Viscosity of Asphalt Binder through Equilibrium and Non-Equilibrium Molecular Dynamics Simulations

Abstract

Viscosity is a curial indicator for evaluating asphalt performance, representing its ability to resist deformation under external forces. The Green–Kubo integral in equilibrium molecular dynamics simulations and the Muller-Plathe algorithm in reverse non-equilibrium molecular dynamics simulations were used to calculate the asphalt viscosity. Meanwhile, the key parameters of both methods were rationalized. The results show that in equilibrium calculations, using a 1/t weighting for the viscosity integral curve results in a well-fitted curve that closely matches the original data. The isotropy of the asphalt model improves for atomic counts exceeding 260,000, rendering viscosity calculations more reasonable. When the viscosity did not converge, it increased linearly with the number of atoms. In non-equilibrium calculations, the number of region divisions had almost no effect on the viscosity value. A momentum exchange period of 20 timesteps exhibits a favorable linear trend in velocity gradients, and an ideal momentum exchange period was found to be between 10 and 20 timesteps. As the model size increased, the linear relationship with the shear rate became more pronounced, and the isotropy of the asphalt system improved. Using an orthogonal simulation box with a side length of 75 Å effectively meets the computational requirements.

1. Introduction

In road engineering, asphalt pavement, with its good smoothness and comfort, has become the primary choice for highways and urban roads. Viscosity is an important indicator to evaluate asphalt performance, reflecting the resistance experienced by asphalt molecules during flow and characterizing the ability to resist deformation under external forces [1]. For example, viscosity has been used to evaluate the degree of aging of asphalt and optimize the dosage of rejuvenators during asphalt regeneration [2,3]. Currently, many techniques are being used to explore the performance of asphalt: Atomic Force Microscopy (AFM) can study the morphology of asphalt at the microscale, and Scanning Electron Microscopy (SEM) is used to study the interfacial properties of modifiers and pure asphalt [4,5]. Although these methods can partially reveal the microstructure of asphalt, they often only stay at the stage of observing phenomena or providing qualitative descriptions. Therefore, it is often challenging to clearly elucidate the micro-mechanisms behind macroscopic phenomena [6].

Molecular Dynamics (MD) simulations have been widely applied to study the viscosity of asphalt to explore the underlying microscopic mechanisms driving macroscopic phenomena [7,8]. MD simulation involves molecular modeling, capturing molecular trajectories and outputting thermodynamic parameters from the microscale to the nanoscale, effectively complementing the limitations of macroscopic experiments. It can assess the compatibility between asphalt and additives and convert energy, volume, density, temperature, and pressure parameters into various performance parameters, such as resistance to rutting, aging, and many other fundamental properties [9,10]. Additionally, MD saves resources, significantly reducing research costs and greatly improving efficiency due to its short simulation periods. However, because of the extremely complex composition of asphalt molecules, some experimental parameters may not be suitable for MD simulations, leading to significant errors. Accurate models are essential to ensure the basis of simulation results, requiring long-term and extensive data collection [11,12].

Currently, at least four methods are used to calculate viscosity in MD simulations [13]. The first method involves using Green–Kubo (GK) integration in equilibrium molecular dynamics (EMD) simulations, which relates the fluctuations of the off-diagonal elements of the stress tensor to viscosity [14]. Viscosity can be calculated through the integration of the velocity autocorrelation function, which is associated with the off-diagonal elements of the stress tensor. These elements describe pressure differences in different directions within the system, and their fluctuations reflect the impact of intermolecular interactions on the viscosity of the fluid. While this method allows for a direct investigation of the effects of finite simulation time and box size on viscosity, its low signal–to–noise ratio often requires a large number of simulations to obtain better statistical data [15,16,17]. The second method is non-equilibrium molecular dynamics (NEMD) simulation, which induces non-equilibrium motion among molecules by introducing a gradient field, such as velocity gradient or shear strain, into the system. NEMD can directly measure the relationship between stress and strain to determine the viscosity of the system [18,19,20]. It can detect signals significantly larger than the noise, making it a direct and effective approach for computing viscosity. However, substantial fluctuations in the pressure tensor require longer simulation times to measure viscosity at low shear rates. In the third method, the reverse non-equilibrium molecular dynamics (rNEMD) based on the Müller-Plathe algorithm is used to estimate the shear viscosity of asphalt [21]. The reason it is called “reverse” is that, compared to traditional non-equilibrium simulations, the rNEMD method applies momentum flux to obtain velocity gradients or shear rates. The advantage of this method is that it involves momentum exchange of particles, fast convergence of shear rates, and it can improve computational speed [16,22]. In this method, the shear viscosity is the proportionality constant between the shear field (shear rate) and the transverse linear momentum flux. The fourth method involves using one or more moving walls to shear the fluid, but this method is rarely used and has almost no related simulation work.

