*Article* **Evaluation of the Rheological Properties of Virgin and Aged Asphalt Blends**

**Tao Liu 1, Weidang Duan 1, Jialin Zhang 1, Qiuping Li 1, Jian Xu 2, Jie Wang 2,\*, Yongchun Qin <sup>2</sup> and Rong Chang 2,\***


**\*** Correspondence: j.wang@rioh.cn (J.W.); r.chang@rioh.cn (R.C.)

**Abstract:** To evaluate the effects of the source and admixture of aged asphalt on the rheological properties of reclaimed asphalt binders, the relative viscosity (Δ*η*), relative rutting factor (ΔG\*/sinδ), and relative fatigue factor (ΔG\*sinδ) were selected as evaluation indicators based on the Strategic Highway Research Program (SHRP) tests to characterize the rheological properties of a reclaimed asphalt binder under medium- and high-temperature conditions. The results of the study showed that the viscosity, rutting factor, and fatigue factor of the reclaimed asphalt binder increased with the addition of aged asphalt; however, the effect of the source and admixture of aged asphalt could not be assessed. The relative viscosity, relative rutting factor, and relative fatigue factor are sensitive to the source, admixture, temperature, and aging conditions, which shows the superiority of these indicators. Moreover, the relative viscosity and relative rutting factor decreased linearly with increasing temperature under high-temperature conditions, while the relative fatigue factor increased linearly with increasing temperature under medium-temperature conditions. In addition, the linear trends of the three indicators were independent of the source and admixture of aged asphalt. These results indicate that the evaluation method used in this study can be used to assess the effects of virgin asphalt and aged asphalt on the rheological properties of reclaimed asphalt binders, and has the potential for application. The viscosity of recycled asphalt increases, and the rutting factor and fatigue factor both increase. The high-temperature stability of reclaimed asphalt is improved, and the fatigue crack resistance is weakened.

**Keywords:** road engineering; reclaimed asphalt binder; rheological properties; SHRP test; variance analysis

#### **1. Introduction**

Reclaimed asphalt pavement (RAP) contains large amounts of aggregate and asphalt, which are potentially usable resources [1–4]. The recycling of RAP contributes to reductions in rock mining and aggregate production, with significant economic and environmental benefits [5,6]. Studies [2,7] have shown that the application of reclaimed asphalt mixtures in pavement subgrade construction can reduce greenhouse gas emissions by 20%, energy consumption by 16%, hazardous waste by 11% (RAP may leach toxic substances such as polycyclic aromatic hydrocarbons (PAHs) in the presence of rainwater in long-term stockpiles), and whole-life costs by 21%.

Asphalt is a viscoelastic material with excellent rheological properties [8,9]. After the blending of virgin and aged asphalt, the reclaimed asphalt binder becomes viscous and produces a large variation in rheology compared to the virgin asphalt [10–12]. RAP comes from a wide range of sources and has a very complex composition. Differences in the asphalt, aggregates, oil-to-rock ratio, gradation, and even admixtures may lead to significant differences in the rheology of the reclaimed asphalt binder or in the road performance of the recycled mix [11,13]. Numerous studies and engineering experiences have shown that

**Citation:** Liu, T.; Duan, W.; Zhang, J.; Li, Q.; Xu, J.; Wang, J.; Qin, Y.; Chang, R. Evaluation of the Rheological Properties of Virgin and Aged Asphalt Blends. *Polymers* **2022**, *14*, 3623. https://doi.org/10.3390/ polym14173623

Academic Editors: Wei Jiang, Quantao Liu, Jose Norambuena-Contreras and Yue Huang

Received: 3 August 2022 Accepted: 31 August 2022 Published: 1 September 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

pavements paved with poorly rheological asphalt are prone to high-temperature rutting, fatigue cracking, and other diseases [14,15]. In addition, the differences in the rheology of reclaimed asphalt binder will also lead to differences in the construction temperature of the mixture, which in turn will cause differences in the fuel required in the plant mix and greenhouse gas emissions [13,16].

Therefore, to optimize the design of reclaimed asphalt mixtures, reduce the reclaimed pavement issues, and prolong the service life, the rheological properties of reclaimed asphalt binders need to be evaluated effectively. At present, there are few studies on this subject, and the effects of different sources and admixtures of aged asphalt on the rheology of reclaimed asphalt binders under different temperatures and aging conditions have not been fully considered [10,17,18]. This study selects the Strategic Highway Research Program (SHRP) asphalt test method to quantitatively evaluate the effects of the source and admixture of aged asphalt on the high-temperature rheology of reclaimed asphalt binders using the relative viscosity, relative rutting factor, and relative fatigue factor.

#### **2. Materials and Method**

#### *2.1. Materials*

This study uses the Abson method (ASTM D 1856) to reclaim asphalt from asphalt mixtures. The asphalt mixture was extracted with a distillation device, and then the solvent in the extraction liquid was removed. The recovered asphalt samples were denoted by A, B, and C. The performance indexes are shown in Table 1. Asphalt with PG 90 was used for the virgin asphalt and denoted by N. The performance indicators of the virgin asphalt are shown in Table 2.

**Table 1.** The technical indicators of aged asphalt.


**Table 2.** The technical indicators of virgin asphalt.


The aged and virgin asphalts were mixed uniformly at 135 ◦C, and the admixtures of the aged asphalts were 15% and 30%, respectively. It is worth mentioning that the admixture refers to the ratio of the mass of aged asphalt to the total mass of reclaimed asphalt binder. The thin-film oven test (TFOT) and pressure aging vessel (PAV) were used to simulate short-term aging and long-term aging, respectively. For the sake of simplicity, the following abbreviations are used in this study: W for unaged, D for short-term aging, and C for long-term aging. For example, A15W represents virgin asphalt mixed with 15% of aged asphalt A, which was not aged; B30D represents virgin asphalt mixed with 30% of aged asphalt B, which underwent short-term aging.

#### *2.2. Asphalt Viscosity Evaluation Test*

The Brookfield rotational viscosity test (T0625) was used to investigate the effects of the source and admixture of aged asphalt on the high-temperature rheology of reclaimed asphalt binders. The viscosities of virgin asphalt and reclaimed asphalt binders were tested under unaged and short-term aging conditions, respectively. Considering the temperature range of 130–160 ◦C for asphalt production and application, the test temperatures were set at 115 ◦C, 125 ◦C, 135 ◦C, 145 ◦C, 155 ◦C, and 165 ◦C.

#### *2.3. Asphalt Viscoelasticity Evaluation Test*

The rheological properties of virgin and reclaimed asphalt binders at high and medium temperatures were investigated using dynamic shear rheometer (DSR) tests (T0628). The test plate was a circular metal plate with a diameter of 25 mm. The test temperatures for the unaged and short-term aged asphalts were 52 ◦C, 58 ◦C, 64 ◦C, 70 ◦C, 76 ◦C, and 82 ◦C, and the test temperatures for long-term aged asphalt were 16 ◦C, 19 ◦C, 22 ◦C, 25 ◦C, 28 ◦C, and 31 ◦C. It should be noted that the test temperatures refer to the American Association of State Highway and Transportation Officials (AASHTO T315) regulations.

#### **3. Results and Discussion**

#### *3.1. Asphalt Viscosity Evaluation Test Results and Analysis*

The viscosities of the asphalts at each test temperature are listed in Table 3. Each asphalt sample was tested 3 times in parallel. The test results satisfy the allowable error of the repeatability test being 3.5% of the average value.


**Table 3.** Viscosity results of virgin and reclaimed asphalt binders.

The viscosities of the virgin and reclaimed asphalt binders decreased gradually with increasing temperature, independent of whether they were aged or not. The viscosity of the reclaimed asphalt was significantly greater than that of the virgin asphalt, while the viscosity of the short-term aged asphalt was significantly greater than that of the virgin reclaimed asphalt.

