# High-Temperature Performance of Asphalt Mixtures: Preliminary Analysis for the Standard Technical Index Based on Gray Relational Analysis Method

Authors: Jian Xu; Yan Gong; Li-Biao Chen; Tao Ma; Jun-Cheng Zeng; Er-Hu Yan; Guang-Shu Xiao
Journal: Advances in Materials Science and Engineering
 (2020)

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## Abstract

Aiming to evaluate the high-temperature performance of asphalt binders and asphalt mixtures and to investigate the reliability of the standard technical indexes to evaluate the performance of the asphalt, six typically used asphalt types were employed in this study. The standard high-temperature rheological test, the multiple stress creep recovery (MSCR) test, and the zero-shear viscosity (ZSV) test were employed to characterize the high-temperature performance and non-Newtonian fluid properties of the asphalt. Meanwhile, the high-temperature performance of the asphalt mixture was evaluated through the rutting tests based on the mixture design of AC-13. In general, the modified asphalt performed better than the unmodified asphalt according to the high-temperature rheological properties tests. The ranking of the six kinds of asphalt was confirmed to be different in various laboratory tests. The test results of the asphalt binders showed that the Tafpack Super- (TPS-) modified asphalt performed best in the MSCR and ZSV tests, while the low-grade asphalt PEN20 had the best technical indexes in the dynamic shear rheometer (DSR) test. Besides, the relation between the asphalt and the asphalt mixture was analyzed by gray relational analysis (GRA) method. The present rutting indicatorG∗/sinδandG∗/1−sinδ⋅tanδ−1 for evaluating the asphalt mixtures’ high-temperature performance might no longer be suitable. The Cross/Williamson model was the most suitable for calculating and fitting the ZSV, which could be used as the key indicator of the high-temperature performance evaluation of the asphalt. This work lays a foundation for the further study of the high-temperature performance evaluation of asphalt binders.


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## Body

## 1. Introduction

At present, the Superpave asphalt performance evaluation system is widely used, which is a main outcome of the Strategic Highway Research Program (SHRP). In the Superpave specification, the shear complex modulusG∗, the phase angle δ, and the rutting factor G∗/sinδ from dynamic shear rheometer (DSR) test are used to evaluate the high-temperature mechanical performance of asphalt binders based on the law of energy dissipation and the rutting resistance [1–4]. For the base asphalt, some researches indicated that the rutting factor G∗/sinδ had a good relation with the rutting resistance of the asphalt mixtures. However, for the modified asphalt, the applicability of traditional rutting factor G∗/sinδ in the evaluation of high-temperature performance has been discussed over the years [5, 6]. It was because, under the uninterrupted sinusoidal alternating loading in DSR test, the influence of the delayed elasticity was neglected, which played an important role in the deformation response of the modified asphalt [7]. In the research of National Cooperative Highway Research Program (NCHRP), the relational coefficient between the rutting factor and the permanent deformation rate of the mixture, measured by the repeated shear test (constant height) (RSCH), was only 0.23.Recently, some new test methods and evaluation indexes have been proposed [8–11]. Considering the viscoelastic properties of asphalt, the elastic and viscous terms of shear complex modulus were defined, respectively. In the United States, Shenoy et al. [12, 13] put forward modified rutting factors G∗/1−sinδ⋅tanδ−1 and G∗/sinδ9 on the basis of the traditional rutting factor, which were proved to be more accurate in judging the rutting resistance of the modified asphalt. At the same time, the repeated creep-recovery test (RCRT) was suggested [14]. Based on RCRT, two new evaluation indexes in the multistress creep recovery (MSCR) test, percent deformation recovery (R) and nonrecoverable deformation compliance Jnr, were also proposed to solve the nonlinearity associated with the modified asphalt [15, 16]. According to the AASHTO TP70 test method [17], two stress levels (0.1 and 3.2 kPa) were adopted as the test loads. The stress sensitivity indexes Rdiff,Jnr,diff were also calculated from the difference in R and Jnr between two stress levels to evaluate the high-temperature performance of asphalt. Since the loading method in MSCR test takes into account the delayed elasticity of asphalt, this test method is applicable for predicting the high-temperature stability of both unmodified asphalt and polymer-modified asphalt [18–21], especially adopting higher stress load to simulate the actual situation [22, 23]. Besides, the zero-shear viscosity (ZSV) test has attracted much attention in Europe. The polymer-modified asphalt is a typical non-Newtonian fluid at high temperatures, and the viscosity decreases with the increasing shear rate. Some researches found that the viscosity of the asphalt tended to have a stable value when the shear rate was very small or extremely large. Thus, in order to exclude the effect of shear rate on the test results, the zero-shear viscosity (ZSV) was proposed as an evaluation index to characterize rutting resistance of the asphalt, especially the polymer-modified asphalt [2, 24–26]. Therefore, in this study, the ZSV and MSCR tests, as well as their technical indexes, were used for the high-temperature performance evaluations on both the base asphalt and modified asphalt.Gray relation analysis (GRA) method has been applied in many fields to explain the problems of the uncertain systems with the multiple input, incomplete information and discrete data. In road engineering fields, it is indicated that GRA is applicable to research the performance of asphalt and its mixture [27]. Based on gray system approach, the relationship between composition of the asphalt and the service performance was analyzed to study the asphalt aging mechanism [28]. Wu et al. [29] predict normal incidence sound absorption coefficient and tire/road noise on asphalt pavements using the gray system model with Fourier residual correction. Zhang et al. [30] evaluated the effects of the regenerant’s viscosity and acid value on the basic, rheological, and chemical properties of a typical rejuvenated asphalt by using GRA method. In Gao’s work [31], the highest gray entropy relation between the dynamic viscosity of bioasphalt and the dynamic stability of the mixture provided a key index of bioasphalt’s high-temperature performance evaluation. These results showed that GRA method could be considered as an alternative to the asphalt performance evaluation and mixture design analysis, of which the analysis results were accurate and reliable.Although a lot of effort has been made to investigate high-temperature performance of various types of asphalt including the base asphalt binders and modifiers in recent years, the existing relationship models between the asphalt and its mixture still have application limitations and accuracy problems. Most of the existing literatures focus on the materials and properties evolution, instead of the development of technical indexes for characterizing the high-temperature performance of both the unmodified and modified asphalt mixtures. Therefore, aiming to evaluate the high-temperature performance of the asphalt binders and their mixtures and to investigate the relation between the performance indexes of the asphalt binders and their mixture, six typically used asphalt types (including three types belonging to base asphalt and three types belonging to polymer-modified asphalt) were employed in this study. The standard high-temperature rheological test, the MSCR test, and the ZSV test were employed to characterize the high-temperature performance. Meanwhile, the high-temperature performance of the asphalt mixture was evaluated through Marshall stability and rutting tests based on the mixture design of AC-13. Furthermore, the relation was determined by the gray relational analysis method. This study will provide a more systematic perspective for the further study of the high-temperature performance evaluation of asphalt binders.


## 2. Materials

### 2.1. Asphalt Binders

Because the applicability of the relationship between different kind of asphalt and asphalt mixture is controversial, the study is to investigate the reliability of the standard technical indexes to evaluate the performance of both the base asphalt and modified asphalt. Thus, in this study, three penetration grades of asphalt (PEN20, PEN50, and PEN90) and three types of polymer-modified asphalt, which are two kinds of SBS- (styrene-butadiene-styrene-) modified asphalt named SBS I-C and SBS I-D (from Jiangsu Sinopec) and TPS- (Tafpack Super-) modified asphalt (from Gansu), were selected to investigate the high-temperature performance. The basic physical properties of six asphalt binders are shown in Table1. Note that since the TPS is a kind of new modifier, one of the objectives of the study is to compare the high-temperature performance of SBS-modified asphalt and that of TPS-modified asphalt.Table 1
The basic physical properties of six asphalt binders.
Asphalt bindersPenetration (25°C, 100 g, 5 g), 0.1 mmPenetration indexn/aSoftening point, °CDuctility (5/15°C), cmViscosity (60°C), Pa·sUnmodifiedPEN20230.8164.210/623315.2PEN50540.2952.41/87393.8PEN9091−0.9447.70.5/˃100198ModifiedSBS I-C740.0366.056/˃10027360SBS I-D57−0.0566.926/˃10049755TPS690.8980.738/˃10082459Note that the date of ductility before “/” was tested at 5°C, and date of ductility after “/” was tested at 15°C.


