5.3 Simulation results
5.3.1 Early stage demixing
The behaviors of PT and ST are well examined [78, 69, 95], more precisely, the quasi-isothermal ST behavior, where the vapor and liquid phase are observed to contrarily cool down or heat up, respectively. This phenomenon becomes clear from Eq. (5.13), where the thermal conductivity is finally responsible for the equi-libration of the system. In contrast, in PT simulations both phases, liquid and vapor, are observed to heat up, but with the vapor temperature temporally lagging behind the liquid temperature. It can be explained by conduction, which is respon-sible for the heating of the vapor by the hotter liquid. This leads to a stagnation of heating of the liquid phase until the vapor is heated up to thermal equilibrium, which is followed by an increase of the overall mean temperature. The results from the PT and ST simulations in this work slightly differ from those obtained in Ref. [78]. This must be addressed to the physically more accurate formulation of the density dependent thermal conductivityκ(ρ), given by Eq. (5.10).
Both theτ = 101 and 100 runs show the stagnation of heating, which is actually characteristic for thermal simulations. The coexistence density of the liquid phase that is reached, corresponds to a higher temperature than the predictedT0. Later
0.01
Figure 5.3: The characteristic lengths of the liquid phaseξl forT0= 0.8, plotted in steps of ∆t= 0.05, additionally connected by lines. PT and ST are highlighted by open dots. The data sets are each vertically shifted by factors of 100.1 due to a better comparability. The factor is also shown true to scale. The black lines indicate power laws to guide the eyes.
in progress, in the simulations with thermostat, the temperature decreases and evolves along the coexistence curve to the desired temperature. Solely, the PT run persists in that heated state.
The demixing process and the growth of the domains can be investigated by means of the characteristic lengthsξ, defined in Eq. (5.17). The lengths are shown in Figs. 5.3 and 5.4 for the liquid and vapor phase, respectively. For the reason of better comparability the curves are each shifted by factors of 100.1. It is due to the initial densityρ0 = 0.5 that per definition the one-component fluid is vapor in the initial state. That means in effectξv(0) = 1, followed by a rapid decrease, whereas the liquid phase initiates at ξl(0) = 0, which further strongly increases.
The initial state also determines the composition that eventually manifests in the separated system [100]. Here, a minor liquid phase in a major vapor phase is expected, which is why the main focus is on the liquid phase, whereas the vapor is only considered as solvent.
The early stage takes place right after the instantaneous quench, and the very early dynamics are dominated by the rise of so called “homophase fluctuations”
[8], which arise from the initial noise of the density field. The point in time of demixing is unaffected by the choice of τ. Therefore, it appears to suggest it also for the rise of fluctuations. However, the further evolution of the ξl-curves in Fig. 5.3 differ. It can be distinguished between the two beforehand mentioned different procedures. First, for ST andτ 1, the initial growth has rather more exponential growth character, which is followed directly by a power law scaling in the late stage. In contrast, for PT and forτ &1, the initial growth seems to scale with an ordinary power law, which is followed by a plateau, where the growth of
Chapter 5 Insights from inside the spinodal: Bridging thermalization time scales
Figure 5.4: The characteristic lengths of the vapor phaseξv forT0= 0.8, plotted in steps of ∆t= 0.05 connected by lines (coding as in Fig. 5.3). The data sets are each vertically shifted by factors of 100.1 due to a better comparability. The black lines are shown to guide the eyes. The inset shows a non-logarithmic plot of theτ= 10−2 curve in the early stage.
ξlstagnates, also followed by the ordinary power law scaling of the late stage. It is obvious that the type of initial growth, whether it is described by an exponential or an ordinary power law, depends on the choice of τ. The change is, as already qualitatively mentioned, placed at τ ≈ 10−1, more precisely, between τ = 10−2 and 10−1.
Since this growth behavior in the early stage is also expected to be strongly affected by the actual temperature of the system, it will now be discussed in more detail by comparison of several quench depths.
The type of growth of the domains in the early stage is found to differ between the simulations. Interfaces become diffuse, unless the interfaces will not be formed before the intermediate stage. These are the simulations with slow thermalization
τ &1.0, which increase the temperature, or apparently those with the
tempera-ture already above the threshold, i.e. T0 & 1.0. In Fig. 5.5 a comparison of the early stages of the characteristic lengths ξl for different quench depths is shown.
As discussed in Appendix A (see Sec. 5.5), theξv are not suitable to realistically capture the growth behavior in the early phase, which is due to the initial rapid decrease of the curves. Therefore, only the lengths of the liquid phases are eval-uated here. Representatively chosen for the fast and slow thermalization are the simulations with τ = 10−4 and τ = 101, respectively. The ξl in the very early stage are used for a least square regression. It turns out that the best fitting regression function strongly depends on both, the actual depth of the quench and thermalization time scale. The fitting parameters, which belong to the overlaying straight curves in Fig. 5.5, are listed in Table 5.1. The corresponding functions, that are used for the regression, are as follows. Three different types of growth can
0.1
Figure 5.5: The early and, if apparent, the intermediate stages ofξl forτ = 10−4 (open symbols) and 101(filled symbols) for different quench depths atT0= 0.6, 0.7, 0.8, 0.9, and 1.0. Overlaying curves (red) belong to least square fits with the parameters presented in Table 5.1.
be distinguished. An exponential increase ξ ∼ eβt, is observed for τ = 10−4 and deep quenched systems withT0 = 0.6, 0.7, 0.8, and 0.9. Asτ becomes larger, the growth rates successively alter and follow an ordinary power law scaling, given by ξ∼tβ. The best fitting values for this range ofτ appear to suggest an exponent β ≈1/2. For higher T0 values, e.g., theT0 = 1.0 run, the growth of fluctuations seems to be unaffected by the choice ofτ. Furthermore, the type of growth of the ξl curves has changed to logarithmic scaling of the form ξ ∼ log10(βt), for both, τ = 10−4 and 101, not least qualitatively, but also supported by the fitted values.