To evaluate the influence of the source and admixture of the aged asphalt on the viscosity of the reclaimed asphalt, the relative viscosity Δ*<sup>η</sup>* was used for the evaluation, which mainly characterizes the influence of the relative viscosity of the reclaimed asphalt on its viscosity for every 1% increase in the content of aged asphalt, calculated as Equation (1):

$$
\Delta\_{\eta} = \frac{\eta\_{\text{mix}.r} - \eta\_{\text{new}.r}}{\chi} \tag{1}
$$

where Δ*<sup>η</sup>* is the dimensionless viscosity of the reclaimed asphalt, *ηmix.r* is the relative viscosity of the reclaimed asphalt, *ηmix.r* = *ηmix*/*ηnew*, *ηmix* is the viscosity of the reclaimed asphalt (Pa·s), *ηnew* is the viscosity of the virgin asphalt (Pa·s), *ηnew.r* is the relative viscosity of the virgin asphalt, *ηnew.r* = 1, and *x* is the amount of aged asphalt blending.

The variance results for the Δ*<sup>η</sup>* and viscosity *η* values of the reclaimed asphalt samples at different test temperatures and under different aging conditions are listed in Table 4. Usually, the significance level α = 0.05.


**Table 4.** Analysis of variance results for reclaimed asphalt with Δ*η* and *η*.

As can be seen from Table 4, the statistical probability *p*-value of *η* is less than 0.05 only for the test temperature and aging conditions, indicating that there is no significant difference in the effects of virgin asphalt and aged asphalt on the viscosity of reclaimed asphalt based on the η index. For Δ*η*, the *p*-values for all four influencing factors are less than 0.05, indicating that assessing the viscoelasticity of reclaimed asphalt with Δ*<sup>η</sup>* can identify the differences in these four factors. Therefore, the high-temperature rheology of the reclaimed asphalt is better assessed using Δ*<sup>η</sup>* than the viscosity index.

The variation in relative viscosity Δ*<sup>η</sup>* of the reclaimed asphalt versus temperature is shown in Figure 1. As can be seen from Figure 1, the Δ*<sup>η</sup>* of the reclaimed asphalt gradually decreases with the increase in temperature, and after short-term aging the Δ*<sup>η</sup>* also gradually decreases. When the temperature is 135 ◦C, the Δ*<sup>η</sup>* values of *A*15*W*, *B*15*W*, and *C*15*<sup>W</sup>* are 2.46, 2.11, and 1.75, respectively, indicating that the relative viscosity increases by 2.46, 2.11, and 1.75 for each 1% increase in the content of aged asphalt under this test condition. However, under the same temperature conditions, the Δ*<sup>η</sup>* of B30W is 2.63, which is not the same as that of B15W, indicating that the viscosity of the reclaimed asphalt produces inconsistent changes under different aged asphalt admixtures, even if the aged asphalt is the same. Moreover, the different reclaimed asphalts at different test temperatures and different aging conditions produced similar test results as described above.

**Figure 1.** The results for the relative viscosity Δ*η* versus temperature.

The regression analysis results show that the reclaimed asphalt Δ*<sup>η</sup>* has a good linear relationship with the test temperature (T). The regression equations are all Δ*<sup>η</sup>* = a*T* + b (a and b are the fitting parameters), as shown in Equations (2) and (3).

$$\Delta\_{\eta} = \begin{cases} -0.0157T + 4.6776, R^2 = 0.92, A15\_W \\ -0.0174T + 5.3464, R^2 = 0.97, A30\_W \\ -0.0144T + 4.1637, R^2 = 0.92, B15\_W \\ -0.0134T + 4.4118, R^2 = 0.87, B30\_W \\ -0.0151T + 3.8304, R^2 = 0.97, C15\_W \\ -0.02T + 4.8268, R^2 = 0.88, C30\_W \end{cases} \tag{2}$$

$$\Lambda\_{\eta} = \begin{cases} -0.0138T + 4.2984, R^2 = 0.91, A15\_D \\ -0.018T + 5.209, R^2 = 0.89, A30\_D \\ -0.0179T + 4.4458, R^2 = 0.98, B15\_D \\ -0.0186T + 4.8137, R^2 = 0.96, B30\_D \\ -0.0146T + 3.6383, R^2 = 0.87, C15\_D \\ -0.0212T + 4.8106, R^2 = 0.98, C30\_D \end{cases} \tag{3}$$

From Equations (2) and (3), it can be seen that the effects of virgin asphalt and aged asphalt on the high-temperature rheology of the reclaimed asphalt can be characterized by the slope and intercept in the linear relationship equation. For example, the slope of *B*30*<sup>W</sup>* is −0.013, which is 23% and 33% smaller than for *A*30*<sup>W</sup>* and *C*30*W*, respectively. In addition, the Δ*<sup>η</sup>* of the reclaimed asphalt is always linearly related to the temperature, independent of the source, the admixture of the aged asphalt, or whether it undergoes short-term aging.

#### *3.2. Asphalt Rutting Factor Test Results and Analysis*

The DSR test results of the asphalts under different test conditions are shown in Figures 2–4.

**Figure 2.** The results of the complex modulus and phase angle versus temperature for the unaged asphalt.

**Figure 3.** The results of the complex modulus and phase angle versus temperature for the short-term aged asphalt.

**Figure 4.** The results of the complex modulus and phase angle versus temperature for the long-term aged asphalt.

The complex modulus G\* can describe the ability of the asphalt to resist deformation, and the δ can reflect the proportional relationship between the elastic and viscous parts of the asphalt. Generally speaking, the larger δ is, the more viscous the asphalt is. From Figures 2–4, it can be seen that as the temperature increases, G\* decreases and δ increases, and the regularity is not related to the source of the aged asphalt, the admixture of the aged asphalt, or whether it has been aged. The G\* of the reclaimed asphalt is greater than that of the virgin asphalt under each test temperature condition, and the δ of the reclaimed asphalt is less than that of the virgin asphalt, whereby the lower the temperature the more significant the result. In addition, the aged asphalt with the higher admixture content has a larger G\* and smaller δ.

Asphalt under the long-term coupling effects of heat, oxygen, light, water, and load will experience serious aging, which will be manifested in the components as a decrease in aromatic content, an increase in asphalt content, and a macroscopic increase in hardness. From a viscoelastic point of view, the viscosity of asphalt decreases, the elasticity increases, and the asphalt changes from the sol–gel state to the gel state, which leads to a higher G\* and lower δ.

Adding a different proportion of aged asphalt to the virgin asphalt can improve the high-temperature performance of recycled asphalt; that is, the ability of the asphalt to resist high-temperature deformation. This is manifested as an increase in G\* and a decrease in δ. According to the changes of G\* and δ, it is considered that adding virgin asphalt to the aged asphalt can restore the rheological properties of the aged asphalt mixture.

#### *3.3. Asphalt Rutting Factor Evaluation Test Results and Analysis*

The road rutting is the irrecoverable deformation of asphalt pavement under the coupling effect of load and high temperature, which can be evaluated by using the rutting factor (G\*/sinδ). The variation curves of the G\*/sinδ with temperature for the virgin and reclaimed asphalts are shown in Figure 5.

As shown in Figure 5, the G\*/sinδ values of the virgin and reclaimed asphalts gradually decreased with the increase in temperature. The G\*/sinδ of the reclaimed asphalt was larger than that of the virgin asphalt. The nonlinear regression analysis showed that the G\*/sinδ had a good exponential relationship with the temperature, and the correlation coefficients were all above 0.90.

