# Study on the Phase Angle Master Curve of the Polyurethane Mixture with Dense Gradation

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## Abstract

**:**

^{−3}Hz. The R

^{2}maximization as the main constraint and the others as the additional constraints were recommended as the error minimization method. The combination of the Christensen Anderson and Marasteanu model (CAM) and kaelble shift factor equation was recommended for fitting the phase angle master curve of the PU mixture. The phase angle master curve of the PU mixture did not follow the “Bell” shape of the asphalt mixture. The PU mixture with smaller temperature susceptibility would still be subject to the PU at higher temperatures and was closer to that of the viscoelastic material. The phase angle master curve construction was analyzed for the first time and proper master curve fitting parameters were recommended for pavement performance predicting and analyzing.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Material

#### 2.2. Methodology

#### 2.2.1. Phase Angle Fitting Model

#### Model 1: Standard Logistic Sigmoid Model

_{r}) is the phase angle in °; f

_{r}is the load frequency at the reference temperature in Hz; α, β, and γ are the fitting parameters. β and γ are the shape parameters of the model curve, which describe the shape of model 1 as depicted in Figure 2. In Figures 2–12 and 20, the x-axis represents the loading frequency in logarithm form, the y-axis represents the phase angle in arithmetic form.

#### Model 2: Generalized Logistic Sigmoidal (GLS) Model

#### Model 3: Christensen Anderson and Marasteanu Model (1999)

_{c}are the fitting parameters and v, and t

_{c}could describe the shape of Model 3 as depicted in Figure 4.

#### Model 4: Modified CAM Model (2001)

_{m}is the phase angle constant at f

_{d}; f

_{d}is the location parameter with a dimension of frequency in Hz; R

_{d}and m

_{d}are shape parameters, which could describe the shape of Model 4 as shown in Figure 5.

#### Model 5: Sigmoidal CAM Model (SCM Model)

_{c}are consistent with the parameters of Model 3, and z is the newly introduced fitting parameter. The parameters v, z, w, and t

_{c}related to the shape of Model 5 are depicted in Figure 6.

#### 2.2.2. Shift Factor Equation

_{r}value is the frequency equivalent of the experimental temperature relative to the reference value. Once the shift factor is known, Equation (7) could be used to calculate the new, lower frequency.

_{r}) is the reduced frequency in reference temperature.

#### Equation (1): Log-Linear Equation

_{T}) is found to vary linearly with temperature below 0 °C, as stated by Christensen and Anderson [40], and this similar relationship has been regarded appropriate for asphalt mixture at low to moderate temperatures [49]. The log-linear equation for calculating the shift factor is

_{T}) is the shift factor, T is the temperature in °C, T

_{r}is the reference temperature (20 °C), and the constant C is calculated from the results of the experiment.

#### Equation (2): Polynomial Equation

#### Equation (3): Arrhenius Equation

_{g}[52] has only one constant that needs figuring out.

#### Equation (4): Williams−Landel−Ferry Equation

_{g}:

_{1}and C

_{2}are the two parameters of regression.

#### Equation (5): Kaelble Equation

_{g},

_{l}’ and C

_{2}’ are the two regression parameters.

#### 2.3. Error Minimization Method

^{2}maximization as in Equation (16), (3) the Sum of Square Error (SSE) minimization as in Equation (17), and (4) Error

^{2}minimization as in Equation (18).

_{i}is the measured phase angle, ${\widehat{x}}_{i}$ is the predicted phase angle, and ${\overline{x}}_{i}$ is the mean value of the measured phase angle.

^{2}of a given model may be calculated using Equation (16):

^{2}value is the absolute difference between the predicted and the experimentally measured phase angle.

#### 2.4. Fitting Process

_{T}) in different shift factor equations were incorporated into the master curve model equation by Equation (9); then, the error minimization methods were separately used as the main constraint for fitting the measured phase angle data and minimizing the error between the predicted and measured phase angle data; the other three error minimization methods were used as the additional constraints. The difference between the phase angle master curves fitted by different error minimization methods was compared and analyzed to determine the more accurate error minimization method for each master curve model. (b) After determining the optimum error minimization method, the effect of the shift factor equation on the fitting results was analyzed by comparing the predicted and measured phase angle, and the most accurate shift factor equation was recommended for each master curve model. (c) Under the determined optimum shift factor equation, the difference between all phase angle master curve models was compared and analyzed to recommend the most accuracy model and shift factor equation for fitting the phase angle data.

