The thermorheological complexity of REX-r-PET could also be due to the presence of long-chain branches. A branched structure is assumed to be related with a more pronounced flattening and will eventually lead to an extra bump in the delta versusplot, in case the LCB character dominates the behavior [ 65 ]. The present result for REX-r-PET does not allow a distinction between different structures, but the minimum observed clearly indicated a second dominating relaxation process that could be attributed to the presence of long-chain branching. In terms of the concentration of LCB, it is reported that LCB-PE metallocene with a sparsely branched structure showed high thermorheological complexity, while LDPE with hyperbranched structures did not [ 66 ]. The findings are quite similar to those found for grafted comb and grafted bottlebrush-like LCB-PS [ 67 ], where the absence of thermorheological complexity in the PS bottlebrush (number of branches greater than 60) is consistent with the results of LDPE, both having branches statistically distributed along the backbone and, therefore, a similar density of branching points. Considering the high level of thermorheological complexity of REX-r-PET, one could expect that hyperbranched structures are not present.
Furthermore, the thermorheological behavior provides us with additional insights into the molecular structures of these samples. Figure 9 d shows the van Gurp Palmen plot, which is the phase shift,, as a function of the complex modulus,*, for virgin PET, r-PET, and REX-r-PET. A thermorheologically simple behavior was observed for virgin PET and r-PET as-dependent phase shift values superimposed at different temperatures, meaning that all the relaxation times have the same temperature dependence [ 64 ], while REX-r-PET exhibits a systematic split between the curves with the temperature, which identifies the thermorheological complex response.
Branched polymers are particularly thermorheologically sensitive. For example, branched polymers exhibit higher activation energies than linear ones of similar weight-average molecular weights, 57 ]. A value forof 50–70 kJ/mol has been reported for linear PET, whereas a drastic increase to a fivefold higher activation energy, Ea, is reported for long-chain branched PET [ 56 ]. In general, it is well-established that LCB polymers have higher values forcompared to linear polymers [ 61 63 ]. The calculated flow activation energy of the investigated samples, Ea, showed a similar trend to that of zero shear viscosity,. The activation energy of virgin PET was 80 KJ/mol, r-PETwas 100 KJ/mol, and REX-r-PET Ea was 350 KJ/mol. The increase of up to four times higher Ea is probably due to the presence of LCB.
Therefore, the viscoelasticity and chain relaxation were greatly affected by the reactive extrusion treatment. The rapid relaxation of the virgin and homogenized PET (r-PET) would be related to their linear structure that led to a Newtonian plateau and a very weak elastic response, while the slower relaxation process of REX-r-PET is probably related to the formation of larger chains (the increase of the intrinsic viscosity is reported in Section 3.1 ) and/or by long-chain branches (LCB) during reactive extrusion with Joncryl [ 58 59 ]. To study this behavior in more depth, the viscoelastic parametersandwere evaluated in terms of a discrete relaxation spectrum modeled from the mechanical spectrum of the virgin PET and REX-r-PET samples using TA Instruments TRIOS® software by applying the following equations:
The modified REX-r-PET sample compared to virgin PET and r-PET shows a pronounced increase in both the complex viscosity ( Figure 9 a) and elasticity ( Figure 9 b). Virgin PET and r-PET display terminal behavior and Newtonian viscosity in the studied frequency range, whereas REX-r-PET is characterized by displaying the onset of the terminal regime, which is shifted to frequencies lower than those accessible experimentally. The corresponding moduli of REX-r-PET slowly decrease with the frequency so that the quasiparallel moduli response indicates a gel-like behavior. The Newtonian plateau was not reached within the measured frequency–temperature window because of the slow relaxation, and a pronounced shear-thinning behavior was observed.
The small amplitude oscillatory shear tests, SAOS, assume that the response of the material is in the linear viscoelastic regime, and the functions of the material, storage modulus,, and loss modulus,(as well as the derived viscoelastic parameters), determined as a function of the frequency, fully describe the material response. Since linear viscoelasticity is based on a rigorous theoretical basis [ 55 57 ], SAOS tests provide a very useful and convenient rheological characterization of polymers of different molecular architectures. The linear viscoelastic parameters of the three PET samples: virgin PET, r-PET, and REX-r-PET, are presented in Figure 9
The REX-r-PET elongational behavior reflected the molecular structure modification as linear behavior was no longer observed, and strain hardening appeared in the range of the imposed extension rates. This behavior is well-known in typical long-chain branched-dominated rheology, including the response of polymers with architectures ranging from star- and H-shaped polymers to comb and pom-pom structures [ 68 77 ] and has also been reported for LCB Poly (ethylene terephthalate) subjected to reactive treatment with the combination of pyromellitic dianhydride and triglycidyl isocyanurate.
