Aging Phenomena during Phase Separation in Fluids: Decay of autocorrelation for vapor-liquid transitions

TL;DR

This study uses molecular dynamics simulations to analyze aging phenomena during vapor-liquid phase separation in fluids, revealing universal power-law decay of autocorrelation functions despite different morphologies and growth mechanisms.

Contribution

It demonstrates that the decay of the order-parameter autocorrelation follows a power-law in both disconnected and percolating morphologies, highlighting universality in coarsening dynamics.

Findings

Power-law decay observed in disconnected morphology

Complex decay behavior in percolating structure

Finite-size scaling reveals strong preasymptotic corrections

Abstract

We performed molecular dynamics simulations to study relaxation phenomena during vapor-liquid transitions in a single component Lennard-Jones system. Results from two different overall densities are presented; one in the neighborhood of the vapor branch of the coexistence curve and the other being close to the critical density. The nonequilibrium morphologies, growth mechanisms and growth laws in the two cases are vastly different. In the low density case growth occurs via diffusive coalescence of droplets in a disconnected morphology. On the other hand, the elongated structure in the higher density case grows via advective transport of particles inside the tube-like liquid domains. The objective in this work has been to identify how the decay of the order-parameter autocorrelation, an important quantity to understand aging dynamics, differs in the two cases. In the case of the disconnected morphology, we observe a very robust power-law decay, as a function of the ratio of the characteristic lengths at the observation time and at the age of the system, whereas the results for the percolating structure appear rather complex. To quantify the decay in the latter case, unlike standard method followed in a previous study, here we have performed a finite-size scaling analysis. Outcome of this analysis shows the presence of a strong preasymptotic correction, while revealing that in this case also, albeit in the asymptotic limit, the decay follows a power-law. Even though the corresponding exponents in the two cases differ drastically, this study, combined with a few recent ones, suggests that power-law behavior of this correlation function is rather universal in coarsening dynamics.

Equations

$$\ell (t)\sim t^{\alpha }.$$

$$C_{\text{ag}}(t,t_w)=⟨\psi (\vec{r},t)\psi (\vec{r},t_w)⟩-⟨\psi (\vec{r},t)⟩⟨\psi (\vec{r},t_w)⟩,$$

$$C_{\text{ag}}(t,t_w)\sim {\left(\frac{\ell }{{\ell }_w}\right)}^{-\lambda }.$$

$$\frac{d}{2}\le \lambda \le d.$$

$$U(r)=4\epsilon \left[{\left(\frac{\sigma }{r}\right)}^{12}-{\left(\frac{\sigma }{r}\right)}^6\right].$$

$$\displaystyle u(r)=U(r)-U(r_{c})-(r-r_{c})\frac{dU}{dr}\Big{|}_{r=r_{c}}.$$

$$\ell =\int P({\ell }_d,t){\ell }_{d}d{\ell }_d,$$

$$C(r=\ell ,t)=f,$$

$$C_{\text{ag}}(t,t_w)=A\text{exp}\left(\frac{-B}{x}\right)x^{-\lambda };x=\ell /{\ell }_w,$$

$$Y=C_{\text{ag}}(t,t_w)\text{exp}\left(\frac{By}{y_w}\right)y_w^{\lambda },$$

$$Y\sim y^{\lambda }.$$

$${\lambda }_i=-\frac{d\text{ln}{C}_{\text{ag}}(t,t_w)}{d\text{ln}x},$$

Full text

I Introduction

Following a quench inside the coexisting curve a homogeneous system coarsens via the formation of domains of particle-rich and particle-poor regions bray ; wadhawan ; onuki ; binderbook1001 ; jones ; lifshitz ; furukawa1 ; furukawa2 ; binder2 ; binder3 ; tanaka1 ; tanaka2 ; das2011 ; das2012 ; roy2012 ; roy2013 ; roy2013jcp ; jung2015 ; nielaba ; watanabe ; jung2016 ; roy2018 ; tanaka2015 ; janke2017 ; azizi ; bouttes ; tanaka2000 ; laradji ; lip ; hugo ; pinaki ; yeungn ; chris ; xia ; ana1 ; hsu ; vul ; raz ; sten ; vin ; jae1 ; jae2 . This coarsening is a self-similar phenomenon bray ; wadhawan ; onuki ; binderbook1001 ; jones with the average size, $\ell$, of the domains typically growing algebraically with time ($t$) bray ; wadhawan ; onuki ; binderbook1001 ; jones :

