Modeling of the ComplexModulus of AsphaltMastic with Biochar Filler Based on the Homogenization and Random Aggregate Distribution Methods

School of Water Resources and Electric Power, Qinghai University, Xining 810016, China College of Architecture and Civil Engineering, Beijing University of Technology, Beijing 100124, China Qinghai University-Tsinghua University Sanjiangyuan University Sanjiangyuan Research Institute, Qinghai University, Xinning 810016, China School of Civil Engineering, Harbin Institute of Technology, Harbin 10086, China Civil Engineering Department, Yancheng Institute of Technology, Yancheng 224051, China College of Architecture and Civil Engineering, Xi’an University of Science and Technology, Xi’an 710054, China School of Highway, Chang’an University, Xi’an 710064, China Department of Civil and Environmental Engineering,3e Hong Kong Polytechnic University, Hung Hong, Kowloon, Hong Kong School of Transportation and Logistics, Dalian University of Technology, Dalian 116024, China


Introduction
e disposal of tons of straw has been a serious issue in the agricultural industry in China [1]. Converting the biomass to biofuel provides an outlet for this problem. However, a large amount of biochar waste may be generated in the meantime.
us, it becomes an increasing issue for the utilization of carbon-based waste. One efficient treatment is to use the biochar as the filler material in the asphalt paving industry.
e utilization of carbonaceous materials as asphalt additives can be dated back to the 1960s [2]. By now, a great number of carbonaceous materials have been introduced, such as carbon black [2], coke dust [3], carbon fiber [4], and carbon nanotubes [5]. It has been evidenced that these carbon-based additives can positively improve the performance of asphalt mixture. Recently, with the development of the biofuel industry, biochar, the by-product from the biofuel manufacture process, has been used as the new carbonaceous modifier. Zhao et al. [6,7] investigated the performances of asphalt binders and asphalt mixtures modified by biochar from switchgrass and found that biochar can improve the resistances of rutting, cracking, and moisture-induced damages. Compared to commercial carbon-based additives, the introduction of biochar seemed to perform better. Çeloglu et al. [8] used the biochar from the walnut crust and apricot seed shell to modify the asphalt binders and found that biochar can increase the binder stiffness and thus have a potential to enhance the rutting resistance as well. Kumar et al. [9] evaluated the performance of asphalt binders with the introduction of biochar from Mesua ferrea seed cover waste and found that biochar can also decrease the aging susceptibility. ey also mentioned that the irregular shape of the biochar particle may also contribute to the physicochemical interaction between biochar particle and asphalt binder. Considering these merits, it should be feasible to use it as the alternative of the conventional filler materials.
To evaluate the effects of using biochar as filler material on the asphalt pavement, characterization of the performance of biochar-modified asphalt mastic is a good way to characterize the effects of the biochar. As the asphalt mastic is the composite only of asphalt binder and filler, therefore, it makes it possible to fully evaluate the interaction between filler and asphalt binder by excluding the effects of aggregate. Besides, as asphalt mastic is the binding material and the viscoelastic component of asphalt concrete, it plays a significant role in the performance of asphalt pavement [10,11]. erefore, it is critical to evaluate the viscoelastic property of asphalt mastic for the evaluation of the performance of asphalt pavement. To do that, complex modulus, composed of both dynamic modulus and the phase angle, is a commonly used parameter to depict the viscoelastic performance of asphalt materials [12]. However, although many studies have been conducted to investigate the effect of biochar on asphalt concrete, most of those investigations are based on experiments. To further explore the essence of biochar on strengthening asphalt pavement, modeling methods are highly required.
Currently, the method to predict the complex modulus of asphalt materials can be divided into numerical methods and analytical methods. e numerical methods based on finite element (FE) model and discrete element methods (DEMs) to perform simulation on the developed microstructural models. Many studies have been conducted to predict the complex modulus of asphalt materials based on the DEM and FE model [13][14][15][16]. However, one of the drawbacks of the numerical methods is the high cost for time and computation, which greatly limited their application. Different from numerical methods, the analysis is based on micromechanical models. In this method, the micromechanical models can predict the mechanical performance of heterogeneous materials based on the volumes of each composition, which provides a more efficient manner to analyze the interaction between particles and matrix. Since the 1920s, many micromechanical models, such as the dilute model (DM), the Mori-Tanaka model (MTM), the selfconsistent model (SCM), and the generalized self-consistent model (GSCM) were introduced to predict the complex modulus of asphalt materials [17][18][19][20][21][22][23]. However, these models only can predict the effective modulus but phase angle for viscoelastic asphalt materials [24][25][26]. Besides, because these models generally cannot consider the effects of physiochemical reinforcement and particle interaction, the predictions always underestimate the modulus of asphalt materials [10,[26][27][28][29]. erefore, to accurately and efficiently predict the complex modulus and time-dependent characteristic of biochar filler asphalt mastic, a more rational micromechanical model is keenly desired. erefore, this study aimed at predicting the viscoelastic performance of biochar-modified asphalt mastic based on computational methods. To achieve these objectives, the following research tasks have been conducted: (i) Measuring the complex modulus of asphalt mastics through dynamic shear rheometer (DSR) tests (ii) Developing a micromechanical model to characterize the viscoelastic performance, especially for dynamic modulus, of biochar asphalt mastic by means of homogenization method (iii) Predicting and proving the effect of biochar on asphalt mastics based on the FE method by random aggregate generation method