### 2.2. Asphalt Mixture Design

In this study, the fine-grained continuous gradation asphalt concrete (AC-13) mixtures with the six kinds of asphalt binder were adopted for the high-temperature performance evaluations, according to the Chinese standard of JTG F40-2004 [32]. Table 2 shows the detailed gradation of AC-13.Table 2
Gradation of aggregate for AC-13.
Sieve size, mm13.29.54.752.361.180.60.30.150.075Selected (%)9782473121.5151186Upper (%)100856850382820158Lower (%)906838241510754Intermediate (%)9576.5533726.51913.5106Based on the gradation of aggregate for AC-13 mixtures, different asphalt contents were selected for applying in the Marshall test to determine the optimum asphalt content (OAC) of the six kinds of asphalt binder. According to the Chinese standard of JTG F40-2004 [32], the testing data of the asphalt mixture Marshall specimen are shown in Table 3, including the asphalt-aggregate ratio, the apparent density, the theoretical maximum density, the air void (VV), the voids in mineral aggregate (VMA), and the voids filled with asphalt (VFA). The Marshall tests mentioned earlier were based on the SBS-modified asphalt binder. The air void (VV) of the Marshall specimen could be calculated through measuring the apparent density and the theoretical maximum density. In order to ensure that the air void of the Marshall specimen was around 4%, the different asphalt binders had various OACs. The final determined OAC of AC-13 mixtures in the study was 4.6 wt% for PEN50 and PEN90 asphalt binder, 4.8 wt% for PEN20 asphalt binder, 4.9 wt% for SBS-modified asphalt binder, and 5.1 wt% for TPS-modified asphalt binder, respectively.Table 3
Testing data of the asphalt mixture Marshall specimen.
Asphalt bindersAsphalt-aggregate ratioApparent densityTheoretical maximum densityVVVMAVFA%g/cm3g/cm3%%%UnmodifiedPEN204.82.4312.5424.3514.2869.5PEN504.62.4332.5474.4214.2168.7PEN904.62.4372.5534.5514.1167.8ModifiedSBS I-C4.92.4142.5224.3014.8371SBS I-D4.92.4192.5284.3314.6670.5TPS5.12.4092.5164.2514.9771.6


## 2.1. Asphalt Binders

Because the applicability of the relationship between different kind of asphalt and asphalt mixture is controversial, the study is to investigate the reliability of the standard technical indexes to evaluate the performance of both the base asphalt and modified asphalt. Thus, in this study, three penetration grades of asphalt (PEN20, PEN50, and PEN90) and three types of polymer-modified asphalt, which are two kinds of SBS- (styrene-butadiene-styrene-) modified asphalt named SBS I-C and SBS I-D (from Jiangsu Sinopec) and TPS- (Tafpack Super-) modified asphalt (from Gansu), were selected to investigate the high-temperature performance. The basic physical properties of six asphalt binders are shown in Table1. Note that since the TPS is a kind of new modifier, one of the objectives of the study is to compare the high-temperature performance of SBS-modified asphalt and that of TPS-modified asphalt.Table 1
The basic physical properties of six asphalt binders.
Asphalt bindersPenetration (25°C, 100 g, 5 g), 0.1 mmPenetration indexn/aSoftening point, °CDuctility (5/15°C), cmViscosity (60°C), Pa·sUnmodifiedPEN20230.8164.210/623315.2PEN50540.2952.41/87393.8PEN9091−0.9447.70.5/˃100198ModifiedSBS I-C740.0366.056/˃10027360SBS I-D57−0.0566.926/˃10049755TPS690.8980.738/˃10082459Note that the date of ductility before “/” was tested at 5°C, and date of ductility after “/” was tested at 15°C.


## 2.2. Asphalt Mixture Design

In this study, the fine-grained continuous gradation asphalt concrete (AC-13) mixtures with the six kinds of asphalt binder were adopted for the high-temperature performance evaluations, according to the Chinese standard of JTG F40-2004 [32]. Table 2 shows the detailed gradation of AC-13.Table 2
Gradation of aggregate for AC-13.
Sieve size, mm13.29.54.752.361.180.60.30.150.075Selected (%)9782473121.5151186Upper (%)100856850382820158Lower (%)906838241510754Intermediate (%)9576.5533726.51913.5106Based on the gradation of aggregate for AC-13 mixtures, different asphalt contents were selected for applying in the Marshall test to determine the optimum asphalt content (OAC) of the six kinds of asphalt binder. According to the Chinese standard of JTG F40-2004 [32], the testing data of the asphalt mixture Marshall specimen are shown in Table 3, including the asphalt-aggregate ratio, the apparent density, the theoretical maximum density, the air void (VV), the voids in mineral aggregate (VMA), and the voids filled with asphalt (VFA). The Marshall tests mentioned earlier were based on the SBS-modified asphalt binder. The air void (VV) of the Marshall specimen could be calculated through measuring the apparent density and the theoretical maximum density. In order to ensure that the air void of the Marshall specimen was around 4%, the different asphalt binders had various OACs. The final determined OAC of AC-13 mixtures in the study was 4.6 wt% for PEN50 and PEN90 asphalt binder, 4.8 wt% for PEN20 asphalt binder, 4.9 wt% for SBS-modified asphalt binder, and 5.1 wt% for TPS-modified asphalt binder, respectively.Table 3
Testing data of the asphalt mixture Marshall specimen.
Asphalt bindersAsphalt-aggregate ratioApparent densityTheoretical maximum densityVVVMAVFA%g/cm3g/cm3%%%UnmodifiedPEN204.82.4312.5424.3514.2869.5PEN504.62.4332.5474.4214.2168.7PEN904.62.4372.5534.5514.1167.8ModifiedSBS I-C4.92.4142.5224.3014.8371SBS I-D4.92.4192.5284.3314.6670.5TPS5.12.4092.5164.2514.9771.6


## 3. Testing Methods and Analysis Models

### 3.1. DSR Test

DSR tests were performed in accordance with the method specified in AASHTO T315 [33]. The complex modulus G∗, the phase angle δ, and the rutting indicator G∗/sinδ were obtained on the virgin asphalt binder samples (except for the correlation analysis in Section 4.3) via Kinexus Ultra+ (Malvern Panalytical) DSR at 64°C, 70°C, 76°C, 82°C, and 88°C, with the sample diameter of 25 mm and the thickness of 1 mm. The loading frequency in the DSR tests was 10 rad/s. For the correlation analysis, the DSR tests were conducted on both the unaged asphalt and the aged asphalt (rolling thin film oven test (RTFOT)). Three repeated tests were employed to ensure the reliability of the results.


### 3.2. MSCR Test

MSCR tests were performed on Kinexus Ultra+ (Malvern Panalytical) DSR at 64°C, 70°C, 76°C, 82°C, and 88°C. The sample was the rolling thin film oven test (RTFOT) residue of asphalt binder, which is applied to simulate the short-term aging condition of asphalt. The percent recovery and nonrecoverable creep compliance were obtained by applying a continuous cyclic load to the samples, with the sample diameter of 25 mm and the thickness of 1 mm. Each creep recovery cycle consisted of a stress loading period (1 s) and then a zero-stress recovery period (9 s). The test process was under two stress levels (0.1 and 3.2 kPa). The first 20 cycles of creep recovery were conducted under the shear load stress of 0.1 kPa, followed by additional 10 cycles under 3.2 kPa. Besides, the RTFOT were conducted at 163°C for 85 min [34]. It is worth noting that three repeated tests were carried out on each type of asphalt binder to ensure the reliability of the results.


### 3.3. ZSV Test

ZSV tests were performed on Kinexus Ultra+ (Malvern Panalytical) DSR at the test temperature of 60°C. The sample was the rolling thin film oven test (RTFOT) residue of asphalt binder, with the same diameter and thickness as those of the samples in MSCR tests (25 mm and 1 mm). The steady-state flow test was used to determine the viscosity at different shear frequencies, which ranged from 10 to 0.01 rad/s. Then the zero-shear viscosity (ZSV) was calculated by the shear rheological fitting curve. For each type of asphalt binder, at least ten repeated tests were conducted within each order of magnitude to ensure the reliability. At present, the commonly used fitting models, which were used to calculate the value of ZSV, include the Cross/Sybilski model, Cross/Williamson model, and Carreau model. The Cross/Sybilski model, Cross/Williamson model, and Carreau model were expressed as follows:(1)ηη0=11+Kωm,(2)η−η∞η0−η∞=11+Kωm,(3)η−η∞η0−η∞=11+Kω2m/2.Equation (1) was for the Cross/Sybilski model, equation (2) was for the Cross/Williamson model, and equation (3) was for the Carreau model. η was the complex viscosity of the asphalt sample, η0 was the ZSV of the asphalt sample, and η∞ was the complex viscosity when the shear frequency is infinite. Note that the value of η∞ cannot be directly obtained from the tests and was replaced with the complex viscosity at 10 rad/s. Kandm are the property coefficients of the asphalt sample.The accelerated loading facility test in the United States showed that the Carreau model was more suitable than the Cross/Williamson model for calculating the value of ZSV. In this paper, the three models were all adopted and compared to characterize rutting resistance of the asphalt sample better.


### 3.4. Rutting Test

In order to evaluate the high-temperature performance of the asphalt mixtures, the rutting test adopted in this paper was carried out in accordance with the method specified in the Chinese standard of JTG E20-2011 [35]. The test pieces of asphalt mixture should have the size of 300×300×50L×b×hmm3. The rutting formed on the test piece surface under the repeated passing action of the wheel along the same trajectory at a frequency of 42 ± 1 cycles/min. The dynamic stability (DS) of the test piece was calculated, as shown in the following equation:(4)DS=t2−t1×Nd2−d1,where d1 and d2were the rutting depths at t1 (45 min) and t2 (60 min), respectively. N was the frequency of the wheel passing through the test pieces. The antirutting property of asphalt mixture was evaluated by the dynamic stability. It is worth noting that three repeated tests were carried out on each type of asphalt binder to ensure the reliability of the results.