In equilibrium, there is no need for complex temperature control methods, the microstructure is stable, and the data sampling range is wide, which can avoid the influence of high shear rates caused by the molecular dynamics simulation scale on the microstructure and viscosity of the system. At the same time, there is no shear rate in the equilibrium state, so it is not necessary to calculate the viscosity and extrapolation at multiple shear rates to obtain the zero-shear viscosity of asphalt. The non-equilibrium method can investigate the changes in the microstructure of asphalt under shear action and qualitatively explore the influence of shear action on various properties of asphalt.

Based on the above methods, researchers have conducted extensive simulations on asphalt viscosity. Li et al. [7] used EMD to calculate the zero-shear viscosity of three asphalt models from the Strategic Highway Research Program (SHRP), finding that the zero-shear viscosities of all asphalts were within the same order of magnitude as experimental values. Ding et al. [8] studied the relationship between asphalt chemical composition and shear viscosity through NEMD. The results showed that asphalt exhibits shear thinning behavior with increasing strain rate, and viscosity increases with molecular weight, aromatic carbon, and heteroatom percentages in asphalt. You et al. [23] analyzed the shear viscosity of the AAA-1 asphalt model using rNEMD and found that the calculated viscosities at 408.15 K and 438.15 K were in close agreement with laboratory results. It is worth noting that these methods involved many parameters that play crucial roles in viscosity calculation. However, few studies have specifically discussed these parameters, resulting in significant uncertainty in calculation results and misleading subsequent researchers.

This work explored the crucial parameters in EMD and rNEMD simulations, including the integration method, the fitting function and system sizes in EMD, as well as the number of region divisions, momentum exchange periods, and system sizes in rNEMD. The results have complementary and reference significance for using molecular dynamics simulations to calculate asphalt viscosity.

2. Molecular Dynamics Simulation

2.1. Equilibrium Viscosity Calculation

2.1.1. Green–Kubo Integral

GK formula is the core theory for calculating viscosity in EMD simulations. This formula integrates the autocorrelation function of the off-diagonal elements of the stress tensor to calculate the viscosity of the system in equilibrium. The specific expression is as follows:

$$\eta ={\displaystyle \frac{V}{k_{B}T}}\underset{0}{\overset{\infty }{{\displaystyle \int }}}\langle P_{ij}(\tau +t)P_{ij}(t)\rangle dt$$

(1)

where $V$ represents the system volume, $k_B$ is the Boltzmann constant, $T$ is the temperature, $\tau$ is the sampling interval, $t$ is the integration time, $P_{ij}$ denotes the off-diagonal elements of the pressure tensor, and $\langle P_{ij}(\tau )P_{ij}(0)\rangle$ denotes the ensemble average [24]. In theory, the autocorrelation function decays to zero over a long time (Figure 1), and the integration in Equation (1) reaches a constant value, corresponding to the calculated viscosity.

2.1.2. Fitting Function

Due to the limited simulation time, it is impossible to obtain viscosity values at infinity. Therefore, it is necessary to predict the trend of the integral curve through function fitting. Particularly for high-viscosity fluids, the convergent value of the integral result cannot be directly observed within a finite simulation time. By using function fitting, the numerical fluctuations caused by the scarcity of samples in the long-time autocorrelation function can be mitigated. Carlos [25], Cheng [26], and Zhang et al. [27] have respectively used a double exponential function of the form given in Equation (2) to fit the integrated results, allowing for the estimation of extrapolated viscosity values and achieving good statistical accuracy.

$$\eta (t)=A\alpha {\tau }_1(1-e^{-t/{\tau }_1})+A(1-\alpha ){\tau }_2(1-e^{-t/{\tau }_2})$$

