# Kinetics Of Vapor-Liquid And Vapor-Solid Phase Separation Under Gravity

Daniya Davis  Bhaskar Sen Gupta<EMAIL_ADDRESS>Department of
Physics, School of Advanced Sciences, Vellore Institute of Technology,
Vellore, Tamil Nadu 632014, India

###### Abstract

We study the kinetics of vapor-liquid and vapor-solid phase separation of a
hydrodynamics preserving three-dimensional one component Lennard Jones system
in the presence of external gravitational field using extensive molecular
dynamic simulation. A bicontinuous domain structure is formed when the
homogeneous system near the critical density is quenched inside the
coexistence region. In the absence of gravity, the domain morphology is
statistically self-similar and the length scale grows as per the existing
laws. However, the presence of gravity destroys the isotropy of the system and
affects the scaling laws. We observe an accelerated domain growth in the
direction of the field at late time which resembles sedimentation process.
Consequently, a new length scale emerges which strongly depends on the field
strength. Similar behavior is observed in the direction perpendicular to the
applied field, with a different growth rate. Finally, the validity of Porod’s
law and Superuniversality in such anisotropic systems is verified in terms of
two-point equal time order parameter correlation function and static structure
factor.

††preprint: APS/123-QED

##  I Introduction

If a homogeneous system is cooled rapidly within the miscibility gap, it
becomes thermodynamically unstable and gradually separates into particle-rich
and particle-poor domains. The separated regions continue to grow over time
until the system reaches a state of local equilibrium or saturation [1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. This process of phase separation has
numerous applications in various fields such as material science,
biochemistry, chemical engineering [14, 15, 16, 17, 18], and the petroleum
industry [19, 20, 21]. It is involved in the separation of various chemicals,
such as the distillation of crude oil, as well as in the extraction of oil,
natural gases and other resources. Phase separation also plays a crucial role
in regulating the morphology and mechanical characteristics of polymers during
their production.

Gaining a comprehensive understanding of the non-equilibrium evolution that
occurs during phase separation in a system holds significant importance and
attracts substantial research interest, both in theoretical and experimental
domains.

It is now well accepted that the domain growth pattern or domain morphology is
characterized by a solitary length parameter that varies with time, denoted as
$\ell(t)$ [18]. This length is obtained from the equal time correlation
function $C(\vec{r},t)$, where $\vec{r}$ is the distance between two spatial
points and t is the time after quench. The average domain size of the system
follows the power law $\ell(t)\sim t^{\alpha}$, where $\alpha$ is the
universal growth exponent [2]. The exponent value is determined by the
corresponding coarsening mechanism that drives phase separation.

The growth behavior observed in a specific system depends on various factors,
including temperature, composition, interfacial tension, and molecular
interactions. In the phase separation process involving solid mixtures,
diffusion of particles dominates and the corresponding growth exponent
$\alpha=1/3$, predicted by Lifshitz and Slyozov [18, 22]. But in the liquid-
liquid or vapor-liquid phase separation, hydrodynamic effect plays a
significant role and the growth exponent changes accordingly. In such systems
the diffusive phase is very transient and we observe a quick transition to the
viscous hydrodynamic regime with $\alpha=1$. Finally, at late times the system
enters into inertial hydrodynamics where the exponent is $\alpha=2/3$ [2, 3].

Numerous studies dedicated their focus on the phase separation of vapor-liquid
systems at both near critical as well as off-critical densities and the
subject is now well understood [23, 24, 25, 26]. Even though infrequently but
occasionally studies have been conducted on the vapor-solid systems and the
kinetics at various densities have been explored [27, 28]. However, the
nonequilibrium phenomena of such systems in the presence of an external field
such as gravity is still in its infancy. Gravitational effect becomes
important for situations when heavier domains situated atop lighter ones
become unstable, leading to an acceleration in domain growth. In both the
systems, one of the phases is heavier than the other as differentiated by
gravity. Here, we investigate the spinodal decomposition and the phase
ordering in vapor-liquid and vapor-solid phase separation under the influence
of gravitational force field. More specifically, our main objective is to
study the effect of gravity on the domain growth laws in different regimes
mentioned above. Finally, we study the final state of the system, where the
two phases are completely separated with an interface between them.

