Dissipative particle dynamics simulation of the interplay between spinodal decomposition and wetting in thin film binary fluids.

Abstract

The dynamics of phase separation of thin film binary fluids is investigated via dissipative particle dynamics simulation. We consider both cases of symmetric and asymmetric interactions between the walls and the two components. In the case of walls interacting symmetrically with the two fluid components, corresponding to a nonwetting case, relatively fast kinetics is observed when the average domain size is smaller than the slit thickness. A crossover to a slow Lifshitz-Slyozov growth is observed at late times. Faster dynamics is observed when the walls act as a slip boundary condition to the velocity field. In the case of asymmetric interactions, such that the system is in the wetting regime, the interplay between wetting kinetics and spinodal decomposition leads to rich dynamics. The phase separation proceeds through three stages. During the first stage, the dynamics is characterized as surface-directed spinodal decomposition, with growth of both average domain size and thickness of the wetting layers. The domain morphology is three dimensional and bicontinuous during the first stage, with kinetics reminiscent of that in bulk systems is observed. The second stage of the phase separation is characterized by the breakup of the bicontinuous domain morphology into small tubular domains bridging the two wetting layers and depletion of the core of the film from the wetting component. During this stage, domains with diameter smaller than thickness of the film shrink and disappear while those with diameter larger that the film thickness grow. The third stage is characterized by growth induced by the backflow of A-component from the wetting layers to the core of the film, leading to the decay in the thickness of the film and growth of the domains bridging the wetting layers. At even later times of the third stage, when the wetting layers become very depleted in the wetting component, growth becomes mediated by diffusion followed by collision of the tubular domains.