(2)

where A and α (0 < α < 1) are empirical fitting parameters, τ₁ and τ₂ are two characteristic decay times (usually differing by an order of magnitude) [25,28]. The double exponential function can directly fit the integrated curve, which is applicable for liquid metals, small molecular liquids, and low-molecular-weight polymer melts. For complex systems with large molecular weights, this fitting method can lead to significant mismatches in the early stages of the curve. Despite these developments, reliably calculating viscosity using GK integration remains a challenging task. Firstly, the methods used in the literature to estimate the integral limit are arbitrary and depend on the data analysis program, potentially yielding different viscosities for the same system. Secondly, these methods do not distinctly determine when the viscosity convergent value is obtained. This issue is more severe for highly viscous asphalt systems because statistical data are often poor within feasible simulation times. Building upon this, Nie et al. [29] proposed a new analytical model to fit the data, which includes a nonlinear decay factor β, expressed as follows:

$$\eta (t)={\eta }_0(1-e^{-{({\displaystyle \frac{t}{\tau }})}^{\beta }})$$

(3)

where ${\eta }_0$ is the fitted viscosity (the convergence plateau of the integral), $\tau$ is the decay time, and $t$ is the integration time. Typically, the sampling frequency, numerical integration limits, number of repeated simulations, fitting functions, and other parameters used to calculate the viscosity for different materials vary. Therefore, selecting appropriate parameters for molecular simulations and data processing is essential to obtain reliable theoretical predictions of viscosity.

2.1.3. Viscosity Calculation Steps

During the simulation, intense pressure fluctuations and the accumulation of computational errors make it difficult to determine the plateau value for viscosity estimation from a single calculation, especially for high-viscosity polymers. To obtain reliable results, the viscosity of asphalt is calculated according to the following steps, and the flowchart can be seen in Figure 2.

(1)

Under 408.15 K and an NVT ensemble, generating N independent trajectories, which is output using the thermo command in LAMMPS and reflects the changes in atomic positions.

(2)

Calculate the asphalt viscosity curves using the GK integration over a total time of 400 ps.

(3)

Compute the average and standard deviation $\sigma (t)$ of the viscosity integral curves for N trajectories.

$$\sigma (t)=\sqrt{{\displaystyle \frac{1}{N-1}}{\displaystyle \sum _{i=1}^N{(\eta {(t)}_i-\langle \eta (t)\rangle )}^2}}$$

(4)

(4)

Fit the standard deviation to a power-law function, using its reciprocal as the fitting weight.

$$\sigma (t)=Q{t}^b$$

(5)

(5)

Utilize Formula (3) to fit the weighted integration and obtain the predicted viscosity value.

(6)

Increase the number of trajectories N and repeat steps (1) to (5) until the viscosity calculated in step (5) falls within the error range compared to the previous iteration.

2.2. Reverse Non-Equilibrium Viscosity Calculation

The rNEMD simulation involves shear strain, which can be directly calculated by defining shear viscosity while observing the effect of shear action on the microstructure of materials [30]. In this method, the shear viscosity is a proportionality constant between the shear rate and the transverse linear momentum flux [21]:

$$j_z(p_x)=-\eta {\displaystyle \frac{\partial v_x}{\partial z}}$$

(6)

where $j_z(p_x)$ is the momentum flux, ${\displaystyle \frac{\partial v_x}{\partial z}}$ is the velocity gradient (the fluid velocity relative to the x-component in the z-direction), and $\eta$ is the shear viscosity. The main features of the algorithm are as follows [14,21,22]:

(1)

The periodic box is divided into N (even) regions along the z-direction, as shown in Figure 3a. Atoms in region 1 (the bottommost region) are propelled in the positive x-direction, while atoms in M = N/2 + 1 are propelled in the negative x-direction.

(2)

Atoms in region 1 have the maximum x-component momentum in the negative x-direction, and atoms in region M have the maximum x-component momentum in the positive x-direction, with these two atoms having the same mass.

(3)

Exchange the x-component of velocities between two corresponding atoms. Since they have the same mass, the exchanged amount is equal to the x-component of momentum.