Fewer attempts were made to understand the effect of gravity using theory and
computer simulations. For example, numerical study on the modified Cahn-
Hilliard equation constructed from the master equation approach was carried
out in two-dimension [29, 30]. An accelerated domain growth was found in the
direction of applied gravitational field, and the growth was linear in time.
The thermal fluctuation and hydrodynamic effects were not taken into account
in this work. In another study, the front and domain growth was studied from
the time-dependent Ginzburg-Landau equation with gravity [31]. The diffusion
coefficient was chosen to be concentration dependent. The growth exponents
were estimated using scaling arguments and was compared with the numerical
results in two dimension. The coarsening dynamics of the binary liquid mixture
with density mismatch between the domains was studied numerically by solving
the coupled Cahn–Hilliard/Navier–Stokes system of equations in the presence of
gravity [32]. The outcome demonstrated that even a slight density mismatch can
have a notable impact on scaling laws due to gravitational effects. The
coarsening was faster in the direction of applied field, and multiple length
scale emerged in this process. Recently, the vapor-liquid phase separation of
a van der Waals fluid described by the Navier-Stokes and the continuity
equation subjected to gravitational force was studied numerically by applying
the lattice Boltzmann method in two dimension. The domain growth along the
gravity direction was characterized by the exponent $\alpha=1$. Surprisingly,
the growth exponent was found to be insensitive to the strength of
gravitational field [33].

To the best of our knowledge, there is no atomistic simulation work on the
kinetics of vapor-liquid phase separation of systems exposed to gravitational
field. On the other hand, the vapor-solid phase separation with gravity
remains completely unexplored. In this paper, we undertake molecular dynamics
simulation study of vapor-liquid and vapor-solid phase separation in a one
component fluid in three dimension with gravity. We consider a more realistic
model from the experimental prospective where hydrodynamics is naturally
incorporated in the system. In the presence of gravitational field, the domain
growth is expected to be anisotropic. The main focus of this paper is to
examine in detail the anisotropic scaling laws at different time regimes under
the influence of various external field strength.

##  II Models And Methods

For the present study, we consider a single component system of volume $V$ in
three-dimension, consisting of $N$ particles interacting via the Lennard Jones
(LJ) interaction potential given by,

$U(r)=4\epsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right]$
(1)

Here $\epsilon$ denotes the interaction strength, $\sigma$ is the particle
diameter and $r=|\vec{r}_{i}-\vec{r}_{j}|$ is the scalar distance between the
two particles $i$ and $j$. To improve computational efficiency the range of
interaction is truncated at $r=r_{c}=2.5\sigma$. The discontinuity in the
potential and the corresponding force due to the insertion of this cut off is
resolved by modifying the potential as follows,

$u(r)=U(r)-U(r_{c})-(r-r_{c})\left(\frac{dU}{dr}\right)|_{r=r_{c}}$ (2)

The distance and temperature are measured in units of $\sigma$ and
$\epsilon/k_{B}$ respectively, where $k_{B}$ is Boltzmann’s constant. For
computational convenience mass of each particle, $\epsilon$, $\sigma$, $k_{B}$
are set to unity. The critical temperature and critical density for this model
exhibiting vapor-liquid transition are $T_{c}=0.94$ and $\rho_{c}=0.32$
respectively [24].

We consider an elongated box with dimensions $L_{x}\times L_{y}\times
L_{z}=72\sigma\times 72\sigma\times 144\sigma$ having reflecting walls at the
top and bottom in the z direction. Periodic boundary conditions are applied
along the horizontal directions. The simulation box consists of 223948
particles which correspond to the near critical density of $\rho=0.3$.

The coarsening dynamics of the system is studied using molecular dynamics
simulation (MD) performed in the canonical ensemble ($NVT$). Since we are
dealing with fluids, it is preferable to use a hydrodynamics preserving
thermostat to keep the temperature at the desired value. Therefore, throughout
the current study, we use Nose-Hoover Thermostat which is well known for
temperature controlling and at the same time preserving the hydrodynamics of
the system [34]. Velocity-Verlet algorithm is used to integrate the equation
of motion for each particle that evaluates the position and velocity during
each time step of the MD simulation [35]. The time step is chosen as $\Delta
t=0.005$, where time is expressed in units of
${(m\sigma^{2}/\epsilon)}^{1/2}$.