[TABLE]

Universality of the exponent $\alpha$ depends, among other parameters, upon bray the conservation and number of components of the appropriate order parameter, relevance of hydrodynamics, etc. In the case of ordering dynamics in a ferromagnet bray or kinetics of phase separation in multi-component mixtures bray ; wadhawan ; binderbook1001 ; lifshitz , with underlying state of the system being solid, space dimension ($d$) and initial composition play only minor role. However, these facts can drastically alter the value of $\alpha$ in fluids, where hydrodynamics is important bray ; wadhawan ; onuki . Despite such weak universality, e.g., compared to equilibrium critical phenomena fisher1967 , significant understanding has been obtained with respect to the values of $\alpha$ in differing situations, via experiments, analytical theories as well as computer simulations bray ; wadhawan ; onuki ; binderbook1001 ; jones ; lifshitz ; furukawa1 ; furukawa2 ; binder2 ; binder3 ; tanaka1 ; tanaka2 ; das2011 ; das2012 ; roy2012 ; roy2013 ; roy2013jcp ; jung2015 ; nielaba ; watanabe ; jung2016 ; roy2018 ; tanaka2015 ; janke2017 ; azizi ; bouttes ; tanaka2000 ; laradji ; lip ; hugo ; pinaki ; yeungn ; chris ; xia ; ana1 ; hsu ; vul ; raz ; sten ; vin ; jae1 ; jae2 .

Another aspect that is important for the deeper understanding of such nonequilibrium processes is the aging property wadhawan ; fisher1988 ; liu1991 ; majumdar1995 ; yeung ; corberi2006 ; henkel ; janke2007 ; das2014 ; das2015 ; ahmad2012 ; das2013 . A crucial quantity in this context is the order-parameter autocorrelation function, $C_{\text{ag}}$. This quantity is a function of the observation time $t$ and the waiting time $t_{w}$, the latter being also often referred to as the age of the system, and is defined as wadhawan ; fisher1988

[TABLE]

the angular brackets representing statistical averaging. Here $\psi(\vec{r},t)$ is an appropriate space ($\vec{r}$) and time dependent order parameter, which, for a vapor-liquid transition, can be defined as the local density fluctuation around a mean value. Decay of $C_{\text{ag}}$ probes the relaxation starting from different ages fisher1988 .

Due to the violation of time translation invariance during growth processes, $C_{\text{ag}}$ is not expected to exhibit scaling as a function of $t-t_{w}$. Fisher and Huse (FH) fisher1988 , from the studies of spin-glass systems, proposed a scaling form of this quantity as

[TABLE]

Here $\ell$ and $\ell_{w}$ are the average domain lengths at times $t$ and $t_{w}$, respectively. FH also provided bounds on the exponent $\lambda$:

[TABLE]

The scaling of $C_{\rm ag}$ as a function of $\ell/\ell_{w}$ has been observed to be valid in a number of phase ordering systems. An interest in the literature has been to learn on the universality of the scaling function. Some recent studies suggest das2014 ; das2015 that even for solids the values of $\lambda$ are far less universal fisher1988 ; liu1991 ; majumdar1995 ; yeung ; corberi2006 ; henkel ; janke2007 ; das2014 ; das2015 than the growth exponent bray ; wadhawan ; onuki ; binderbook1001 ; jones ; lifshitz ; furukawa1 ; furukawa2 ; binder2 ; binder3 ; tanaka1 ; tanaka2 ; das2011 ; das2012 ; roy2012 ; roy2013 ; roy2013jcp ; jung2015 ; nielaba ; watanabe ; jung2016 ; roy2018 ; tanaka2015 ; janke2017 ; azizi ; bouttes ; tanaka2000 $\alpha$, the former being very strongly dependent upon $d$ and order parameter conservation. It is then expected that in fluids, where growth exponent itself is a strong function bray ; furukawa1 ; furukawa2 ; binder2 ; binder3 of the space dimension and the overall composition, situation will be far more complex.

