# Study on Styrene-Butadiene-Styrene Modified Asphalt Binders Relaxation at Low Temperature

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{1}to t

^{β}, where the β exponent is the fifth parameter of the analyzed model. Since the values of parameter β achieved at the end of modeling were within the range of 0.47 to 0.54, during the second attempt the model has been simplified and four parameters were used with a defined value of time exponent β = 0.5 (achieving time domain t

^{0.5}). Using modified equations in the four-parameter model allowed to achieve a better fit for relaxation curves prepared by means of empirical method than in the case of classic generalized Maxwell model with six parameters.

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Experimental Methods

#### 2.2.1. Penetration and Softening Point Tests

#### 2.2.2. RTFOT-Short-Term Aging Simulation

#### 2.2.3. The Relaxation Test Using Ductilometer

#### 2.2.4. Mathematical Description of Relaxation Curves

_{i}) as well as viscous (Newtonian liquid) ones (characterized by dynamic viscosity η

_{i}) is presented in Figure 3.

- σ(t)—stress at t moment;
- ε
_{0}—maximum strain obtained as a result of stretching the samples (strain with a constant value for 20 min to observe relaxation); - t-time;
- E
_{i}—modulus of elasticity of Hooke’s element i; - η
_{i}—dynamic viscosity of Newton’s element i; - ${\tau}_{i}=\frac{{\eta}_{i}}{{E}_{i}}$ is the time of relaxation of Maxwell element i.

## 3. Results and Discussion

#### 3.1. Penetration and Softening Point

_{R&B}value.

#### 3.2. Relaxation Test Results

^{2}]. Brass forms that were used guaranteed that cross-sectional area of every tested sample was 10

^{−4}m

^{2}. Due to the small deformation of the samples, the assumption concerning the invariant cross-section was adopted [36,37].

#### 3.3. Relaxation Curves Mathematical Description

^{2}coefficient of determination, root mean square error–RMS Error and sum of the squared deviations–SSq) for exemplary, unaged asphalt binder with a 4.5% SBS copolymer are summarized in Table 2, Table 3 and Table 4.

^{2}coefficient of determination and lowest values of RMS Error and SSq sum of squares. Despite the fact that in each of the used models, high values of coefficient of determination were observed (in all cases R

^{2}> 0.99), the differences between them are significant which is demonstrated by two other parameters (e.g., there is a 16 times difference in RMS Error values and 160 times difference in SSq values). It should be kept in mind that increasing the number of parameters in a model results in achieving a better fitting of curves obtained via the analytical and empirical methods. At the same time a large quantity of parameters means that arrangement of their values shows a visible diversification. That is why, an attempt was made to modify the Equation (1) by changing the domain of time from t

^{1}to t

^{β}. For n = 2 Equation (1) will take the following form:

^{2}, RMS Error and SSq statistical values for exemplary, unaged asphalt binder containing a 4.5% of SBS copolymer.

^{2}, RMS Error and SSq statistical values for exemplary, unaged asphalt binder with a 4.5% SBS copolymer.

^{2}values available in Table 2, Table 3, Table 4 and Table 7, it has been found that the modified generalized Maxwell model with four parameters, which was described by Equation (5), allows for acquisition of a very good fitting of relaxation curves obtained via the analytical and empirical methods, conceding slightly to the model with five parameters described by Equation (4) and generalized Maxwell model with eight parameters. That is why, further in this study, analysis has been performed on values of E

_{1}, η

_{1}, E

_{2}, η

_{2}(assuming that E

_{1}> E

_{2}) of a modified generalized Maxwell model described by Equation (5). These values are presented in Figure 10, Figure 11, Figure 12 and Figure 13 and Table 8, respectively.