In order to evaluate the effects of the source and admixture of the aged asphalt on the rutting factor of the reclaimed asphalt, the dimensionless rutting factor ΔG\*/sin<sup>δ</sup> was evaluated, and the calculation can be found in Equation (4):

$$\Delta\_{\rm G\*/\sin\delta} = \frac{\rm G\*/\sin\delta\_{\rm mix.r} - \rm G\*/\sin\delta\_{\rm new.r}}{\rm x} \tag{4}$$

where ΔG\*/sin<sup>δ</sup> is the dimensionless rutting factor of the reclaimed asphalt, G\*/sinδ*mix.r* is the relative rutting factor of the reclaimed asphalt, G\*/sinδ*mix.r* = (G\*/sinδ*mix*)/(G\*/sinδ*new*), G\*/sinδ*mix* is the rutting factor of the reclaimed asphalt, G\*/sinδ*new* is the rutting factor of the virgin asphalt, G\*/sinδ*new.r* is the relative rutting factor of the virgin asphalt, G\*/sinδ*new.r* = 1, and *x* is the amount of aged asphalt mixing.

**Figure 5.** The results of the rutting factor versus temperature for unaged (**a**) and short-term aged (**b**) asphalts.

The variance results for ΔG\*/sin<sup>δ</sup> and G\*/sinδ for reclaimed asphalt at different test temperatures and under different aging conditions are shown in Table 5 with the significance level of *α* = 0.05.

**Table 5.** Analysis of variance results for reclaimed asphalt ΔG\*/sin<sup>δ</sup> and G\*/sinδ.


As can be seen from Table 5, the statistical probability *p*-value of G\*/sinδ is only less than 0.05 under one test temperature, indicating that using G\*/sinδ as an indicator to evaluate the high-temperature stability of the reclaimed asphalt under different aging conditions is unable to distinguish the difference between the virgin asphalt and aged asphalt. The four *p*-values of ΔG\*/sin<sup>δ</sup> are less than 0.05, indicating that using ΔG\*/sin<sup>δ</sup> as an indicator to evaluate the high-temperature performance of the reclaimed asphalt under different temperature and aging conditions can distinguish the differences between virgin asphalt and aged asphalt. Therefore, it is more reasonable to use ΔG\*/sin<sup>δ</sup> as an indicator.

The variation in ΔG\*/sin<sup>δ</sup> versus temperature for the reclaimed asphalt is shown in Figure 6. It can be found that the ΔG\*/sin<sup>δ</sup> of the reclaimed asphalt decreases gradually with the increase in temperature. The ΔG\*/sin<sup>δ</sup> values for A30W, B30W, and C30W at the test temperature of 58 ◦C were 16.0, 11.8, and 9.2, respectively, indicating that the relative rutting factors of the aged asphalt increased by 16.0, 11.8, and 9.2 for each 1% increase in the admixture of aged asphalt. Under the same temperature conditions, the relative rutting factor of A15W was 14.9, which was not the same as that of A30W, indicating that every 1% increase in the admixture of aged asphalt produced inconsistent changes in the high-temperature performance of the reclaimed asphalt, even if the aged asphalt was from the same source. Different reclaimed asphalts at different test temperatures and different aging conditions will produce similar test results as above, which are similar to the relative viscosity test results.

**Figure 6.** Variations of ΔG\*/sin<sup>δ</sup> for reclaimed asphalt versus temperature.

The results of the regression analysis show that ΔG\*/sin<sup>δ</sup> has a good linear relationship with the test temperature (T). The regression equations are all ΔG\*/sin<sup>δ</sup> = a*T* + b (a and b are fitting parameters), as shown in Equations (5) and (6):

$$\mathbf{A}\_{\rm G\*/sin\delta} = \begin{cases} -0.5518T + 44.855, \, R^2 = 0.92, \, A15\_W \\ -0.4383T + 40.44, \, R^2 = 0.92, \, A30\_W \\ -0.4957T + 39.797, \, R^2 = 0.92, \, B15\_W \\ -0.3548T + 31.908, \, R^2 = 0.91, \, B30\_W \\ -0.3799T + 29.819, \, R^2 = 0.83, \, C15\_W \\ -0.2861T + 24.891, \, R^2 = 0.92, \, C30\_W \end{cases} \tag{5}$$
 
$$\Delta\_{\rm G\*}/\sin\delta = \begin{cases} -0.4626T + 42.694, \, R^2 = 0.95, \, A15\_D \\ -0.4702T + 46.508, \, R^2 = 0.76, \, A30\_D \\ -0.5171T + 43.036, \, R^2 = 0.93, \, B15\_D \\ -0.3696T + 35.064, \, R^2 = 0.81, \, B30\_D \\ -0.4229T + 33.405, \, R^2 = 0.87, \, C15\_D \\ -0.3281T + 33.405, \, R^2 = 0.86, \, C30\_D \end{cases} \tag{6}$$

From Equations (5) and (6), it can be seen that the ΔG\*/sin<sup>δ</sup> of the reclaimed asphalt is always linearly related to the test temperature, independent of the source and admixture of aged asphalt, and will not change after short-term aging.

#### *3.4. Asphalt Fatigue Factor Evaluation Test Results and Analysis*

An increase in rutting factor enhances the ability of the asphalt to resist permanent deformation under high-temperature conditions; however, a high rutting factor can lead to the asphalt being susceptible to cracking under low- and medium-temperature conditions. Therefore, the fatigue factor G\*sinδ was introduced to characterize the ability of the asphalt to resist fatigue cracking under medium-temperature conditions after long-term aging. The variation curves of G\*sinδ values with temperature for virgin and reclaimed asphalts after long-term aging are shown in Figure 7.

As can be seen from Figure 7, the G\*sinδ gradually decreases as the temperature increases, and the G\*sinδ of the reclaimed asphalt is larger than that of the virgin asphalt. The higher the amount of aged asphalt admixture, the larger the G\*sinδ. The results of the nonlinear regression analysis showed that the G\*sinδ was exponentially related to the temperature, and the correlation coefficients were all above 0.95.

Here, ΔG\*sin<sup>δ</sup> was selected as the dimensionless fatigue factor indicator for the reclaimed asphalt, and the effect of the relative fatigue factor of the reclaimed asphalt on its resistance to fatigue cracking under medium-temperature conditions was evaluated.

The variance results for the reclaimed asphalt ΔG\*sin<sup>δ</sup> and G\*sinδ are shown in Table 6.

**Figure 7.** Variation of ΔG\*sin<sup>δ</sup> for asphalt versus temperature.



As can be seen from Table 6, the effect of the aged asphalt admixture on the fatigue resistance of reclaimed asphalt cannot be evaluated using G\*sinδ as an indicator. The three *p*-values of ΔG\*sin<sup>δ</sup> are less than 0.05, indicating that the effects of the source and admixture on the rheological properties of the reclaimed asphalt can be assessed using ΔG\*sin<sup>δ</sup> as an indicator. Therefore, ΔG\*sin<sup>δ</sup> is suitable for characterizing the mid-temperature rheology of aged asphalt. The variation in ΔG\*sin<sup>δ</sup> of the reclaimed asphalt versus temperature is shown in Figure 8.

**Figure 8.** Variation of ΔG\*sin<sup>δ</sup> for reclaimed asphalt versus temperature.

It can be seen that the ΔG\*sin<sup>δ</sup> of the reclaimed asphalt increases gradually with the increase in temperature. Even if the source of the aged asphalt is the same, the fatigue resistance of the reclaimed asphalt will vary with every 1% increase in admixture. The different reclaimed asphalts at different temperatures produced similar test results, as described above, which were similar to the relative viscosity and relative rutting factor test results.