In this work we present first results on the aging dynamics in connection with vapor-liquid phase separation with overall density close to the vapor branch of the coexistence curve. In addition, we also revisit the same problem for overall density close to the critical value das2013 . This is in the wake of important development das2014 ; das2015 ; das2018 in the techniques for analyzing aging data. For the sake of completeness, certain differences between the low and high density cases, with respect to structure and growth, though known, are briefly mentioned below.

For a density close to the critical value the vapor and liquid domains percolate through the system. On the other hand, in the case of very low overall density the morphology consists of disconnected droplets of liquid phase. While hydrodynamics is important in both the cases, it manifests differently in the two situations, leading to different growth laws bray ; furukawa1 ; furukawa2 ; binder2 ; binder3 ; das2011 ; das2012 ; roy2012 ; roy2013 ; roy2013jcp – overall coarsening rate being much faster in the higher density case. Thus, more complex picture may be expected with respect to the decay of autocorrelation. The objective here is to see if at least a power-law decay of the function is an universal feature and if so, how the exponents in the two cases differ from each other. For this purpose, we take help of a very effective finite-size scaling method, devised with the objective of analyzing simulation results for two-time quantities.

The rest of the paper is organized as follows. In section II we describe the model and provide details on various methods. Results are presented in section III. Finally, section IV concludes the paper with a summary and outlook.

II Model and Methods

We consider a single component system das2011 ; das2012 ; roy2012 in which particles interact with each other via the Lennard-Jones (LJ) potential allen1987

[TABLE]

Here $\sigma$ is the interparticle diameter, $\varepsilon$ is the interaction strength and $r$ is the scalar distance between two particles. For the sake of computational convenience we have truncated the potential and shifted it to zero at a distance $r=r_{c}$ ($=2.5\sigma$). This exercise makes the force discontinuous at the cut-off distance. To avoid this problem a new term is added so that the simulated potential allen1987 reads

[TABLE]

The phase behavior for the vapor-liquid transition in this model in various space dimensions have been studied das2017 ; errington ; jia2017 via Monte Carlo simulations wild ; landau . In this paper we are interested in $d=3$. In this dimension the critical values of the temperature ($T$) and the overall number density ($\rho$) are errington ; jia2017 respectively $T_{c}\simeq 0.94\varepsilon/k_{B}$ and $\rho_{c}\simeq 0.32$, where $k_{B}$ is the Boltzmann constant. For the study of aging dynamics related to the growth in droplet morphology we will consider $\rho=0.08$, while for the percolating case simulations will be performed with $\rho=0.3$. Of course, both the densities fall inside the coexistence curve at the considered values of $T$, numbers for which we will mention in appropriate places.

Kinetics for the above mentioned densities have been studied via molecular dynamics (MD) simulations allen1987 ; frenkel . We present results at multiple temperatures, values of which were controlled via the application of a suitable thermostat. For accurate preservation of hydrodynamics, a requirement for studying phase separation in fluids, one should work in microcanonical ensemble. However, a thermostat is needed to study kinetics for temperature driven phase separation. Though not perfectly, a number of thermostats are known to preserve hydrodynamics well. Examples are dissipative particle dynamics thermostat groot , Nosé-Hoover thermostat (NHT) frenkel ; nose , Lowe-Andersen thermostat koopman , etc. From the point of view of temperature control, we observe that NHT is a better one and will use it for the present study.

All our simulations were performed in cubic boxes of side $L\sigma$. Results are presented after averaging over a minimum of 80 independent initial configurations for coarsening in low density situation and this number is 10 for the high density case. These configurations were prepared at $T=10\varepsilon/k_{B}$, far above the critical value. Time in our MD simulations is counted in units of $t_{0}=\sqrt{m\sigma^{2}/\varepsilon}$, where $m$ is the mass of each particle. For the sake of convenience, from here on we set $\sigma$, $\varepsilon$, $k_{B}$, and $m$ to unity.

