From the data of , it is clear that a dimensionless slope of O(10 −5 ) is within the experimental error and can thus be considered as zero. Nondimensional slopes smaller than O(10 −4 ) are then reported in grey in to highlight that these values can be actually considered nil. From the analysis of the slopes, it is clear that when the system is in the phase separation region (T1 and T2), the microstructure does not evolve in time at a constant temperature; conversely, when the system is in the gel-formation region (T4), the microstructure evolves in time at a constant temperature and G’ increases faster than G” indicating that the elasticity builds more rapidly than the viscous stiffening. At an intermediate temperature, T3, which corresponds to the transition zone from the phase separation to the gel region, we see that G’ increases with time while G” remains unchanged, the complex viscosity also increases very slowly at a rate of the order of the experimental error. This suggests that the gel is at the beginning of its formation; at this early stage the material is mainly viscous, such that the increase of the number of physical bonds with time induces an elasticity increase, but it is not enough to stiffen the system.
Some of the data of , in the time sweep region, show an average increase with time, superimposed to the oscillations. This slight increase with time is more evident at higher temperatures (T3 and T4) and suggests that the microstructure evolves at a constant temperature. To better quantify this, the mean dimensionless slope of the evolution with time of η*, G’, and G” is evaluated and reported in . The slopes are calculated with a linear regression, Equation (1), by interpolating the time sweep data not affected by the initial transient consequent to the temperature overshoot shown in a. The first-order coefficient, b, is then normalized with the mean value of the data used for the interpolation:
We first notice the very good data reproducibility in the sol phase region, for T < T A , and in the phase separation one, for T A < T < T B . We also notice that the temperature during the time sweep experiments passes through a very small overshoot before reaching the target value around which oscillates with a period of about 170 s and an amplitude of 0.05 °C, well within the specifications of the Peltier cell used. Interestingly, the complex viscosity also oscillates after an undershoot following both the temperature oscillations and overshoot, respectively. The same happens to both G’ ( b) and G” ( c), and we will analyse this later in Section 3.3 : “Activation energy (Ea) evaluation”.
Time sweep tests at the four selected temperatures were run after a thermal ramp at 1 °C/min, from 30 °C to the selected temperature. After each time sweep test, the system was cooled back to 30 °C. At 30 °C, before running any new thermal ramp, the system was sheared at γ ˙ =10 s −1 for 600 s to erase any flow history. The viscosity measured at 30 °C was always the same (62.34 ± 2.74 Pa·s), confirming the full reversibility of HPMC thermogelation. In , we show the complex viscosity measured during each thermal ramp followed by the time sweep experiment vs. time. The temperatures vs. time during the time sweep experiments are also shown as a reference.
The spectra at T2 and T3 are compatible both with a viscoelastic fluid formed by a more elastic polymer-rich phase immersed in a viscoelastic polymer-poor phase, and with a viscoelastic fluid where a 3D network is forming. To try to discern between these two hypotheses, we took advantage of the intrinsic temperature oscillations during the time sweep tests (upper part of a) that allowed us to estimate the Ea of the system during the different phases of the thermogelation process.
Let us now consider that to take most of the spectra points immediately after the temperature has reached the set point, the frequency sweep experiments are run by decreasing the frequency, so the data at low frequencies are those taken at the end of the experiment when an increase in moduli may have taken place. Thus, the moduli behaviour shown at small frequencies may be affected by the microstructure change in time at a constant temperature.
At T1, a sol behaviour is still dominant, and the viscoelastic weakening induced by the temperature increase is evident both from the shift of the crossover frequency towards higher values and from the reduction of G’ and G”. However, the terminal zone at low frequencies of G’ is not clearly visible, either because of the instrument sensibility or because some elasticity is starting to build-up. At T2, the liquid–liquid phase separation, according to the literature [ 32 , 34 ], is occurring and we notice that the moduli increase with the increase of temperature. A clear elasticity is now evidenced by the values of G’ at small frequencies, which depart from the typical terminal zone of a sol phase. This may be due to either the viscoelasticity of the polymer-rich phase or the build-up of a gel network. At T3, the elasticity at small frequencies is more evident and G’ overcomes G” indeed. By increasing the frequency G’ crosses G” and more classical sol behaviour is again observed with a second crossover that can be guessed at higher frequencies. At these two temperatures, T2 and T3, data indicate that the system is mainly a viscoelastic fluid with a classic moduli crossover at the high frequency, but also with an elastic solid-like fingerprint, with a long characteristic relaxation time. Finally, at T4, we have the clue of the gel formation as G’ is everywhere larger than G” and the moduli are both practically horizontal at small frequencies. However, we should emphasise that at T3 and T4 the microstructure evolves in time at a constant temperature, in particular, as shown in and , at T3 G’ increases with time, while G” remains mainly constant, and at T4 G’ increases with time more rapidly than G”.
The gel build-up can be investigated with the frequency sweep tests carried out at the four selected temperatures. The elastic and viscous moduli vs. the angular frequency are shown in . As a reference, the frequency sweep results at 20 °C (in the sol phase) and at 90 °C (when the gel is formed) are also shown. In the former case, the classical spectra of a viscoelastic solution are obtained; in particular, the moduli crossover is at about 10 rad/s and the terminal zone is approached for angular frequencies smaller than 0.1 rad/s. Similarly, in the latter case, the spectra typical of a gel are obtained, in fact, the viscoelastic moduli are essentially parallel and independent of the frequency, with G’ much larger than G”, indicating a well-entangled network.