The regression analysis showed that the linear relationship between the ΔG\*sin<sup>δ</sup> and temperature T is correlated well, and their regression equations are both ΔG\*sin<sup>δ</sup> = a*T* + b (a and b are fitting parameters), the calculation equation for which is shown in (7):

$$
\Delta\_{\text{G}\ast\sin\delta} = \begin{cases}
0.1657T + 6.8668, R^2 = 0.67, A15\_{\text{C}} \\
0.823T - 4.9943, R^2 = 0.88, A30\_{\text{C}} \\
0.1941T + 4.1691, R^2 = 0.85, B15\_{\text{C}} \\
0.7204T - 5.2205, R^2 = 0.89, B30\_{\text{C}} \\
0.1282T + 4.2875, R^2 = 0.85, C15\_{\text{C}} \\
0.4313T - 0.9126, R^2 = 0.83, C30\_{\text{C}}
\end{cases}
(7)
$$

From Equation (7), it can be seen that the effects of the source and admixture of the aged asphalt on the fatigue resistance of the reclaimed asphalt can be similarly expressed by the slope and intercept in the linear relationship equation.

#### **4. Conclusions**

The addition of the aged asphalt increases the viscosity, rutting factor, and fatigue factor of the reclaimed asphalt, indicating that the high-temperature stability of the reclaimed asphalt is enhanced but the fatigue cracking resistance is attenuated.

The effect of the aged asphalt on the viscosity of the reclaimed asphalt can be evaluated using Δ*η*. The variance results showed that the value of Δ*<sup>η</sup>* depends on the source and the admixture of aged asphalt. At high temperatures, Δ*<sup>η</sup>* decreases linearly with increasing temperature, and its linear trend is independent of the source, the admixture, and whether it has been aged or not.

The effect of the aged asphalt on the viscoelasticity of reclaimed asphalt can be evaluated using ΔG\*/sin<sup>δ</sup> and ΔG\*sinδ. The variance results showed that ΔG\*/sin<sup>δ</sup> and ΔG\*sin<sup>δ</sup> depend on the source and admixture of aged asphalt. At high temperatures, ΔG\*/sin<sup>δ</sup> decreases linearly with increasing temperature, and at medium temperatures, ΔG\*sin<sup>δ</sup> increases linearly with increasing temperature; both linear trends are independent of the source and admixture of aged asphalt.

**Author Contributions:** Conceptualization, T.L.; methodology, W.D.; software, Y.Q.; validation, J.Z.; formal analysis, Q.L.; resources, J.W.; data curation, J.X.; writing—original draft preparation, R.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by [the basic scientific research of central institute: 2021-9045a].

**Institutional Review Board Statement:** Not applicable for studies not involving humans or animals.

**Informed Consent Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Bagdat Teltayev 1,\*, Erik Amirbayev <sup>1</sup> and Boris Radovskiy <sup>2</sup>**

<sup>2</sup> Radnat Consulting, Irvine, CA 90292, USA; b.radovskiy@att.net

**\*** Correspondence: bagdatbt@yahoo.com; Tel.: +7-701-760-6701

**Abstract:** This paper investigates the viscoelastic properties of oxidized neat bitumen and three polymer-modified binders at low temperatures. The earlier proposed interrelated expressions for the relaxation modulus and for the creep compliance of bitumen binders are further developed. The results of creep testing of the binders on a bending beam rheometer at the six temperatures from −18 ◦C to −36 ◦C are presented. The results were analyzed using the equations developed for the relaxation modulus and the relaxation time spectrum. Viscosities at the low temperatures of tested binders were estimated. Approximate interrelations between the loss modulus and the relaxation spectrum were presented. The method for the determination of the glass transition temperature of a binder in terms of the relaxation time spectrum is proposed. The glass transition temperatures of tested binders were determined by the proposed method and compared with ones determined by the standard loss modulus-peak method.

**Keywords:** neat and polymer-modified bitumens; low temperatures; creep compliance; relaxation modulus; relaxation time spectrum; glass transition temperature

#### **1. Introduction**

Bitumens are widely used in road paving because of their good adhesion to mineral aggregates and their viscoelastic properties. In paving applications, the bitumen should be resistant to climate and traffic loads, for which reason its rheological properties play a key role. It has to be stiff enough at high temperatures to resist rutting at local pavement temperature around 60 ◦C while it must remain soft and viscoelastic enough at low temperatures (from −20 ◦C to −45 ◦C) to resist thermal cracking. Those requirements are almost opposite, and most of the available neat bitumens would not provide all the needed characteristics together because bitumen is brittle in cold winters and softens readily in hot summers. Moreover, asphalt pavement at intermediate temperatures should be resistant to fatigue cracking from tensile and shear stresses under the action of repeated loading caused by traffic.

In order to enhance neat bitumen properties and widen the service temperature, bitumens are often modified by the addition of polymers. Polymer modification improves mechanical properties, decreases thermal susceptibility and permanent deformation (rutting), and increases resistance to low-temperature cracking. The most commonly used additives are copolymers, such as styrene butadiene styrene (SBS), ethylene vinyl acetate (EVA), styrene- ethylene-butylene-styrene (SEBS). The wide use of this type of polymer for modification is due to its thermoplastic nature at higher temperatures and its ability to form networks upon cooling. Particularly, when SBS is blended with bitumen, the elastomeric phase of the SBS copolymer absorbs the maltenes (oil fractions) from the bitumen and swells up to nine times its initial volume [1]. At SBS concentrations 5–7% by mass, a continuous polymer network (phase) is formed throughout the modified binder, significantly improving the bitumen properties at high and intermediate temperatures.

**Citation:** Teltayev, B.; Amirbayev, E.; Radovskiy, B. Evaluating the Effect of Polymer Modification on the Low-Temperature Rheological Properties of Asphalt Binder. *Polymers* **2022**, *14*, 2548. https://doi.org/10.3390/ polym14132548

Academic Editors: Wei Jiang, Quantao Liu, Jose Norambuena-Contreras and Yue Huang

Received: 16 May 2022 Accepted: 15 June 2022 Published: 22 June 2022

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Until now, the researchers have not developed a convincing opinion on the positive and significant effect of modifying bitumens with polymers at low temperatures. Lu et al. [2] tested three bitumens blended with 6% SBS, EVA, SEBS, or EBA. They concluded that the effect of modification on the low-temperature properties of bituminous binders was small. Authors of [3,4] concluded that modified bitumens have better resistance to low temperature cracking compared to the unmodified ones while Peng [5] found that at −12 ◦C and −18 ◦C, the low-temperature stability of the modified bitumen is significantly increased, although at −24 ◦C it is slightly reduced. Lu et al. in work [6] reported that the glass transition temperature of a bitumen defined as the temperature of the peak loss modulus is reduced by polymer modification; but the results of creep tests performed using a BBR at temperatures of−15 ◦C, −20 ◦C, 25 ◦C, and −30 ◦C showed that polymer modification does not give beneficial effect; in some cases, especially for the limiting temperature at 0.3 m value, even adverse effect is found for polymer modification. In work [7], dense asphalt concrete samples prepared with bitumens modified with various amounts of SBS polymer were subjected to restrained cooling tests and at a standard cooling rate of 10 ◦C/h, no significant effect of modification was noted.

Although considerable research was undertaken in this area, the polymer-modified bitumen has still to be comprehensively characterized, due to the complex nature and interaction of the bitumen and polymer system. The present work focuses on the low-temperature rheological properties of the polymer-modified bitumen binders for road pavements.

The low-temperature transverse cracking of asphalt concrete pavements is major pavement distress commonly observed in regions affected by cold weather. Thermal cracking is induced by a rapid drop in temperature that tends to cause contraction and results in tensile stresses that may eventually reach the tensile strength of the material causing its fracture. The field performance data from test roads in Alberta, Manitoba, and Ontario (Canada), and Pennsylvania (USA) indicated that the binder is mostly responsible for the cold-weather cracking of asphalt pavements [8–12]. It became clear that only a fundamental understanding based on sound theory such as binder rheology might provide confidence for moving forward into a low-temperature cracking problem [13–20].

