# Effect of Chemical Composition of Bio- and Petroleum-Based Modifiers on Asphalt Binder Rheology
## 3 Results
### Rheological Characterization of Modified Binders
#### 3.2.3 Frequency Sweep Test  
In this section, complex modulus master curves, black space diagrams, and rheological parameter, GR, are presented for, RTFO, PAV, 2 PAV, and 3 PAV aging conditions. Complex Shear Modulus Master Curves  
Complex modulus master curves were determined at a reference temperature of 15 \({}^{\circ}\)C. Figure 6a-d show the progression of complex modulus at different aging conditions, for limited low reduced-frequency range. This range was selected to represent low-temperature (non-load associated) cracking conditions. Following are the observations from Figure 6.  
Figure 5: \(\Delta\)T\({}_{\text{c}}\) values of different modified binders, base binder and unmodified binder for aging conditions (**a**) pressure aging vessel (PAV); (**b**) 2 PAV; and (**c**) 3 PAV. 1. In Figure 6a, after RTFO aging, modified binders are difficult to differentiate based on their shear modulus values. All the binders have similar master curves representing similar stiffnesses. 2. As the aging progressed to PAV condition, modified binder containing K separates from the rest of the binders at the lower frequency range (Figure 6b) representing a stiffer behavior. Other binders (A, C, D, E, G, and S5) show similar behavior. 3. Upon further aging (2 PAV), modified binder containing G slightly separates from the rest of the binders towards lower moduli (Figure 6c), showing a softer behavior. K is again the stiffest and separates from the rest (A, C, D, E, and S5) in this condition. 4. At 3 PAV, modified binders containing G and K have clear distinction in moduli, G being the softest and K being the stiffest. All other binders have similar variation in the complex shear modulus as shown in Figure 6d. Black space diagrams were plotted for RTFO, PAV, 2 PAV, and 3 PAV aging conditions. A black space diagram shows the variation of complex shear modulus (G*) with phase angle (\(\delta\)). A reference temperature of 15 \({}^{\circ}\)C was used to obtain these curves. A higher \(\delta\) at a fixed temperature and G* represents a material more prone to flow in a viscous manner and a lower \(\delta\) represents a more elastic behavior [42]. Materials with higher \(\delta\) are less likely to crack in a brittle way. In other words, a lower \(\delta\) indicates that more energy would be stored, and faster accumulation of stress would be observed during repeated deformation [3]. A good correlation has been demonstrated between isothermal phase angle (especially at 50 \({}^{\circ}\)C) and test-pavement cracking severity [43]. Phase angle (at G* = 8967 kPa) has  
Figure 6: Complex shear modulus curves at reference temperature of 15 \({}^{\circ}\)C for aging conditions (**a**) Rolling Thin-Film Oven (RTFO); (**b**) PAV; (**c**) 2 PAV; and (**d**) 3 PAV. been found to be more repeatable measurement than G* and capable of identifying phase-incompatible asphalt binders [3]. Figure 7a-d show complete black space diagrams for specific aging conditions, whereas the inset figure highlights the effect on \(\delta\) for G* ranging from 10 MPa to 11 MPa. The motivation for the selection of this G* range was the intermediate-temperature PG criteria [3]. However, a different range of G* selection does not affect the trends in \(\delta\) and could represent other loading conditions in the field. Some observations from the black space diagrams are the following:  
1. For the selected range of G* at RTFO (Inset Figure 7a) aged condition, K-modified binder has the lowest \(\delta\) with clear distinction compared to other binders. The remaining binders have higher \(\delta\) values than K-modified binder and are close to each other. C-modified binder has highest \(\delta\). 2. As the aging progresses to PAV aging, the separation in \(\delta\) can be distinguished clearly. K-modified binder is separated from rest of the binders with lower \(\delta\) (Inset Figure 7b), while G-modified binder has the highest \(\delta\) followed by C, S5, E, D, and A, respectively (close to each other). 3.