## 3. Results

#### 3.1. Analyzing the Influence of the Solver Method on the Fitting of the Phase Angle Master Curve

#### 3.1.1. Model 1 Fitting Results

#### 3.1.2. Model 2 Fitting Results

#### 3.1.3. Model 3 Fitting Results

#### 3.1.4. Model 4 Fitting Results

#### 3.1.5. Model 5 Fitting Results

#### 3.1.6. Comparison of Different Shift Factor Equations

^{2}maximization error minimization method as the main constraint.

#### 3.2. Analyzing the Influence of Shift Factor Equation on the Fitting of Phase Angle Master Curve

#### 3.2.1. Model 1 Fitting Results

#### 3.2.2. Model 2 Fitting Results

#### 3.2.3. Model 3 Fitting Results

#### 3.2.4. Model 4 Fitting Results

#### 3.2.5. Model 5 Fitting Results

#### 3.3. Comparing Master Curves with Recommended Models and Shift Factor Equations

#### 3.3.1. Comparing Polynomial Shift Factor Equation Fitting Results

#### 3.3.2. Comparing Kaelble Shift Factor Equation Fitting Results

#### 3.4. Comparing the Master Curves under Different Models

## 4. Discussion

#### 4.1. Analyzing the Influence of the Solver Method on the Fitting of the Phase Angle Master Curve

#### 4.1.1. Model 1 Fitting Results

^{−3}Hz, the curves had obvious differences. The master curves fitted by different shift factor equations showed different shapes, and the master curve peak values of different shift factor equations exhibited significant difference.

^{−3}Hz, the master curves under the same shift factor equation tend to be the same, and the differences are minimized. For the PU mixture used in road pavement under different vehicles, the loading frequency below 10

^{−3}Hz would be rare.

^{−3}Hz. Based on the four parameter values (R

^{2}, Se/Sy, error

^{2}, SSE) fitted by four error minimization methods, the R

^{2}maximization method would produce the highest R

^{2}value, and the lowest Se/Sy, error

^{2}, SSE values, and the ultimate phase angle value at the lowest frequency were different according to different error minimization methods; the ultimate value produced by the R

^{2}maximization method located in the middle and did not show extreme high or low values. The R

^{2}maximization method is adopted to obtain the regression parameters of all equations for Model 1 in the following discussion.

#### 4.1.2. Model 2 Fitting Results

^{−3}Hz, the difference between different master curves became insignificant, and all the lines converged.

^{−3}Hz. Therefore, the error minimization methods had an insignificant influence on the phase angle master curve at higher loading frequency. By comparing the R

^{2}, Se/Sy, error

^{2}, and SSE values fitted by different error minimization methods, the R

^{2}maximization method had relatively lower Se/Sy, error

^{2}, and SSE values, and higher R

^{2}value. The ultimate value of the R

^{2}maximization method did not show extremely higher or lower compared with the other methods. In the following discussion, the R

^{2}maximization method was used for analyzing the phase angle master curve.

#### 4.1.3. Model 3 Fitting Results

^{−3}Hz, the phase angle master curves were confluent.

^{2}maximization method based on the identical initial values. Therefore, for Model 3, the R

^{2}maximization used as the main constraint was selected for regressing the parameters in the following discussion.

#### 4.1.4. Model 4 Fitting Results

^{2}, Se/Sy, error

^{2}, SSE) fitted by four error minimization methods also had little difference. The R

^{2}maximization method is adopted for comparison in the following discussion, with conclusion the same as the above.

#### 4.1.5. Model 5 Fitting Results

^{−3}Hz.

^{−3}Hz. Analyzing the fitting parameter values by five different shift factor equations, the R

^{2}maximization method produced relatively good fitting parameter values compared with the other methods. The R

^{2}maximization method is recommended for equation parameter regression in the following section.

#### 4.1.6. Comparison of Different Shift Factor Equations

^{−3}Hz, the effect of the error minimization method on the construction of the phase angle master curve was insignificant. The R

^{2}maximization (>0.99) method was recommended as the main constraint for the regression procedure, and the Se/Sy minimization (<0.05), SSE minimization (<0.05), and Error

^{2}minimization (<0.05) were adopted as the additional constraints.