### 3.5. GRA Method

Gray relational analysis (GRA) method is a mathematical analysis method to measure the relationship between the factors and the system behaviors based on the finite and irregular data [36, 37]. In the principle of GRA, the relation among the data array curves is usually judged based on the similarity among geometric shapes. The gray relational coefficient can be calculated quantitatively by the normalization and comparison of the data arrays. The gray relational degree is subsequently calculated by taking the arithmetic average of the gray relational coefficients at different time point. A larger relational degree means a higher similarity among the data.The detailed calculating methods of the gray relational coefficient and gray relational degree are as follows.Suppose that there was a matrix, composed of a series of data arrays:(5)X00r,r=1,2,3,…,n,X10r,r=1,2,3,…,n,X20r,r=1,2,3,…,n,…Xk0r,r=1,2,3,…,n.X was the gray relational factor set, and the k arrays were k factors. n was the dimension of different k factors, which was the same value in this paper. X00 was the main array (or the reference array) and Xm0m=1,2,3,…,k were the subarrays (or the comparison arrays).As different values in the matrix have a great difference, the normalization of the arrays is necessary for comparing different factors directly and eliminating the dimension divergence. In this paper, the equalization method was utilized for normalizing treatment. The main array and the subarrays were processed as follows:(6)Xm1r=Xm0rXm¯,(7)Xm¯=1Nm∑r=1NmXm0r.Xm1m=0,1,2,3,…,k were the normalized arrays and Xm¯ was the arithmetic average value of the original arrays.The gray relational coefficientγmr between X00 and Xm0 was expressed as follows:(8)γmr=minmminrX01r−Xm1r+ξmaxmmaxrX01r−Xm1rX01r−Xm1r+ξmaxmmaxrX01r−Xm1r.The gray relational degreeΓm indicated the relational degree between the main array and the subarrays, which was calculated as follows:(9)Γm=∑r=1nβrγmr,and ξ was the distinguishing coefficient, which was 0.2 in this paper [31], according to the previous research. βr was the weighting of gray relational coefficient.Gray relational entropy is a kind of gray entropy to characterize the distances between the points and each reference point. The density value of distribution of the subarraysPm was calculated as follows:(10)Rm=γmr,r=1,2,3,…,n;m=0,1,2,3,…,k,(11)Pm≜γmr∑r=1nγmr,m=0,1,2,3,…,k,(12)HRm≜∑r=1nPmInPm,m=0,1,2,3,…,k.Rm is a matrix composed of the gray relational coefficient γmr. The gray relational entropy of the sequence Xm was introduced as follows:The entropy relational degreeEXm of the sequence Xm was calculated as follows:(13)EXm≜HRmHm,where Hm was the maximum value of the gray relational entropy.The entropy relational degree was used to characterize the correlation between the subarrays and the main arrays. The larger the gray relational degree of the sequenceXm was, the closer the geometry of the sequence Xm to that of the reference sequence would be. Therefore, using the GRA model, the degree of correlation between the high-temperature performance of asphalt binders and the dynamic stability of the mixtures was determined.When studying the relationship between the asphalt and mixture, the steps used to calculate the gray correlation degree and the entropy relational degree are listed as follows. Firstly, the test results of asphalt binders, including the penetration, softening point, viscosity, rutting factor, modified rutting factor, percent recovery, nonrecoverable creep compliance, and zero-shear viscosity, were the subarrays. The DS of asphalt mixtures was the main array. Then, the main array and subarrays were both normalized. Finally, the gray relational coefficients and gray relational degree could be calculated according to equations (8)∼(13).


## 3.1. DSR Test

DSR tests were performed in accordance with the method specified in AASHTO T315 [33]. The complex modulus G∗, the phase angle δ, and the rutting indicator G∗/sinδ were obtained on the virgin asphalt binder samples (except for the correlation analysis in Section 4.3) via Kinexus Ultra+ (Malvern Panalytical) DSR at 64°C, 70°C, 76°C, 82°C, and 88°C, with the sample diameter of 25 mm and the thickness of 1 mm. The loading frequency in the DSR tests was 10 rad/s. For the correlation analysis, the DSR tests were conducted on both the unaged asphalt and the aged asphalt (rolling thin film oven test (RTFOT)). Three repeated tests were employed to ensure the reliability of the results.


## 3.2. MSCR Test

MSCR tests were performed on Kinexus Ultra+ (Malvern Panalytical) DSR at 64°C, 70°C, 76°C, 82°C, and 88°C. The sample was the rolling thin film oven test (RTFOT) residue of asphalt binder, which is applied to simulate the short-term aging condition of asphalt. The percent recovery and nonrecoverable creep compliance were obtained by applying a continuous cyclic load to the samples, with the sample diameter of 25 mm and the thickness of 1 mm. Each creep recovery cycle consisted of a stress loading period (1 s) and then a zero-stress recovery period (9 s). The test process was under two stress levels (0.1 and 3.2 kPa). The first 20 cycles of creep recovery were conducted under the shear load stress of 0.1 kPa, followed by additional 10 cycles under 3.2 kPa. Besides, the RTFOT were conducted at 163°C for 85 min [34]. It is worth noting that three repeated tests were carried out on each type of asphalt binder to ensure the reliability of the results.


## 3.3. ZSV Test

ZSV tests were performed on Kinexus Ultra+ (Malvern Panalytical) DSR at the test temperature of 60°C. The sample was the rolling thin film oven test (RTFOT) residue of asphalt binder, with the same diameter and thickness as those of the samples in MSCR tests (25 mm and 1 mm). The steady-state flow test was used to determine the viscosity at different shear frequencies, which ranged from 10 to 0.01 rad/s. Then the zero-shear viscosity (ZSV) was calculated by the shear rheological fitting curve. For each type of asphalt binder, at least ten repeated tests were conducted within each order of magnitude to ensure the reliability. At present, the commonly used fitting models, which were used to calculate the value of ZSV, include the Cross/Sybilski model, Cross/Williamson model, and Carreau model. The Cross/Sybilski model, Cross/Williamson model, and Carreau model were expressed as follows:(1)ηη0=11+Kωm,(2)η−η∞η0−η∞=11+Kωm,(3)η−η∞η0−η∞=11+Kω2m/2.Equation (1) was for the Cross/Sybilski model, equation (2) was for the Cross/Williamson model, and equation (3) was for the Carreau model. η was the complex viscosity of the asphalt sample, η0 was the ZSV of the asphalt sample, and η∞ was the complex viscosity when the shear frequency is infinite. Note that the value of η∞ cannot be directly obtained from the tests and was replaced with the complex viscosity at 10 rad/s. Kandm are the property coefficients of the asphalt sample.The accelerated loading facility test in the United States showed that the Carreau model was more suitable than the Cross/Williamson model for calculating the value of ZSV. In this paper, the three models were all adopted and compared to characterize rutting resistance of the asphalt sample better.


## 3.4. Rutting Test

In order to evaluate the high-temperature performance of the asphalt mixtures, the rutting test adopted in this paper was carried out in accordance with the method specified in the Chinese standard of JTG E20-2011 [35]. The test pieces of asphalt mixture should have the size of 300×300×50L×b×hmm3. The rutting formed on the test piece surface under the repeated passing action of the wheel along the same trajectory at a frequency of 42 ± 1 cycles/min. The dynamic stability (DS) of the test piece was calculated, as shown in the following equation:(4)DS=t2−t1×Nd2−d1,where d1 and d2were the rutting depths at t1 (45 min) and t2 (60 min), respectively. N was the frequency of the wheel passing through the test pieces. The antirutting property of asphalt mixture was evaluated by the dynamic stability. It is worth noting that three repeated tests were carried out on each type of asphalt binder to ensure the reliability of the results.


## 3.5. GRA Method

Gray relational analysis (GRA) method is a mathematical analysis method to measure the relationship between the factors and the system behaviors based on the finite and irregular data [36, 37]. In the principle of GRA, the relation among the data array curves is usually judged based on the similarity among geometric shapes. The gray relational coefficient can be calculated quantitatively by the normalization and comparison of the data arrays. The gray relational degree is subsequently calculated by taking the arithmetic average of the gray relational coefficients at different time point. A larger relational degree means a higher similarity among the data.The detailed calculating methods of the gray relational coefficient and gray relational degree are as follows.Suppose that there was a matrix, composed of a series of data arrays:(5)X00r,r=1,2,3,…,n,X10r,r=1,2,3,…,n,X20r,r=1,2,3,…,n,…Xk0r,r=1,2,3,…,n.X was the gray relational factor set, and the k arrays were k factors. n was the dimension of different k factors, which was the same value in this paper. X00 was the main array (or the reference array) and Xm0m=1,2,3,…,k were the subarrays (or the comparison arrays).As different values in the matrix have a great difference, the normalization of the arrays is necessary for comparing different factors directly and eliminating the dimension divergence. In this paper, the equalization method was utilized for normalizing treatment. The main array and the subarrays were processed as follows:(6)Xm1r=Xm0rXm¯,(7)Xm¯=1Nm∑r=1NmXm0r.Xm1m=0,1,2,3,…,k were the normalized arrays and Xm¯ was the arithmetic average value of the original arrays.The gray relational coefficientγmr between X00 and Xm0 was expressed as follows:(8)γmr=minmminrX01r−Xm1r+ξmaxmmaxrX01r−Xm1rX01r−Xm1r+ξmaxmmaxrX01r−Xm1r.The gray relational degreeΓm indicated the relational degree between the main array and the subarrays, which was calculated as follows:(9)Γm=∑r=1nβrγmr,and ξ was the distinguishing coefficient, which was 0.2 in this paper [31], according to the previous research. βr was the weighting of gray relational coefficient.Gray relational entropy is a kind of gray entropy to characterize the distances between the points and each reference point. The density value of distribution of the subarraysPm was calculated as follows:(10)Rm=γmr,r=1,2,3,…,n;m=0,1,2,3,…,k,(11)Pm≜γmr∑r=1nγmr,m=0,1,2,3,…,k,(12)HRm≜∑r=1nPmInPm,m=0,1,2,3,…,k.Rm is a matrix composed of the gray relational coefficient γmr. The gray relational entropy of the sequence Xm was introduced as follows:The entropy relational degreeEXm of the sequence Xm was calculated as follows:(13)EXm≜HRmHm,where Hm was the maximum value of the gray relational entropy.The entropy relational degree was used to characterize the correlation between the subarrays and the main arrays. The larger the gray relational degree of the sequenceXm was, the closer the geometry of the sequence Xm to that of the reference sequence would be. Therefore, using the GRA model, the degree of correlation between the high-temperature performance of asphalt binders and the dynamic stability of the mixtures was determined.When studying the relationship between the asphalt and mixture, the steps used to calculate the gray correlation degree and the entropy relational degree are listed as follows. Firstly, the test results of asphalt binders, including the penetration, softening point, viscosity, rutting factor, modified rutting factor, percent recovery, nonrecoverable creep compliance, and zero-shear viscosity, were the subarrays. The DS of asphalt mixtures was the main array. Then, the main array and subarrays were both normalized. Finally, the gray relational coefficients and gray relational degree could be calculated according to equations (8)∼(13).


