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

Sutapa Roy1,2, Arabinda Bera3,4, Suman Majumder3,5 and Subir K. Das3,4,∗ 1
Max-Planck-Institut für Intelligente Systeme, Heisenbergstr. 3, 70569
Stuttgart, Germany
2 Institut für Theoretische Physik IV, Universität Stuttgart, Pfaffenwaldring
57, 70569 Stuttgart, Germany
3 Theoretical Sciences Unit, Jawaharlal Nehru Centre for Advanced Scientific
Research, Jakkur P.O, Bangalore 560064, India
4 School of Advanced Materials, Jawaharlal Nehru Centre for Advanced
Scientific Research, Jakkur P.O, Bangalore 560064, India
5 Institut für Theoretische Physik, Universität Leipzig, Postfach 100920,
D-04009, Leipzig, Germany

###### 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.

##  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 :

$\displaystyle\ell(t)\sim t^{\alpha}.$ (1)

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

$\displaystyle
C_{\text{ag}}(t,t_{w})=\langle\psi(\vec{r},t)\psi(\vec{r},t_{w})\rangle-\langle\psi(\vec{r},t)\rangle\langle\psi(\vec{r},t_{w})\rangle,$
(2)

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

$\displaystyle
C_{\text{ag}}(t,t_{w})\sim\left(\frac{\ell}{\ell_{w}}\right)^{-\lambda}.$ (3)

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

$\displaystyle\frac{d}{2}\leq\lambda\leq d.$ (4)

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

$\displaystyle
U(r)=4\varepsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right].$
(5)

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

$\displaystyle u(r)=U(r)-U(r_{c})-(r-r_{c})\frac{dU}{dr}\Big{|}_{r=r_{c}}.$
(6)

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

$\displaystyle\ell=\int P(\ell_{d},t)~{}\ell_{d}~{}d\ell_{d},$ (7)

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

$\displaystyle C(r=\ell,t)=f,$ (8)

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 .

Figure 1: Representative snapshots obtained during the evolutions of the
single component Lennard-Jones systems, following quenches of high temperature
homogeneous configurations to $T=0.6$. We have presented pictures or two
different overall densities, viz., $\rho=0.08$ (left frame) and $0.3$ (right
frame), both of which, for the chosen temperature, correspond to state points
inside the vapor-liquid coexistence curve. The times at which the snapshots
were recorded are mentioned in appropriate places. For both the cases we have
chosen $L=64$. The quantitative results in each of the cases will be presented
for bigger system sizes.

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.

Figure 2: Plots of average domain size, $\ell$, versus time, on a log-log
scale, for (main frame) droplet ($\rho=0.08$) and (inset) percolating
($\rho=0.3$) morphologies. The data for $\rho=0.08$ are from $T=0.6$ and for
$\rho=0.3$ the temperature is $T=0.7$. The continuous lines represent power-
laws, exponents of which are mentioned. Here and in subsequent figures all
results for $\rho=0.08$ and $0.3$ are presented from $L={{200}}$ and $100$,
respectively, except for $(\rho,T)=(0.3,0.6)$. In the latter case we have used
$L=96$. The arrow in the inset marks the onset of finite-size effects.

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.

Figure 3: (a) Plots of the autocorrelation function, $C_{\text{ag}}(t,t_{w})$,
versus $t-t_{w}$, for the droplet morphology. Results from a few different
values of $t_{w}$ are presented. (b) Here we have plotted the correlation
functions, on a log-log scale, versus $t/t_{w}$. The continuous line
represents a power-law decay, exponent for which is mentioned in the figure.
(c) Same as (b), but here the results are presented versus $\ell/\ell_{w}$.
Again, the continuous line is for a power-law decay. All results correspond to
$T=0.6$.

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.

Figure 4: (a) Log-log plot of $C_{\text{ag}}(t,t_{w})$ as a function of
$\ell/\ell_{w}$, for a few values of $t_{w}$ that are mentioned in the figure.
These results are for the percolating morphology, i.e., from $\rho=0.3$, with
$T=0.7$. (b) Same as (a), but in this part we have shown data for only one
value of $t_{w}$, on a semi-log scale. The solid line here represents an
exponential decay.

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

$\displaystyle
C_{\text{ag}}(t,t_{w})=A\text{exp}\left(\frac{-B}{x}\right)x^{-\lambda};~{}x=\ell/\ell_{w},$
(9)

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

$\displaystyle
Y=C_{\text{ag}}(t,t_{w})\text{exp}\left(\frac{By}{y_{w}}\right)y_{w}^{\lambda},$
(10)

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

$\displaystyle Y\sim y^{\lambda}.$ (11)

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 $B$.

Figure 5: (a) Finite-size scaling plot of the autocorrelation data for the
percolating morphology. Here we have presented data from $T=0.7$. The solid
line represents a power-law. Onset of finite-size effects has been marked by
an arrow. Inset: Plot of instantaneous exponent $\lambda_{i}$ as a function of
$\ell_{w}/\ell$. The solid line is a guide to the eye. (b) Same as (a) but for
$T=0.6$.

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

$\displaystyle\lambda_{i}=-\frac{d~{}\text{ln}C_{\text{ag}}(t,t_{w})}{d~{}\text{ln}x},$
(12)

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.

<EMAIL_ADDRESS>
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