Elongation rheology tests were also performed to explore the viscoelastic properties of PET. The extensional viscosity curves of virgin and recycled PETs at different elongation rates are represented in Figure 10 . For virgin PET ( Figure 10 a) and r-PET ( Figure 10 b), the tensile tests were very difficult to perform, because the samples tended to drop during measurements due to their low viscosity. The results confirmed the Newtonian behavior of both samples, as the curves fit the predicted Trouton relationship of three times the complex viscosity data.
σ ( t ; ω , γ 0 ) = γ 0 ∑ n o d d G n ′ ( ω , γ 0 ) sin n ω t + G n ″ ( ω , γ 0 ) cos n ω t
(9)
G n ′ and G n ″ aren
th-order harmonic coefficients. The linear response reduces to the first–order harmonics (n
= 1), and higher–order harmonic coefficients or phase differences accounts for the nonlinearities, where the relative harmonic intensity ratios I n / 1 ≡ I ( n ω ) I ( ω ) or phase angles ∅ ′ n ≡ ∅ n − n ∅ 1 are widely used as indicators of nonlinearity.Oscillatory shear tests can be divided into two regimes. One regime is a linear viscoelastic response (SAOS) that was addressed in Section 3.8.1 , and the second regime is the nonlinear material response (large amplitude oscillatory shear, LAOS)) that will be discussed here. From an experimental point of view, the objective of these nonlinear oscillatory experiments is to investigate the evolution of the nonlinear response with increasing deformation and to quantify the nonlinear material functions. Furthermore, a great effort has been made in the last decades to establish sound relationships between these nonlinearities and the molecular structures of polymers. For that purpose, several quantitative methods have been described for analyzing nonsinusoidal waveforms of shear stresses. Fundamentally, LAOS analytical methods are based upon the principle of Fourier Transform Rheology (FTR). Under shear strainand a strain rate, shear stress can be expressed as a Fourier series of elastic and viscous stress.whereandareth-order harmonic coefficients. The linear response reduces to the first–order harmonics (= 1), and higher–order harmonic coefficients or phase differences accounts for the nonlinearities, where the relative harmonic intensity ratiosare widely used as indicators of nonlinearity.
Q n ( γ 0 , ω ) can be defined in the limit of small-strain amplitudes Q n 0 ( ω ) . The parameter, which is only frequency dependent, can be defined for every harmonic through the following equation Equation (10):Q n ( γ 0 , ω ) = I n / 1 γ 0 n − 1 with Q n 0 ( ω ) = lim γ 0 → 0 Q n ( ω )
(10)
Q n 0 ( ω ) gives information about the inherent nonlinear material properties of a sample as the trivial scaling I n / 1 α γ 0 n − 1 is eliminated.Additionally, for every harmonic, an intrinsic nonlinear parameter,can be defined in the limit of small-strain amplitudes. The parameter, which is only frequency dependent, can be defined for every harmonic through the following equation Equation (10):gives information about the inherent nonlinear material properties of a sample as the trivial scalingis eliminated.
79,Q ≡ I 3 / 1 / γ 0 2 with lim γ → 0 Q ≡ Q 0
(11)
The intrinsic nonlinearity parameter 3Q, or simply Q, that is derived from the third harmonic, written as in Equation (5), has been reported to be useful in evaluating structural features such as the topology of polymer melts [ 78 80 ], the droplet size distribution of emulsions [ 81 ], and recently, the morphology of polymer blends [ 82 83 ].
