3.1. Steady Viscous Behavior
Steady viscosity curves of Gf/AVJ suspensions showed that the viscosity increased with the increase of Gf amount (
Figure 2). On the other hand, the viscosity of each Gf/AVJ suspension decreased with shear rate indicating (non-Newtonian) shear-thinning behavior. The decrease in viscosity of AVJ (black square symbols) has been previously reported [
10], and is due to breaking of structural units due to hydrodynamic forces generated during shear [
13]. This is a typical non-Newtonian behavior also found in fruit-based products [
14,
15,
16].
The dependence of the viscosity
with shear rate
was fitted with the power law model (Ostwald-de Waele),
In Equation (1)
and
are consistency and flow indexes, respectively. As can be seen, good correlation between experimental and predicted values was found for the liquid phase (AVJ) and Gf/AVJ suspensions (
Table 1). It is worthy to note that the “strange” units of the consistency index
can become a handicap for the comparison of the value of this magnitude taken by different materials. Fortunately, in this specific case,
-values coincide
for all the suspensions and, therefore, we can affirm that the consistency of Gf/AVJ suspensions increases with Gf content. In addition, the abrupt shear-thinning behavior characterized by a relatively small
-value is not affected by Gf concentration. Consequently, the shear-thinning behavior of Gf/AVJ suspensions should be justified by the shear-thinning behavior shown by the carrier liquid (AVJ). In other words, Gf/AVJ suspensions are non-Newtonian systems resulting from the dispersion of a solid phase (Gf) in a non-Newtonian carrier liquid (AVJ). Therefore, the fact that
-value was independent on Gf content allow us to conclude that the shear-thinning behavior of the liquid phase (AVJ) is the main cause of the shear-thinning behavior observed in Gf/AVJ suspensions. AVJ is a dispersion of high molecular weight insoluble polysaccharides entangled at rest. Increasing shear disentangles polysaccharide molecules and tend to align them along the flow direction. Consequently, the flow field distortion is less at high shear and the viscosity is lower.
The presence of Gf particles gives place to an additional flow field distortion (higher viscosity) at each shear rate value. The ability of Gf particles and aggregates to distort the flow field can be quantified with the intrinsic viscosity
of the suspensions [
17]. Krieger-Dougherty equation [
18] relates the viscosity of the suspension with the volume fraction
occupied by the solid phase,
In Equation (2)
is the viscosity of AVJ and
is the maximum packing fraction achieved by Gf particles dispersed in Gf/AVJ suspensions. Although Krieger-Dougherty equation was formulated assuming Newtonian behavior of the suspension, its utility for the estimation of the flow field distortion in non-Newtonian suspensions has shown also valid [
19]. Experimental relative viscosity values corresponding to four different shear rate values
against Gf volume fraction overlapped within the experimental error, indicating that the shape and size of Gf aggregates are not significantly affected by the shear rate. To confirm this result, Krieger-Dougherty equation was fitted to these experimental data. The resulting values of the parameters of Krieger-Dougherty equation confirmed that the intrinsic viscosity practically does not vary when the shear rate increases
. This means that the shape and size of Gf aggregates is not affected by the mechanical action due to shear. In addition, the maximum packing fraction was the same independently the shear rate value
. In conclusion, the shear-thinning behavior observed in Gf/AVJ suspensions is mainly due to the shear thinning behavior shown by AVJ solvent.
3.2. Viscoelastic Behavior
The viscoelastic flow of G/AVJ suspensions was characterized using Small Amplitude Oscillatory Shear (SAOSstrain) tests. With this rheological technique the linear viscoelastic response of the microstructure-at-rest developed by the suspensions is analyzed. The importance of this study is justified by the fact that help to determine the stability of the storage suspension against sedimentation. With this rheological study, the variation of the deformation
imposed to the suspension is sinusoidal,
Therefore, firstly an amplitude
sweep is applied to the sample maintaining constant the angular frequency
of the oscillation. The aim of this first part of the SAOSstrain test is to determine the maximum
-value delimiting the linear viscoelastic (LVE) behavior of the suspension.
The LVE response of suspensions to the sinusoidal deformation is an out of phase sinusoidal shear stress
,
In Equation (4)
is the stress amplitude and
is the phase difference between input (deformation) and output (stress) signals. From
Equation (4) two viscoelastic moduli are defined,
is the elastic or storage modulus and
is the viscous or loss modulus. The accomplishment of LVE behavior condition implies that both moduli must be independent on
. Results corresponding to amplitude of deformation sweep tests showed that the viscoelastic moduli do not depend on
(LVE behavior) when the amplitude of the oscillatory shear is lower than
.