## 4. Results and Discussion

### 4.1. High-Temperature Rheological Properties of Asphalt

#### 4.1.1. DSR Test Results

The high-temperature rheological properties of six asphalt binders including the complex modulusG∗, the phase angle δ, and the rutting indicator G∗/sinδ are shown in Figure 1, respectively. The high-temperature grades of the polymer-modified asphalt were 82, 70, 64, 76, 82, and 82°C for PEN20, PEN50, PEN90, SBS I-C, SBS I-C, and TPS-modified asphalt. It was found that the phase angle curves had different forms in Figure 1(b), which was related to whether the modifiers were added and the properties of various modifiers. For the unmodified asphalt, the phase angle curves both rose with the increasing of the temperature. Meanwhile, for the modified asphalt, the trend of rising had slowed down, and there was a certain degree of decline in the temperature range of 76–88°C. The phase angle was an important feature to characterize the viscosity change of asphalt under the external shear force, which reflected that the addition of SBS and TPS was beneficial for the asphalt to maintain a relatively good shear strain recovery ability at higher temperatures. Among them, TPS-modified asphalt was the preferred except in the temperature range of 82–88°C, at which SBS I-D-modified asphalt had the smaller phase angle.Figure 1
High-temperature rheological properties of six asphalt binders: (a) complex modulus, (b) phase angles, (c) rutting indicator, and (d) rheological parameters at the PG grade temperatures.
(a)(b)(c)(d)Besides, as shown in Figures1(a) and 1(c), the complex modulus G∗ and the rutting indicator G∗/sinδ value of low-grade asphalt PEN20 were significantly larger than the other five kinds of asphalt in the interval of 64–82°C. When the temperature was higher than 82°C, the rutting indicator G∗/sinδ of TPS-modified asphalt was the best, while the high-temperature performances of PEN90 and PEN50 asphalt were always weak. The rheological properties of the six kinds of asphalt at the PG grade temperatures were summarized in Figure 1(d). It can be observed that the SBS I-D presented the lowest complex modulus G∗ and rutting indicator G∗/sinδ, and PEN20 had the largest phase angle δ. Since the difference between the SBS I-D and TPS-modified asphalt was less than 20%, TPS had similar high-temperature responses to SBS, and it could be considered for replacing the traditional SBS with proper regard to the high-temperature performances.


#### 4.1.2. MSCR Test Results

The MSCR test results of the six asphalt binders at different temperature are shown in Figure2. These results indicated the significant viscoelastic-plastic properties of the asphalt. Figure 2(a) illustrates the nonrecoverable deformation compliance Jnr of the six asphalt binders at 3.2 kPa. It could be found that the nonrecoverable deformation compliance Jnr values all had an exponential relationship with the temperature, indicating that the high-temperature stability of asphalt would be significantly weakened as the temperature increased. The difference in Jnr between two stress levels (0.1 kPa and 3.2 kPa) was an important indicator to evaluate the high-temperature performance related to the external stress response. The larger Jnr diff value indicated the greater stress sensitivity of the resistance to cumulative deformation. As shown in Figure 2(b), TPS-modified asphalt showed the greatest stress sensitivity, followed by SBS I-D-modified asphalt. It was concluded that the modified asphalt had greater stress sensitivity of the resistance to cumulative deformation than the base asphalt.Figure 2
Multiple stress creep recovery (MSCR) test results of six asphalt binders.
(a)(b)(c)(d)Figure2(c) shows the creep and recovery rate of the asphalt at 3.2 kPa. The curves had different forms, which were automatically divided into two shapes according to whether the modifiers were added. Under the creep load condition, the PEN90 asphalt had a significant degree of deformation and a low deformation recovery rate, which indicated that the PEN90 asphalt performed soft viscosities compared with the other types of asphalt. Note that the R3.2 value of the PEN90 asphalt was lower than zero, indicating that the destruction of the structure made the asphalt completely lose its elasticity and become a fluid. Figure 2(d) shows the change in the stress sensitivity index Rdiff of the asphalt with temperature. The large Rdiff value indicated the recovery capability of asphalt was sensitive to the applied stress. It was reconfirmed that the six asphalt binders all exhibited greater sensitivity to the stress at the high temperature, as shown in Figure 2(b). Generally speaking, the TPS-modified asphalt performed best in the MSCR test, followed by the SBS I-D-modified asphalt, because they had the least deformation and the most substantial deformation recovery ability. It is worth noting that the low-grade PEN20 asphalt also had the low deformation, while its deformation recovery ability was poor.


#### 4.1.3. ZSV Test Results

The ZSV test results of the six asphalt binders at 60°C are shown in Figure3 as well as the complex viscosity η-frequency ω curves, the fitted K,m, and ZSV values using the Cross/Sybilski model, Cross/Williamson model, and Carreau model.Figure 3
Zero-shear viscosity (ZSV) test results of six asphalt binders.
(a)(b)(c)(d)As shown in Figure3, there were substantial differences in K,m, and ZSV values between different kinds of asphalt using three different models, especially the base asphalt and modified asphalt. In order to present the data more clearly and in detail, the standard deviation σ for data is shown in Table 4.Table 4
FittingK,m, and ZSV values in three ZSV models.
Asphalt bindersKσKmσmZSV (kPa·s)σZSV (kPa·s)PEN20Sybilski5.06 × 10−42.33 × 10−124.864.541.7180.4Williamson2.21 × 10−67.97 × 10−100.080.0613483.064521.4Carreau392.95297.346.04 × 10−42.07 × 10−36694.85424.38PEN50Sybilski1.56 × 10−34.673.831.111.5646.74Williamson1.45 × 10−65.35 × 10−90.070.068106.443400.58Carreau753.61195.477.83 × 10−42 × 10−33926.12812.54PEN90Sybilski3.3 × 10−33.89 × 10−29.292.932.0810.24Williamson8.62 × 10−91.41 × 10−110.340.25630.8136.71Carreau11.638.161.44 × 10−48.29 × 10−3599.178.32SBS I-CSybilski12435.680−5147.3705622.27704.7Williamson4.640.251.510.1210841.59183.54Carreau8.780.910.880.0510718.52108.85SBS I-DSybilski18747.940−3965.6708845.341155.74Williamson3.590.181.980.1816295.35253.91Carreau4.620.731.390.1416365.79220.56TPSSybilski0.02026.44017.462.92Williamson5.450.432.120.3125233.52729.53Carreau6.281.711.630.2825453.75670.56It has been proved that the parameterm mainly reflected the sensitivity of the asphalt to shear stress, and the closer the Kandm values were to zero, the closer the property of the material was to Newtonian fluid based on the Cross/Sybilski model [38]. Thus, the large m value indicated that the asphalt was sensitive to the shear frequency. As shown in Figure 3(b), the K value of SBS I-D-modified asphalt was the highest, which reflected that the SBS I-D-modified asphalt was more similar to the non-Newtonian fluid. Thus, it was necessary to adopt ZSV value to analyze the high-temperature performance.Based on the fittedm value from Cross/Williamson model in Figure 3(c), the viscosity values of the modified asphalt were more sensitive to the shearing frequency than those of the base asphalt. The TPS-modified asphalt had the most considerable m value, indicating that the viscosity test result was most affected by the shear frequency. In the results of the SBS-modified asphalt, it was found that the fitted m from the Cross/Sybilski model were negative values, which may be caused by the small range of the selected shear frequency in this study. As shown in in Figure 3(d), the ZSV value obtained by the Cross/Sybilski model fluctuated greatly between different types of asphalt; in particular, the standard errors of ZSV of the base asphalt binders were large. In order to distinguish the high-temperature performance of different asphalt by using ZSV test, it is necessary to perform relational analysis with the calculation results through three different calculation models.In addition, it could be concluded that the SBS I-D-modified asphalt performed best to resist high-temperature deformations, followed by the SBS I-C-modified asphalt, due to the large and harmonious ZSV value in three different models. The ZSV of the PEN90 asphalt was significantly less than the other five types of asphalt, which was similar to the MSCR and DSR test results. Thus, the PEN90 asphalt performance was significantly soft-viscoelastic.In summary, for the SBS-modified asphalt with high viscosity, the high-temperature performance could be well characterized by using three ZSV models, of which the results coincided with each other. Meanwhile, for the asphalt with low viscosity but large complex modulusG∗ (such as the low-grade PEN20 asphalt) and the asphalt with especially high viscosity (such as TPS-modified asphalt), the ZSV value by one ZSV model did not reflect their low stress sensitivity and good rutting resistance. This result was consistent with the results of previous studies [16].