γ ( t ) and γ ˙ (t). Ewoldt et al. [Tn
), expressing elastic and viscous stresses as:σ ′ ( γ / γ 0 ) = γ 0 ∑ n , o d d e n ( ω , γ 0 ) T n ( γ / γ 0 )
(12)
σ ″ ( γ ˙ / γ 0 ˙ ) = γ 0 ˙ ∑ n , o d d ν n ( ω , γ 0 ) T n ( γ ˙ / γ 0 ˙ )
(13)
To interpret the higher harmonics of a FT rheological series [ 84 ], orthogonal stress decomposition is used to separate the nonlinear stress into elastic and viscous contributions based on the symmetry of stress with respect toand(t). Ewoldt et al. [ 85 ] extended this method with the Chebysev polynomials of the first type (), expressing elastic and viscous stresses as:
e
1= G
′
andv
1= G
″/
ω
). Any deviation from linearity, i.e., then
= 3 harmonic, is interpreted depending on the signs of e3 and v3. A positive third-order contribution results in higher elastic (or viscous) stress at the maximum strain (or strain rate) than is represented by the first-order contribution alone. Thus, depending on the sign of the third-order coefficients, the following physical interpretation can be suggested (see Ewoldt et al. [e 3 = − G 3 ′ { > 0 strain-stiffening = 0 linear elastic < 0 strain-softening v 3 = − G 3 ″ ω { > 0 shear-thickening = 0 linear viscous < 0 shear-thinning
(14)
The first-order Chebyshev coefficients (e1 and v1) defined the viscoelastic properties in the linear region (i.e.,and). Any deviation from linearity, i.e., the= 3 harmonic, is interpreted depending on the signs of eand v. A positive third-order contribution results in higher elastic (or viscous) stress at the maximum strain (or strain rate) than is represented by the first-order contribution alone. Thus, depending on the sign of the third-order coefficients, the following physical interpretation can be suggested (see Ewoldt et al. [ 85 ] for further details):
γ
0, in the linear region,G
′ andG
′′ remain constant values, but, as the applied amplitude of strain is increased from small to large, a transition between the linear and nonlinear regime is observed so that in the nonlinear regime, both virgin PET and REX-r-PET, display moduli that decrease with the increasing strain (Nonlinear responses obtained at a constant frequency of 0.1 Hz and T = 260 °C of virgin PET and REX-r-PET are shown in Figure 11 a. At a small, in the linear region,′ and′′ remain constant values, but, as the applied amplitude of strain is increased from small to large, a transition between the linear and nonlinear regime is observed so that in the nonlinear regime, both virgin PET and REX-r-PET, display moduli that decrease with the increasing strain ( Figure 11 a). It is interesting to note that the stress patterns of the molten virgin PET and r-PET differ from that of the REX-r-PET (see insert in Figure 11 a, which corresponds to the stress signals at 200% of the strain; r-PET data are not included for clarity).
e
3/e
1 andv
3/v
1 in the nonlinear responses in the case of filled and vulcanized polyisoprene, respectively [As observed (in the Figure 11 a inset), the unmodified virgin PET displays a weak distortion, whereas REX-r-PET displays a “backward-tilted” shape stress. The distorted directions are considered to be related to specific polymer structures. Previous studies revealed that “forward-tilted stress” tends to appear in the case of polymer melts and solutions with a linear chain structure [ 86 87 ], whereas “backward-tilted stress” was reported for suspensions and polymer melts with branched chains. This behavior is generally attributed to the effect of branched structures during the flow alignment of polymer chains occurring at the larger strains. Branching is considered an obstacle and leads to an extra resistance to the flow. As a result, a stress shoulder appears at higher times—that is, the stress tilts backwards, delayed with respect to the symmetry axis. Interestingly enough, the distorted directions were reported to be related to the relative magnitudes ofandin the nonlinear responses in the case of filled and vulcanized polyisoprene, respectively [ 88 ]. Therefore, as a first approach, nonlinearity is very sensitive to the different structures of these materials. To further distinguish the differences in the topological structure of the PET samples, the analysis of the nonlinear region is divided in terms of the MAOS (medium amplitudes oscillatory shear) and LAOS (large amplitudes oscillatory shear) regimes.