We have applied periodic boundary conditions in all directions. Unless otherwise mentioned, for $\rho=0.08$ we have simulated systems with $L=200$ and for $\rho=0.3$ we have considered $L=100$. For these linear dimensions our systems contain 640000 and 300000 particles, respectively. Given the nature of the problem, that requires long runs and good statistics, these numbers are significantly large, certainly for MD simulations.

From the snapshots gathered via MD simulations, average length scale, which is necessary to quantify $\lambda$, can be obtained by exploiting the scaling properties of various morphology characterizing functions bray . These are two-point equal-time correlation function, $C(r,t)$; its Fourier transform, referred to as the structure factor; domain size distribution function; etc. E.g., the first moment of the domain size distribution function, $P(\ell_{d},t)$, provides the average size as das2011 ; das2012 ; roy2012

[TABLE]

where $\ell_{d}$ is the size of a domain that can be obtained from the separation between two interfaces along any Cartesian direction. Equivalently, $\ell$ can also be obtained from the decay of $C(r,t)$, defined as $C(r,t)=\langle\psi(\vec{r},t)\psi(\vec{0},t)\rangle-\langle\psi(\vec{r},t)\rangle\langle\psi(\vec{0},t)\rangle$, via bray ; das2011 ; das2012 ; roy2012 ; dasroy2015

[TABLE]

where $f$ is a constant. Note here that, in order to calculate the correlation functions, $\psi$, at a space point $\vec{r}$, has been assigned the number $+1$ if the local density is higher than the critical value and we have set $\psi$ to $-1$ when this density is less than the critical number das2011 ; das2012 ; roy2012 . This method essentially maps the configuration to that of Ising model landau . In this work we have calculated $\ell$ from Eq. (7).

III Results

In Fig. 1 we show two snapshots, one each for $\rho=0.08$ and $0.3$, obtained during the evolutions of the LJ systems at $T=0.6$. The morphology in the low density case consists of disconnected liquid drops in the vapor background roy2012 . On the other hand, for the high density system we have bicontinuous structure made of elongated liquid and vapor domains das2011 .

In absence of hydrodynamics, growth in systems with conserved order parameter, which is also the case in the present problem, occurs with $\alpha=1/3$, referred to as the Lifshitz-Slyozov (LS) exponent lifshitz . This is due to diffusive transport of material. While this mechanism is responsible for growth in fluids as well, dominant contribution in this case comes from hydrodynamic mechanisms. In fluids the growth depends upon the overall density furukawa1 ; furukawa2 ; binder2 ; binder3 ; tanaka1 ; tanaka2 ; das2011 ; das2012 ; roy2012 ; roy2013 ; roy2013jcp . This is because the above mentioned disconnected and percolating morphologies provide different mechanisms, as discussed below.

In the high density case, advective transport through tube-like regions is expected to provide an exponent bray $\alpha=1$, following a brief period of LS regime. This high value of $\alpha$ can be obtained via a balance between interfacial free energy density and viscous stress. Thus, this is referred to as the viscous hydrodynamic growth bray . At a much later time, there is expected to be a further crossover from $\alpha=1$ to $2/3$, the latter being referred to as the inertial hydrodynamic regime bray . The corresponding exponent can be obtained by equating the interfacial energy density with the kinetic energy density. In this paper, for the high density part our focus will be on the viscous hydrodynamic regime.

Manifestation of hydrodynamics is different in the disconnected case binder2 ; binder3 . If the background vapor density is reasonably high, i.e., temperature is close to the critical value, the droplets may exhibit diffusive motion and the growth will primarily occur via coalescence among these mobile clusters. In this case, solution of a dynamical equation siggia , obtained by equating the rate of change of droplet density with the collision frequency, provides the Binder-Stauffer (BS) binder2 ; binder3 growth law $\alpha=1/d$. In $d=3$, the space dimension of our interest, value of $\alpha$ is $1/3$, same as the LS law. However, the amplitudes in the two cases are expected to be different.