Material Preparation.
e biochar used in this study was prepared from rice straw. e process of biochar preparation is shown in Figure 1. e dry rice straw was completely burned in an oven at 500°C for an hour. e residue ash was then collected and ground in a high-speed mixer for 30 seconds to obtain homogeneous biochar powder used as the mineral filler in this study.
A scanning electron microscope (SEM), FEI Quanta 250 FE-SEM, was used to examine the microstructure differences between conventional mineral filler, i.e., granite filler, and biochar filler. Figure 2 presents the SEM images of biochar and granite filler. It can be observed that the granite particles have a smooth fractured surface with an irregular shape, while the biochar particles are characterized with special porous structures with an uneven surface. Research also showed that this special structure can improve the antiaging properties of asphalt materials [30].
To prepare the asphalt mastics, asphalt binder with a penetration grade of 60/70 (Pen 60/70) was designed. e proportion of asphalt mastic was designed based on the stone mastic asphalt (SMA) due to the high asphalt binder and filler contents. In this study, SMA10, a commonly used asphalt mixture with a maximum aggregate size of 10 mm, was selected [31]. In this mixture, the mineral filler ratio in the gradation and the binder content in the mixture are 9% and 6%, respectively, which corresponds to a mass ratio of 58.5 : 41.5 of mineral filler to asphalt binder. e granite filler in the asphalt mastic was partially substituted by biochar filler with a volume fraction of 0%, 40%, 80%, and 100%. e physical properties of asphalt binder, granite filler, and biochar filler are presented in Table 1.
e mixture proportions designed in volume and mass compositions in each asphalt mastic are presented in Table 2. e biochar and granite filler materials were mixed with the hot asphalt binder at 150°C for 3 minutes until the mixture was prepared homogeneously.

Laboratory Tests.
To characterize the viscoelastic properties of asphalt mastic, frequency sweep tests were conducted using an Anton Paar MCR 702 direct shear rheometer (DSR). e two standardized DSR configurations, i.e., 8 mm and 25 mm diameter plates for asphalt binder and asphalt mastic, with applied sinusoidal strain were used for the frequency sweep test of asphalt mastic as shown in Figure 3. Following AASHTO-T315 [32], an 8 mm plate was used for the test at a temperature lower than 25°C and a 25 mm plate for temperature higher than 25°C. Tests were performed at the frequencies from 100 Hz to 0.1 Hz at the temperature from 0°C to 60°C with a 10°C interval. All the tests were conducted at the strain level where specimens behave linearity.

