% did not significantly influence their values. Next, the effect of ultrasonic homogenization time at an amplitude of 100% on the dependence of G\({}^{\prime}\) and G\({}^{\prime\prime}\) with frequency was studied. The investigated samples were submitted to a previous treatment with Ultraturrax T50. For the sake of comparison, the viscoelastic properties of a sample prepared only by rotor-stator homogenization were also determined as a control. The latter exhibited a mechanical spectrum typical of weak gels, like the fiber dispersions submitted to different ultrasonic amplitude percentages, but exhibited slightly higher values of G\({}^{\prime}\) and G\({}^{\prime\prime}\). It is worth noting that this fact allows for a connection of the results obtained for samples U and LU. When the ultrasound time was 10 min, i.e., more energy was applied, a change in the shape of the mechanical spectrum occurred. G\({}^{\prime}\) was still greater than G\({}^{\prime\prime}\), but both moduli increased their dependence on frequency and, in addition, the difference in values between them decreased. The change of the microstructure was more pronounced when the ultrasonic homogenization time was 20 min (L-20 min) to such an extent that the sample exhibited predominantly liquid behavior, with the values of G\({}^{\prime\prime}\) exceed those of G\({}^{\prime}\). [PERSON] et al. (2016) [50] studied the influence of ultrasound treatment on the molecular weight (M\({}_{\rm w}\)) and rheological properties of wheat bran arabinoxylans and found decreasing values of M\({}_{\rm w}\) and G\({}^{\prime}\) with the extension of the ultrasound treatment time. According to these authors, this result was due to the intense cavitation and mechanical forces generated by the ultrasound treatment, which led to the disruption of the arabinoxylan molecular chains. From this analysis, it can be deduced that a similar phenomenon could have occurred in flaxseed fiber at high ultrasound exposure times. The G\({}^{\prime}\) curves have been fitted quite well (R\({}^{2}\) > 0.99) to the Equation (1), which was previously used by other authors [51; 52]. The values obtained for parameters a and b are shown in Table 2.  
Figure 2: Mechanical spectra for the aqueous dispersions of flaxseed fiber as a function of different treatments of ultrasounds.