## 3 Results and Discussion  
Rheological Properties of Aqueous Dispersions of Flaxseed Fiber Submitted to Ultrasonic Homogenization  
A linear viscoelastic range (LVR) and the corresponding critical oscillatory stress amplitude values can be determined by means of stress sweeps at constant frequency. A linear viscoelastic range comprises the stress amplitude values for which the viscoelastic functions remain constant or do not vary significantly with increasing stress and strain values, with critical stress (\(\tau_{\mathrm{c}}\)) being the value of stress amplitude at which these functions cease to be constant. In this case, the storage modulus (G\({}^{\prime}\)), related to the elastic component, and the loss modulus (G\({}^{\prime\prime}\)), associated with the viscous component, have been chosen as control viscoelastic functions. Figure 1 shows the variation of G\({}^{\prime}\) and G\({}^{\prime\prime}\) with stress amplitude as a function of the ultrasonic treatment applied to the aqueous dispersions of fiber. In addition, the stress sweep for the sample corresponding to 0 min in an ultrasound (L-0 min) is also included. All samples, except LU-20 min-100%, show a greater elastic than viscous character (G\({}^{\prime}>\) G\({}^{\prime\prime}\)). The aqueous flaxseed dispersions subjected to ultrasonic homogenization for 2 min and the increasing amplitude values exhibit similar viscoelastic moduli and critical stresses values (Table 2). This result indicated that, under these conditions, the extension of the linear viscoelastic region is not affected. In this same fiber, the amplitude of the LVR is also not affected by the homogenization rate or the geometry employed during rotor-stator homogenization, as previously reported [44]. Under different treatments of homogenization (non-homogenized, slightly homogenized, or highly homogenized), other dispersions of fibers such as those from apple pulp, a carrot, or a potato also did not exhibita significant effect of homogenization on the value of critical stress [47]. The application of pre-shear and a higher ultrasonic time, as can be observed in Figure 1, provoked a significant decrease in the values of G\({}^{\prime}\) and G\({}^{\prime\prime}\), with a greater decrease occurring the longer the exposure time. At the same time, there is a clear and significant reduction in the extension of LVR, with the value of critical stress going from about 4 Pa for LU-10 min-100% to 0.4 Pa for LU-20 min-100% (Table 2). A smaller value of this parameter is associated with a lower structural strength and a lower degree of molecular association [48]. It is worth noting that the sample L-0 min exhibited a critical stress value similar to that of the samples submitted to different ultrasonic amplitudes, 12.6 Pa. Therefore, it is deduced that the most relevant effect on the extension of LVR is the time of exposure to ultrasonic homogenization. Once the linear viscoelastic region has been determined, the influence of frequency is studied. It is worth noting the relevance of this test since the small amplitude oscillatory tests allow for understanding of the viscoelastic behavior of the samples without an alteration of the structure. First, the influence of amplitude percentage (A%) on mechanical spectra was investigated (Figure 2). All the samples exhibited storage modulus (G\({}^{\prime}\)) values greater than their loss modulus (G\({}^{\prime\prime}\)) values in the whole frequency range studied, which pointed out a higher solid character than liquid. The small difference in values between G\({}^{\prime}\) and G\({}^{\prime\prime}\) and the slight