Figure 1: The representative snapshots of our phase separating vapor-liquid
system at $T=0.8$ in the absence of external gravitational field. The blue
dots represent the particles.

The system is first equilibrated at a high temperature of $T=10$ to get a
homogeneous configuration of uniform density. It is then quenched to $T=0.8$,
well below the critical temperature $T_{c}$ of the vapor-liquid transition to
obtain the vapor-liquid system. For the study of vapor-solid phase separation,
the system is quenched to $T=0.1$. A spatially uniform external gravitational
force $g$ is applied downward in the z-direction after the quench.

Representative configurations obtained from the MD simulation with time, shown
in Fig. 1 demonstrate the evolution morphology of the vapor-liquid system
under consideration. The formation of percolating domains clearly suggest that
the system is near the critical density. As discussed in the previous section,
the standard parameter to analyze the phase ordering dynamics is the average
lengthscale $\ell(t)$. To compute $\ell(t)$, we introduce the two point equal-
time correlation function defined as

$C(\vec{r},t)=\langle\psi(0,t)\psi(\vec{r},t)\rangle-\langle\psi(0,t)\rangle\langle\psi(\vec{r},t)\rangle$
(3)

where the angular brackets denote statistical averaging and $\psi(\vec{r},t)$
is the order parameter calculated by mapping the system to Ising model.
$\psi(\vec{r},t)$ is assigned the value +1 if the local density over a box
size of $(2\sigma)^{3}$ located at $\vec{r}$ is greater than the critical
density $\rho_{c}$ and -1 otherwise [37, 36]. The lengthscale $\ell(t)$ is
estimated from the first zero value of the correlation function
$C(r=\ell,t)=0$. Of course, there are other methods to calculate the domain
size of the system like inverse of first moment of the structure factor,
domain size distribution of the system, etc. The Fourier transform of the
correlation function gives the structure factor given by

$S(\vec{k},t)=\int d\vec{r}\ exp(i\vec{k}.\vec{r})\ C(\vec{r},t)$ (4)

The presence of external gravitational force results in the anisotropic growth
of the domains in different spatial directions. Hence the correlation function
is calculated separately in each spatial direction [38] rather than being
spherically averaged.

##  III Results

In this section we discuss the results for both the vapor-liquid and vapor-
solid phase separating systems influenced by gravity.

###  III.1 Vapor-Liquid phase separation

We focus on the isothermal vapor-liquid phase separation triggered by a sudden
thermal quench below the critical temperature $T_{c}$ in the presence of an
external gravitational field. We begin our analysis with the zero gravity case
which is considered as a reference point. In Fig. 1 we show the time evolution
of the system at $T=0.8$, near the critical density $\rho=0.3$ without any
external field ($g=0$). During the coarsening process, the system exhibits a
bi-continuous domain morphology. Eventually, a complete transition into vapor
phase and liquid phase is observed. Note that the order parameter
$\psi(\vec{r},t)$ is the prime measure to distinguish between the vapor and
liquid phase.

Figure 2: The morphological evolution of the vapor-liquid system with time
under the influence of gravity value $g=0.003$ is shown in the left column. In
the right column, we show the same results for $g=0.05$.

Next we investigate the effect of gravity on the coarsening process of our
system. Results are presented for a chosen set of $g$ values in the range
(0.003, 0.05). Fig. 2 shows the representative snapshots for the domain
evolution of the systems for $g=0.003$ and 0.05. From Fig. 2 it is conspicuous
that the onset of phase separation occurs immediately after quenching, and the
domains evolve with time. Here, along with surface tension and viscous forces,
the gravitational field also plays an important role in determining the domain
morphology and coarsening behavior. Due to coarsening process, particle rich
liquid phase and particle deficit vapor phase grow with time. This facilitates
the gravitational force to overcome the interfacial tension, and the denser
phase moves downward due to gravity, analogous to sedimentation. Eventually a
single interface between the lighter and heavier phase is formed.

A systematic change in the rate of structural evolution of the system is
observed with increasing gravitational force. At low $g$ values the rate of
phase separation as well as sedimentation is much slower. This is finely
evident in Fig. 2 for $g=0.003$. Contrary to this, for higher $g$ values, the
system phase separates at a faster rate and the denser phase accumulates at
the bottom of the simulation box by creating completely distinguishable liquid
and vapor domains.