Master Curve Construction
e master curves were generated to present the complex moduli obtained from different temperatures to a given temperature based on the time-temperature superposition principle. e Williams-Landel-Ferry (WLF) formula was applied to shift the complex moduli to the reference temperature of 25°C and then the Christensen-Anderson (CA) model [33,34] was further used to fit the shifted data. e WLF formula and the CA model are shown in the following equations: where a T is the shift factor, T is the test temperature, T 0 is the reference temperature, and C 1 and C 2 are constants.
where G * (ω) is complex shear modulus, G g is glass modulus assumed to be 1 GPa, ω r is the reduced frequency at the defining temperature (rad/s), ω c is crossover frequency at the defining temperature (rad/s), ω is the frequency (rad/s), and R is the rheological index.  Advances in Materials Science and Engineering Table 3 presents the shift factors of WLF and parameters of the CA model of fitted master curves. Figure 4 presents a typical master curve construction of mastic (0%). It can be observed that the CA model can fit both the dynamic modulus and phase angle curves well with high consistency. Figure 5 illustrates the micromechanical model used in this paper. Figure 5(a) is a commonly used GSCM. is model is composed of three layers including the asphalt binder, aggregate, and a homogenized material layer [10]. One of the drawbacks of this model is that it cannot take the coating layer on the surface of the filler particle into the model. However, this coating layer can significantly affect the mechanical performance of asphalt mastic [30]. erefore, this model generally underestimates the modulus of asphalt mixes. To solve this problem, a four-phase micromechanical model including the coating layer was therefore proposed to predict the complex modulus of asphalt mastic named modified generalized self-consistent model (MGSCM). is micromechanical model was proposed by Doghri [35,36], which is composed of four different phases including effective homogeneity matrix, real matrix, coating layer, and inclusion compared with the traditional GSCM model as illustrated in Figure 5. Peng et al. [23] have developed this method for investigating the upper limit and lower limit of dynamic modulus of the asphalt concrete. In this MGSCM model, the inclusions, mineral filler, and biochar filler were assumed as elastic components with an elastic modulus and Poisson's ratio of 60 GPa and 0.15, respectively. To describe the viscoelastic properties of asphalt binder, the complex modulus master curves of asphalt binder were expressed as the Prony series model as presented in Table 4.

Modeling
e Prony series parameters were determined by minimizing the storage modulus and loss modulus [37]. Figure 6 displays the master curves of asphalt binder expressed by the CA model and Prony series model. It is worth noting that the Prony series model can fit the CA model master curve with high accuracy within a wide frequency range from 10 −4 Hz to 10 4 Hz. e process to determine the coating layer's properties may follow the flowchart presented in Figure 7 due to the complexity of physical measurement on the coating layer. e coating layer was assumed as an elastic component due to the relatively high stiffness modulus compared with asphalt binder. A Poisson's ratio of 0.15 was assumed because of its marginal effect on complex modulus perdition. e thickness of the coating layer was then determined when the complex modulus at low frequency reached the minimum relative error while this thickness was then used to determine the elastic modulus of the coating layer when the complex modulus at high frequency reached the minimum relative error. e complex modulus master curves of the asphalt mastic (0%) and asphalt mastic (100%) were predicted. Both models including GSCM and MGSCM were used to predict

Advances in Materials Science and Engineering
Homogenized material: C r Asphalt binder: C 0 Aggregate: Homogenized material: C r Asphalt binder: C 0 Aggregate: C 1 Coating layer:   Advances in Materials Science and Engineering the complex modulus of asphalt mastics. e MSC Digimat software system was used to conduct the modeling. e simulation process can be completed in seconds to obtain the complex moduli within the whole frequency range from 10 −4 Hz to 10 4 Hz. Figure 8 presents the complex modulus curves representing both dynamic modulus and phase angle of asphalt mastics with different biochar volume fractions. It can be observed that with the increasing volume substitution of biochar filler, the dynamic moduli of mastics increase but the phase angle decreases regarding frequency.

Complex Moduli.
is means that the biochar can stiffen the asphalt mastics. It is expected that the porous structure of biochar enhances the absorption of asphalt binder and then further increases the stiffness of asphalt mastics. To quantify the effect of biochar on the dynamic moduli and phase angles, their relative errors at low, high, and the whole frequency ranges are listed in Table 5. It is worth noting that a significant increase in dynamic modulus can be observed with the biochar substitutions from 0% to 100% and a 50% modulus increase can be achieved with a 100% biochar substitution. Besides, compared the differences in different frequency ranges, it can be found that the increase of dynamic modulus is substantially higher than the dynamic modulus at low frequency, and with the increase of substitution, this tendency becomes more noticeable, which means that the stiffening effect of biochar is strengthened with the biochar increase. Based on the performance of biochar mastic, it can be concluded that biochar filler can exhibit low deformation at long-term loading and high temperature, which indicates a better rutting resistance of asphalt pavement constructed by biochar filler than the conventional mineral filler.