In Fig. 2 we show plots of average domain size, versus time, for both types of morphology das2011 ; roy2013jcp . For $\rho=0.08$ (main frame), we have presented data from $T=0.6$, whereas the temperature for $\rho=0.3$ (inset) is $0.7$. For the higher density case, related previous study das2013 was at $T=0.7$, which we revisit. Nevertheless, for aging we will also present results for $T=0.6$, for this density, at the end. From this figure, it can be clearly identified that growth is much faster when the overall density is higher. Due to this reason, for the low density case one needs to perform simulations over very long times to access significant scaling regime in the growth dynamics. Thus, the case of disconnected morphology is computationally more difficult.

On a double-log scale, the late time growth for the disconnected morphology is consistent with $\alpha=1/3$. For the percolating case, on the other hand, an $\alpha=1$ regime can be identified from the plot presented in double-linear scale. The slow growth towards the end is related to finite-size effects. Note that for very low overall density nucleation is delayed. This is reflected in the reasonably long “no-growth” period for the disconnected pattern. The sharp rise of data set in this case in an intermediate time scale corresponds to the onset of instability and does not belong to the scaling regime of growth. Further details on the exponents and related aspects of growth, for this case as well as for the percolating morphology, was provided in earlier works das2011 ; das2012 ; roy2012 ; roy2013 ; roy2013jcp . Next we move to our primary interest that is in the aging property.

In Fig. 3(a) we show $C_{\text{ag}}(t,t_{w})$, with the variation of $t-t_{w}$, for $\rho=0.08$ and $T=0.6$. Results from a few values of $t_{w}$ are included. Clearly, unlike the equilibrium situation, there is no overlap of data from different $t_{w}$, implying violation of time translation invariance. In Fig. 3(b) we plot $C_{\text{ag}}(t,t_{w})$, on a log-log scale, versus $t/t_{w}$. In this case, it appears that data for different $t_{w}$ nicely superimpose on top of each other. This collapse confirms the scaling behavior proposed by FH [see Eqs. (1) and (3)]. The solid line in the figure represents a power-law decay with exponent $2.2$. The simulation data, over two decades in $t/t_{w}$, are consistent with this line. Data sets from higher values of $t_{w}$ also fall on the master curve. However, given that these do not cover large range of $t/t_{w}$, we abstain from presenting them.

In Fig. 3(c) we show again the same data sets, but here $C_{\text{ag}}$ is plotted as a function of $\ell/\ell_{w}$. Again, all the data sets reasonably overlap with each other. Note that we have chosen $t_{w}$ in such a way that we are practically in the scaling regime of the growth. This can be verified from the main frame of Fig. 2. Again, appearance of the data sets is linear, over the whole range, in the log-log scale, re-confirming a power-law decay of the scaling function. This master function appears consistent with the continuous line that has a power-law exponent $\lambda=6.6$. Noting that $\ell$ and $t$ are connected to each other via Eq. (1), the exponent values in Fig. 3(b) and Fig. 3(c) are consistent with $\alpha\simeq 1/3$, confirming again the expected BS growth exponent in $d=3$.

Our scaled data cover a range less than a decade when plotted versus $\ell/\ell_{w}$. Given that the scaling in growth appears late and the corresponding exponent is small, to reach a decade in $\ell/\ell_{w}$, without encountering finite-size effects and with acceptable statistics, it will be necessary to run many simulations with $L\gtrsim 400$, i.e., with systems containing about five million particles, till about $t=10^{5}$. This has not been possible with the resoures available to us. But the presented $\ell/\ell_{w}$ range is comparable with the latest available studies of simplier models das2018 , e.g., the Ising model. We emphasize again that our data cover a range of two decades when plotted versus $t/t_{w}$, in addition to the correlation function falling by four decades.

The value $\lambda\simeq 7$ certainly satisfies the FH lower bound fisher1988 , which is 1.5 in $d=3$. But the number violates the upper bound, value of which is 3. At this point a brief discussion on the bounds will be useful liu1991 . A domain of size $\ell$ at time $t$ consisted of many domains of much smaller sizes at the waiting time $t_{w}$. Since the average value of the order parameter within $\ell$ should be $\ell^{-d/2}$ when many domains reside within this area, one obtains the lower bound to be “$d/2$”. In the special case, when no growth occurs, the upper bound “$d$” is arrived at. FH themselves warned about the strictness of the upper bound.