In a pioneering work, Monismith et al. [21] developed a calculation method for the thermally induced stress in the longitudinal direction of asphalt pavement as in an infinite viscoelastic beam. Boltzmann's superposition principle and the constitutive equation for linear viscoelastic material were applied to relate time-dependent stresses and strains. This approach was then widely used to estimate the development of thermal stresses [12,22–28]. It turned out that the problem of tensile strength determination for the binder and asphalt concrete was much more complicated, particularly including a ductile-to-brittle failure domain related to the peak to the tensile strength vs. temperature curve [16,24,29–32].

For many years, researchers have attempted to develop tests that can be incorporated into the binder pass/fail low-temperature specifications. Mandatory binder stiffness (S at 60 s < 300 MPa) and its slope (m value at 60 s, *d* log *S*/*d* log *t* = m > 0.30) at the designed low pavement temperature were included in the standard specifications AASHTO M 320 and ASTM D 6816. Later proposed rheological low-temperature parameters include the difference Δ*T* between *TS*=<sup>300</sup> and *Tm*=0.30 [33]; stiffness at m-value = 0.3; Glover parameter that is a combination of storage modulus and a real part of complex viscosity and serves as a surrogate parameter for ductility [34]; Glover–Rowe parameter (G–R) which is the same parameter expressed in other terms [35].

Bitumen contains up to several thousand individual chemical components ranging from non-polar saturated alkanes to polar hetero-hydrocarbons [36]. Phenomenologically, it, therefore, seems natural to view a bitumen as a continuum of molecules with a gradual transition in molecular weight and polarity and with the corresponding continuous spectrum of relaxation times. The change in bitumen binder properties associated with the degree of packing does not occur instantaneously with change in temperature, but requires the passage of time. As the temperature is reduced, then the time scale for these rearrangements increases. In some temperature ranges, this time is of the order of the

time scale of the experimental measurement, from minutes to hours. If the temperature is reduced still further, this time scale is increased so much that no further change in order can be observed over practical time scales. The temperature at which this transition occurs is the glass transition temperature.

It was shown that the glass transition temperature of bitumen binders is a reasonable predictor of the temperature close to which the asphalt pavement will thermally fracture [12,37,38]. Since low-temperature cracking is widespread pavement distress, the ability to determine bitumen binders is an important tool for asphalt researchers [18,29,39–45]. The objective of this paper is to increase the understanding of the low-temperature behavior of bitumen binders, particularly to estimate the glass transition temperature in terms of the relaxation time spectrum.

#### **2. Theoretical Background**

#### *2.1. Linear Viscoelastic Rheological Characterization*

To describe the rheological properties of binders we used the earlier proposed model [24,46,47] that includes expressions for tensile relaxation modulus *E*(*t*) and for the tensile creep compliance *D*(*t*):

$$E(t) = E\_{\mathcal{S}} \left[ 1 + \left( \frac{E\_{\mathcal{S}} t}{3\eta} \right)^b \right]^{-(1 + 1/b)} \text{ .} \tag{1}$$

$$D(t) = \frac{1}{E\_{\mathcal{S}}} \left[ 1 + \left( \frac{E\_{\mathcal{S}}t}{\Im \eta} \right)^{\beta} \right]^{1/\beta},\tag{2}$$

$$\beta = \left[\frac{1}{b} - \frac{\ln(\pi)}{\ln(2)} + 2\right]^{-1} \text{ .} \tag{3}$$

where *t* is a time, s; *Eg* is tensile instantaneous modulus (tensile glassy modulus), Pa; *η* is the steady-state shear viscosity, Pa·s; *b* is a constant (0 < *b* < 1) governing the shape and width of the relaxation time spectrum.

The model [24,46,47] includes expressions (1) and (2) for tensile relaxation modulus *E*(*t*) and for the tensile creep compliance *D*(*t*). The constant *β* in Equation (2) was shown to be dependent on the thermal susceptibility of the bitumen binder based on SHELL testing data for 46 bitumens [24]. Equations (2) and (3) were derived by means of linear viscoelasticity in our monograph [46]. Expressions (1) and (2) for tensile relaxation modulus and for the tensile creep compliance match each other (Figure 1).

**Figure 1.** The convolution product of relaxation modulus and creep compliance.

As it is seen, Equations (1) and (3) include three parameters and all of them have physical meanings.

The relaxation and creep functions *E*(*t*) and *D*(*t*) are connected by the relation in form of the exact convolution integral [48]:

$$\mathcal{C}(t) = \int\_0^t E(\xi) \, D(t - \xi) \, d\xi \, = \text{ t.} \tag{4}$$

The convolution product *C*(*t*) of Equation (4) was calculated using Equations (1) and (2) at 3*η*/*Eg* = 0.01 s for *b* = 0.2, 0.3, and 0.4. The results plotted in Figure 1 numerically confirm that the expressions (1) and (2) for relaxation modulus and creep compliance match each other.

The mean relative deviation of the convolution product *C*(*t*) from the equality line in Figure 1 is around 1.4% for the time scale of twelve logarithmic decades. Supposing isotropy and incompressibility, Equations (1) and (2) for shear relaxation modulus *G*(*t*) and for the shear creep compliance *J*(*t*) can be written in the forms:

$$G(t) = G\_{\mathcal{S}} \left[ 1 + \left( \frac{G\_{\mathcal{S}}t}{\eta} \right)^b \right]^{-(1+1/b)} \text{.} \tag{5}$$

$$\mathbf{J}(t) = \frac{1}{\mathcal{G}\_{\mathcal{S}}} \left[ 1 + \left( \frac{\mathcal{G}\_{\mathcal{S}} t}{\eta} \right)^{\mathcal{B}} \right]^{1/\mathcal{B}},\tag{6}$$

where *Gg* is the shear glassy modulus, MPa.

Complex modulus in shear can be determined from creep compliance Equation (6) using the Schwarzl and Struik method [49]:

$$|G^\*|(\omega) = \frac{G\_{\mathcal{S}}}{\Gamma(1 + m(\omega))} \left[ 1 + \left(\frac{G\_{\mathcal{S}}}{\eta \omega}\right)^{\mathcal{J}} \right]^{-\frac{1}{\mathcal{J}}}, \delta(\omega) = \frac{\pi}{2} m(\omega),\tag{7}$$

where |*G*∗(*ω*)| is the norm of complex modulus; *δ*(*ω*) is phase angle; *ω* is the angular frequency, rad/s; Γ(*x*) is the gamma function, and

$$m(\omega) = \frac{(G\_{\mathcal{S}}/\eta\omega)^{\beta}}{1 + (G\_{\mathcal{S}}/\eta\omega)^{\beta}}.\tag{8}$$

Here the Van der Poel-Koppelmann [13,50] conversion from time to frequency domain *t* → 1/*ω* was applied to Equation (6) to derive Equation (7) using [49].

#### *2.2. Spectrum of Relaxation Times*

According to Bernstein's theorem [51], every monotonic function can be written as a sum of exponential decay functions exp(−*t*/*τ*). The relaxation modulus of a viscoelastic liquid *E*(*t*) is a continuous, decreasing function and thus it can be expressed in form of the integral transform [52]:

$$E(t) = \int\_0^\infty H(\tau) \exp(-t/\tau) \frac{d\tau}{\tau} = \int\_{-\infty}^\infty H(\tau) \exp(-t/\tau) d\ln\tau,\tag{9}$$

where *H*(*τ*) is the distribution function of relaxation times *τ*, shortly the relaxation– time spectrum.