### 4.2. High-Temperature Properties of Asphalt Mixtures

As shown in Table5, the dynamic stability (DS) of the TPS-modified asphalt mixture was 9965 times/mm, while the DS of SBS I-D, PEN20, SBS I-C, PEN50, and PEN90 asphalt mixture were 8150, 5415, 4603, 1986, and 580 times/mm, respectively. The DS results concluded that the TPS-modified asphalt mixture performed best for resisting the rutting deformations.Table 5
Rutting testing data of six asphalt mixtures.
ParametersUnmodifiedModifiedPEN20PEN50PEN90SBS I-CSBS I-DTPSDS, times/mm54151986580460381509965σ, times/mm601.1186.745.85801352.91135.3σ denotes the standard deviation.


### 4.3. GRA for the Technical Indexes

In order to comprehensively analyze the high-temperature performance of the asphalt and its mixtures, the high-temperature performance technical indexes were normalized based on the maximum test values of the different series, as shown in equation (14). It is worth noting that the DSR and MSCR data were all at the temperature of 64°C. Aiming to ensure the reliability of the conclusions, DSR data were also obtained from the RTFOT residue of asphalt binders to study the correlation between rutting factor and dynamic stability under different experimental conditions.(14)Normalizedxik=xikxmaxk,where x was the technical index, such as softening point, viscosity, rutting indicator, R, and ZSV. k was the type of the asphalt binders, which includes the unmodified asphalt and modified asphalt.As shown in Figure4, the normalized high-temperature performance technical indexes of the asphalt were drawn in hexagonal envelope diagram. The asphalt with the largest hexagonal envelope area should have the best high-temperature overall performance. Thus, the high-temperature performance ranking of the asphalt was TPS-modified asphalt, followed by SBS I-D, SBS I-C, PEN20, PEN50, and PEN90. The presented analysis results had a good match with the viscosity at 60°C and the Carreau modeled ZSV test, while they were different from the traditional DSR and MSCR results. In general, the modified asphalt performed better than the unmodified asphalt according to the high-temperature rheological properties tests. Despite the excellent performance in some test results, such as the low deformation in MSCR, the hexagonal envelope area analysis showed that the PEN20 asphalt presented the unbalanced high-temperature performance.Figure 4
High-temperature performance distributions of six asphalt binders in hexagonal envelope diagram.Besides, in the previous research, dynamic stability (DS) was a useful indicator to characterize the high-temperature performance of asphalt mixtures [39–41]. The DS ranking of the asphalt mixtures was TPS-modified asphalt, followed by SBS I-D, PEN20, SBS I-C, PEN50, and PEN90, which was also different from the result by the envelope area analysis.In order to create the relationship between the performance evaluation indexes of the asphalt and the performance of the asphalt mixture, GRA was adopted and the results of different asphalt binders and asphalt mixtures were defined asXm0 (where m=0,1,2,…,13) and listed in Table 6. The DS of the asphalt mixture were selected for the main arrays, while the high-temperature performance indexes of asphalt were taken as the subarrays. In this section, the normalized results were calculated based on equations (6) and (7), as shown in Table 6.Table 6
Data preprocessing for GRA.
ParametersXmUnmodifiedModifiedPEN20PEN50PEN90SBS I-CSBS I-DTPSOriginal sequenceDS (times/mm)X0054151986580460381509965Penetration (25°C) (0.1 mm)X10235491745769Softening point (°C)X2064.252.447.76666.980.7Viscosity (60°C) (Pa·s)X303315394198273604975582459G∗/sinδ (64°C) (OB)X4015.115.091.093.204.404.63G∗/1−sinδ⋅tanδ−1 (64°C) (OB)X5018.945.751.215.959.410.30Jnr3.2 (kPa−1) (64°C)X600.23911.39763.430.33280.13690.0473Jnr0.1 (kPa−1) (64°C)X700.20091.06352.1750.03890.02970.0046R3.2 (%) (64°C)X8023.612.61.591.388.496.7R0.1 (%) (64°C)X9032.325.720.894.397.399.6ZSV-Sybilski (Pa·s)X1001.711.562.085622.278845.3417.46ZSV-Williamson (Pa·s)X11013483.18106.44630.8110841.616295.425233.5ZSV-Carreau (Pa·s)X1206694.853926.12599.1710718.516365.825453.8G∗/sinδ (64°C) (RTFOT)X13027.598.811.985.206.855.87Normalized sequenceDS (times/mm)X011.060.390.110.901.591.95Penetration (25°C) (0.1 mm)X110.380.881.481.210.931.13Softening point (°C)X211.020.830.761.051.061.28Viscosity (60°C) (Pa·s)X310.120.010.011.001.833.03G∗/sinδ (64°C)X412.710.910.200.570.790.83G∗/1−sinδ⋅tanδ−1 (64°C)X512.200.670.140.691.101.20Jnr3.2 (kPa−1) (64°C)X610.261.503.690.360.150.05Jnr0.1 (kPa−1) (64°C)X710.341.823.720.070.050.01R3.2 (%) (64°C)X810.450.240.031.741.691.85R0.1 (%) (64°C)X910.520.420.341.531.581.62ZSV-Sybilski (Pa·s)X1010.000.000.002.333.660.01ZSV-Williamson (Pa·s)X1111.080.650.050.871.312.03ZSV-Carreau (Pa·s)X1210.630.370.061.011.542.40G∗/sinδ (64°C) (RTFOT)X1312.940.940.210.550.730.63In order to further analyze whether the rutting indicatorG∗/sinδ was the suitable candidate for the high-temperature performance evaluation of the asphalt, both the unaged samples (marked as OB) and RTFOT aged samples (marked as RTFOT) were analyzed in this part. In addition, the modified rutting indicator G∗/1−sinδ⋅tanδ−1 was also introduced. The absolute difference between the two sequences was calculated, and then the gray relational coefficients and gray relational degree were calculated, based on equations (8) and (9). The detailed results are shown in Table 7.Table 7
Gray relational coefficients between asphalt indexes and dynamic stability.
ParametersUnmodifiedModifiedGray relational degreePEN20PEN50PEN90SBS I-CSBS I-DTPSγ10.520.610.350.720.530.480.53γ20.970.630.540.850.590.530.68γ30.440.670.890.890.770.410.68γ40.310.590.920.700.480.400.57γ50.390.730.980.790.600.500.67γ60.480.400.170.580.340.280.38γ70.510.340.170.470.330.280.35γ80.550.850.910.470.900.900.76γ90.590.980.780.541.000.700.77γ100.410.660.880.340.260.280.47γ110.990.750.940.980.730.920.88γ120.641.000.950.890.950.630.84γ130.280.580.900.690.460.360.55The gray entropy relational degree was calculated by using equations (10)∼(13). Figure 5 shows the gray relational degree and entropy relational degree between asphalt binder indexes and dynamic stability of the corresponding asphalt mixtures. The gray relational degree was to take the arithmetic average of the gray relational coefficients of each type of asphalt, which would cause the loss of the tendency information of each local point. But the entropy relational degree could effectively evaluate the correlation on the entire sequence correlation degree rather than the influence of the single-point correlation coefficient. According to the entropy relational criterion, the larger the gray relational degree and entropy relational degree values of the high-temperature performance indexes were, the stronger the correlation between the high-temperature performance of the asphalt binder and the dynamic stability of asphalt mixtures would be. For the gray relational degree, Γ11>Γ12>Γ9>Γ8>Γ2>Γ3>Γ5>Γ4>Γ13>Γ1>Γ10>Γ6>Γ7, while, for the entropy relational degree, EX11>EX12>EX1>EX9>EX2>EX8>EX5>EX3>EX7>EX4>EX6>EX13>EX10. Although the two indicators were arranged in different rank order, the X11 sequence, representing the Cross/Williamson modeled ZSV value, was the most relevant to the main array. Since the DS value would well reflect the high-temperature performance of the asphalt mixture, the ZSV test results fitted by the Cross/Williamson model had the highest correlation with the DS and would best characterize the high-temperature performance. The Cross/Williamson model was the most suitable for calculating and fitting the ZSV. Besides, it was found that the rutting indicator G∗/sinδ of the unaged asphalt (named X4) and the RTFOT asphalt (named X13) had a similar trend, since the values of the gray relational degree were close. The results indicated that the correlation between the rutting indicator G∗/sinδ in the asphalt’s DSR tests and the DS value of asphalt mixture was very poor. The present rutting indicator for evaluating the asphalt mixtures’ high-temperature performance might no longer be suitable. Although the correlation between the modified rutting factor (named X5) and the DS value had improved, the accuracy of its evaluation was still lower than the MSCR and ZSV test results. However, the conclusion should be further proved based on more laboratory tests for the asphalt binder and its mixture.Figure 5
Gray relational degree and entropy relational degree between asphalt indexes and DS.


