Numerical investigation on phase separation in polymer-modified bitumen: effect of thermal condition

The finite element software COMSOL Multiphysics has been used to implement the model described in the previous sections. The geometry of the simulation domain is a rectangle of 2.5 mm × 2.0 mm meshed with triangular elements. This domain is basically the same size as the microscopic images (magnification 50×) [11, 12] that are used for calibration of the model parameters. As the boundary condition, the contact angle on the four sides of the rectangle is 90°. In order to present the initial state of the simulated PMB, normally distributed values are randomly located in the domain. The used mean value of the initial values is 0.05, and the standard deviation is 0.005. This means that the polymer content of the simulated PMB is 5% by weight with certain variation, if neglecting the polymer–bitumen density difference.

The phase separation behaviour is simulated under five different thermal conditions, numbered 1–5 as in Fig. 1. They represent three different temperature levels (180, 160 and 140 °C) and two cooling rates (fast cooling 8 °C/min and slow cooling 2 °C/min) between 180 and 140 °C. Of the investigated temperatures, 180 °C stands for a normal PMB storage temperature; 160 °C corresponds to a lower temperature for long-time PMB storage in order to avoid polymer degradation; and 140 °C may help to understand the effect of cooling during the construction process. All the simulations are run to apply the thermal conditions for 4800 s.

Four different PMBs (numbered 1–4) are simulated in this paper with the intention of representing the four PMBs experimentally studied in [11, 12]. The four PMBs were prepared in laboratory with different base bitumen binders of penetration grade 70/100 from various sources, but they had the same styrene–butadiene–styrene copolymer modifier and the same polymer content (5% by weight of the blend). The modifier was a linear triblock copolymer. Its weight average molecular weight was 189,000 g/mol; the styrene content was about 30%; and the fraction of tri- versus diblock copolymer was 0.8.

The needed parameters for simulation include the maximum mobility coefficients (M _p0 and M _b0), activation energies (E _p and E _b), gradient energy coefficient (κ), segment numbers of the chains (N _p and N _b) and the constants (α, β, k and c) for interaction and dilution parameters. In order to get the parameter values for the PMBs, the theoretical ranges of the model parameters are firstly discussed before fixing the specific values. After this, the specific values are calibrated within the theoretical ranges of the model parameters by the comparison between the experimental and numerical results.

Since some of the needed PMB parameters are not commonly measured for the time being, the used ranges of M _p0, M _b0, E _p, E _b and κ in this paper are estimated based on the reported parameter values for general phase-field models and other materials like polymer blends and alloys [27‐30]. For N _p and N _b, it is postulated that the hypothetical chains for bitumen have the same length as the polymer chains. This does not mean that the bitumen has the same molecular size as the polymer chains. But the complexity of bitumen composition, due to its diversity in chemical species, has the same contribution to the configurational entropy as the polymer chains. In other words, N _b is not the actual value of the bitumen molecular weight but a parameter representing the complexity of the bitumen composition. By setting N _p = N _b = N, the parameter N χ scales up into the widely reported theoretical range where N χ ≤ 2.0 results in a single well and N χ > 2.0 leads to a double well [31‐34]. Thus, the values of α and β can be obtained according to the estimated N χ values. In this sense, the parameters α and β discussed in this paper also indicate the influences of the molecular size and its distribution (they have in fact the values of N α and N β). As for the dilution parameter, the values of k and c can be estimated on the basis of the reported swelling ratio values of the polymer modifiers in PMBs [26, 35, 36].

Microscopy observation results from [11, 12] are used as the experimental corroboration for model calibration in this paper. As presented in Fig. 2, the microscopy images display the microstructure of the four PMBs after one hour isothermal annealing at three different temperatures. It can be seen from the microscopy observation (MO) results in Fig. 2 that the four PMBs show different phase separation behaviours. PMB1 and PMB4 separate into two phases at 180 °C (the lighter polymer-rich phase and the darker bitumen-rich phase), while PMB 2 and PMB3 remain homogeneous at this scale. At 140 °C, all the PMBs show two-phase structures but the patterns are different. For each of the PMBs, the microstructure changes as the temperature varies between 180 and 140 °C.