Modeling.
e complex moduli of asphalt mastic (0%) and mastic (100%) were predicted by both GSCM and MGSCM models. e experimental and the predicted master curves for both mastics are displayed in Figures 9 and  10, and the corresponding errors are presented in Table 6. It can be seen that both models can well capture the trend of complex modulus for both mastics as indicated by high R 2 values of around 1. However, the GSCM substantially underestimates the dynamic modulus for both granite and biochar filler asphalt mastics with relative errors of 32% and 56% for mastic (0%) and mastic (100%), respectively. It is believed that the existence of the coating layer in the real mastic increases the experimental modulus of mastics. As mentioned before, due to the physiochemical reinforcement between filler particles and asphalt binder, the modulus of the coating layer will increase significantly and then further increase the overall modulus of asphalt mastic. However, this factor cannot be considered in the GSCM model. erefore, to give a correct prediction, it is required to take the coating layer into the micromechanical model. To this end, the coating layer was introduced into the MGSCM model. From Figures 9 and 10, we can find that, with the introduction of the coating layer, significant improvement is achieved for the predicted master curves for both mastics and only the relative errors for both dynamic modulus and phase angle predictions on both mastics are narrowed down to less than 5%. erefore, the results demonstrate that the MGSCM considering the coating layer can give an accurate prediction for the viscoelastic performance of asphalt mastics. e properties of the coating layer are presented in Table 7. In the model, a thicker and high modulus coating layer was applied to mastic (100%) with biochar filler than the conventional filler mastic (0%) with granite filler. is is also consistent with the SEM test result, which indicates the biochar particles are characterized with special porous structures with an uneven surface.
is porous structure indicates the biochar filler can have a thicker coating layer than the conventional granite filler with a smooth surface.

Dynamic Modulus Simulation based on the FE Method.
e viscoelastic property of asphalt materials is derived from that of asphalt binder. e complex modulus of the composites can be directly obtained from laboratory tests [25]. Besides, from the continuum mechanics aspect, asphalt mixtures were regarded as a representative volume element (RVE) composed of asphalt binder, aggregate particles, and air voids.
us, the complex modulus is transferred to a calculable parameter of the resultant function of their mechanical properties, volume contents, and space location [19].
As the aggregate particle is an elastic material, the viscoelastic property of asphalt mastic is determined by asphalt matrix. For FE simulation, the linear transformation is required for the viscoelastic matrix [26]. Prony series model was adopted for linear transformation. e deviation is as follows: firstly, the sinusoidal external strain load is expressed as By replacing ε(t) with progressive relaxation modulus, the following stress expression can be obtained: where e(t) � (E(t)/E ∞ ) − 1 is the relaxation formula after regulation and τ is the relaxation time.
erefore, the complex modulus can be further expressed as follows: where E s (ω) is storage modulus and E l (ω) is loss modulus. By rewriting equation (6), the following equation is obtained: erefore, the complex modulus is expressed as the progressive relaxation moduli at different frequencies, which can be used as a viscoelastic constitutive model in the FE simulation [38].

FE Results.
To prove the assumption of the coating layer in the MGSCM, FE simulation was also conducted. e simulations based on mastic (0%) and mastic (100%) were performed. In the FE simulation, the complex moduli of asphalt mastic within the frequency range from 10 −2 Hz to        10 3 Hz were predicted based on the steady-state dynamic (SSD) method, which is an efficient and accurate method to predict the complex modulus of asphalt materials [39,40].
To consider the coating layer effect, the microstructural models with a coating layer and without a coating layer were developed based on a random aggregate distribution algorithm. e developed models for the two mastics were presented in Figure 11. e coating layer thicknesses of 1 μm and 1.35 μm were assigned to the mastic (0%) and mastic (100%), respectively. e corresponding predicted complex  modulus master curves are presented in Figure 12. It can be observed that the FE simulation based on the microstructural model without a coating layer underestimates the dynamic shear moduli within the whole frequency range for both mastics, but significant improvement was achieved for both mastics after a thin coating layer was added. is result verifies the existence of the coating layer. Besides, since the coating layer of mastic (100%) is thicker than that of mastic (0%), it also proves the assumption that the biochar filler particle is attributed to a thicker coating layer than the granite filler particle in the MGSCM.

Summary and Conclusions
In this study, biochar converted from straw was used as an alternative mineral filler in asphalt mastic. e complex moduli of mastics with different biochar contents were measured by laboratory tests firstly.
en, the complex moduli were predicted by the proposed micromechanical model based on homogeneous theory. Further, the complex moduli of asphalt mastics were predicted and analyzed by FE simulations based on microstructure models. Based on the outcome of this study, the following conclusions can be drawn: (i) e modified MGSCM by considering the coating layer can effectively predict the complex modulus of biochar-modified asphalt mastic (ii) e predictions based on FE simulation by random aggregate generation method have a good correlation with the predictions based on MGSCM and experimental results, which verifies the accuracy of MGSCM in the complex modulus prediction of asphalt mastic, and this method can be further extended to other areas (iii)Biochar filler can improve the modulus of mastic by developing a thick coating layer to improve the modulus of mastics, which would contribute to the rutting performance of asphalt pavement

Data Availability
All the data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.