The differing behavior in the early stage can be clarified with a view on the temperature-density plane of the phase diagram, which is shown for all quenches in Fig. 5.6 for the simulations with τ = 101 and 10−4 . Note that the τ 1 simulations are very similar to the corresponding ST runs, for which, as recently described in Ref. [78], both phases evolve in different directions, forced by the caloric vdW EOS in Eq. (5.13). The volume expansion of the vapor phase causes a temperature decrease of the phase, and the liquid temperature increases, caused by the increase ofρl. Eventually, thermal conductivity forces the system to thermal equilibrium. However, the τ & 1 simulations show contrary behavior, because the mean temperature of the system is allowed to deviate from T0, which is an effect of the slow thermalization. Therefore, the mean temperature is expected to finally evolve in the same direction of that phase that accretes most of the mass. Therefore, it depends on the initialρ0. In our setup, the liquid becomes the mass rich phase, whereas the vapor phase still tends to decrease the temperature.
Therefore, the vapor phase gets heated through thermal conduction, which affects
1Asymptotic standard error expresses the accuracy in accordance to the calculated characteristic lengths.
Chapter 5 Insights from inside the spinodal: Bridging thermalization time scales
Table 5.1: The parametersβ(with asymptotic standard error1δβ) are obtained via least square fitting of the ξl in Fig. 5.5 within suggested best fit intervals. The expected types of growth are exponential (exp), power law scaling (scale) or logarithmic (log). The appearance of an intermediate stage is synonymous to the plateau in theξcurve.
T0 log10(τ) type fit interval β(±δβ) plateau
1 Asymptotic standard error expresses the accuracy in accordance to the calculated characteristic lengths.
0.4
Figure 5.6: Phase diagram for several quench depths for the thermalization time scales τ = 101 (open) andτ= 10−4 (filled) up to the simulation timetmax= 50.
0
Figure 5.7: Liquid mass fractionsµl for several quench depths over time t for the ther-malization time scalesτ= 101(filled symbols) andτ = 10−4(open symbols).
the vapor temperature to generally lag behind the liquid phase in relation of the total heating of the system. A stagnation of heating in the liquid phase is observed for all initial temperatures except for the T0 = 1.0 simulations. Note that the exceptional role of the latter also manifests in the differing type of scaling of ξl, which is logarithmic. It can be explained by the prolongation of the dynamics with higher system temperatures [78, 69], in the way that thermal conductivity already directly affects the demixing in the very early stage. Moreover, T0 = 1.0 is above this temperature, where the heat stagnation can be observed in the remaining simulations.
The equilibration of the temperature imbalance in the early stage can be ex-pected to be mainly driven by thermal conductivity, which is certainly true for τ 1 and for largeT0. For theτ &1 range it is found, that also convective heat transfer plays a decisive role for spinodal decomposition [102]. Convective heat transfer is also responsible for the τ dependent reduction of the growth rate in Fig. 5.5. This can be seen from the liquid mass fraction, which is defined by
µl= Ml
Mtot, (5.18)
where Ml is the total mass of the liquid phase. It is due to our definition of the threshold densityρc = 2/3, that the liquid mass fraction initializes with µl = 0, which is followed by a strong increase. A comparison of the liquid mass fractions for several quenches is shown in Fig. 5.7. Again, only the curves forτ = 10−4 and τ = 101 are exemplarily shown, for τ 1 andτ &1, respectively.
For the latter, a comparison with the temporal evolution in the phase diagram reveals an unexpected coinciding feature. Mass becomes not only reduced, but a backflow from liquid to vapor is observed in the µ curves. It takes place at
Chapter 5 Insights from inside the spinodal: Bridging thermalization time scales
exactly the same time, when the stagnation of heating is observed for the deep quenched simulationsT0 = 0.6,0.7,0.8 in the phase diagram. The strength of this convection driven mass flow increases with the depth of the quench. The µl of the deep quenches further undergo local minima, followed by a slow increase until equilibrium is reached, whereas forT0= 0.9,1.0 the backflow diminishes and only a halted increase is observed. The minima are found to take place at exactly the same time, where the liquid density starts to decrease in the phase diagram (see Fig. 5.6) at the reflection point. Moreover, the further evolution of the µ curves coincide with the time interval, where the plateau appears in the characteristic lengths.
However, for high temperatures the morphological evolution slows down and droplet nucleation dominates spinodal decomposition and the diffuse interfaces are formed simultaneously with the separation of the phases [78]. This leads to the instantaneous formation of spherical-shaped droplets. The mass backflow in the evolution ofµvanishes, e.g., at theT0 = 1.0 curve, as it can be seen in Fig. 5.7.
Additionally, a prolonging behavior of the dynamics is also observed for larger τ, which must be mainly due to the rapid increase in temperature. In contrast, the thermal relaxation, back to the desired temperatureT0, in these systems is slow.
Therefore, the temperature is allowed to exceed a certain threshold value, which is atTth≈1.0, as it can be seen in Fig. 5.6.