If relaxation modulus *E*(*t*) is given, the spectrum *H*(*τ*) can be found from the integral Equation (9) by inverting the Laplace transform. Applying the Widder's inverse Laplace transformation [53] leads to the following asymptotic formula for relaxation–time spectrum:

$$\begin{array}{l} H\_{\boldsymbol{n}}(\boldsymbol{\tau}) = \frac{(-1)^{\boldsymbol{n}}(n\boldsymbol{\tau})^{\boldsymbol{n}}}{(n-1)!} E^{(\boldsymbol{n})}(n\boldsymbol{\tau}),\\ E^{(\boldsymbol{n})}(n\boldsymbol{\tau}) = \begin{bmatrix} \frac{d^{\boldsymbol{n}}E(\boldsymbol{t})}{dt^{\boldsymbol{n}}} \end{bmatrix}\_{\boldsymbol{t}=n\boldsymbol{\tau},} \end{array} \tag{10}$$

where *n* is the degree of approximation (*n* = 1, 2, 3. . . ).

As *n* becomes infinite [53], the right side of Equation (10) tends to the exact relaxation– time spectrum *H*(*τ*). The convergence rate depends on the relaxation modulus *E*(*t*).

Substituting Equation (1) to Equation (10) leads to the following expressions for the first and second approximations of the spectrum:

$$H\_1(\tau) = \frac{E\_\mathcal{S} (1+b) \left(\frac{E\_\mathcal{X}\tau}{3\eta}\right)^b}{\left[1 + \left(\frac{E\_\mathcal{X}\tau}{3\eta}\right)^b\right]^{2+1/b}}\tag{11}$$

$$H\_{2}(\tau) = \frac{E\_{\mathcal{S}}(1+b)\left[(2+b)\left(\frac{2E\_{\mathcal{S}}\tau}{3\eta}\right)^{b} + 1 - b\right] \left(\frac{2E\_{\mathcal{X}}\tau}{3\eta}\right)^{b}}{\left[1 + \left(\frac{2E\_{\mathcal{X}}\tau}{3\eta}\right)^{b}\right]^{3+1/b}}.\tag{12}$$

Figure <sup>2</sup> presents an example of calculated spectra for *Eg* = 2.460 × 109 Pa, *<sup>b</sup>* = 0.1914, *<sup>η</sup>* = 4.247 × <sup>10</sup><sup>6</sup> Pa·s in the first, second and third approximations. The maximum spectrum density in the first approximation is only 1.6% smaller than in the second and 1.8% smaller than in the third approximation.

**Figure 2.** Comparison of calculated spectra.

Thus, the precision of simple first approximation Equation (11) is acceptable for our purposes. Moreover, Equation (11) has three intriguing features associated with it. First, analytically taking the integral one can obtain exactly

$$\int\_0^\infty \frac{H\_1(\tau)}{\tau} d\tau = E\_{\mathbb{S}'} \tag{13}$$

as it follows from Equation (9) at *t* = 0. The area between the curve *H*1(*τ*) and axis ln *τ* equals the instantaneous modulus, as it should.

Secondly, analytically taking the integral one can obtain exactly

$$\int\_{0}^{\infty} H\_{1}(\tau)d\tau = 3\eta\_{\prime} \tag{14}$$

as it should be [48]. The area between the curve *H*1(*τ*) and axis *τ* equals the elongational viscosity *η<sup>e</sup>* = 3*η* [54].

Thirdly, differentiation leads to

$$d \log H\_1(\tau) / d \text{ log } \tau = \frac{1 + 2b}{1 + \left(\frac{E\_\chi \tau}{3\eta}\right)^b} - 1 - b \tag{15}$$

It follows from Equation (15) that at *τ* = 0 the slope *d* log *H*1(*τ*)/*d* log *τ* = *b* while when *τ* → ∞ the slope *d* log *H*1(*τ*)/*d* log *τ* = −(1 + *b*). Obviously, the shape parameter *b* is related to the slopes of the relaxation spectrum. The slope *d* log *H*1(*τ*)/*d* log *τ* = *b* describes the low-relaxation time wing of the spectrum while the slope *d* log *H*1(*τ*)/*d* log *τ* = −(1 + *b*) corresponds to the high-relaxation time wing (Figure 3).

**Figure 3.** Spectrum and geometrical meaning of parameter *b*.

Spectrum density *H*1(*τ*) peaks at the modal relaxation time:

$$\pi\_{\rm m} = \frac{3\eta}{E\_{\rm g}} \left( \frac{b}{1+b} \right)^{1/b} \text{ \textsuperscript{\rm m} \,\tag{16}$$

#### **3. Materials and Methods**

#### *3.1. Binders*

Four asphalt binders were tested. A neat bitumen of penetration grade BND 100/130 produced by direct oxidation from Siberian crude oil by Pavlodar petrochemical plant is commonly used in Kazakhstan paving industry. The second binder was the base bitumen BND 100/130 modified by the reactive ethylene terpolymer Elvaloy 4170 (Du Pont, NY, USA) in the amount of 1.4% by weight. The third binder was the base bitumen modified by the cationic bitumen emulsion of Butanol NS 198 (BASF, Ludwigshafen, Germany) in the amount of 3% by weight. The fourth binder was the base bitumen compounded with a flux (vacuum residue) from the same plant (flow time 82 s at 80 ◦C) in the amount of 20% by weight and modified by the polymer SBS L 30-01 (Sibur Co., Moscow, Russia)in the amount of 5% by weight.

Elvaloy 4170 is a chemically active copolymer of ethylene (71%) with butyl acrylate (20%) and glycidyl methacrylate (9%). Butanol NS 198 is a cationic, high molecular weight styrene butadiene dispersion designed for use in asphalt modification and waterproofing. The content of solid polymers in Butanol is 64%. SBS L 30-01 A represents a linear block for copolymer of styrene (30%) and butadiene (70%). The molecular weights of the polymers Elvaloy and Butanol are in the range of 60,000–80,000 and 80,000–90,000, respectively. The structural formulas of SBS and Elvaloy polymers are shown in Figure 4.

**Figure 4.** The structural formulas of (**a**) SBS and (**b**) Elvaloy polymers.

#### *3.2. Preparation of Compounded and Modified Binders*

Modification of the bitumen with polymers Elvaloy and Butanol was carried out in accordance with the normative documents of Kazakhstan [55] and [56], respectively. The polymers Elvaloy and Butanol were gradually added to the heated neat bitumen at 170 ◦C using a laboratory mixing device. Continuous mixing process of polymer–bitumen binders lasted for two hours, the next twelve hours the Elvaloy modified binder was conditioned at constant temperature of 170 ◦C.

Compounding of the base bitumen with the flux was performed by means of mixing with the rate of 450–500 rotations per minute at the constant temperature of 120 ◦C for 30 min [57]. Then, the compounded bitumen was gradually heated up to 180 ◦C and polymer SBS was gradually added. During the first two hours and next four hours, a mixing rate was equal to 1200 and 1800 rotations per minute, respectively.

#### *3.3. Conventional Properties of Binders*

Conventional properties of the binders were defined in the Research Laboratory of Kazakhstan Highway Research Institute according to the test specification ST PK 1373−2013 and they are presented in Table 1.

**Table 1.** Conventional properties of binders.


#### *3.4. Rheological Testing*

The binders were tested at low temperatures (−18 ◦C, −24 ◦C, −27 ◦C, −30 ◦C, −33 ◦C, and −36 ◦C) on the ATS Bending Beam Rheometer (BBR) according to the standard ASTM D 6648.

Prior to the rheological testing, the binders were aged using a Rolling Thin-Film Oven (RTFO) according to the test specification ASTM D2872 to simulate the short-term aging during asphalt mix manufacture. Then the binders were further aged in the Pressurized Aging Vessel (PAV) to simulate the long-term aging (ASTM D6521).

The binder samples for the testing had a shape of a beam with dimensions 6.25 × 12.5 × 125 mm. The duration of specimen conditioning prior to the testing was set to one hour. In the BBR creep test a constant load *P* = 0.98 N was applied at the midpoint of the simply supported binder beam for 240 s. The mid-span deflection *d*(*t*) was constantly recorded. The creep stiffness *S*(*t*) was automatically calculated from the equation:

$$S(t) = \frac{PL^3}{4wh^3 \, d(t)},\tag{17}$$

where *L* is the span of the beam (102 mm); *w* is the width of the beam (12.5 mm); *h* is the height of the beam (6.25 mm); *d*(*t*) is maximum deflection of the beam at time *t*.