#### 4.2. Analyzing the Influence of Shift Factor Equation on the Fitting of Phase Angle Master Curve

^{2}, trend line slope, and residual sum of squares (RSS) were adopted to evaluate the fitting results of different shift factor equations under difficult models. The trend line slope and trend line R

^{2}were generally used as statistics to evaluate the correlation. The linear relationships’ R

^{2}value provides information about how well each model fits the data and how much of the data’s variability can be accommodated by that model. Higher R

^{2}and lower RSS values indicate better prediction accuracy. When the trend line slope values were higher than 1, this meant that the prediction procedure would overestimate the measured data; when the trend line slope values were lower than 1, this represented an underestimation of the phase angle. If the trend line slope is closer to 1, the prediction method is more accurate. The bias is minimized when the slope is 1 and the intercept is 0. The model’s accuracy was determined by calculating the slope of the linear connection between the predicted and measured values, with the intercept set close to zero. The R

^{2}value of the linear relationship could evaluate the variability of the measured results by each model and indicate the overall accuracy of the model. The closer the R

^{2}value is to 1, the higher the the degree of correlation between the predicted and measured results. The RSS index represents the bias between the predicted and measured results; the smaller the RSS index, the more accurate the prediction model.

#### 4.2.1. Model 1 Fitting Results

^{2}value and the RSS value in Table 1, the prediction accuracy of different shift factor equations was ranked as Polynomial > WLF > Arrhenius > Log-linear > Kaelble, and the value of the trend line R

^{2}varied from 0.9201 to 0.96584. For Model 1, the Polynomial shift equation had the strongest R

^{2}value and the lowest RSS value and showed the best goodness-of-fit statistics. This means that the Polynomial shift factor equation could provide better prediction accuracy compared with the measured values.

^{2}value, the closer the predicted date spots to the LOE. The shift factor equations with higher trend line R

^{2}values predict more accurate data compared with the measured results. All the trend line slopes were smaller than 1, which means that the shift factor equations combined with Model 1 underestimated the measured results. The higher the trend line R

^{2}value, the closer the trend line slope to 1.

#### 4.2.2. Model 2 Fitting Results

^{2}and RSS values in Table 2, the prediction precision of different shift factor equations was ranked as Kaelble > Polynomial > WLF > Arrhenius > Log-linear, and the value of the trend line R

^{2}varied from 0.95745 to 0.9789, which was higher than that of Model 1. The RSS values followed the opposite trend and also proved the same rank of different shift factor equations. The Kaeble shift factor equation showed the best goodness-of-fit statistics with the strongest R

^{2}value and the smallest RSS value, which also proved that the Kaelble shift factor equation could provide better accuracy predictions than the other equations.

#### 4.2.3. Model 3 Fitting Results

^{2}values and RSS values in Table 3, and the order was the same as that in Model 2. The trend line R

^{2}values ranged from 0.95847 to 0.9792, which was similar to Model 2. The Kaelble shift factor equation had the highest trend line R

^{2}value and the smallest RSS value. These results suggested that the Kaelble shift factor equation had the best goodness-of-fit statistics and could predict the phase angle more precisely than the other equations under Model 3.

^{2}value was closer to the LOE than the other equations. The trend line slope of all shift factor equations also followed the same order as the trend line R

^{2}values. The trend line slopes were all smaller than 1, which means that all the shift factor equations under Model 3 slightly underestimated the measured data.

#### 4.2.4. Model 4 Fitting Results

^{2}values and RSS values in Table 4, the prediction precision of all shift factor equations was as follows: Kaelble > Polynomial > WLF > Log-linear > Arrhenius and the corresponding trend line R

^{2}values increased from 0.96108 to 0.98577. The RSS values also proved the rank order. The Kaelble shift factor equation had the strongest trend lineR

^{2}value and the smallest RSS value. The equation could predict more precise results than the other equations. The data spots would be closer to the LOE with the higher trend line R

^{2}value and smaller RSS value. The trend line slope of the Kaelble shift factor equation approached 1 more prominently than the other shift factor equations, and all the trend line slopes of different shift factor equations were slightly smaller than 1, which means that the shift factor equations underestimated the measured phase angle.

#### 4.2.5. Model 5 Fitting Results

^{2}values and RSS values in Table 5 ranked as Kaelble > Polynomial > WLF > Arrhenius > Log-linear, which was the same as the results for Models 1 and 2. The trend line R

^{2}values ranged from 0.97729 to 0.95163, and the RSS values also proved the equation order. For Model 5, the Kaelble shift factor equation could provide more precise predictions than the other equation with the best goodness-of-fit statistics.