_{1}and η

_{1}(Figure 10 and Figure 11), simple correlations have been observed only in the case of unaged asphalt binders. It has been found that increasing the content of SBS copolymer in modified bitumen lowers values of parameters E

_{1}and η

_{1}. In case of binders subjected to the RTFOT aging process, correlations between parameters E

_{1}and η

_{1}have a different character. When increasing the content of SBS copolymer in modified asphalt binder in the 0 to 4.5% range, an increase of E

_{1}and η

_{1}values has been observed. Further, increase of the SBS copolymer content in modified binder (in the 4.5 to 9% range) results in a significant decrease of E

_{1}and η

_{1}values. The effect of the aging process simulated by RTFOT method on E

_{1}and η

_{1}values is also interesting. In the case of 50/70 penetration grade bitumen, aging results in a decrease of values of both parameters. On the other hand, in the case of all analyzed binders containing SBS copolymer, an increase of values of these parameters has been observed. In case of parameters E

_{2}and η

_{2}(Figure 12 and Figure 13), simple correlations have been observed both in the case of unaged asphalt binders and those aged using the RTFOT method. It has been found that increasing the content of SBS copolymer in modified asphalt binder lowers values of parameters E

_{2}and η

_{2}. On the other hand, as a result of aging using the RTFOT method, an increase of values of E

_{2}and η

_{2}parameters in all analyzed binders has been observed. The above-described observations show that one of the elements of the Maxwell model (in which an elasticity element with a higher value of modulus of elasticity E

_{1}is present) exhibits a higher sensitivity to changes caused by aging of binders than the latter one (in which an elasticity element with a lower value of modulus of elasticity E

_{2}is present).

_{1}and τ

_{2}, respectively, calculated on the basis of modelling results compared in Figure 10, Figure 11, Figure 12 and Figure 13. A general trend has been observed by which values of relaxation time decrease with an increase of SBS copolymer content in the asphalt binder. Decrease of relaxation time should be considered as a positive phenomenon, allowing for a faster reduction of tensile stress in asphalt binder (e.g., arising from a sudden temperature drop of the asphalt pavement). It should be remembered that asphalt binder properties are primarily responsible for asphalt pavement resistance against low temperature cracking. Influence of the aging process in case of 50/70 penetration grade bitumen and modified binders on relaxation time values is varied. Shortening of relaxation times in case of 50/70 penetration grade bitumen and their increase in case of analyzed binders containing SBS copolymer has been observed. This is caused by a different structure of 50/70 bitumen and modified binders. In the case of modified bitumen, different degrees of polymer saturation in asphalt binder can be differentiated. In highly modified bitumens, a continuous polymer phase occurs, that is why the binder has a more homogenous structure. Aging process has an effect on both the change of base bitumen properties as well as degradation of copolymer. These effects, however, can be varied, depending on the content of SBS copolymer in the modified asphalt binder. This can explain the differences observed in Figure 14 and Figure 15.

## 4. Conclusions

- The larger the SBS copolymer content in the binder the faster the relaxation phenomenon occurs;
- Simultaneously with the change in the structure of the binder with 7.5 and more percentage of SBS copolymer in modified binder weight, differences in relaxation curves become negligible regardless of the polymer content in the sample (7.5% or 9% in this research)-increasing the SBS content above this limit will not affect the relaxation phenomenon;
- Aging simulated by Rolling Thin Film Oven Test method causes the relaxation phenomenon to slow down;
- Relaxation of the asphalt binder occurs the fastest immediately after the stop of the elongation process, then it slows down. In none of the tested binders, the tensile stress decreased to zero in established measurement conditions (time of relaxation of 1200 s);
- With the increase in the content of the SBS copolymer, the influence of the short-term aging on the final value of the recorded force in the test conducted at −16 °C has been reduced;
- SBS copolymer reduces the susceptibility of binders to altering their rheological properties occurring under the influence of aging—the divergences between relaxation curves of unaged samples and those subjected to RTFOT aging simulation decrease with a higher content of the elastomer;
- SBS copolymer reduces the susceptibility of binders to altering their rheological properties occurring under the influence of short-term aging—the divergences between relaxation curves of unaged samples and those subjected to RTFOT aging simulation decrease with a higher content of the elastomer;
- Phenomenon of asphalt binders relaxation at low temperatures can be described mathematically using a generalized Maxwell model. Out of five variants of this model, the modified generalized Maxwell model with four parameters and a constant value of time exponent (t
^{0.5}) was analyzed in detail. This allowed to achieve a better fitting of relaxation curves prepared by means of empirical and analytical methods than in the case of, e.g., using a generalized Maxwell model with six parameters and a linear time scale (t^{1}); - Values of parameters of the modified generalized Maxwell model used in this study (E
_{1}, E_{2}, η_{1}, η_{2}and relaxation times τ_{1}and τ_{2}calculated on their basis) are dependent on SBS copolymer content in modified asphalt binder. Furthermore, the short-term aging process (RTFOT) also has a significant effect on the values of the examined binders.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Combination of elastic and viscous elements in generalized Maxwell models with 4, 6 and 8 parameters [14].