It is also evident that the completely separated phases at long time are
affected by the external field. For better understanding we calculate the
density of the vapor and the liquid phase separately. In Fig. 3 we show the
distribution of the densities P($\rho$) for the fully separated vapor and
liquid phase for all $g$ values. For better visualization, the peak of the
distributions is normalized to 1. The peak position of the distribution
represents the average density. When the strength of the gravitational field
is low, the vapor and liquid phases are closer to the zero gravity case. But
as the field strength in increased, we observe the density of the vapor phase
gradually decreases and the liquid phase becomes tightly packed. The average
vapor and liquid phase density for the $g=0$ case is 0.02 and 0.71
respectively. When the gravitational field is present on the system, all the
particles experience an additional force towards the bottom. As the strength
of the field is increased, the gravitational pull becomes stronger and hence
more particles are attracted towards the bottom and become the part of liquid
phase making the domain denser. For $g=0.05$, which is the highest
gravitational strength chosen in our simulation, the average vapor and liquid
phase density are found to be 0.0001 and 0.8 respectively.

Figure 3: The density distribution profile for the fully separated vapor and
liquid phases are shown in (a) and (b) respectively.

Figure 4: (a) The scaling plots of $C(r_{z},t)$ vs $r_{z}/\ell_{z}(t)$ for
$g=0.01$ and 0 at different times are shown in the main figure and inset
respectively. (b) The same scaling plots are shown for different $g$ values at
a fixed time.

To characterize the domain morphology and study the domain growth we compute
the two-point equal time correlation function $C(r_{z},t)$ along the direction
of gravity, from Eq. 3. The same is plotted alongside with scaled length
$r_{z}/\ell_{z}(t)$ in Fig. 4(a) for $g=0$ and 0.01. An excellent data
collapse is observed for different times. This exercise is repeated for the
other $g$ values (not shown) and the quality of overlap is found to be the
same. This is the well known Porod law and remains vindicated in the presence
of gravity [39]. This also suggests that even in the presence of gravity, the
system belongs to the same dynamical universality class [22]. Further, we find
the correlation functions corresponding to different $g$ values overlap with
each other when plotted at a fixed time. This is shown in Fig. 4(b).
Therefore, the superuniversality (SU) property applies to our system [40, 41].

To gain a qualitative understanding of the growth of average domain size with
time it is sufficient to calculate the length scale $\ell(t)$. Due to the
expected anisotropic growth in the presence of gravity, we compute the length
scale separately in all three directions. We begin with the z-component of the
length scale, $\ell_{z}(t)$ along which the gravitational field is applied.
The time dependence of $\ell_{z}(t)$ is shown in Fig. 5 for different $g$
values. Our simulation is able to access the viscous hydrodynamic regime after
an initial transient period. In the absence of gravity $(g=0)$ we expect a
linear behavior of $\ell_{z}(t)$. The slight deviation from this can be
attributed to the nonzero offsets at the crossovers, which can be subtracted
from $\ell(t)$ to recover the proper linear growth [42]. A dramatic change in
the growth behavior is observed under the influence of gravity. At the early
stage and $g>0$, the $\ell_{z}(t)$ exhibits linear growth, typical of the
viscous regime and independent of $g$ value, suggesting that a universal
scaling exists in the gravity direction. After that, a crossover to a
considerably faster growth ($\alpha>1$) regime is observed. This crossover
happens at an earlier time for the systems with higher $g$ value. This is
consistent with the observations in Fig. 2. The fastest growth rate observed
in our simulations corresponds to $\alpha=3$.

This accelerated domain growth in the presence of external drive can be
comprehended as follows. After the initial coarsening the liquid domains
become moderately larger and heavier. A transition to gravity induced flow
arises when heavy domains become unstable, i.e. the gravitational force on the
denser phases overcomes the interfacial tension that keeps them suspended.
This phenomenon can be likened to sedimentation. Once the instability sets in,
additional length scales emerge due to accelerated growth in the direction of
gravity.

Figure 5:  The time evolution of the average domain size $\ell_{z}(t)$ are
shown for different $g$ values. The dashed line represents the guideline for
the slope.