^{2}values were, the closer the data spots were to the LOE. The trend line describes the relationship between the measured and predicted values. Except the trend line slope of the WLF shift factor equation, which was higher than 1, the slopes of the other four shift factor equations followed the same order as the R

^{2}values. The slopes were slightly lower than 1, which indicated that the four shift factor equations would somewhat underestimate the measured results.

#### 4.3. Comparing Master Curves with Recommended Models and Shift Factor Equations

#### 4.3.1. Comparing Polynomial Shift Factor Equation Fitting Results

^{2}value of Model 4 was the highest (0.97464), and the corresponding RSS value was the smallest (1.87013), which means that Model 4 had the best fitting result and the smallest errors between the measured and predicted data values. The trend line R

^{2}values of the other four models were close, which ranged from 0.96435 to 0.96584, and the RSS values ranged from 2.50653 to 2.73905, which means that the other four models had similar fitting results, and the errors between the measured and predicted data values also had little difference.

^{2}values of models used for PU mixtures were much higher than those for asphalt mixtures. For example, in literature [56], the trend line R

^{2}values ranged from 0.851 to 0.858 for the SLS, GLS, and SCM models. Therefore, Model 4 exhibited the best-fit statistic, and Model 4 is capable to fit the measured data much more closely in comparison with the other models, which is in agreement with the conclusion of literature [56].

#### 4.3.2. Comparing Kaelble Shift Factor Equation Fitting Results

^{2}value was the smallest, and the corresponding RSS value was the biggest. This means that Model 1 with Kaelble shift factor equation had the worst fitting results and the lowest prediction precision compared with the other models. Model 4 had the strongest R

^{2}value (0.98577), the highest trend line slope (0.98378), and the smallest RSS value (1.09921), which suggested that Model 4 could have the best-fit statistic and produced better prediction accuracy than the other models under the Kaelble shift factor equation. The other three models had similar fitting results, e.g., trend line R

^{2}ranged from 0.97729 to 0.97792, trend line slope ranged from 0.97563 to 0.98029, and the RSS values ranged from 1.59565 to 1.75707.

^{2}, slope, and RSS values. However, the shape of the master curve of Model 4 with Kaelble shift factor did not comply with the changing regularity of the mixture which was explained in Section 4.1.4. Model 3 with Kaelble shift factor equation was recommended for the construction of the phase angle master curve of the PU mixture.

#### 4.4. Comparing the Master Curves under Different Models

^{−3}Hz, the phase angle master curve of Model 5 was extremely higher than that of the other models, and the GLS and CAM model had the same trend, while Model 1 had lower values. The master curves of the models excluding Model 4 would decrease monotonously with the increase in the loading frequency as expected, the master curve of Model 4 would first decrease and then increase with the increase in the loading frequency. When the loading frequency was between 10

^{−3}and 10

^{3}Hz, the difference between all the master curves became small. The shape of Model 1 was similar to that of the results of the dense PU mixture in [57].

## 5. Conclusions

- (1)
- When the loading frequency was higher than 10
^{−3}Hz, the master curves fitted by different error minimization methods under the same master curve model and shift factor equation exhibited little difference. The R^{2}error minimization method, including R^{2}(>0.99) as the main constraint, Se/Sy (<0.05), SSE (<0.05), and Error^{2}(<0.05) as the additional constraints, was used for parameter regression. - (2)
- The LOE method, linear fitting method, and Pearson linear correlation analysis proved to be effective methods to evaluate the precise fitting of the phase angle from the test data.
- (3)
- According to the linear fitting and statistics analysis, the Kaelble shift factor equation had the strongest correlation with the measured result for Models 2, 3, 4, and 5. Then, for Model 1, the Polynomial shift factor equation could provide more accurate predictions than the other equations.
- (4)
- The combination of Model 3 and the Kaelble shift factor equation was recommended for fitting the phase angle master curve of the PU mixture.
- (5)
- The trend line R
^{2}values of the linear fitting results for the PU mixtures were much higher than those with the same master curve models for the asphalt mixture, which indicated that the PU mixture had lower temperature sensitivity than the asphalt mixture. - (6)
- For all the phase angle master curves, Model 5 had the highest values, Models 2 and 3 had the same trend, Model 1 had the lowest value, and all four models decreased monotonously with the increase in the loading frequency. Model 4 first decreased and then increased with the increase in the loading frequency. When the loading frequency ranged between 10
^{−3}and 10^{3}Hz, all the master curve models exhibited insignificant differences. - (7)
- The phase angle master curve did not show the “Bell” shape as that of the asphalt mixture and did not have the highest phase angle values at intermediate loading frequency. For the asphalt mixture, at higher loading frequency or lower temperature, the asphalt mixture exhibited elastic properties and was mainly subject to the asphalt. With the increase in temperature or the decrease in loading frequency, the asphalt became soft, and the asphalt mixture exhibited viscosity and was subject to the aggregate skeleton. For the PU mixture, the viscous property component increased with the increase in temperature or the decrease in loading frequency, but the PU mixture was still subject to the PU, which was different from that of the asphalt mixture. The phase angle behavior of the PU mixture complied more with the characteristic of the viscoelastic material compared with the asphalt mixture.