After the LVE region was detected, the behavior of Gf/AVJ suspensions at short and long experimental times was tested varying the frequency of the SAOSstrain maintaining constant the amplitude into the LVE regime. This second rheological test is named frequency sweep. More concretely, to be sure that responses of Gf/AVJ suspensions to frequency sweep rheological tests were recorded in the LVE region, an amplitude
was maintained constant during oscillatory shear. Results of frequency sweeps in the LVE regime obtained with Gf/AVJ suspensions are shown in
Figure 3. As can be seen, Gf/AVJ is a viscoelastic gel in all cases [
20]. This qualification results from the fact that booth moduli are practically independent on the frequency. This is an indication of the existence of a relatively strong microstructure, which gives the suspensions a gel appearance. This gel-microstructure is mainly built by polymeric molecules dispersed in the AVJ. We arrive to this conclusion because
and
dependence with angular frequency is qualitatively similar for AVJ solvent and Gf/AVJ suspensions. Certainly, the value of both moduli increases with Gf concentration, which is an indication of some additional effect due to the increase of solid phase.
For the analysis of
and
experimental data Jeffreys model (
Figure 4) will be used. Jeffreys mechanical analog is the simplest equivalent encompassing the entire spectrum of mechanical behaviors [
21]. Therefore, it is indicated for the analysis of the general LVE response of non-Newtonian fluids.
The constitutive equation that corresponds to the mechanical analog shown in
Figure 4 is,
In
Equation (7)
is the steady-state shear viscosity and
is the elastic modulus. With the definitions
and
for relaxation and retardation times, respectively,
Equation (7) can be re-written as,
Jeffreys material functions can be expressed in terms of the experimentally accessible magnitudes
and
, enhancing its utility for the physical interpretation of the LVE behavior of Gf/AVJ suspensions. Substituting Eqs. (3-6) in
Equation (8), the following relationships are obtained,
To obtain the function
, the experimental dependence of the steady-state viscosity with shear rate given by
Equation (1) was re-written using the relationship
with
,
Therefore, the dependence of Jeffreys-based material functions
and characteristic times
with angular frequency can be finally obtained. The results are shown in Figures 5a–c.
As can be seen,
is highly dependent on frequency and Gf content. More specifically, decreases with frequency and increases with Gf content. Although as in the ideal Jeffreys model
represents the viscosity of the completely unstructured material, i.e., theoretically it should be independent of the frequency, we can see that it decreases with frequency. However, it is worthy to note that
by some orders of magnitude. Results from
Figure 5a correspond to very low shear rate range after using the relationship
. Specifically, the shear rate interval
is accessible with the use of oscillatory rheological tests. Therefore, these results can be combined with results obtained in the shear rate range
using continuous rheological tests to obtain information on the viscosity dependence with the shear rate in a much wider range. As can be seen in
Figure 2, the results obtained using both methods are consistent.
The shear modulus
increases only very slightly with frequency (
Figure 5b). This means that Gf/AVJ suspensions stiffness only a little bit when the experimental time decreases. It is worthy to note that this material function raised a saturation value when Gf concentration was
, i.e., the microstructure achieved the stiffer state with this Gf concentration.
The relaxation time
and the retardation time
decrease with frequency (
Figure 5c). As the retardation time
is much lower than the relaxation time
, liquid-like behavior is dominant in the full frequency range. On the other hand,
increases with Gf content suggesting liquid-like behavior due to higher presence of Gf particles in the suspension. In other words, the presence of higher Gf content in the suspension gives place to higher dissipation of energy (viscous effect, liquid-like behavior). Note that, from data shown in
Figure 3 the relative increase of the viscous modulus
is higher than the relative increase of the elastic modulus
when the amount of Gf in suspensions increases. Therefore, it is apparent that the elastic behavior of AVJ solvent due to entanglement of polysaccharide molecules is diminished by the presence of Gf particles probably due to breaking of links, reducing the extent and stiffness of the network formed by polysaccharide molecules. From a sensorial point of view, this means that increasing the addition of Gf to AVJ changes the texture of the suspension from chewy to creamy.