In order to calibrate the model parameters for the four PMBs, the model described in the previous sections is implemented to reproduce these microscopy observation results. The numerical simulations represent the same condition as the experimental procedure in [11, 12]. The calibrated model parameters are shown in Table 1, together with M _p0 = 3.2 × 10⁻¹⁴ m⁵/(J s), E _p = 3.6 × 10⁴ J/mol and κ = 4.50 × 10⁻⁵ J/m. With these values, the numerical simulation (NS) results for the four PMBs are also shown in Fig. 2, following the microscopy images. It is indicated that the phase separation behaviours of the PMBs are well reproduced by the model with the calibrated parameter values, including the stability differences between the PMBs and the microstructure differences between different temperatures. Thus, it is believed that the listed parameters in Table 1 can describe the phase separation behaviours of the four PMBs properly.

With the calibrated parameter values, the variations of the mobility coefficients, interaction and dilution parameters with temperature are plotted as in Figs. 3, 4 and 5. It can be seen in Fig. 3 that the base bitumen binders have higher mobility coefficients than the polymer within the discussed temperature range. The mobility coefficients decrease as the temperature decreases. Figure 4 shows that the interaction parameters decrease as the temperature increases, which means that the polymer and base bitumen binders are more interactive with each other at higher temperatures. It is indicated in Fig. 5 that the dilution parameters increase as the temperature increases. This means that the relative amount of the interactive molecules in the base bitumen binders becomes higher (and the polymer swells more) when the temperature rises. All the PMBs have different temperature dependencies for each of the discussed parameters, showing the different material properties among the PMBs. Based on these numerical results, it is thus believed that the calibrated parameter values generally describe reasonable material properties. This also confirms that the parameter values listed in Table 1 can accurately represent the four investigated PMBs. However, it is worth mentioning that all the listed values are only valid for the studied temperature range, i.e. 140–180 °C. Beyond this range, more discussions are still needed.

The calibrated parameter values also give the free energy curves by Eq. 4. Consequently, a theoretical phase diagram of the four PMBs can be computed on the basis of the free energy curves. By plotting the free energy minimum points at different temperatures, the computed phase diagram of the four PMBs (the binodal curves) is obtained and presented in Fig. 6. This phase diagram reveals the phase separation behaviours of the four PMBs within the studied temperature range and can serve as the analytical solutions for the simulations. It will be discussed with the numerical simulation results in the following sections of this paper.

The simulation results with Therm.Cond.1–3 (180, 160 and 140 °C) are shown in Figs. 7, 8 and 9, respectively. The results disclose the influence of temperature level on phase separation behaviour of the simulated PMBs. It can be seen in Fig. 7 that PMB1 and PMB4 separate into two phases at 180 °C (warmer colours for the polymer-rich phase and cooler colours for the bitumen-rich phase), while PMB 2 and PMB3 have homogeneous structures even after the high-temperature storage. According to Fig. 4, the interaction parameters of PMB1 and PMB4 are greater than 2.0 at 180 °C. Thus, their free energy curves are double wells at 180 °C, indicating their thermodynamic instability under Therm.Cond.1. In contrast, PMB2 and PMB3 have interaction parameters less than 2.0, leading to their single-well free energy curves and thermodynamic stability under Therm.Cond.1.

For the unstable PMB1 and PMB4, different patterns are shown in the simulation results with Therm.Cond.1. The polymer-rich phase forms threads in PMB1 but droplets in PMB4. This means the different base bitumen binders in PMB1 and PMB4 result in different swelling ratios of the polymer modifier, in spite of the same polymer content. According to Fig. 5, PMB1 has a higher dilution parameter than PMB4, showing a higher number of interactive molecules in the base bitumen of PMB1 than PMB4. This leads to the higher swelling ratio of the polymer in PMB1. In addition, Fig. 7 gives the same values of equilibrium phase composition for the PMBs as indicated in Fig. 6.