I
3/1) can be used as a representative nonlinear parameter, helpful to detect the boundary of linear-to-nonlinear transition. In this regime (50–300%), the third-harmonic intensity is the only higher harmonic contribution, and the parameterI
3/1 scaled quadratically with the strain amplitude as expected [Q
0 in the MAOS regime can be applied to detect different polymer architectures, e.g., linear, 3-arm star, comb with many branches, and long-chain branching architectures.Q
parameter with the strain amplitude for the three samples: virgin PET, r-PET, and REX-r-PET.Q
has a constant value (Q
0) at relatively small strain amplitudes, while it becomes a function of the strain at larger strain amplitudes. At the investigated frequency of 0.1 Hz, theQ
0 value for REX-r-PET is much higher than the values of virgin PET and r-PET, for whichQ
0 is very similar. Ahirwal et al. [Q
0 parameter increased monotonically as a function of the long-chain branched PP weight fraction in the PP blends.Under MAOS, using the FT rheology method, the third-harmonic intensity normalized by the first-harmonic intensity () can be used as a representative nonlinear parameter, helpful to detect the boundary of linear-to-nonlinear transition. In this regime (50–300%), the third-harmonic intensity is the only higher harmonic contribution, and the parameterscaled quadratically with the strain amplitude as expected [ 89 ]. According to the experimental and theoretical findings, Hyun and Wilhelm [ 90 ] suggested that the intrinsic nonlinearityin the MAOS regime can be applied to detect different polymer architectures, e.g., linear, 3-arm star, comb with many branches, and long-chain branching architectures. Figure 11 b clearly differentiates the evolution of theparameter with the strain amplitude for the three samples: virgin PET, r-PET, and REX-r-PET.has a constant value () at relatively small strain amplitudes, while it becomes a function of the strain at larger strain amplitudes. At the investigated frequency of 0.1 Hz, thevalue for REX-r-PET is much higher than the values of virgin PET and r-PET, for whichis very similar. Ahirwal et al. [ 91 ] obtained results for the branched PP and branched PE comparable to those obtained for the PET samples here. They found that theparameter increased monotonically as a function of the long-chain branched PP weight fraction in the PP blends.
The structural differences of PET samples can be more evidently characterized under LAOS (in our case, from 300 to 900%). By decomposing the nonlinear stress waveforms based on symmetry arguments and using a Chebyshev polynomial analysis, the contribution of higher-order harmonics can be useful to gain advanced understanding in terms of the elastic and viscous nonlinearities described, respectively, by the intracycle strain stiffening (or softening) and intracycle strain rate thickening (or thinning) indices. The analysis could find application in the evaluation of molecular architecture and branching characteristics of polymer melts.
3, and the elastic nonlinear stiffening ratio (ε
3, defined by Equation (11), are plotted as a function of the applied strain. On the one hand, viscous nonlinearity between the samples was found to differ especially at the lower frequencies. Figure 11 c,d shows the comparisons among intracycle nonlinear coefficients of the investigated samples. The viscous nonlinear thickening ratio ( Figure 11 c), ν, and the elastic nonlinear stiffening ratio ( Figure 11 d),, defined by Equation (11), are plotted as a function of the applied strain. On the one hand, viscous nonlinearity between the samples was found to differ especially at the lower frequencies.3 < 0). On the other hand, elastic nonlinearity was found to be more sensitive to high frequencies, because REX-r-PET showed the strain thickening (e
3 > 0) increasing with the frequency (frequency effect not shown to avoid data overlapping). Figure 11 c shows the strain dependence of the thickening ratio obtained at 0.1 Hz for the three samples. The virgin PET and r-PET samples were characterized by quasilinear behavior, whereas REX-r-PET was characterized by strong strain rate thinning (ν< 0). On the other hand, elastic nonlinearity was found to be more sensitive to high frequencies, because REX-r-PET showed the strain thickening (> 0) increasing with the frequency (frequency effect not shown to avoid data overlapping).Figure 11 d shows the different stiffening ratios obtained at 5 Hz for the three samples. As a general trend, viscous and elastic nonlinear behavior were analogous to the response previously described in shear and elongational rheological tests. Under shear, REX-r-PET showed pseudoplastic behavior in contrast to the quasi-Newtonian response obtained for virgin PET and r-PET. Additionally, under melt elongational experiments, REX-r-PET showed typical strain hardening, as the sample hardens when the strain increases at a constant strain rate, a behavior not present in virgin PET and r-PET. Similarly, during the LAOS test, the stiffening behavior of PET REX can be understood when considering the ability of the branched structure to stretch during the oscillatory flow and re-stretch in the reverse direction.Want more information on rPET, RPET sheet? Click the link below to contact us.