The black arrows in Figure 3a represent the velocity gradient and direction caused by momentum exchange at both ends. When the periodic transfer of momentum is repeated, the total exchanged momentum after a simulation time t is $P_{tot}={\displaystyle {\sum }_{exch}(p_{x,M}-p_{x,1})}$. Therefore, the momentum flux can be expressed as follows:

$$j_z(p_x)=-{\displaystyle \frac{P_{tot}}{2t{L}_x{L}_y}}$$

(7)

where $p_{x,M}$ and $p_{x,1}$ are the x-components of the exchanged momenta in regions 1 and M, respectively. As shown in Figure 3b, $L_x$ and $L_y$ are the lengths of the simulation box in the x and y directions, $A=L_x\times L_y$ is the area through which momentum is being exchanged, and 2 in the denominator arises from the periodicity of momentum exchange. When non-physical momentum transfer occurs between the first and center slabs, the system balances this momentum flow by physical friction, causing it to flow in the opposite direction [22]. Once a steady state is reached, the rate of non-physical momentum exchange is counteracted by the rate at which momentum flows back through friction, resulting in a uniform and approximately linear velocity distribution in the z-direction. This balanced state ensures the stable transfer of momentum within the system [31].

2.3. Modeling and Settings

To gain a clear observation of the microscale behavior of asphalt components, the simplified asphalt molecular models were adopted as depicted in Figure 4, which were previously utilized by Nie [32] to investigate asphalt properties, effectively reproducing the thermodynamic characteristics. The model comprises four representative molecules: asphaltene (C₅₃H₅₅NOS), resin (C₁₈H₁₀S₂), aromatics (C₁₂H₁₂), and saturate (C₂₂H₄₆) [33]. The elemental mass percentages are C 83.67%, H 10.42%, O 0.54%, N 0.48%, and S 4.81%. Other information is shown in

Table 1. Overall components of the asphalt model.

Asphalt Molecule	Chemical Formula	Mass (g/mol)	Mass Ratio (%)
Asphaltene	C53H55NOS	754.04	26
Aromatics	C12H12	156.22	8
Resin	C18H10S2	290.38	17
Saturate	C22H46	310.59	49

. All models are established using the Amorphous Cell module in the Materials Studio software 2019, where molecules of each component are randomly dispersed within a cubic box with periodic boundary conditions in all directions.

The CVFF force field was employed to describe intermolecular interactions, which can be used for the study of organic molecular systems and has been successfully applied to the molecular modeling of asphalt [29,34]. The Particle–Particle–Particle–Mesh (PPPM) method was used to compute long-range Coulomb forces between molecules [35]. The van der Waals force cutoff distance was set to 12.5 Å, which not only saves computational time for long-range interactions between molecules but also eliminates interactions between atoms and the periodic boundary box while ensuring the accuracy of interaction energy calculations. Temperature control was achieved using a velocity-rescaling thermostat [34]. The timestep was 0.5 fs. Initially, the asphalt model underwent energy minimization using the steepest descent algorithm. Subsequently, it was relaxed for 2 ns in NPT and NVT ensembles. Then, viscosity was calculated using the NVT ensemble. Since the model did not involve bond breaking and formation, all intermolecular interactions were of a physical nature.

3. Results and Discussions

3.1. Density

For the established asphalt molecular model, it is essential to initially compare it with experimental results to ensure the reliability of the outcomes. At present, in MD simulations, parameters used to validate asphalt models primarily include density [36], solubility parameters [37,38], atomic radial distribution functions, and glass transition temperatures [39]. Among these, density is the most common indicator to assess the model reasonableness, which can be obtained through relaxation in the NPT ensemble [10]. Generally, it is considered reasonable if the error between simulated and measured asphalt density falls within a 5% range, indicating the rational construction of the model [40]. The initial pristine asphalt density was set at 1.00 g/cm³, and the assembled model underwent structural optimization through an energy minimization process. At a temperature of 408.15 K, the model was sequentially relaxed for 500 ps and 1000 ps in the NVT ensemble and then in the NPT ensemble to eliminate initial configuration internal stresses [41,42]. Figure 5 illustrates the change in density over time between 500 ps and 1500 ps, and it can be clearly seen that the asphalt density gradually stabilizes during the relaxation process, ultimately reaching 0.891 g/cm³.

The relationship between asphalt density and temperature can be represented as follows:

$$\rho =kT+C$$

(8)

where ρ is the asphalt density, T is the temperature, k is the slope, and C is the intercept. To validate whether the model conforms to a linear relationship, 10 different temperatures ranging from 273.15 K to 463.15 K were selected, and the corresponding densities were obtained through relaxation; the results are shown in Figure 6. Zhang [43] determined the density of AH-70 asphalt using standard test methods for density of semi-sold and solid. Lv [44] and Li [45] respectively calculated the densities of three-dimensional periodic asphalt and AAA-1 model asphalt systems using molecular dynamics simulations. The discrete data points were extrapolated to the entire temperature range through linear fitting, and the fitted regression equations and correlation coefficients are shown in

Table 2. Equation and correlation coefficient between asphalt density and temperature.