^{−3}Hz. Therefore, the phase angle master curve models introduced from the asphalt mixture were only suitable for the PU mixture when the loading frequency was higher than 10

^{−3}Hz. There are few studies conducted about the phase angle master curve of the PU mixture; this paper analyzed the influence factors during the fitting process of the phase angle of the PU mixture and contributed to the construction of the phase angle master curve and prediction of the characteristic of the PU mixture at extreme temperatures or loading frequencies, which could be used to predict the road performance of the PU mixture and determine the proper pavement structure combined with the PU mixture layer.

^{−3}Hz. More phase angle master curves and shift factor equations should be compared in the future for a better understanding of the viscoelastic property of the PU mixture.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 7.**The master curve of Model 1 fitting results with different error minimization methods. (

**a**) Arrhenius equation; (

**b**) Kaelble equation; (

**c**) Log-linear equation; (

**d**) Polynomial equation; (

**e**) WLF equation.

**Figure 8.**The master curve of Model 2 fitting results with different shift factor equations. (

**a**) Arrhenius equation; (

**b**) Kaelble equation; (

**c**) Log-linear equation; (

**d**) Polynomial equation; (

**e**) WLF equation.

**Figure 9.**The master curve of Model 3 fitting results with different shift factor equations. (

**a**) Arrhenius equation; (

**b**) Kaelble equation; (

**c**) Log-linear equation; (

**d**) Polynomial equation; (

**e**) WLF equation.

**Figure 10.**The master curve of Model 4 fitting results with different shift factor equations. (

**a**) Arrhenius equation; (

**b**) Kaelble equation; (

**c**) Log-linear equation; (

**d**) Polynomial equation; (

**e**) WLF equation.

**Figure 11.**The master curve of Model 5 fitting results with different shift factor equations. (

**a**) Arrhenius equation; (

**b**) Kaelble equation; (

**c**) Log-linear equation; (

**d**) Polynomial equation; (

**e**) WLF equation.

**Figure 12.**The master curve of different model fitting results with the R

^{2}minimization error minimization method. (

**a**) the model 1 fitting result; (

**b**) the model 2 fitting result; (

**c**) the model 3 fitting result; (

**d**) the model 4 fitting result; (

**e**) the model 5 fitting result.

**Figure 13.**Comparison of predicted and measured phase angle for different shift factor equations under model 1.

**Figure 14.**Comparison of predicted and measured phase angle for different shift factor equations under model 2.

**Figure 15.**Comparison of predicted and measured phase angle for different shift factor equations under model 3.

**Figure 16.**Comparison of predicted and measured phase angle for different shift factor equations under 4.

**Figure 17.**Comparison of predicted and measured phase angle for different shift factor equations under model 5.

**Figure 18.**Comparison of predicted and measured phase angle with different master curve models (Polynomial shift factor equation).

**Figure 19.**Comparison of predicted and measured phase angle with different master curve models (Kaelble shift factor equation).