It is also important to mention here that Yeung, Rao and Desai (YRD) yeung later provided a stricter lower bound: $\lambda\geq(d+\beta)/2$, where $\beta$ is related to the structure factor as wadhawan ; yeung1988 $S(k\rightarrow 0,t)\sim k^{\beta}$, $k$ being the wave number. While the FH bound typically applies to growth with nonconserved bray order parameter ($\beta=0$), the YRD bound is more general. This is in the sense that $\beta$ is (positive) nonzero when one deals with conserved bray ; lifshitz order parameter. We have checked that for the droplet morphology in Fig. 1 value of $\beta$ is $\lesssim 1$ (see also Ref. paul ). This implies that the observed value of $\lambda$ satisfies both FH and YRD lower bounds. However, $\lambda=7$ is a number much higher than both the bounds. The huge differences can only be attributed to the dynamical aspect. Here we just mention that this value of $\lambda$ is in reasonable agreement das2015 with the corresponding number obtained for LS mechanism in $d=3$ for which also one has $\alpha=1/3$. Next, we present results for the percolating morphology.

Turning to the high density case, we show log-log plots of the $C_{\text{ag}}(t,t_{w})$ versus $\ell/\ell_{w}$ data for $\rho=0.3$ and $T=0.7$ in Fig. 4(a). Few different values of $t_{w}$, all lying in the linear growth regime, are used. Again, collapse of data appear nice. Deviations from the master curve towards the end, appearing earlier for larger $t_{w}$, are due to finite-size effects das2014 ; das2015 ; das2018 . This feature was absent in the low density case for which finite-size effects did not appear yet, due to much slower growth. Furthermore, in contrary to the low density case, the master curve, on the double-log scale, for the present density, does not appear linear. A continuous downward bending is clearly visible. This bending may imply ahmad2012 ; das2013 a faster than power-law decay. Thus, in Fig. 4(b) we show a log-linear plot das2013 , for a representative value of $t_{w}$, chosen in such a way that the chances of its falling outside the linear growth regime is minimal. In this plot the data indeed appear linear, and parallel to the continuous line that, of course, has an exponential form. However, given that the general theoretical prediction is that of a power-law, further analysis should be performed, if possible. A reason for the continuous bending could be the presence of corrections to the power-law scaling.

Similar bending was observed in $C_{\text{ag}}(t,t_{w})$ for coarsening during ferromagnetic ordering das2014 ; das2018 , as well as in kinetics of phase separation in solid binary mixtures das2015 . In both the cases it turned out that $C_{\text{ag}}(t,t_{w})$ can be reasonably accurately written as ($A$ and $B$ being constants) das2014 ; das2015 ; das2018

[TABLE]

such that one obtains a power-law only in the asymptotic limit $x=\infty$. Here also we assume the validity of the similar exponential correction factor and perform finite-size scaling (FSS) analysis das2014 ; das2015 ; das2018 ; landau ; fisher2 ; barber to check for consistency of Eq. (9) with the simulation data.

For that purpose we introduce a scaling function $Y$ that reads das2018

[TABLE]

where $y=L/\ell$ and $y_{w}=L/\ell_{w}$. The transformation of variables: $x\rightarrow y$ is motivated by the fact that an appropriate FSS variable is $L/\ell$, analogous to $L/\xi$ in critical phenomena landau ; fisher2 ; barber , $\xi$ being the equilibrium correlation length. Given that $y$ is a dimensionless variable, it is expected that $Y$ should be independent of system size. Thus, an $Y$ versus $y$ plot should have collapse of data from various different system sizes. It can be easily verified that in the thermodynamic limit of $y\rightarrow\infty~{}(L>>\ell)$, i.e., when there are no finite-size effects, Eq. (10) will provide the behavior in Eq. (9) if

[TABLE]

In the other limit, i.e., when $y\rightarrow 0~{}(t>>t_{w})$, we expect $C_{\text{ag}}(t,t_{w})$ to vanish, which must lead to the null value of $Y$ as well. Observation of the latter will require long simulations which we avoid.