Asphalt System	Regression Equation between Density and Temperature	Correlation Coefficient
This Work	ρ = −0.0007 T + 1.157	0.9938
Exp. AH-70 [43]	ρ = −0.0006 T + 1.022	0.9968
Sim. Lv [44]	ρ = −0.0003 T + 1.068	0.9881
Sim. Li [45]	ρ = −0.0006 T + 1.084	0.9895

. It can be observed that the correlation coefficients of density with temperature for all systems were greater than 0.988, indicating a strong linear correlation between asphalt model density and the corresponding temperature.

Based on the above results, the effect of temperature on asphalt density can be described by the slope of the regression equation. When the slope of the regression equation is negative, it indicates that asphalt density decreases with increasing temperature. This phenomenon can be explained from a molecular thermodynamic perspective, including changes in molecular motion speed and range of activity. An increase in temperature intensifies the thermal motion of asphalt molecules, increasing their kinetic energy and expanding their activity space, thereby reducing their density.

From the comparison of simulation and experimental results, it can be seen that the actual asphalt ratio calculation model has a higher density, which has been verified in many studies. Zheng [37] calculated the density of asphalt using MD at a temperature of 273.15 K, and the results showed that the simulated density of asphalt was slightly lower than the measured density, with a maximum error of 5.8%, which is basically consistent with the real asphalt system. Xu et al. [38] calculated the density values of two different asphalt models at different temperatures, and the results showed that the highest density of asphalt at 283.15 K was 0.92 g/cm³, and the density decreased to 0.85~0.88 g/cm³ at 333.15 K. The reason why the density of the asphalt molecular model was lower than that of the actual material is that the model is relatively simple and established under ideal conditions, with more carbon and hydrogen elements and fewer heteroatoms such as sulfur and nitrogen. In this work, the average error between the calculated asphalt density and the experimental results was 4.93%, indicating that the constructed asphalt model is reasonable.

3.2. Equilibrium Viscosity Calculation

3.2.1. Integral Method

Viscosity is a collective property that cannot be improved in statistical accuracy by averaging all particles in the system. Therefore, when using the GK method to calculate viscosity, obtaining reliable data typically requires longer integral trajectories. However, in a single simulation, the low sample count and large fluctuations in the long-time autocorrelation function of pressure result in inconsistent integral plateaus for the same asphalt model in multiple calculations. Figure 7 provides an example of using the GK integral in viscosity calculation, generating five independent curves at 408.15 K for a duration of 400 ps. It can be observed that the five integral curves deviate from each other and fail to converge to the same value, with discrepancies becoming more pronounced at longer times. Furthermore, even if a stable region of the curve is identified, there is no unified standard for determining the viscosity value reasonably. Currently, researchers have different methods to address this issue. Zhang et al. [15] extracted a value from the first plateau of the integral curve after system relaxation. Li and Greenfield [46] studied asphalt viscosity using the GK method and took the average value of a specific region in the integral as an estimate. Van-Oanh et al. [47] adopted the method of averaging multiple calculations to smooth the integral curve but did not specify how to determine the estimated value. It should be noted that these methods may yield different viscosities for the same integral curve. Thus, a reasonable approach is to run multiple independent trajectories in parallel and average the integrals to reduce errors from a single simulation, as shown in Figure 7. Considering the availability of large-scale parallel computing servers (LAMMPS), running multiple independent trajectories not only saves computational time but is also easily achievable.

3.2.2. Fitting Function

For all initial configuration models, 30 independent calculations were performed, generating 30 integral trajectories, each with a length of 400 ps. Subsequently, the average of multiple trajectories was computed, and the viscosity was calculated using the Formula (1). The average and standard deviations at 408.15 K are shown in Figure 8a. As expected, the long-term accumulation of random noise in the autocorrelation function causes the standard deviation to increase with simulation time. To ensure the validity of the results, the average integral was fitted with a double exponential function, as shown in Formula (3), and extrapolated to the limit time. Here, the weight function $t^b$ was used to fit the average integral, where b was obtained by fitting the standard deviation using Formula (5). Through calculation, within the finite integral time, the parameter b fitted was close to −1. Therefore, $1/t$ was adopted for the viscosity curve in the simulations. Figure 8b displays the residuals of the average and fitted line, indicating a good fit and a high accuracy of the fitting function.