Only the observations at 8, 15, 30, 60, 120, and 240 s were employed in the present study.

#### **4. Results and Discussion**

*4.1. Stiffness and Viscosity*

As an example, the results of testing are shown in Figure 5 for the Elvaloy modified bitumen binder.

**Figure 5.** Time-dependent stiffness at different temperatures for Elvaloy modified binder.

To produce a master curve at a selected reference temperature *Tr*, Equation (2) combined with the Arrhenius time-temperature superposition function was used:

$$\mathcal{S}(t) = \frac{1}{D(t)} = E\_{\mathcal{S}} \left[ 1 + \left( \frac{E\_{\mathcal{S}}t}{\Im \eta} \right)^{\beta} \right]^{-1/\beta},\tag{18}$$

$$\eta = \eta\_r \exp\left[\frac{\Delta H\_a}{R}\left(\frac{1}{273 + T} - \frac{1}{273 + T\_r}\right)\right],\tag{19}$$

where *η<sup>r</sup>* is a viscosity at a reference temperature, Pa·s; Δ*Ha* is the flow activation energy, J/mol; *R* is the universal gas constant equal to 8.314 J/(mol·K).

The parameter *β* is related to *b* as before by Equation (3).

Based on our previous study [24,46], the instantaneous tensile modulus was assumed *Eg* = 2460 MPa for all tested binders. The reference temperature was selected close to the midrange of testing temperatures *Tr* = −30 ◦C. Using the Mathcad software, a nonlinear minimization algorithm was implemented to determine simultaneously the parameters *ηr*, Δ*Ha*, and *b* by minimizing the sum of squared deviations of data points from the master curve *S*(*t*) Figure 6.

**Figure 6.** Master curve of stiffness as a function of time for the Elvaloy modified bitumen binder at *Tr* = −30 ◦C.

The obtained values of the parameters are given in Table 2. Figure 7 shows the viscosity as a function of temperature calculated using Equation (19).

**Table 2.** Values of the parameters for binders.


The slopes of the viscosity-temperature relationships to the temperature axis for the binders modified by the polymers Elvaloy and Butanol are almost equal and they are smaller than the slopes for the neat bitumen and the bitumen compounded by the flux and modified by the polymer SBS. This indicates the lower temperature susceptibility of the Elvaloy and Butanol-modified binders. In the range of the testing temperatures, the Elvaloy-modified binder has the smallest viscosity while the Butanol-modified binder (at temperatures higher than −27 ◦C) and the flux compounded and SBS-modified −27 ◦C) have the greatest one. The ability to estimate the viscosity of a binder at subzero temperatures indirectly from conventional BBR testing is a useful feature of the paper.

**Figure 7.** Viscosity–temperature relationships for the binders.

#### *4.2. Glass Transition Temperature in Terms of Loss Modulus*

Several researchers have shown that the glass transition temperature *Tg* of a bitumen binder is associated with the low-temperature cracking of a pavement [12,37,38]. The transition to a glassy state increases the brittleness of the binder, reducing its ability for stress relaxation and resulting in higher cracking susceptibility of an asphalt pavement. Researchers measured the glass transition temperature of bitumen binders by using three different techniques: dilatometry, calorimetry and rheological method-peak in the loss modulus versus temperature.

The classic method for the determination of the glass transition temperature is dilatometry. The temperature dependence upon cooling of the specific volume is determined by a suitable technique, and the temperature at the change in slope is taken as *Tg* at a given cooling rate [8,39–41]. Because of the need for precise measurements of small changes in volume with decreasing temperature, the dilatometric method is a difficult procedure to perform. Calorimetry was extensively employed, a peak in heat capacity being observed at the temperature *Tg*, depending on the heating rate [29,42,43].

Last year, the rheological dynamic measurements were conducted to estimate the glass transition temperature of bitumen binders. The data are collected over the temperature range at constant frequency and the loss modulus G (or E) peak temperature is taken as *Tg*. The standard ASTM 1640 [58] recommends the testing frequency 1 Hz (ω = 6.28 rad/s). This standard admits other frequencies but they should be reported. Anderson and Marasteanu [41] used frequency 0.1 rad/s, Reinke and Engber [44] used 0.1–1.0 rad/s, Planche et al. [43] used 5 rad/s, Sun et al. used 10 rad/s [45]. Changing the time scale by a factor of 10 will generally result in a shift of about 8 ◦C for a typical amorphous material [58]. Anderson and Marasteanu [59] compared dilatometry, calorimetry, and the peak in the G" and concluded that all methods give estimates of the glass transition temperature that are in relatively good agreement, given the different nature of the measurements and the time scale of the measurements.

In the present study, *Tg* is defined based on BBR testing as the temperature where the tensile loss modulus *E* peaks at frequency 1 rad/s. The tensile loss modulus *E* can be determined from tensile creep compliance *D*(*t*) using the Schwarzl and Struik method [49] from the equation

> *<sup>E</sup>*(*ω*) = *Eg* sin( *<sup>π</sup>* <sup>2</sup> *mE*(*ω*)) <sup>Γ</sup>(1+*mE*(*ω*)) 1 + - *Eg* <sup>3</sup>*ηω <sup>β</sup>* <sup>−</sup> <sup>1</sup> *β* , *mE*(*ω*) = (*Eg*/3*ηω*) *β* 1+(*Eg*/3*ηω*) *β* . (20)

where

Loss modulus *E*(*ω*) was calculated from Equation (20) using Equation (19) and the parameters are shown in Table 2 at the frequency *ω* = 1 rad/s. When the loss modulus *E* is plotted versus temperature, the resulting curve exhibits a peak value (Figure 8). The temperature at this peak value can be interpreted as a glass transition temperature *Tg*. The temperatures at which the calculated *E* reached a maximum value were *Tg* = −45.4 ◦C for the neat bitumen, *Tg* = −52 ◦C for the Elvaloy-modified binder, *Tg* = −46.3 ◦C for the Butanol-modified binder and *Tg* = −46.2 ◦C for the flux-compounded and SBS-modified binder. Changing the "testing" frequency by a factor of 10 results in a *Tg* shift of 5.5–7 ◦C.

**Figure 8.** Calculation of the glass transition temperatures for the tested binders using the loss modulus peak method at frequency *ω* = 1 rad/s.

At low polymer concentration (not more than about 5%), the properties of the base bitumen and the compatibility of polymer with bitumen are very important. Modification with 3% Butanol primarily aims to improve the high-temperature properties of the bitumen and does not show a positive effect at subzero temperatures compared with the base neat bitumen in this study. Considerable improvement of high-temperature properties (softening point from 45 ◦C up to 76.5 ◦C) of the base bitumen, keeping its low-temperature properties, was achieved by means of compounding by flux and modifying by polymer SBS. Modification with 1.4% Elvaloy, which is intended to enhance the high-temperature properties as well, also improved the low-temperature properties. Modification with Elvaloy lowered the temperature susceptibility, lowered the low-temperature viscosity and reduced the glass transition temperature. Particularly, these results are important for countries with a sharp-continental climate, including Kazakhstan, which is the ninth-largest in the world where half of the territory requires the bitumen binder of Superpave grade PG 58−40.