## 4.1. High-Temperature Rheological Properties of Asphalt

### 4.1.1. DSR Test Results

The high-temperature rheological properties of six asphalt binders including the complex modulusG∗, the phase angle δ, and the rutting indicator G∗/sinδ are shown in Figure 1, respectively. The high-temperature grades of the polymer-modified asphalt were 82, 70, 64, 76, 82, and 82°C for PEN20, PEN50, PEN90, SBS I-C, SBS I-C, and TPS-modified asphalt. It was found that the phase angle curves had different forms in Figure 1(b), which was related to whether the modifiers were added and the properties of various modifiers. For the unmodified asphalt, the phase angle curves both rose with the increasing of the temperature. Meanwhile, for the modified asphalt, the trend of rising had slowed down, and there was a certain degree of decline in the temperature range of 76–88°C. The phase angle was an important feature to characterize the viscosity change of asphalt under the external shear force, which reflected that the addition of SBS and TPS was beneficial for the asphalt to maintain a relatively good shear strain recovery ability at higher temperatures. Among them, TPS-modified asphalt was the preferred except in the temperature range of 82–88°C, at which SBS I-D-modified asphalt had the smaller phase angle.Figure 1
High-temperature rheological properties of six asphalt binders: (a) complex modulus, (b) phase angles, (c) rutting indicator, and (d) rheological parameters at the PG grade temperatures.
(a)(b)(c)(d)Besides, as shown in Figures1(a) and 1(c), the complex modulus G∗ and the rutting indicator G∗/sinδ value of low-grade asphalt PEN20 were significantly larger than the other five kinds of asphalt in the interval of 64–82°C. When the temperature was higher than 82°C, the rutting indicator G∗/sinδ of TPS-modified asphalt was the best, while the high-temperature performances of PEN90 and PEN50 asphalt were always weak. The rheological properties of the six kinds of asphalt at the PG grade temperatures were summarized in Figure 1(d). It can be observed that the SBS I-D presented the lowest complex modulus G∗ and rutting indicator G∗/sinδ, and PEN20 had the largest phase angle δ. Since the difference between the SBS I-D and TPS-modified asphalt was less than 20%, TPS had similar high-temperature responses to SBS, and it could be considered for replacing the traditional SBS with proper regard to the high-temperature performances.


### 4.1.2. MSCR Test Results

The MSCR test results of the six asphalt binders at different temperature are shown in Figure2. These results indicated the significant viscoelastic-plastic properties of the asphalt. Figure 2(a) illustrates the nonrecoverable deformation compliance Jnr of the six asphalt binders at 3.2 kPa. It could be found that the nonrecoverable deformation compliance Jnr values all had an exponential relationship with the temperature, indicating that the high-temperature stability of asphalt would be significantly weakened as the temperature increased. The difference in Jnr between two stress levels (0.1 kPa and 3.2 kPa) was an important indicator to evaluate the high-temperature performance related to the external stress response. The larger Jnr diff value indicated the greater stress sensitivity of the resistance to cumulative deformation. As shown in Figure 2(b), TPS-modified asphalt showed the greatest stress sensitivity, followed by SBS I-D-modified asphalt. It was concluded that the modified asphalt had greater stress sensitivity of the resistance to cumulative deformation than the base asphalt.Figure 2
Multiple stress creep recovery (MSCR) test results of six asphalt binders.
(a)(b)(c)(d)Figure2(c) shows the creep and recovery rate of the asphalt at 3.2 kPa. The curves had different forms, which were automatically divided into two shapes according to whether the modifiers were added. Under the creep load condition, the PEN90 asphalt had a significant degree of deformation and a low deformation recovery rate, which indicated that the PEN90 asphalt performed soft viscosities compared with the other types of asphalt. Note that the R3.2 value of the PEN90 asphalt was lower than zero, indicating that the destruction of the structure made the asphalt completely lose its elasticity and become a fluid. Figure 2(d) shows the change in the stress sensitivity index Rdiff of the asphalt with temperature. The large Rdiff value indicated the recovery capability of asphalt was sensitive to the applied stress. It was reconfirmed that the six asphalt binders all exhibited greater sensitivity to the stress at the high temperature, as shown in Figure 2(b). Generally speaking, the TPS-modified asphalt performed best in the MSCR test, followed by the SBS I-D-modified asphalt, because they had the least deformation and the most substantial deformation recovery ability. It is worth noting that the low-grade PEN20 asphalt also had the low deformation, while its deformation recovery ability was poor.


### 4.1.3. ZSV Test Results

The ZSV test results of the six asphalt binders at 60°C are shown in Figure3 as well as the complex viscosity η-frequency ω curves, the fitted K,m, and ZSV values using the Cross/Sybilski model, Cross/Williamson model, and Carreau model.Figure 3
Zero-shear viscosity (ZSV) test results of six asphalt binders.
(a)(b)(c)(d)As shown in Figure3, there were substantial differences in K,m, and ZSV values between different kinds of asphalt using three different models, especially the base asphalt and modified asphalt. In order to present the data more clearly and in detail, the standard deviation σ for data is shown in Table 4.Table 4
FittingK,m, and ZSV values in three ZSV models.
Asphalt bindersKσKmσmZSV (kPa·s)σZSV (kPa·s)PEN20Sybilski5.06 × 10−42.33 × 10−124.864.541.7180.4Williamson2.21 × 10−67.97 × 10−100.080.0613483.064521.4Carreau392.95297.346.04 × 10−42.07 × 10−36694.85424.38PEN50Sybilski1.56 × 10−34.673.831.111.5646.74Williamson1.45 × 10−65.35 × 10−90.070.068106.443400.58Carreau753.61195.477.83 × 10−42 × 10−33926.12812.54PEN90Sybilski3.3 × 10−33.89 × 10−29.292.932.0810.24Williamson8.62 × 10−91.41 × 10−110.340.25630.8136.71Carreau11.638.161.44 × 10−48.29 × 10−3599.178.32SBS I-CSybilski12435.680−5147.3705622.27704.7Williamson4.640.251.510.1210841.59183.54Carreau8.780.910.880.0510718.52108.85SBS I-DSybilski18747.940−3965.6708845.341155.74Williamson3.590.181.980.1816295.35253.91Carreau4.620.731.390.1416365.79220.56TPSSybilski0.02026.44017.462.92Williamson5.450.432.120.3125233.52729.53Carreau6.281.711.630.2825453.75670.56It has been proved that the parameterm mainly reflected the sensitivity of the asphalt to shear stress, and the closer the Kandm values were to zero, the closer the property of the material was to Newtonian fluid based on the Cross/Sybilski model [38]. Thus, the large m value indicated that the asphalt was sensitive to the shear frequency. As shown in Figure 3(b), the K value of SBS I-D-modified asphalt was the highest, which reflected that the SBS I-D-modified asphalt was more similar to the non-Newtonian fluid. Thus, it was necessary to adopt ZSV value to analyze the high-temperature performance.Based on the fittedm value from Cross/Williamson model in Figure 3(c), the viscosity values of the modified asphalt were more sensitive to the shearing frequency than those of the base asphalt. The TPS-modified asphalt had the most considerable m value, indicating that the viscosity test result was most affected by the shear frequency. In the results of the SBS-modified asphalt, it was found that the fitted m from the Cross/Sybilski model were negative values, which may be caused by the small range of the selected shear frequency in this study. As shown in in Figure 3(d), the ZSV value obtained by the Cross/Sybilski model fluctuated greatly between different types of asphalt; in particular, the standard errors of ZSV of the base asphalt binders were large. In order to distinguish the high-temperature performance of different asphalt by using ZSV test, it is necessary to perform relational analysis with the calculation results through three different calculation models.In addition, it could be concluded that the SBS I-D-modified asphalt performed best to resist high-temperature deformations, followed by the SBS I-C-modified asphalt, due to the large and harmonious ZSV value in three different models. The ZSV of the PEN90 asphalt was significantly less than the other five types of asphalt, which was similar to the MSCR and DSR test results. Thus, the PEN90 asphalt performance was significantly soft-viscoelastic.In summary, for the SBS-modified asphalt with high viscosity, the high-temperature performance could be well characterized by using three ZSV models, of which the results coincided with each other. Meanwhile, for the asphalt with low viscosity but large complex modulusG∗ (such as the low-grade PEN20 asphalt) and the asphalt with especially high viscosity (such as TPS-modified asphalt), the ZSV value by one ZSV model did not reflect their low stress sensitivity and good rutting resistance. This result was consistent with the results of previous studies [16].