An interesting fact about the FSS analysis for the aging phenomena is that one can get away without simulations of different system sizes das2015 ; das2018 . As seen in Fig. 4 (a), where we have presented data for a fixed value of $L$, finite-size behavior for higher values of $t_{w}$ appear earlier in $\ell/\ell_{w}$, analogous to quicker emergence of the effects for smaller system sizes if $t_{w}$ were kept fixed das2015 ; das2018 . This is because different “effective” system sizes are available for growth when relaxation is studied by starting from different ages of the system. Thus, $t_{w}$ essentially serves the purpose of $L$ and so, we will look for collapse by using data from different values of $t_{w}$, by fixing $L$. For an optimum collapse we will adjust the values of $\lambda$ and $B$.

In Fig. 5(a) we show FSS exercise for percolating morphology data from $T=0.7$. The data collapse is very good and obtained for $\lambda=4$. For large $y$, the master curve is quite consistent with the behavior in Eq. (11). The deviation from this power-law behavior is marked on the figure. This occurs when $\ell\simeq 0.37L$. As we have previously demonstrated das2012 with growth data, this fraction of $L$ corresponds to the onset of finite-size effects das2012 . Consistency of $Y$ with the power-law all the way till the finite-size effects emerge is suggestive of the fact that the exponential factor indeed nicely sums up the corrections, as in a number of other systems, in addition to pointing to the possibility of a power-law decay of $C_{\text{ag}}(t,t_{w})$ in the asymptotic limit, as opposed to an exponential decay das2013 . Results from similar exercise for $T=0.6$ is presented in the main frame of Fig. 5 (b) which we describe below.

In this case best collapse is obtained from $\lambda=4.3$. Again, for large $y$ behavior of $Y$ is consistent with Eq. (11). Appearance of finite-size effects is in agreement with that of higher temperature data in Fig. 5 (a). If there exists any temperature dependence in $\lambda$, further studies will be needed to confirm that. Given that the values quoted in Figs. 5 (a) and 5 (b) differ from each other by a little more than $5\%$, we are not inclined to make a comment on this issue. Since with the variation of temperature we expect only change in the amplitude of growth law, it will be surprizing if the aging exponent changes due to the variation in amplitude only. Nevertheless, we do not rule out such a possibility, though very unlikely.

The fact on the full form can be further appreciated from the plot of instantaneous exponent das2014 ; huse

[TABLE]

versus $1/x$, in the insets of Fig. 5. The linear behavior in these data sets, when incorporated in Eq. (12), in fact provides Eq. (9). Extrapolation of these data to $x=\infty$, discarding the finite-size affected parts, also lead to essentially the same values of $\lambda$ that we have obtained from the FSS analyses. The quoted values of $\lambda$ for this morphology satisfy the YRD bound $3.5$. Note here that for percolating structure in $d=3$, value yeung1988 of $\beta$ is $4$.

IV Conclusion

We have presented results on aging phenomena, in the context of vapor-liquid phase transitions, from molecular dynamics simulations of a single component Lennard-Jones fluid das2011 ; das2012 ; roy2012 . Effects of hydrodynamics on aging have been investigated for two different coarsening mechanisms: (i) growth via advective transport through tube-like liquid domains in interconnected morphology; (ii) equilibration through diffusive coalescence process that occurs in disconnected pattern where isolated liquid droplets are immersed in the matrix of the vapor phase.

We have quantified the aging dynamics via calculation of the two-time order-parameter autocorrelation function wadhawan ; fisher1988 $C_{\text{ag}}(t,t_{w})$, where $t$ and $t_{w}$ are respectively the observation and waiting times. It has been demonstrated that in both the cases $C_{\text{ag}}(t,t_{w})$ exhibits nice scaling with respect to $\ell/\ell_{w}$, the ratio of characteristic lengths at $t$ and $t_{w}$. For the disconnected domain pattern very robust power-law decay emerges from the beginning and the estimated value of the exponent $\lambda$ is $\simeq 7$.