**Figure 6.**Results obtained from the force-ductility-relaxation test at −16 °C conducted for one sample of the asphalt binder modified with 3% of SBS copolymer.

**Figure 7.**Relaxation curves of unaged bituminous binders obtained from the force-ductility-relaxation test conducted at −16 °C.

**Figure 8.**Relaxation curves of RTFOT aged bituminous binders obtained from the force-ductility-relaxation test conducted at −16 °C.

**Figure 9.**Values of time exponent (parameter β) in a modified generalized Maxwell model with five parameters.

**Figure 10.**Values of modulus of elasticity (parameter E

_{1}) in a modified generalized Maxwell model with four parameters.

**Figure 11.**Values of dynamic viscosity (parameter η

_{1}) in a modified generalized Maxwell model with four parameters.

**Figure 12.**Values of modulus of elasticity (parameter E

_{2}) in a modified generalized Maxwell model with four parameters.

**Figure 13.**Values of dynamic viscosity (parameter η

_{2}) in a modified generalized Maxwell model with four parameters.

**Figure 14.**Relaxation time τ

_{1}= η

_{1}/E

_{1}in modified generalized Maxwell model with four parameters.

**Figure 15.**Relaxation time τ

_{2}= η

_{2}/E

_{2}in modified generalized Maxwell model with four parameters.

Bitumen | Penetration at 25 °C (mm/10) | Softening Point (°C) |
---|---|---|

50/70 before RTFOT | 54.6 ± 0.7 | 50.3 ± 0.5 |

50/70 after RTFOT | 37.1 ± 0.5 | 55.4 ± 0.3 |

3%SBS before RTFOT | 63.7 ± 0.5 | 53.0 ± 0.7 |

3%SBS after RTFOT | 42.2 ± 0.9 | 56.4 ± 0.3 |

4.5%SBS before RTFOT | 68.5 ± 0.6 | 83.3 ± 0.8 |

4.5%SBS after RTFOT | 44.4 ± 0.8 | 83.8 ± 4.1 |

6%SBS before RTFOT | 72.3 ± 0.4 | 99.0 ± 2.4 |

6%SBS after RTFOT | 51.1 ± 0.6 | 90.7 ± 1.4 |

7.5%SBS before RTFOT | 76.9 ± 0.6 | 101.6 ± 0.8 |

7.5%SBS after RTFOT | 59.9 ± 0.3 | 95.4 ± 2.0 |

9%SBS before RTFOT | 83.4 ± 0.8 | 104.5 ± 0.7 |

9%SBS after RTFOT | 65.7 ± 0.7 | 101.0 ± 0.9 |

**Table 2.**Values of parameters in a generalized Maxwell model with four parameters for unaged asphalt binder containing 4.5% SBS.

Model with 4 Parameters | |
---|---|

R^{2} | 0.99204 |

RMS Error | 5909.93 |

SSq | 2.093·10^{11} |

Modulus of elasticity E_{1}, Pa | (2114 ± 6)·10^{4} |

Dynamic viscosity η_{1}, Pa·s | (1612 ± 9)·10^{6} |

Modulus of elasticity E_{2}, Pa | (1487 ± 4)·10^{4} |

Dynamic viscosity η_{2}, Pa·s | (2317 ± 7)·10^{7} |

**Table 3.**Values of parameters in a generalized Maxwell model with six parameters for unaged asphalt binder containing 4.5% SBS.