Figure 6: The time evolution of the average domain size $\ell_{x}(t)$ and
$\ell_{y}(t)$ for different $g$ values are shown in (a) and (b) respectively.
The dashed lines represent the guideline for the slope.

Next we focus on the characteristic domain growth in the horizontal
directions. Due to symmetry in the system, the lengthscale in the x and y
directions is expected to analogous. These results are shown in Fig. 6. For
the $g=0$ case, the growth follows approximately a linear behavior. In the
presence of gravity ($g>0$), again we see an accelerated growth ($\alpha>1$)
after a transient period.

To gain a comprehensive understanding of the relative growth rate in different
directions, we compute the instantaneous slope of the length scale defined as
$\alpha_{\mathrm{I}}=d(\mathrm{ln}\ell(t)/d(\mathrm{ln}t)$ for the highest
gravity value of $g=0.05$ [43]. The results are shown in Fig. 7. The exponent
is found to be higher in the z direction at the intermediate time. Therefore,
growth rate is faster along the direction of gravity and the system is
anisotropic. Over the interim period, the exponent experiences a gradual
increase. This phenomenon can be attributed to the expanding size of the
liquid domains over time, wherein the gravitational force becomes increasingly
influential, thereby prompting accelerated growth. At late time, the growth
exponent converges to the value $\alpha\sim 3.0$ for all the directions and
the system restores the isotropic nature.

The intermediate accelerated growth rate in the z direction can be understood
in the following way. Subsequent to the initial phase separation period,
denser domains commence precipitating at the lower portion of the simulation
box. This circumstance leads to a faster growth rate in the vertical direction
compared to the horizontal direction. As the sedimentation process nears its
conclusion, the liquid domains spread horizontally, thereby hastening the
growth of domains along the x and y directions. Eventually, the growth rate
becomes uniform in all dimensions and the system regains its isotropic
characteristics.

Figure 7: The instantaneous slope of the length scale vs time for $g=0.05$.

For further quantification of the structural details we focus on the structure
factor given by Eq. 4. The structure factor $S(k,t)$ is has a scaling form,

$S(\vec{k},t)\equiv\ell^{d}\tilde{S}(\vec{k}\ell(t))$ (5)

in systems with self-similar patterns where $\tilde{S}$ is another time-
independent master scaling function. The power-law behavior observed in the
tail of $S(\vec{k},t)$ has the following form,

$S(\vec{k},t)\sim k^{-(d+1)}$ (6)

where d is the dimension of the ordering system. The $k$-dependence of Eq. 6
is referred to as the Porod’s law [22]. In Fig. 8 we show the scaled structure
factor $S(k_{z},t)\ell_{z}^{-3}$ vs $k_{z}\ell_{z}$ for different $g$ values.
We can see an agreeable data collapse from the figure. The dashed line
represents the power law $k_{z}^{-4}$. The strong agreement of our large
$k_{z}$ data with the Porod’s law is clearly evident. These results are
consistent with the observations in Fig. 4b and substantiate the validation of
superuniversality.

Figure 8: The scaled structure factor $S(k_{z},t)\ell_{z}^{-3}$ vs
$k_{z}\ell_{z}$ is plotted for different $g$ values. The straightline is a
guideline for the Porod law $S(k_{z})\sim k^{-4}$.

###  III.2 Vapor-Solid phase separation

In this subsection we focus on the kinetics of vapor-solid phase separation
under the influence of gravity. We begin the simulation by preparing a
homogeneous system at high temperature of $T=10$. The vapor-solid phase
separation is initiated by quenching the system at density $\rho=0.3$ to a
very low temperature $T=0.1$ at time $t=0$. The sufficiently low temperature
guarantees solid phase formation [28]. The system is then allowed to evolve
via MD simulation. The time evolution of our phase separating system under the
influence of a chosen gravitational field is shown in Fig. 9. For better
visualization of the solid phase formation, we show the surface mesh plot
rather than the particles. The interconnected solid domains formation is
clearly observed in the figure. The effect of gravity forces the domains to
move downward. As we analyze the evolution snapshots under various
gravitational strengths (not presented here) it is observed that the rate of
sedimentation varies according to the applied field strength.

Figure 9: The representative snapshots for the vapor-solid phase separating
system in the presence of external gravitational field of $g=0.003$.