As the temperature decreases, PMB1 and PMB4 keep the two-phase structures but PMB2 and PMB3 become thermodynamically unstable through a stability–instability transition, shown in Figs. 8 and 9. A lower temperature can only affect the microstructure and composition of the equilibrium phases in PMB1 and PMB4. But the temperature drop from 180 to 140 °C changes the stability of PMB2 and PMB3. At 140 °C, PMB2 and PMB3 have interaction parameters greater than 2.0 according to Fig. 4. This results in the double wells on their free energy curves and causes their phase separations under Therm.Cond.3. PMB2 and PMB3 show different patterns in Fig. 9: a bicontinuous structure for PMB2 but a droplet pattern with a continuous matrix for PMB3. This can be attributed to the higher swelling ratio of the polymer in PMB2 than PMB3 at 140 °C, which is controlled by the dilution parameters as presented in Fig. 5. With Therm.Cond.2, Fig. 8 displays the intermediate states of the PMBs during the thermodynamic transition between 180 and 140 °C.

In Fig. 9, it is interesting to see that PMB3 and PMB4 show similar microstructures at 140 °C, although their phase structures are completely different with each other at other temperatures (as in Figs. 7, 8). This can be interpreted by the computed phase diagram Fig. 6. At 180 °C, PMB3 lies in the one-phase regime of its phase diagram, but PMB4 is most possibly in its unstable regime. At 160 °C, the bimodal points of PMB3 are far away from those of PMB4. However, they come to almost the same locations at 140 °C. The different material parameters of PMB3 and PMB4 (as in Table 1) decide the different temperature dependencies of their phase separation behaviour. All these differences are expressed by the model and presented in the numerical simulation results. The same also applies for PMB1 and PMB2, and probably all PMBs.

The effect of temperature level on PMB phase separation behaviour is discussed in the previous section. However, PMB phase separation is essentially a time-dependent microstructure evolution process. Consequently, the changing rate of the temperature may also have its influence on PMB phase separation behaviour. Therm.Cond.4 and Therm.Cond.5 aim to investigate the effect of cooling rate on phase separation behaviour of the simulated PMBs. The simulation results are shown in Figs. 10, 11, 12, 13 and 14.

In Fig. 10, it can be seen that PMB1 and PMB4 approximately follow a similar evolution routine as under the previous thermal conditions, although the fast cooling has its impacts on the separation process and the final PMB microstructure. But a homogenization process occurs in PMB2 and PMB3 during the fast cooling between 0 and 300 s. This homogenization process is due to the transition of the PMBs from a thermodynamically stable state to a thermodynamically unstable state during the cooling. After the homogenization, PMB2 and PMB3 start to separate and form the binary structures at 140 °C.

Since the model presented in this paper uses fixed material property parameters for a fixed temperature (e.g. 140 °C), it is not unexpected that the same temperature leads to the same equilibrium phase composition and polymer swelling ratio. As a consequence, the final patterns of the PMB binary structures in Fig. 10 are quite similar as those in Fig. 9, though not exactly the same. The final values of the local polymer volume fraction in the equilibrium phases in Fig. 10 are the same as the final values in Fig. 9, essentially controlled by the phase diagram of the four PMBs as Fig. 6. However, the process can be more complicated in reality due to the potential effects of the interfacial tension, material viscosity and rheology.

Figures 11, 12, 13 and 14 show the PMB structure evolution processes under Therm.Cond.5 (with a lower cooling rate). According to Figs. 11 and 14, both PMB1 and PMB4 start to separate from the very beginning of the simulated process. The results in Figs. 11 and 14 indicate the influence of the bimodal point change (Fig. 6) of the PMBs during the cooling. After the cooling termination at 1200 s, the PMBs reach the equilibrium states shown in Fig. 6 but the final microstructures are slightly affected by the cooling rate.