Model with 6 Parameters | |
---|---|

R^{2} | 0.99966 |

RMS Error | 1226.73 |

SSq | 9.014·10^{9} |

Modulus of elasticity E_{1}, Pa | (1534 ± 3)·10^{4} |

Dynamic viscosity η_{1}, Pa·s | (2245 ± 6)·10^{6} |

Modulus of elasticity E_{2}, Pa | (1268 ± 3)·10^{4} |

Dynamic viscosity η_{2}, Pa·s | (252 ± 2)·10^{6} |

Modulus of elasticity E_{3}, Pa | (1243 ± 2)·10^{4} |

Dynamic viscosity η_{3}, Pa·s | (2794 ± 6)·10^{7} |

**Table 4.**Values of parameters in a generalized Maxwell model with eight parameters for unaged asphalt binder containing 4.5% SBS.

Model with 8 Parameters | |
---|---|

R^{2} | 0.99995 |

RMS Error | 466.15 |

SSq | 1.301·10^{9} |

Modulus of elasticity E_{1}, Pa | (1357 ± 2)·10^{4} |

Dynamic viscosity η_{1}, Pa·s | (2375 ± 4)·10^{6} |

Modulus of elasticity E_{2}, Pa | (1180 ± 2)·10^{4} |

Dynamic viscosity η_{2}, Pa·s | (2979 ± 4)·10^{7} |

Modulus of elasticity E_{3}, Pa | (1105 ± 3)·10^{4} |

Dynamic viscosity η_{3}, Pa·s | (378 ± 2)·10^{6} |

Modulus of elasticity E_{4}, Pa | (558 ± 3)·10^{4} |

Dynamic viscosity η_{4}, Pa·s | (327 ± 5)·10^{5} |

**Table 5.**Values of parameters in a generalized Maxwell model described with Equation (4) for unaged asphalt binder with 4.5% SBS.

Model with 5 Parameters | |
---|---|

R^{2} | 0.99988 |

RMS Error | 725.71 |

SSq | 3.155·10^{9} |

Modulus of elasticity E_{1}, Pa | (3014 ± 6)·10^{4} |

Dynamic viscosity η_{1}, Pa·s | (2666 ± 6)·10^{5} |

Modulus of elasticity E_{2}, Pa | (1405 ± 4)·10^{4} |

Dynamic viscosity η_{2}, Pa·s | (951 ± 6)·10^{6} |

β | (5374 ± 8)·10^{−4} |

Bitumen | β |
---|---|

50/70 before RTFOT | 0.5147 ± 0.0006 |

50/70 after RTFOT | 0.4731 ± 0.0014 |

3%SBS before RTFOT | 0.5267 ± 0.0007 |

3%SBS after RTFOT | 0.5038 ± 0.0008 |

4.5%SBS before RTFOT | 0.5374 ± 0.0008 |

4.5%SBS after RTFOT | 0.5216 ± 0.0008 |

6%SBS before RTFOT | 0.5067 ± 0.0009 |

6%SBS after RTFOT | 0.5089 ± 0.0008 |

7.5%SBS before RTFOT | 0.5116 ± 0.0008 |

7.5%SBS after RTFOT | 0.5223 ± 0.0007 |

9%SBS before RTFOT | 0.5241 ± 0.0007 |

9%SBS after RTFOT | 0.5175 ± 0.0007 |

**Table 7.**Values of parameters in a generalized Maxwell model described with Equation (5) for unaged asphalt binder containing 4.5% SBS.

Model with 4 Parameters and a Constant Value β = 0.5 | |
---|---|

R^{2} | 0.99983 |

RMS Error | 854.85 |

SSq | 4.378·10^{9} |

Modulus of elasticity E_{1}, Pa | (3287 ± 3)·10^{4} |

Dynamic viscosity η_{1}, Pa·s | (2569 ± 6)·10^{5} |

Modulus of elasticity E_{2}, Pa | (1240 ± 4)·10^{4} |

Dynamic viscosity η_{2}, Pa·s | (760 ± 2)·10^{6} |

**Table 8.**Values of modulus of elasticity (parameters E

_{1}, E

_{2}) and dynamic viscosity (parameters η

_{1}, η

_{2}) in a modified generalized Maxwell model with four parameters.