## 4.1.1. DSR Test Results

The high-temperature rheological properties of six asphalt binders including the complex modulusG∗, the phase angle δ, and the rutting indicator G∗/sinδ are shown in Figure 1, respectively. The high-temperature grades of the polymer-modified asphalt were 82, 70, 64, 76, 82, and 82°C for PEN20, PEN50, PEN90, SBS I-C, SBS I-C, and TPS-modified asphalt. It was found that the phase angle curves had different forms in Figure 1(b), which was related to whether the modifiers were added and the properties of various modifiers. For the unmodified asphalt, the phase angle curves both rose with the increasing of the temperature. Meanwhile, for the modified asphalt, the trend of rising had slowed down, and there was a certain degree of decline in the temperature range of 76–88°C. The phase angle was an important feature to characterize the viscosity change of asphalt under the external shear force, which reflected that the addition of SBS and TPS was beneficial for the asphalt to maintain a relatively good shear strain recovery ability at higher temperatures. Among them, TPS-modified asphalt was the preferred except in the temperature range of 82–88°C, at which SBS I-D-modified asphalt had the smaller phase angle.Figure 1
High-temperature rheological properties of six asphalt binders: (a) complex modulus, (b) phase angles, (c) rutting indicator, and (d) rheological parameters at the PG grade temperatures.
(a)(b)(c)(d)Besides, as shown in Figures1(a) and 1(c), the complex modulus G∗ and the rutting indicator G∗/sinδ value of low-grade asphalt PEN20 were significantly larger than the other five kinds of asphalt in the interval of 64–82°C. When the temperature was higher than 82°C, the rutting indicator G∗/sinδ of TPS-modified asphalt was the best, while the high-temperature performances of PEN90 and PEN50 asphalt were always weak. The rheological properties of the six kinds of asphalt at the PG grade temperatures were summarized in Figure 1(d). It can be observed that the SBS I-D presented the lowest complex modulus G∗ and rutting indicator G∗/sinδ, and PEN20 had the largest phase angle δ. Since the difference between the SBS I-D and TPS-modified asphalt was less than 20%, TPS had similar high-temperature responses to SBS, and it could be considered for replacing the traditional SBS with proper regard to the high-temperature performances.


## 4.1.2. MSCR Test Results

The MSCR test results of the six asphalt binders at different temperature are shown in Figure2. These results indicated the significant viscoelastic-plastic properties of the asphalt. Figure 2(a) illustrates the nonrecoverable deformation compliance Jnr of the six asphalt binders at 3.2 kPa. It could be found that the nonrecoverable deformation compliance Jnr values all had an exponential relationship with the temperature, indicating that the high-temperature stability of asphalt would be significantly weakened as the temperature increased. The difference in Jnr between two stress levels (0.1 kPa and 3.2 kPa) was an important indicator to evaluate the high-temperature performance related to the external stress response. The larger Jnr diff value indicated the greater stress sensitivity of the resistance to cumulative deformation. As shown in Figure 2(b), TPS-modified asphalt showed the greatest stress sensitivity, followed by SBS I-D-modified asphalt. It was concluded that the modified asphalt had greater stress sensitivity of the resistance to cumulative deformation than the base asphalt.Figure 2
Multiple stress creep recovery (MSCR) test results of six asphalt binders.
(a)(b)(c)(d)Figure2(c) shows the creep and recovery rate of the asphalt at 3.2 kPa. The curves had different forms, which were automatically divided into two shapes according to whether the modifiers were added. Under the creep load condition, the PEN90 asphalt had a significant degree of deformation and a low deformation recovery rate, which indicated that the PEN90 asphalt performed soft viscosities compared with the other types of asphalt. Note that the R3.2 value of the PEN90 asphalt was lower than zero, indicating that the destruction of the structure made the asphalt completely lose its elasticity and become a fluid. Figure 2(d) shows the change in the stress sensitivity index Rdiff of the asphalt with temperature. The large Rdiff value indicated the recovery capability of asphalt was sensitive to the applied stress. It was reconfirmed that the six asphalt binders all exhibited greater sensitivity to the stress at the high temperature, as shown in Figure 2(b). Generally speaking, the TPS-modified asphalt performed best in the MSCR test, followed by the SBS I-D-modified asphalt, because they had the least deformation and the most substantial deformation recovery ability. It is worth noting that the low-grade PEN20 asphalt also had the low deformation, while its deformation recovery ability was poor.


## 4.1.3. ZSV Test Results

The ZSV test results of the six asphalt binders at 60°C are shown in Figure3 as well as the complex viscosity η-frequency ω curves, the fitted K,m, and ZSV values using the Cross/Sybilski model, Cross/Williamson model, and Carreau model.Figure 3
Zero-shear viscosity (ZSV) test results of six asphalt binders.
(a)(b)(c)(d)As shown in Figure3, there were substantial differences in K,m, and ZSV values between different kinds of asphalt using three different models, especially the base asphalt and modified asphalt. In order to present the data more clearly and in detail, the standard deviation σ for data is shown in Table 4.Table 4
FittingK,m, and ZSV values in three ZSV models.
Asphalt bindersKσKmσmZSV (kPa·s)σZSV (kPa·s)PEN20Sybilski5.06 × 10−42.33 × 10−124.864.541.7180.4Williamson2.21 × 10−67.97 × 10−100.080.0613483.064521.4Carreau392.95297.346.04 × 10−42.07 × 10−36694.85424.38PEN50Sybilski1.56 × 10−34.673.831.111.5646.74Williamson1.45 × 10−65.35 × 10−90.070.068106.443400.58Carreau753.61195.477.83 × 10−42 × 10−33926.12812.54PEN90Sybilski3.3 × 10−33.89 × 10−29.292.932.0810.24Williamson8.62 × 10−91.41 × 10−110.340.25630.8136.71Carreau11.638.161.44 × 10−48.29 × 10−3599.178.32SBS I-CSybilski12435.680−5147.3705622.27704.7Williamson4.640.251.510.1210841.59183.54Carreau8.780.910.880.0510718.52108.85SBS I-DSybilski18747.940−3965.6708845.341155.74Williamson3.590.181.980.1816295.35253.91Carreau4.620.731.390.1416365.79220.56TPSSybilski0.02026.44017.462.92Williamson5.450.432.120.3125233.52729.53Carreau6.281.711.630.2825453.75670.56It has been proved that the parameterm mainly reflected the sensitivity of the asphalt to shear stress, and the closer the Kandm values were to zero, the closer the property of the material was to Newtonian fluid based on the Cross/Sybilski model [38]. Thus, the large m value indicated that the asphalt was sensitive to the shear frequency. As shown in Figure 3(b), the K value of SBS I-D-modified asphalt was the highest, which reflected that the SBS I-D-modified asphalt was more similar to the non-Newtonian fluid. Thus, it was necessary to adopt ZSV value to analyze the high-temperature performance.Based on the fittedm value from Cross/Williamson model in Figure 3(c), the viscosity values of the modified asphalt were more sensitive to the shearing frequency than those of the base asphalt. The TPS-modified asphalt had the most considerable m value, indicating that the viscosity test result was most affected by the shear frequency. In the results of the SBS-modified asphalt, it was found that the fitted m from the Cross/Sybilski model were negative values, which may be caused by the small range of the selected shear frequency in this study. As shown in in Figure 3(d), the ZSV value obtained by the Cross/Sybilski model fluctuated greatly between different types of asphalt; in particular, the standard errors of ZSV of the base asphalt binders were large. In order to distinguish the high-temperature performance of different asphalt by using ZSV test, it is necessary to perform relational analysis with the calculation results through three different calculation models.In addition, it could be concluded that the SBS I-D-modified asphalt performed best to resist high-temperature deformations, followed by the SBS I-C-modified asphalt, due to the large and harmonious ZSV value in three different models. The ZSV of the PEN90 asphalt was significantly less than the other five types of asphalt, which was similar to the MSCR and DSR test results. Thus, the PEN90 asphalt performance was significantly soft-viscoelastic.In summary, for the SBS-modified asphalt with high viscosity, the high-temperature performance could be well characterized by using three ZSV models, of which the results coincided with each other. Meanwhile, for the asphalt with low viscosity but large complex modulusG∗ (such as the low-grade PEN20 asphalt) and the asphalt with especially high viscosity (such as TPS-modified asphalt), the ZSV value by one ZSV model did not reflect their low stress sensitivity and good rutting resistance. This result was consistent with the results of previous studies [16].


## 4.2. High-Temperature Properties of Asphalt Mixtures

As shown in Table5, the dynamic stability (DS) of the TPS-modified asphalt mixture was 9965 times/mm, while the DS of SBS I-D, PEN20, SBS I-C, PEN50, and PEN90 asphalt mixture were 8150, 5415, 4603, 1986, and 580 times/mm, respectively. The DS results concluded that the TPS-modified asphalt mixture performed best for resisting the rutting deformations.Table 5
Rutting testing data of six asphalt mixtures.
ParametersUnmodifiedModifiedPEN20PEN50PEN90SBS I-CSBS I-DTPSDS, times/mm54151986580460381509965σ, times/mm601.1186.745.85801352.91135.3σ denotes the standard deviation.