As for PMB2 and PMB3, Fig. 6 reveals that their stability–instability transitions both occur between 170 (300) and 160 °C (600 s). Before the transitions, Figs. 12 and 13 show that they form more homogenous structures than the initial ones. After the homogenization, the separation processes start and the PMBs gradually form the binary structures. As the phase separations start from more homogenized structures under Therm.Cond.5 than Therm.Cond.4, it takes longer time for the PMBs to reach the equilibrium states under Therm.Cond.5. By 2100 s (900 s after the cooling terminated), PMB3 has not displayed a two-phase structure in Fig. 13. At the end of the simulated process, PMB2 and PMB3 reach the equilibrium states shown in Fig. 6. However, the cooling rate only slightly affects the final patterns of the PMBs’ binary structures in this paper.

With the simulation results, the polymer-rich phase of a PMB can be defined as the area where the local polymer volume fraction is higher than the mean of the initial values (5% in this paper). Thus, the polymer swelling ratio can be defined as the ratio between the area fraction of the polymer-rich phase and the mean of the initial values. By post-processing the numerical results, the swelling ratio values during the whole simulated process can be obtained, as shown in Fig. 15 for the thermodynamically unstable cases. Because of the normal distribution of the initial values, all the curves in Fig. 15 start from the swelling ratio value of 10, representing the sufficient swelling of the polymer in the beginning. At the end of the simulated process, it is the temperature that determines the final value of polymer swelling ratio for each of the PMBs. A higher temperature leads to a higher polymer swelling ratio, while Therm.Cond.3, 4 and 5 tend to finalize with the same value.

However, the variations of the swelling ratio values at the early stage can provide explicit information on the process of reaching the equilibrium state from the initial state. It can be seen in Fig. 15a that the polymer swelling ratios of both PMB1 (blue lines) and PMB4 (green lines) gradually decrease until the final values are reached under all the studied thermal conditions. The differences between the thermal conditions lie on the time needed to reach the equilibrium state. Among Therm.Cond.1, 2 and 3, PMB1 and PMB4 show different effects of the temperature on the changing rate of the polymer swelling ratio at the beginning stage. This might depend on their different material property parameters. But all the curves with cooling follow the middle paths between 180 and 140 °C. For both PMBs, the slow cooling curves follow the 180 °C curves for longer time and reach the equilibrium at 140 °C later than the fast cooling curves.

As for PMB2 (red lines) and PMB3 (orange lines), Fig. 15b gives only the curves under Therm.Cond.3, 4 and 5, since neither PMBs show definite instability at 180 or 160 °C (no typical polymer-rich phase). Under Therm.Cond.3, PMB3 presents a representative curve for unstable PMBs in Fig. 15b, similar as those of PMB1 and PMB4 in Fig. 15a. However, with cooling implemented, the PMB goes through a stability–instability transition around 160 °C. This homogenizes the PMB structure in the beginning of the simulated process and makes the polymer swelling ratio stay on a plateau for some time (longer for slow cooling than fast cooling). The phase separation starts in PMB3 after the homogenization and transition. The bumps on the orange curves in Fig. 15b reflect the starting of the two-phase structures in PMB3 under Therm.Cond.4 and 5. After the bumps, the curves reach almost the same polymer swelling ratio as Therm.Cond.3 (the slow cooling curve still needs some time to reach it). PMB2 is supposed to have experienced the same process. But the dilution parameter of PMB2 is very high at 140 °C according to Fig. 5. This results in the very small difference between its initial polymer swelling ratio and the equilibrium one at the temperature. As a consequence, the polymer swelling ratio plateaus and bumps are not explicitly shown on the red curves in Fig. 15b. However, the numerical simulation results in the previous sections have indicated the phase separation behaviour of PMB2.