A qualitative understanding of domain growth is obtained from the
characteristic lengthscale $\ell(t)$. Due to the anticipated anisotropy in the
system, the lengthscale is calculated separately in the gravity direction and
the horizontal direction, as elaborated in the preceding section. The
$\ell(t)$ is calculated from the domain length distribution in a particular
direction, a conventional approach employed in the dynamics of phase ordering
[25, 44, 45, 46].

Figure 10: The time evolution of the average domain size $\ell_{z}(t)$ and
$\ell_{x}(t)$ for different $g$ values are shown in (a) and (b) respectively.
The dashed lines represent the guideline for the slope.

In Fig. 10a we show the time variation of $\ell_{z}(t)$ in the z direction for
a chosen set of $g$ values mentioned in the inset. According to Lifshitz-
Slyozov law, in solid mixtures the spinodal decomposition takes place through
the transport mechanism of diffusion and hence the growth exponent is
$\alpha=1/3$. In the absence of gravity, we observe that the growth exponent
aligns with the Lifshitz-Slyozov law. However, when the influence of gravity
comes into play, the growth dynamics undergo a dramatic change. Similar to
vapor-liquid phase separation, following the initial transient period where
the growth law is unaffected by gravity, a crossover to an accelerated growth
is observed. The Fig. 10a illustrates that the domain growth exponent exhibits
a systematic variation with increasing field strength. The increase in the
growth exponent stems from the collective motion of domains in the field
direction, leading to the rapid growth of domains.

An important point to note here, distinct from the previous section is that
the saturation length scale increases systematically. This can be comprehended
as follows. The snapshots displayed in Fig. 9 clearly demonstrate that, over
an extended period, the solid domains remain bicontinuous, and a complete
phase separation into a single solid domain is never attained. On the other
hand, there exists bicontinuous vapor phases as well, in between the solid
domains. With increasing field strength the solid domains undergo a
compressive force towards the bottom of the system. As a result the
bicontinuous solid domains compactify and coalesce into a more interconnected,
larger-sized domain. Consequently, with gravity the saturation length scale
increases.

For the horizontal direction, we show the length scale in Fig 10b in the x
direction. Due to the obvious symmetry in the system, we abstain from showing
results in the y direction. In the zero-gravity scenario, as anticipated, the
growth exponent is found to be $\alpha=1/3$. Similarly, in the initial regime
of finite gravity cases, the exponent is expected to remain the same. With
time, as the domains increase in size, the influence of gravity becomes more
pronounced. In the horizontal direction, the growth of the domain also
escalates gradually, owing to the larger size of solid domains in the overall
direction, not limited to the gravitational axis alone. Consequently, an
accelerated domain growth is observed.

Figure 11: The instantaneous slope of the length scale vs time for $g=0.05$.

To quantify the growth rate, we show in Fig. 11 the instantaneous slope of the
length scale $\alpha_{\mathrm{I}}=d(\mathrm{ln}\ell(t)/d(\mathrm{ln}t)$
associated with the highest field strength $g=0.05$. A faster growth rate is
discernible in the direction of gravity at the intermediate stage. It is
noteworthy that the growth rate in this scenario surpasses that observed in
vapor-liquid phase separation influenced by gravity. This disparity arises due
to the increased weight of the solid domains compared to the liquid domains.

Figure 12: The scaling plots of $C(r_{z},t)$ vs $r_{z}/\ell_{z}(t)$ for
$g=0.01$ at different times are shown in (a). In (b) the same scaling plots
are shown for different $g$ values at a fixed time $t=100$.

The meaningful quantification of growth dynamics and extraction of the
mentioned length scale depend on the satisfaction of the scaling property by
the correlation function $C(r,t)$. In Fig. 12a, we present a scaling plot of
$C(r,t)$ at a selected value of $g=0.01$ for various times. The result clearly
illustrates the scaling behavior of the correlation functions in the region
$r_{z}/\ell_{z}<1.0$, indicating that the domain growth is inherently self-
similar. The lack of scaling in $C(r,t)$ at larger distance is attributed to
the asymmetry in the composition concerning negative and positive values of
the order parameter. This asymmetry increases with time as the domains
compactify in the presence of gravity. We performed the same analysis for
different $g$ values (not depicted here), and consistently observed a similar
behavior, reinforcing the validity of the length scale analysis.