First, we evaluated the activation energy (Ea) of the system in the sol phase, for T < TA, using data recorded during the thermal ramp at 1 °C/min ( ). It can be calculated according to an Arrhenius-like equation (see Equation (2)) from the variation of the complex viscosity with the temperature. Analogously, we can estimate the magnitude of the temperature dependence of the viscoelastic moduli by evaluating Ea of G’ and G” by fitting Equation (2) through the data of the moduli vs. temperature:
where p can be ƞ*, G’, or G”; k is a pre-exponential factor; Ea is the activation energy; R is the gas constant; and T is the absolute temperature (K).
From the data of , we obtained the positive Ea value reported in , i.e., the larger the temperature, the smaller the complex viscosity. The activation energies calculated from G’ and G” are also positive, and that calculated from G’ is larger than that calculated from G”. This implies that in the sol phase the elastic modulus is more sensitive to the temperature variation; it weakens more rapidly than the viscous modulus.
Similarly, we can estimate the activation energies from the variation of the complex viscosity and of the moduli with the temperature in the phase separation branch of , i.e., for TA < T < TB, using the Arrhenius-like equation (Equation (2)). We obtain positive values, reported in , that are one order of magnitude larger than those of the sol phase. Additionally, in this case, Ea calculated from G’ is larger than that calculated from G”. These are apparent activation energies, however, since in the phase separation branch by varying the temperature we simultaneously probe two mechanisms: the typical variation of viscoelastic parameters with temperature, similarly to what is observed in the sol phase, and the change of the microstructure related to the formation of polymer-rich and polymer-depleted regions; the more phase-separated, the weaker the system. Let us emphasise that we also run the thermal ramp at a slower velocity and it superimposes to the one here discussed at 1 °C/min. This suggests that during the phase separation branch, the microstructure is at its equilibrium, at any temperature.
To decouple the two mechanisms, we took advantage of the intrinsic temperature oscillation around its set-up value to estimate Ea from the time sweep experiments at the four selected temperatures. To this end, we first interpolated the temperature oscillatory signal vs. time by using third-order polynomial curves between successive data points. a shows, as an example, the interpolation of temperature data at T2 = 69.5 °C. Then, with the T(t) function obtained, we fitted Equation (2) through ƞ*, G’, and G” data vs. time ( b–d). In order to have an estimation of Ea not affected by the average increase of the moduli and of the complex viscosity with time, particularly evident at T3 and T4 (see and ), we fitted the data after having subtracted from them the values due to this increase in time, calculated with Equation (1). We fitted the same range of data used to calculate the average evolution in time of the rheological parameters (orange points in ). As shown in , the fitting is more than satisfactory in all cases. The values of the activation energies calculated at the four characteristic temperatures are listed in . Let us emphasise that the oscillation period of the temperature is about 170 s, while the time sweep test is run at 10 rad/s, and each experimental point is collected in one oscillatory cycle of 0.628 s, during which the variation of the temperature can be considered negligible. Moreover, as the microstructure was in its equilibrium state during the thermal ramp at 1 °C/min, we can assume that it is also in its equilibrium state at each temperature during the oscillations. Furthermore, the amplitude of the temperature oscillation is 0.05 °C, and it is too small to induce a measurable change of the rheological properties with the temperature if the microstructure does not change. Indeed, this small temperature variation in the sol region (T < TA) does not cause a measurable data fluctuation. Consequently, with these time sweep tests, we will actually probe the sole response of the system to a change of microstructure induced by the change of temperature.
At temperature T1, we measure positive activation energies for the three rheological parameters, and they are very comparable to the respective values calculated from the whole phase separation branch (see ). At T2, well within the phase separation region, once again the activation energies are all positives and about double those previously estimated from the full branch. This seems to suggest that a more phase-separated system is more sensitive to temperature variations. At temperature T3, in a region approaching the beginning of the gel formation, the responses of ƞ*, G’, and G” differ. Compared to the value at T2, the Ea calculated from the complex viscosity remains almost unvaried, the Ea calculated from G’ significantly decreases, remaining positive, while the Ea calculated from G” measurably increases. We might explain this behaviour by imagining that the polymer-rich phase is sufficiently concentrated that a minimum increase of its concentration induced by the increase of temperature is accompanied by a significant increase of elasticity that partially compensates for the reduction of the modulus due to the progress of the phase separation. Similarly, the viscous response, dominated by the polymer-poor phase, is more sensitive to the concentration variation induced by the temperature changes. Finally, at temperature T4 in the gel phase, we detect a change of sign of the Ea calculated from G’ and ƞ*, but not from G”. This implies that by increasing the temperature, G’ and ƞ* increase while G” still decreases. This is quite in agreement with the behaviour shown in , where the inverse thermogelation is depicted. We are thus probing the gel response to the temperature where the higher the temperature, the more elastic the gel. At T4, the viscous modulus keeps reducing with the temperature because the physical bond density is still too low to limit the viscous dissipation after an imposed deformation.
To summarize the main results obtained from this analysis, we first highlight the change of sign of the Ea of G’ and η* passing from T3 to T4, which is in perfect agreement with a transition from a phase separation region, where the behaviour of the system is still liquid-like, to a gel region, where the system is solid-like. Since the gelation investigated is an inverse thermogelation, the negative value of the Ea at T4 is perfectly sound. Data calculated at T3 suggest that the increase of elasticity recorded during the frequency sweep test at this temperature can be reasonably related to the phase separation into a polymer-rich elastic phase and a polymer-poor one, more than to the onset of the 3D gel network. In fact, in the latter case, the Ea calculated from G’ should have been negative. The increase of Ea passing from the data at T1 to those at T2 can be explained considering that at T1 the phase separation involves the system marginally and thus the sol behaviour still affects the system response significantly. More generally, data suggest that the more phase-separated the system, the more sensitive to the temperature variation. This is confirmed by the Ea calculated from G” at T3.
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