3.2.3. Atomic Number

Restricted by limited computational resources, researchers commonly use smaller volume boxes for molecular dynamics simulations, which, however, is not conducive to thorough mixing of asphalt molecules [48]. It is generally believed that the more atoms, the more accurate the viscosity value obtained, while larger models often demand an incredibly high computational capacity. In order to compare the effect of the atomic number on the viscosity in MD simulations, this study established six models, as shown in

Table 3. Details and viscosity values of asphalt models.

Model	Side Length (Å)	Density (g/cm3)	Number of Molecules				Number of Atoms	Viscosity (cp)
			Asphaltene	Resin	Aromatic	Saturate
M-1	25	0.858	3	5	5	15	1183	8.52
M-2	50	0.895	26	38	44	120	9898	12.52
M-3	75	0.891	86	129	148	406	33,224	16.52
M-4	100	0.893	204	306	350	962	78,712	35.36
M-5	125	0.893	399	598	683	1879	153,769	43.43
M-6	150	0.893	689	1033	1181	3247	265,705	80.94

. Figure 9 presents the relationship between asphalt atom count and viscosity, which reveals a strong linearity (Pearson correlation coefficient of 0.9830).

Visualizing asphalt molecules, it can be observed that an increase in the number of atoms does not lead to significant changes in the microstructure. As per Figure 10a, even the M-1 model with very few molecules also fully complies with colloid theory [49], where asphaltene is at the center, surrounded by adsorbed resins and aromatics, and saturates randomly fill the interstices as solvents. Additionally, resins and aromatics can form stacked structures around the asphaltene through π-π conjugation, thus forming larger aggregates. In Figure 10b, in the M-2 model with more molecules, the π-π conjugation effect is similarly evident, and the increase in molecular number leads to larger and more dispersed stacks.

According to the results of the mean square displacement (MSD) components in Figure 11a, molecules in the M-1 exhibit significant fluctuations in displacement. Meanwhile, Figure 11b shows that the difference in displacement of molecules in the three directions decreases in the larger M-6 model, indicating enhanced isotropy of the molecular system. The results suggest that the viscosity of asphalt calculated using the M-6 model is smoother, with less directional variation. Li et al. [45] have suggested using models with an atom count between 260,000 and 375,000 to analyze asphalt viscosity, which is consistent with the results of this study.

3.3. Reverse Non-Equilibrium Viscosity Calculation

When using the Muller-Plathe algorithm to calculate non-equilibrium viscosity, it is necessary to investigate the influence of the number of region divisions in the z-axis direction, momentum exchange period, atomic number, and other factors on results. This section conducts a research analysis on the three parameters mentioned above and aims to determine the set parameters for asphalt viscosity calculation.

3.3.1. Number of Regional Divisions

The rNEMD method divides the simulation system into multiple (even) subregions in the z-direction, so determining the number of subregions is the first step in calculating viscosity. Here, the simulation box was divided into 10, 20, 30, 40, 50, and 60 different numbers of regions. Figure 12a illustrates the influence of the subregion numbers on the shear viscosity of asphalt. It can be observed that the viscosities calculated with different regions vary little, with a relative error of approximately 1.02%, allowing the error due to this variable to be neglected. The corresponding velocity distribution curves are shown in Figure 12b, where it can be seen that fewer divisions (10, 20, and 30) result in a deviation of velocity distribution relative to the middle position and different velocity gradients appearing on either side. Simultaneously, too few region divisions can lead to a lower number of sampling points, resulting in a poor linear determination coefficient. On the other hand, too many region divisions may cause the thickness of the plates to be smaller than the geometric size of asphalt molecules, which can affect the statistical independence between different divided regions and lead to significant non-linearity in the middle and at both ends of the curve. According to Formula (6), since the momentum flux is known, the shear viscosity can be directly calculated from the uniform velocity gradient. In the face of this situation, consideration of the influence of the non-linear shear strain field is required during calculation and increases the complexity. Based on the above analysis, the entire system was divided into 40 regions in the z-direction, with each region having a thickness of 3.75 Å, which can be used for subsequent simulations. As shown in Figure 13, from the perspective of configuration, the aggregation behavior of asphaltene molecules constantly changes between face-to-face stacking, offset stacking, and T-shaped stacking, and their planes do not clearly face the direction of shear flow but rather undergo translational or rolling motion with shear flow.