#### *4.3. Glass Transition Temperature in Terms of Relaxation Time Spectrum* 4.3.1. Interrelations between Loss Modulus and Relaxation Spectrum

To increase the understanding of the glass transition temperature of the binder it would be of interest to express *Tg* in terms of the relaxation time spectrum. In the previous section, the widely accepted definition of the glass transition temperature was used, which consisted of taking the loss modulus (G) maximum as a function of temperature at a given frequency *ω*. Loss modulus is related to the relaxation–time spectrum by an exact equation [48].

$$G''(\omega) = \int\_0^\infty \frac{H(\tau)}{\tau} \cdot \frac{\omega \tau}{1 + \omega^2 \tau^2} d\tau. \tag{21}$$

The kernel of the loss modulus function

$$f(\mathbf{r}) = \frac{\omega \mathbf{r}}{1 + \omega^2 \mathbf{r}^2} \tag{22}$$

is a crude approximation to the Dirac delta function *δD*(*τ*). The function *f*(*τ*) reaches a maximum at *τ* = 1/*ω*. Since the integral

$$\int\_0^\infty \frac{\omega \tau}{1 + \omega^2 \tau^2} d\tau = \frac{\pi}{2}$$

is not unity as it is for the delta function *δD*(*τ*), the function *f*(*τ*) must be multiplied by 2/*π* before being replaced by the delta function. Then

$$\frac{2}{\pi} \cdot \frac{\omega \tau}{1 + \omega^2 \tau^2} \approx \delta(1 - 1/\omega). \tag{23}$$

If Equation (23) is substituted into Equation (21), and using the sifting property of the Dirac delta function yields the relation:

$$G''(\omega) \approx \frac{\pi}{2} H\left(\frac{1}{\omega}\right). \tag{24}$$

Applying the Van der Poel-Koppelmann conversion from the frequency domain to the time domain *ω* → 1/*τ* leads to the equation:

$$H(\tau) \approx \frac{2}{\pi} G''(\omega)|\_{\omega \to 1/\tau}. \tag{25}$$

It follows from Equation (25) that a relaxation time spectrum *H*(*τ*) is approximately proportional to a loss modulus function *G*(*ω*). Their shapes are similar and their maxima represent the concentration of the relaxation time spectrum or correspond to the relaxation frequency spectrum in a certain region of the logarithmic *τ* or *ω* scales.

#### 4.3.2. Glass Transition Temperature

Calculated spectrum densities at the modal relaxation time *τ<sup>m</sup>* = 1/*ω<sup>m</sup>* = 1 s using Equation (11) and the parameters given in Table 2 are presented in Figure 9. The curves in Figures 8 and 9 look very similar with allowance for approximations in the derivation of Equations (11) and (25).

For a given modal frequency, e.g., *ω<sup>m</sup>* = 1 rad/s, loss modulus *G*(*ω*) at a certain temperature has a maximum, e.g., *Tg* = −45.4 ◦C for bitumen (Figure 8). According to time–temperature superposition principle, it equivalently means that for the constant temperature −45.4 ◦C the loss modulus *G*(*ω*) has a maximum at the modal frequency *ω<sup>m</sup>* = 1 rad/s. Similarly, in the relaxation time domain, the relaxation time spectrum *H*(*τ*) has a maximum at the modal relaxation time *τ<sup>m</sup>* = 1/*ω<sup>m</sup>* [Equation (16)].

Substituting Equation (19) into Equation (16) leads to the relation:

$$\tau\_{\rm ll} = \frac{3\eta\_r}{E\_\mathcal{g}} \left(\frac{b}{1+b}\right)^{1/b} \exp\left[\frac{\Delta H\_\mathcal{d}}{R} \left(\frac{1}{273+T} - \frac{1}{273+T\_I}\right)\right] \tag{26}$$

where the modal relaxation time *τ<sup>m</sup>* corresponds to the arbitrary temperature *T*.

The ratio of the modal relaxation time *τmg* at a temperature *Tg* to the modal relaxation time *τmr* at a reference temperature *Tr* is

$$\frac{\tau\_{\rm mg}}{\tau\_{\rm arr}} = \exp\left[\frac{\Delta H\_a}{R}\left(\frac{1}{273 + T\_\S} - \frac{1}{273 + T\_r}\right)\right],\tag{27}$$

It follows from (27) that the fixed *τmg* glass transition temperature *Tg* can be calculated by equation:

$$T\_{\mathcal{S}} = \left[\frac{1}{273 + T\_r} + \frac{R}{\Delta H\_d} \ln\left(\frac{\tau\_{mg}}{\tau\_{mr}}\right)\right]^{-1} - 273,\tag{28}$$

where *τmr* is the modal relaxation time at a reference temperature *Tr*:

$$\pi\_{mr} = \frac{3\eta\_r}{E\_\mathcal{g}} \left(\frac{b}{1+b}\right)^{1/b} \text{ \textit{\textsuperscript{\textsuperscript{\textsuperscript{\textsuperscript{\textsuperscript{\textsuperscript{\textsuperscript{\textsuperscript{\textsuperscript{\boxscript{\boxscript{\boxfrown{\boxscript{0}}}}}}}}}}}{1+b}}\right)^{1/b} \textit{\textsuperscript{\textsuperscript{\boxminus}}}}\tag{29}$$

For the fixed *τmg* = 1 s, using the parameters shown in Table 2, the calculated from Equation (28) glass transition temperature equals *Tg* = −45.8 ◦C for bitumen, *Tg* = −52.4 ◦C for the Elvaloy-modified binder and *Tg* = −46.7 ◦C for the Butanol-modified binder, which almost coincides with *Tg* determined from the peak of loss modulus at frequency *ω* = 1 rad/s (Figure 8). The calculated dependence of glass transition temperature on modal relaxation time is presented for the tested binders in Figure 10.

**Figure 9.** Definition of the glass transition temperatures for the binders using relaxation spectrum density at the modal relaxation time *τ<sup>m</sup>* = 1 s.

The ability to estimate the glass transition temperature in terms of relaxation time spectrum is important for understanding the behavior of binders at low temperatures. The molecular mobility of liquids including bitumen binders expresses itself in a relaxation– time spectrum. The broader the molecular weight distribution is the broader the relaxation spectrum becomes [60,61]. When a liquid-cooled down, its volume reduces due to the translational molecular readjustments rather than due to their oscillating motions. When a temperature reaches the glass transition region, the speed of molecular adjustment becomes slower and no further change in order can be observed over a given time scale. Physically, the glass transition occurs at the temperature *Tg* when the root-mean-squared displacement of the particle of average molecular weight becomes smaller than the average size of that particle [62], on a given relaxation timescale (of the order of *τ* = 1 s, for example). Relaxation at a temperature lower than *Tg* occurs mostly due to the oscillating motion of molecules. Thus, the glass transition is the transformation of a disordered state with molecular mobility to an immobilized state of a similar structure by means of decreasing temperature. The transformation of liquid (e.g., a bitumen binder) is caused by a continuous increase in the modal relaxation time up to the given scaling time. It can be the time scale of practical observation or an experiment. Relating the bitumen binder properties or performance to the particular scaling time value requires more research.

**Figure 10.** Dependence of glass transition temperature on modal relaxation time.

It is known that the addition of a polymer to a bitumen increases its viscosity, and one would expect a significant decrease in its low-temperature characteristics. However, the results of this work showed that when the bitumen is modified with the given amounts of selected polymers, a positive effect can be obtained. This can be explained by the lower glass transition temperatures of the elastic parts of the polymers (butyl acrylate in Elvaloy, butadiene in SBS and Butanol).

#### **5. Conclusions**

The main conclusions and findings based on the analysis presented in this paper are as follows:


**Author Contributions:** Conceptualization and writing—original draft preparation, investigation, B.R.; conceptualization and writing—original draft preparation, supervision, investigation, B.T.; investigation, E.A. All authors have read and agreed to the published version of the manuscript.

**Funding:** The work was performed under grant IRN AP 08857446 from the Committee of Science of the Ministry of Education and Science of the Republic of Kazakhstan. Agreement No. 230 dated 12 November 2020.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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