On the other hand, for the connected case there is no simple power-law, at least for $\ell/\ell_{w}<\infty$. Even though standard method of analysis indicates that the decay could be exponentially fast, an advanced analysis via a recently devised finite-size scaling technique das2014 ; das2018 is supportive of an asymptotic power-law decay. For small values of $\ell/\ell_{w}$, this method suggests, there exist corrections that can be consolidated in an exponential factor, similar to the observations in studies related to ordering in ferromagnets das2014 ; das2018 and phase separation in solid binary mixtures das2015 .

Interestingly, the exponent $\lambda\simeq 7$, for the droplet morphology, even though differs from the interconnected counterpart of the present work, is in nice agreement with that for solid binary mixture das2015 in the same space dimension $d=3$. In the latter case, albeit due to a different mechanism, the growth exponent $\alpha$ is the same as the disconnected morphology part of this work. Nevertheless, one should be careful about drawing conclusion on expectation of similar values of $\lambda$ if $\alpha$ remains same. Value of $\alpha$ remains unaltered bray ; lifshitz for solid binary mixtures even in $d=2$. But we have observed a drastically different value of $\lambda$ there das2015 . Furthermore, the growth exponent for the percolating morphology case here is in nice agreement with certain active matter systems belmonte ; das2017jcp . However, aging exponents differ, despite being in the same space dimension das2017jcp . Thus, the universality in the value of $\lambda$ is weaker than $\alpha$ and the issue is more complex. Nevertheless, it appears that at least the asymptotic power-law behavior is rather universal with respect to the decay of autocorrelation in nonequilibrium processes. Also, interestingly enough, the pre-asymptotic corrections are nicely summed up in an exponential factor for apparently different systems. These include coarsening in ferromagnets, solid binary mixtures, and now in fluids.

For both the morphologies, the Fisher-Huse (FH) lower bound is satisfied. Of course, the Yeung, Rao and Desai bounds appear stricter. Nevertheless, for disconnected morphology the value of $\lambda$ is far higher than any of the bounds. From the construction of the lower and higher bounds of FH it is expected that the value of $\lambda$ should be smaller for faster growth. Nevertheless, drawing general conclusion is difficult, given that both structure and dynamics play important roles that are till now unclear to us. It perhaps makes better sense to compare the percolating morphology outcome of this work with that of solid binary mixtures das2015 in $d=3$. This is because in both the cases one has similar structure, viz., value of $\beta$ is $4$. In that case we should expect a slower decay of the autocorrelation in fluids. This indeed is the case, making the conclusion of the work more plausible. Furthermore, in the above mentioned active matter problem das2017jcp even though the growth exponent and space dimension are same as the percolating case of this work, $\lambda$ is smaller in the active matter case. This could probably be justified by the fact that $\beta$ in the latter case is smaller. Another general trend that we observe is that for same growth mechanism and structure, with the increase of space dimensionality value of $\lambda$ increases. To make general conclusions from these observations we need to study more cases and require appropriate theoretical considerations.

An important question one may ask here: whether there will be a jump from $\lambda\simeq 7$ to $\simeq 4$, or this change will be realized in a continuous manner with the increase of density. Answer to this question, we believe, will be decided by whether the growth exponent $\alpha$ has a smooth dependence on density or not. As long as the density is less than the percolation threshold, the growth will occur via coalescence mechanism and the value of $\alpha$ will be $1/3$, keeping $\lambda$ high. However, with the increase of density it is possible that the onset of disconnected morphology will be delayed. As density increases towards the percolation value, initially there may be random non-spherical structures, providing the possibility of connectedness at early time. This feature will go away at late time, as long as the value of density remains below the above mentioned threshold.

The scaling method without the need for performing simulations with different system sizes is important. This becomes possible due to the fact that for different values of $t_{w}$ (or $\ell_{w}$) different effective system sizes are available for the relaxation to occur. In this sense the ratio $\ell_{w}/L$, rather than $L$, is only important. This outcome certainly helps avoiding time consuming simulations like molecular dynamics for fluid phase separation.

Acknowledgement: Parts of the simulation results are obtained by using the package LAMMPS lammps . AB acknowledges research fellowship from the Council of Scientific and Industrial Research, India.

$*$das@jncasr.ac.in