Bitumen | E_{1} | η_{1} | E_{2} | η_{2} |
---|---|---|---|---|

50/70 before RTFOT | 56.18 ± 0.04 | 521.64 ± 0.97 | 19.97 ± 0.05 | 1665.58 ± 4.89 |

50/70 after RTFOT | 44.04 ± 0.06 | 369.80 ± 1.31 | 40.81 ± 0.08 | 3316.41 ± 7.30 |

3%SBS before RTFOT | 38.77 ± 0.03 | 317.43 ± 0.66 | 15.01 ± 0.04 | 920.71 ± 1.61 |

3%SBS after RTFOT | 48.60 ± 0.04 | 416.59 ± 0.88 | 36.68 ± 0.05 | 2715.84 ± 3.62 |

4.5%SBS before RTFOT | 32.87 ± 0.03 | 256.87 ± 0.59 | 12.40 ± 0.04 | 759.92 ± 1.56 |

4.5%SBS after RTFOT | 54.19 ± 0.05 | 458.83 ± 1.01 | 31.34 ± 0.06 | 2233.23 ± 3.75 |

6%SBS before RTFOT | 26.15 ± 0.02 | 162.41 ± 0.31 | 12.73 ± 0.02 | 515.08 ± 0.25 |

6%SBS after RTFOT | 37.39 ± 0.03 | 293.97 ± 0.60 | 18.64 ± 0.04 | 1211.42 ± 1.92 |

7.5%SBS before RTFOT | 25.18 ± 0.01 | 155.77 ± 0.24 | 9.70 ± 0.02 | 407.80 ± 0.24 |

7.5%SBS after RTFOT | 25.52 ± 0.02 | 167.14 ± 0.26 | 11.89 ± 0.02 | 558.27 ± 0.37 |

9%SBS before RTFOT | 15.54 ± 0.01 | 98.33 ± 0.16 | 6.79 ± 0.01 | 262.72 ± 0.10 |

9%SBS after RTFOT | 23.50 ± 0.01 | 142.01 ± 0.21 | 11.20 ± 0.02 | 467.33 ± 0.22 |

Bitumen | τ_{1} | τ_{2} |
---|---|---|

50/70 before RTFOT | 9.28 | 83.41 |

50/70 after RTFOT | 8.40 | 81.27 |

3%SBS before RTFOT | 8.19 | 61.33 |

3%SBS after RTFOT | 8.57 | 74.04 |

4.5%SBS before RTFOT | 7.81 | 61.29 |

4.5%SBS after RTFOT | 8.47 | 71.27 |

6%SBS before RTFOT | 6.21 | 40.46 |

6%SBS after RTFOT | 7.86 | 64.99 |

7.5%SBS before RTFOT | 6.18 | 42.05 |

7.5%SBS after RTFOT | 6.55 | 46.94 |

9%SBS before RTFOT | 6.33 | 38.68 |

9%SBS after RTFOT | 6.04 | 41.71 |

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**MDPI and ACS Style**

Dziadosz, S.; Słowik, M.; Niwczyk, F.; Bilski, M.
Study on Styrene-Butadiene-Styrene Modified Asphalt Binders Relaxation at Low Temperature. *Materials* **2021**, *14*, 2888.
https://doi.org/10.3390/ma14112888

**AMA Style**

Dziadosz S, Słowik M, Niwczyk F, Bilski M.
Study on Styrene-Butadiene-Styrene Modified Asphalt Binders Relaxation at Low Temperature. *Materials*. 2021; 14(11):2888.
https://doi.org/10.3390/ma14112888

**Chicago/Turabian Style**

Dziadosz, Sylwia, Mieczysław Słowik, Filip Niwczyk, and Marcin Bilski.
2021. "Study on Styrene-Butadiene-Styrene Modified Asphalt Binders Relaxation at Low Temperature" *Materials* 14, no. 11: 2888.
https://doi.org/10.3390/ma14112888