Equation | Fitting Equation | R^{2} | RSS |
---|---|---|---|

Arrhenius | Y = 0.95353 × X + 0.22017 | 0.96271 | 2.77061 |

Kaelble | Y = 0.90081 × X + 0.61604 | 0.92021 | 5.53531 |

Log-linear | Y = 0.94968 × X + 0.24469 | 0.95881 | 3.04802 |

Polynomial | Y = 0.94916 × X + 0.24175 | 0.96584 | 2.50653 |

WLF | Y = 0.95157 × X + 0.20955 | 0.96376 | 2.67867 |

Equation | Fitting Equation | R^{2} | RSS |
---|---|---|---|

Arrhenius | Y = 0.95353 × X + 0.22017 | 0.96271 | 2.77061 |

Kaelble | Y = 0.90081 × X + 0.61604 | 0.92021 | 5.53531 |

Log-linear | Y = 0.94968 × X + 0.24469 | 0.95881 | 3.04802 |

Polynomial | Y = 0.94916 × X + 0.24175 | 0.96584 | 2.50653 |

WLF | Y = 0.95157 × X + 0.20955 | 0.96376 | 2.67867 |

Equation | Fitting Equation | R^{2} | RSS |
---|---|---|---|

Arrhenius | Y = 0.95329 × X + 0.22095 | 0.96207 | 2.81832 |

Kaelble | Y = 0.9771 × X + 0.11388 | 0.9792 | 1.59565 |

Log-linear | Y = 0.94557 × X + 0.26792 | 0.95847 | 3.04782 |

Polynomial | Y = 0.95999 × X + 0.18318 | 0.96511 | 2.6209 |

WLF | Y = 0.95853 × X + 0.18611 | 0.96325 | 2.7579 |

Equation | Fitting Equation | R^{2} | RSS |
---|---|---|---|

Arrhenius | Y = 0.95517 × X + 0.21258 | 0.96108 | 2.90655 |

Kaelble | Y = 0.98378 × X + 0.07795 | 0.98577 | 1.09921 |

Log-linear | Y = 0.95141 × X + 0.22902 | 0.96121 | 2.87395 |

Polynomial | Y = 0.95584 × X + 0.19652 | 0.97464 | 1.87013 |

WLF | Y = 0.93769 × X + 0.26129 | 0.97083 | 2.07863 |

Equation | Fitting Equation | R^{2} | RSS |
---|---|---|---|

Arrhenius | Y = 0.95939 × X + 0.17299 | 0.95809 | 3.1671 |

Kaelble | Y = 0.98029 × X + 0.09528 | 0.97729 | 1.75707 |

Log-linear | Y = 0.95075 × X + 0.23697 | 0.95163 | 3.61417 |

Polynomial | Y = 0.97043 × X + 0.12692 | 0.96435 | 2.73905 |

WLF | Y = 1.0891 × X + 0.05371 | 0.96013 | 3.87499 |

**Table 6.**The linear fitting results of different master curve models with the Polynomial shift factor equation.

Model | Fitting Equation | R^{2} | RSS |
---|---|---|---|

Model 1 | Y = 0.94916 × X + 0.24175 | 0.96584 | 2.50653 |

Model 2 | Y = 0.96169 × X + 0.17361 | 0.96506 | 2.63534 |

Model 3 | Y = 0.95999 × X + 0.18318 | 0.96511 | 2.6209 |

Model 4 | Y = 0.95584 × X + 0.19652 | 0.97464 | 1.87013 |

Model 5 | Y = 0.97043 × X + 0.12692 | 0.96435 | 2.73905 |

**Table 7.**The linear fitting results of different master curve models with the Kaelble shift factor equation.

Model | Fitting Equation | R^{2} | RSS |
---|---|---|---|

Model 1 | Y = 0.90081 × X + 0.61604 | 0.92021 | 5.53531 |

Model 2 | Y = 0.97563 × X + 0.12184 | 0.9789 | 1.61379 |

Model 3 | Y = 0.9771 × X + 0.11388 | 0.9792 | 1.59565 |

Model 4 | Y = 0.98378 × X + 0.07795 | 0.98577 | 1.09921 |

Model 5 | Y = 0.98029 × X + 0.09528 | 0.97729 | 1.75707 |

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## Share and Cite

**MDPI and ACS Style**

Zhao, H.; Wang, X.; Cui, S.; Jiang, B.; Ma, S.; Zhang, W.; Zhang, P.; Wang, X.; Wei, J.; Liu, S.
Study on the Phase Angle Master Curve of the Polyurethane Mixture with Dense Gradation. *Coatings* **2023**, *13*, 909.
https://doi.org/10.3390/coatings13050909

**AMA Style**

Zhao H, Wang X, Cui S, Jiang B, Ma S, Zhang W, Zhang P, Wang X, Wei J, Liu S.
Study on the Phase Angle Master Curve of the Polyurethane Mixture with Dense Gradation. *Coatings*. 2023; 13(5):909.
https://doi.org/10.3390/coatings13050909

**Chicago/Turabian Style**

Zhao, Haisheng, Xiufen Wang, Shiping Cui, Bin Jiang, Shijie Ma, Wensheng Zhang, Peiyu Zhang, Xiaoyan Wang, Jincheng Wei, and Shan Liu.
2023. "Study on the Phase Angle Master Curve of the Polyurethane Mixture with Dense Gradation" *Coatings* 13, no. 5: 909.
https://doi.org/10.3390/coatings13050909