## 4.3. GRA for the Technical Indexes

In order to comprehensively analyze the high-temperature performance of the asphalt and its mixtures, the high-temperature performance technical indexes were normalized based on the maximum test values of the different series, as shown in equation (14). It is worth noting that the DSR and MSCR data were all at the temperature of 64°C. Aiming to ensure the reliability of the conclusions, DSR data were also obtained from the RTFOT residue of asphalt binders to study the correlation between rutting factor and dynamic stability under different experimental conditions.(14)Normalizedxik=xikxmaxk,where x was the technical index, such as softening point, viscosity, rutting indicator, R, and ZSV. k was the type of the asphalt binders, which includes the unmodified asphalt and modified asphalt.As shown in Figure4, the normalized high-temperature performance technical indexes of the asphalt were drawn in hexagonal envelope diagram. The asphalt with the largest hexagonal envelope area should have the best high-temperature overall performance. Thus, the high-temperature performance ranking of the asphalt was TPS-modified asphalt, followed by SBS I-D, SBS I-C, PEN20, PEN50, and PEN90. The presented analysis results had a good match with the viscosity at 60°C and the Carreau modeled ZSV test, while they were different from the traditional DSR and MSCR results. In general, the modified asphalt performed better than the unmodified asphalt according to the high-temperature rheological properties tests. Despite the excellent performance in some test results, such as the low deformation in MSCR, the hexagonal envelope area analysis showed that the PEN20 asphalt presented the unbalanced high-temperature performance.Figure 4
High-temperature performance distributions of six asphalt binders in hexagonal envelope diagram.Besides, in the previous research, dynamic stability (DS) was a useful indicator to characterize the high-temperature performance of asphalt mixtures [39–41]. The DS ranking of the asphalt mixtures was TPS-modified asphalt, followed by SBS I-D, PEN20, SBS I-C, PEN50, and PEN90, which was also different from the result by the envelope area analysis.In order to create the relationship between the performance evaluation indexes of the asphalt and the performance of the asphalt mixture, GRA was adopted and the results of different asphalt binders and asphalt mixtures were defined asXm0 (where m=0,1,2,…,13) and listed in Table 6. The DS of the asphalt mixture were selected for the main arrays, while the high-temperature performance indexes of asphalt were taken as the subarrays. In this section, the normalized results were calculated based on equations (6) and (7), as shown in Table 6.Table 6
Data preprocessing for GRA.
ParametersXmUnmodifiedModifiedPEN20PEN50PEN90SBS I-CSBS I-DTPSOriginal sequenceDS (times/mm)X0054151986580460381509965Penetration (25°C) (0.1 mm)X10235491745769Softening point (°C)X2064.252.447.76666.980.7Viscosity (60°C) (Pa·s)X303315394198273604975582459G∗/sinδ (64°C) (OB)X4015.115.091.093.204.404.63G∗/1−sinδ⋅tanδ−1 (64°C) (OB)X5018.945.751.215.959.410.30Jnr3.2 (kPa−1) (64°C)X600.23911.39763.430.33280.13690.0473Jnr0.1 (kPa−1) (64°C)X700.20091.06352.1750.03890.02970.0046R3.2 (%) (64°C)X8023.612.61.591.388.496.7R0.1 (%) (64°C)X9032.325.720.894.397.399.6ZSV-Sybilski (Pa·s)X1001.711.562.085622.278845.3417.46ZSV-Williamson (Pa·s)X11013483.18106.44630.8110841.616295.425233.5ZSV-Carreau (Pa·s)X1206694.853926.12599.1710718.516365.825453.8G∗/sinδ (64°C) (RTFOT)X13027.598.811.985.206.855.87Normalized sequenceDS (times/mm)X011.060.390.110.901.591.95Penetration (25°C) (0.1 mm)X110.380.881.481.210.931.13Softening point (°C)X211.020.830.761.051.061.28Viscosity (60°C) (Pa·s)X310.120.010.011.001.833.03G∗/sinδ (64°C)X412.710.910.200.570.790.83G∗/1−sinδ⋅tanδ−1 (64°C)X512.200.670.140.691.101.20Jnr3.2 (kPa−1) (64°C)X610.261.503.690.360.150.05Jnr0.1 (kPa−1) (64°C)X710.341.823.720.070.050.01R3.2 (%) (64°C)X810.450.240.031.741.691.85R0.1 (%) (64°C)X910.520.420.341.531.581.62ZSV-Sybilski (Pa·s)X1010.000.000.002.333.660.01ZSV-Williamson (Pa·s)X1111.080.650.050.871.312.03ZSV-Carreau (Pa·s)X1210.630.370.061.011.542.40G∗/sinδ (64°C) (RTFOT)X1312.940.940.210.550.730.63In order to further analyze whether the rutting indicatorG∗/sinδ was the suitable candidate for the high-temperature performance evaluation of the asphalt, both the unaged samples (marked as OB) and RTFOT aged samples (marked as RTFOT) were analyzed in this part. In addition, the modified rutting indicator G∗/1−sinδ⋅tanδ−1 was also introduced. The absolute difference between the two sequences was calculated, and then the gray relational coefficients and gray relational degree were calculated, based on equations (8) and (9). The detailed results are shown in Table 7.Table 7
Gray relational coefficients between asphalt indexes and dynamic stability.
ParametersUnmodifiedModifiedGray relational degreePEN20PEN50PEN90SBS I-CSBS I-DTPSγ10.520.610.350.720.530.480.53γ20.970.630.540.850.590.530.68γ30.440.670.890.890.770.410.68γ40.310.590.920.700.480.400.57γ50.390.730.980.790.600.500.67γ60.480.400.170.580.340.280.38γ70.510.340.170.470.330.280.35γ80.550.850.910.470.900.900.76γ90.590.980.780.541.000.700.77γ100.410.660.880.340.260.280.47γ110.990.750.940.980.730.920.88γ120.641.000.950.890.950.630.84γ130.280.580.900.690.460.360.55The gray entropy relational degree was calculated by using equations (10)∼(13). Figure 5 shows the gray relational degree and entropy relational degree between asphalt binder indexes and dynamic stability of the corresponding asphalt mixtures. The gray relational degree was to take the arithmetic average of the gray relational coefficients of each type of asphalt, which would cause the loss of the tendency information of each local point. But the entropy relational degree could effectively evaluate the correlation on the entire sequence correlation degree rather than the influence of the single-point correlation coefficient. According to the entropy relational criterion, the larger the gray relational degree and entropy relational degree values of the high-temperature performance indexes were, the stronger the correlation between the high-temperature performance of the asphalt binder and the dynamic stability of asphalt mixtures would be. For the gray relational degree, Γ11>Γ12>Γ9>Γ8>Γ2>Γ3>Γ5>Γ4>Γ13>Γ1>Γ10>Γ6>Γ7, while, for the entropy relational degree, EX11>EX12>EX1>EX9>EX2>EX8>EX5>EX3>EX7>EX4>EX6>EX13>EX10. Although the two indicators were arranged in different rank order, the X11 sequence, representing the Cross/Williamson modeled ZSV value, was the most relevant to the main array. Since the DS value would well reflect the high-temperature performance of the asphalt mixture, the ZSV test results fitted by the Cross/Williamson model had the highest correlation with the DS and would best characterize the high-temperature performance. The Cross/Williamson model was the most suitable for calculating and fitting the ZSV. Besides, it was found that the rutting indicator G∗/sinδ of the unaged asphalt (named X4) and the RTFOT asphalt (named X13) had a similar trend, since the values of the gray relational degree were close. The results indicated that the correlation between the rutting indicator G∗/sinδ in the asphalt’s DSR tests and the DS value of asphalt mixture was very poor. The present rutting indicator for evaluating the asphalt mixtures’ high-temperature performance might no longer be suitable. Although the correlation between the modified rutting factor (named X5) and the DS value had improved, the accuracy of its evaluation was still lower than the MSCR and ZSV test results. However, the conclusion should be further proved based on more laboratory tests for the asphalt binder and its mixture.Figure 5
Gray relational degree and entropy relational degree between asphalt indexes and DS.


## 5. Conclusion

Aiming to evaluate the relation between the high-temperature performance of the asphalt binders and their mixture, six typically used asphalt types were employed in this study. The DSR, MSCR, and ZSV tests were conducted on the asphalt binders, and the rutting tests were conducted on the asphalt mixture. Besides, to analyze the gray relational degree and entropy relational degree of the technical index, the correlation between the asphalt binder and the dynamic stability of the mixture was studied through the GRA method. Results in this paper provided a more systematic perspective for the further study of the high-temperature performance evaluation of asphalt binders.Conclusions are listed as follows:(1)
In general, the modified asphalt performed better than the unmodified asphalt according to the high-temperature rheological properties tests. Despite the excellent performance in some test results, such as the lowest deformation in MSCR test, the hexagonal envelope area analysis showed that the PEN20 asphalt presented the unbalanced high-temperature performance.(2)
The ranking of the six kinds of asphalt binders from various tests was different. The TPS-modified asphalt performed best in the MSCR and ZSV tests, while the low-grade asphalt PEN20 had the best technical indexes in the DSR test.(3)
The Cross/Williamson model fitted zero-shear viscosity of the asphalt and the dynamic stability of the asphalt mixture had the highest gray relational degree and entropy relational degree, which indicated that the Cross/Williamson modeled ZSV value had the highest correlation with the DS value. Thus, the Cross/Williamson model was the most suitable for calculating and fitting the ZSV, which could be used as the key indicator of the high-temperature performance evaluation of the asphalt.(4)
Furthermore, according to the preliminary results on the six kinds of asphalt and its mixtures, the correlation between the rutting indicator in the asphalt’s DSR tests and the DS value of asphalt mixture was very poor. The present rutting indicatorG∗/sinδandG∗/1−sinδ⋅tanδ−1 for evaluating the asphalt mixtures’ high-temperature performance might no longer be suitable. However, the conclusion should be further proved based on more laboratory tests for the asphalt binder and its mixture.



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*Source: 5818036-2020-11-11.xml*