Furthermore, to assess the validity of Porod law and superuniversality in our
system, we plot in Fig. 12b the $C(r,t)$ for all the chosen g values at a
fixed time. For zero gravity case, the correlation function exhibits a linear
behavior at small distance as $C(z,t)\simeq 1-az+\cdots$ due to the scattering
from sharp interfaces, popularly known as Porod law. Therefore, the scaling of
$C(r,t)$ displayed in Fig. 12b showing a linear trend for $r_{z}/\ell_{z}<1.0$
affirms the validity of Porod law in the presence of gravity. Additionally,
the overlap in the correlation functions for different g values confirms the
existence of superuniversality in the system. Note that, with increasing field
strength, the asymmetry in the order parameter increases, resulting in the
lack of scaling at large distance.

Figure 13: Color maps of the height fields for the vapor-solid interface with
respect to its mean for different $g$ values. The colors represent the local
interface height in relation to its mean.

Finally, we explore the influence of gravity on the vapor-solid interface at
the final stage of the system that has undergone phase separation over a long
period. In contrast to vapor-liquid phase separation, where the two phases
exhibit a smooth interface, in the current scenario, with the accumulation of
the solid phase at the bottom, the interface takes on a rough texture. The
gravitational force is anticipated to play a role in modulating the structure
of this interface. The resulting height variation is shown in Fig. 13 using a
color map. Here we show the top view of the typical interfaces for all the $g$
values under consideration. For the low gravity case a pronounced roughness
across the interface is visible. As the gravitational strength is increased
the color map becomes homogeneous. This is a clear indication that the height
fluctuation decrease with increasing gravity strength, symbolizing a reduction
in the surface roughness.

To quantify the interface roughness, we compute the standard deviation of the
surface height distribution given by

$w=\sqrt{\langle h(x,y)^{2}\rangle-\langle h(x,y)\rangle^{2}}$ (7)

The results are shown in Fig. 14 with gravity. The interface roughness
diminishes with a rise in gravitational strength, aligning with a power-law
relationship of $g^{-1/4}$. Therefore, the power-law behavior suggests that
the roughness of the interface approaches zero as gravitational strength
becomes exceedingly large.

Figure 14: The standard deviation of the vapor-solid interface roughness as a
function of gravity. The dotted line is the guideline for the slope.

##  IV Summary and Discussion

In summary, we have employed molecular dynamics simulations to study the
effect of gravity on the kinetics of vapor-liquid as well as vapor-solid phase
separation of a one component Lennard Jones fluid. The phase separation was
initiated by the thermal quench of the initially prepared high temperature
homogeneous fluid inside the coexistence region. The hydrodynamics of the
fluid was preserved by using Nose-Hoover thermostat. The density of the liquid
being close to the critical value, we observe a bicontinuous domain structure
of all the phases. To characterize the domain morphology and study the domain
growth we resort to the two-point equal time order parameter correlation
function. In the absence of gravity, our systems clearly exhibit the domain
growth law consistent with the existing literature. However, when the systems
are subjected to gravity, they become anisotropic, and a faster domain growth
is observed along the direction of the field. The growth mechanics resemble
the sedimentation process. After an initial period, under the influence of
gravity, the scaling law breaks down and a new length scale emerges which
strongly depends on the field strength. We examined the validity of the Porod
law and superuniversality in terms of the correlation function and the static
structure factor. Our results demonstrated that the Porod’s law and
superuniversality remained vindicated in the presence of gravity. Finally, we
studied the equilibrium state of the phase separated systems at long time. At
this point, the complete understanding of the scaling exponents with
gravitational field is missing and it will certainly be worthwhile to
investigate. Our model can easily be extended to study the effect of gravity
on the coarsening dynamic in many other segregating systems e.g. liquid-liquid
phase separation, and it will be worth exploring these.

### Acknowledgements

B. Sen Gupta acknowledges Science and Engineering Research Board (SERB),
Department of Science and Technology (DST), Government of India (no.
CRG/2022/009343) for financial support. D. Davis acknowledges VIT for doctoral
fellowship. Discussions with Parameshwaran A. are gratefully acknowledged.

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