3.3.2. Momentum Exchange Period

The momentum exchange period affects the intensity of shear forces, manifested as the magnitude of the velocity gradient in the system. To investigate the influence of this parameter on the velocity gradient, different momentum exchange periods were used for analysis, and Figure 14 illustrates the velocity distribution. From the graph, shorter periods (two and five timesteps) result in larger velocity gradients and lead to pronounced non-linear velocities, especially near the boundaries (relative positions of 0, 0.5, and 1). Different velocity gradients in different regions can affect viscosity. Those near the boundaries overestimate viscosity, while gradients in linear regions underestimate viscosity. When momentum is exchanged every 10 timesteps, nonlinearity at the boundaries decreases, but there is significant fluctuation in the middle region. In contrast, a momentum exchange period of 20 timesteps demonstrates better linearity across the entire velocity curve. Additionally, it is important to note that if momentum exchange is too slow, for instance, every 50 or more timesteps, the velocity gradients become smaller relative to the noise, necessitating an increase in the number of samples. Thus, obtaining accurate viscosity results requires appropriate velocity gradients. The exchange period should not be too fast to avoid non-linear effects nor too slow to cause significant errors. Based on the aforementioned analysis, an exchange period of 10 to 20 timesteps is suitable.

3.3.3. System Size

In atomic-scale simulations, the prediction of material properties is influenced by the size of the simulation system due to the presence of periodic boundaries and limited sampling. Therefore, this work established six models of different sizes and compared the effects on shear viscosity of asphalt. Detailed model information is provided in Table 3.

First, the velocity gradients of models with different sizes under the same momentum exchange period were computed. As shown in Figure 15a, smaller model systems result in higher shear rates, and the velocity distribution exhibits significant nonlinearity. As the model size increases to a side length of 75 Å, a better linear velocity becomes evident. Moreover, larger model sizes demonstrate a more pronounced linear relationship in velocity gradients. The results of rNEMD calculations for asphalt systems that meet the velocity distribution requirements are depicted in Figure 15b. Under the same momentum exchange period, the M-3 asphalt system corresponds to a shear rate of 3.16 × 10⁻⁶ fs⁻¹ and a viscosity of 5.86 ± 0.64 cp. With further expansion, the M-5 system corresponds to a shear rate of 4.63 × 10⁻⁷ fs⁻¹, resulting in a shear viscosity of 13.96 ± 1.05 cp. Finally, the largest system yields a shear viscosity of 14.71 ± 1.14 cp at a shear rate of 3.59 × 10⁻⁷ fs⁻¹. It is evident that under the same momentum exchange period, as the model size increases, the velocity gradient of the system decreases, leading to an increase in viscosity. This phenomenon is likely due to the shear thinning effect of asphalt rather than errors in the calculation method. Based on the linearity of the velocity field, it is reasonable to select the smallest model size with a stable velocity gradient, i.e., an initial size of a 75 Å cubic box.

4. Conclusions

This work investigated the parameter settings for asphalt viscosity calculations using the Green–Kubo integral in EMD and the Muller-Plathe algorithm in rNEMD, yielding the following conclusions:

(1)

The Pearson correlation coefficient between asphalt density and temperature exceeds 0.988, indicating a strong linear correlation. At 408.15 K, the asphalt density is 0.891 g/cm³, with an average error of less than 5% compared to experimental values.

(2)

In EMD simulations, employing a 1/t weight for viscosity curve calculation results in a well-fitted curve that closely aligns with the original data, demonstrating high precision in the fitting function. The isotropy of the asphalt model improves for atomic counts exceeding 260,000, rendering viscosity calculations more reasonable.

(3)

In rNEMD simulations, the number of regions within a certain range has a negligible impact on asphalt viscosity calculation results, with errors being negligible. A momentum exchange period of 20 timesteps exhibits a favorable linear trend in velocity gradients. Using a momentum exchange period within the range of 10 to 20 timesteps is suitable. Larger model sizes demonstrate a more pronounced linear relationship in velocity gradients, and using an orthogonal simulation box with a side length of 75 Å meets the computational requirements effectively.