Full text

Page 1

Dissipative particle dynamics simulation of the interplay between spinodal
decomposition and wetting in thin film binary fluids
Michael J. A. Hore1and Mohamed Laradji1,2,a?
1Department of Physics, The University of Memphis, Memphis, Tennessee 38152, USA
2MEMPHYS-Center for Biomembrane Physics, University of Southern Denmark, Odense Dk-5230, Denmark
?Received 21 October 2009; accepted 10 December 2009; published online 12 January 2010?
The dynamics of phase separation of thin film binary fluids is investigated via dissipative particle
dynamics simulation. We consider both cases of symmetric and asymmetric interactions between the
walls and the two components. In the case of walls interacting symmetrically with the two fluid
components, corresponding to a nonwetting case, relatively fast kinetics is observed when the
average domain size is smaller than the slit thickness. A crossover to a slow Lifshitz–Slyozov
growth is observed at late times. Faster dynamics is observed when the walls act as a slip boundary
condition to the velocity field. In the case of asymmetric interactions, such that the system is in the
wetting regime, the interplay between wetting kinetics and spinodal decomposition leads to rich
dynamics. The phase separation proceeds through three stages. During the first stage, the dynamics
is characterized as surface-directed spinodal decomposition, with growth of both average domain
size and thickness of the wetting layers. The domain morphology is three dimensional and
bicontinuous during the first stage, with kinetics reminiscent of that in bulk systems is observed. The
second stage of the phase separation is characterized by the breakup of the bicontinuous domain
morphology into small tubular domains bridging the two wetting layers and depletion of the core of
the film from the wetting component. During this stage, domains with diameter smaller than
thickness of the film shrink and disappear while those with diameter larger that the film thickness
grow. The third stage is characterized by growth induced by the backflow of A-component from the
wetting layers to the core of the film, leading to the decay in the thickness of the film and growth
of the domains bridging the wetting layers. At even later times of the third stage, when the wetting
layers become very depleted in the wetting component, growth becomes mediated by diffusion
followed by collision of the tubular domains. © 2010 American Institute of Physics.
?doi:10.1063/1.3281689?
I. INTRODUCTION
Spinodal decomposition, a phenomenon referring to the
spontaneous demixing of multicomponent systems following
a quench from a homogeneous state into a multiphase coex-
istence state, has been the subject of extensive studies during
the past three decades.1–3Immediately after a quench of a
two-component system, long wavelength fluctuations in the
composition become unstable leading the system to initially
break up into small domains and their subsequent growth.
The late time dynamics of spinodal decomposition in bulk
systems is characterized by the emergence of a single time-
dependent length scale that displays a simple power-law de-
pendence on time, ??t??tn,4where the growth exponent, n,
is an indicator of the physical mechanism that governs the
phase separation process of the system. For example, the
growth exponent in alloys is n=1/3 and is an indication that
domain growth is mediated by the evaporation-condensation
mechanism.5In binary fluids, the velocity field plays a major
role on spinodal decomposition. In the case where the fluid
mixture has a globular domain structure, the dynamics is
mediated by the viscous Brownian motion of the domains
and their coalescence, leading to a growth exponent
n=1/3.4,6,7In contrast, when the domain structure of the
binary fluid is bicontinuous, fast domain growth with an ex-
ponent n=1 is observed, and results from a necking-down
instability of the tubular domains.8
While most of the experimental, theoretical, and compu-
tational studies of spinodal decomposition have been carried
out on bulk systems, in a typical experimental setting, the
binary mixture is a thin film sandwiched between two sub-
strates or with a free surface. The substrate typically interacts
more favorably with one of the two components of the mix-
ture leading to wetting of the substrate?s? by the preferred
component. The interplay between wetting, confinement, and
spinodal decomposition leads to dynamics that is richer than
its bulk counterpart, as has been shown in several experi-
mental studies9–22and computational studies.23–31Reviews
of spinodal decomposition in thin film binary mixtures can
be found in Refs. 32 and 33.
When one of the two components wets the substrate?s?,
both experiments and simulations have shown that the kinet-
ics of the phase separation of an AB binary mixture proceeds
through two main stages. During the first stage, right after a
critical quench, instability of the long wavelength fluctua-
tions in the composition leads to their amplification and for-
a?Author to whom correspondence should be addressed. Electronic mail:
mlaradji@memphis.edu.
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mation of small segregated domains with bicontinuous mor-
phology. Simultaneously, thin wetting layers of the preferred
?A? component rapidly form next to the substrates and
thicken with time. The thickness of the wetting layers grows
algebraically with time, l?t?with ?=1 in the case of binary
fluids, due to hydrodynamic flow, and ?=1/3 in the case of
binary alloys due to diffusion.33In two-dimensional systems
with hydrodynamic flow, the exponent is ?=2/3.34During
this stage, transverse composition profiles show damped os-
cillations as the distance from the substrate is increased. This
phenomenon is known as surface-directed spinodal decom-
position ?SDSD?.32,33As time proceeds, the preferred com-
ponent becomes increasingly depleted from the core of the
thin film, leading to a breakup of the interconnected domain
structure in the core into tubes of the wetting component
bridging the upper and lower wetting layers. The crossover
between the two stages occurs when the average domain size
is about the thickness of the film. During the early part of the
second stage, tubes with diameters smaller than the film
thickness decay while the larger tubes start to thicken at the
expense of material in the wetting layer and coalescence be-
tween the tubes.13The second stage is therefore also charac-
terized by the decay in the thickness of the wetting layers. In
the case of a film with a free surface, Wang and Composto
showed in their study of a poly?methyl methacrylate?-
poly?styrene-ran-acrylonitrile? thin film that during late
times, the capillary fluctuations of the free surface amplify
leading to roughening of the surface and eventually a
breakup of the middle layer into droplets.20
While most of the experiments of spinodal decomposi-
tion in thin films deal with binary fluids, for the sake of
tractability, the majority of the theoretical and computational
studies of these systems ignore hydrodynamic interactions,
although these are well known to play a crucial role on the
dynamics.23–29,31Das et al.30,35,36recently investigated spin-
odal decomposition with wetting of a binary fluid sand-
wiched between two substrates using molecular dynamics of
a generalized Lennard-Jones model. Among their results is
that during late stages, when the wetting layers are bridged
through the thin film core by cylindrical domains, the aver-
age diameter of these domains grows with time as t2/3.
Tanaka13found, in the case of a planar slit pore, that the
average domain size scales linearly with time at late times,
but slowly crosses over to a slower regime at even later
times. Wang and Composto, on the other hand, found that the
kinetics is characterized by two main characteristic lengths
corresponding to the average lateral domain size which
scales as t0.4, and the average interdomain distance which
scales as t1/3. Also recently, Bucior et al.37used molecular
dynamics to study a fluid mixture of hexadecane and carbon
dioxide confined between two walls. However, these authors
considered the case where the walls interact symmetrically
with the two components. Hence, the interplay between spin-
odal decomposition and wetting could not be inferred in the
study of Bucior et al. More recently, Náraigh and
Thiffeault30used Model H ?Ref. 38? to numerically investi-
gate a binary fluid in contact with a substrate and with a free
surface. The latter study, however, only considered the case
where the substrate interacts symmetrically with the two
components and focused primarily on the interplay between
capillary roughening of the free surface and spinodal decom-
position. The computational studies of spinodal decomposi-
tion with wetting in thin film fluid mixtures have not fully
analyzed the various growth regimes nor inferred the mecha-
nisms of domain growth in these systems. In order to further
elucidate the kinetics of spinodal decomposition in thin film
binary fluids, a systematic set of simulations based on the
dissipative particle dynamics ?DPD? approach is performed.
During the past 15 years, DPD has been shown to be a reli-
able computational approach for studying fluid systems since
it correctly describes hydrodynamic interactions. Although
both cases of nonwetting and wetting are considered, the
bulk of this article deals with the rich case of wetting. The
results are found to be largely in agreement with the experi-
ments of Tanaka13and of Wang and Composto.20
II. MODEL
The dynamics of phase separation of thin film binary
mixture is investigated via the DPD approach.39–41In DPD,
molecules are coarse grained into soft particles that interact
with each other via pairwise conservative, dissipative, and
random forces, respectively, given by
Fij
?C?= ?ij??rij?r ˆij,
?1?
Fij
?D?= − ?ij?2?rij??r ˆij· vij?r ˆij,
?2?
and
Fij
?R?= ?ij??t?1/2??rij??ijr ˆij,
?3?
where i and j are indices of two fluid particles, riand viare
the position and velocity of particle i, respectively, and rij
=rj−ri, vij=vj−vi. In Eqs. ?2? and ?3?, ??r? is a weight func-
tion given by the form used in conventional DPD,
??r? =?
0
In Eq. ?1?, ?ijis the amplitude of the conservative force of
the pair ?i,j?. In Eq. ?2?, ?ijis the dissipative strength for the
pair ?i,j?, and in Eq. ?3?, ?ijis the amplitude of the random
noise for the pair ?i,j?. In Eq. ?3?, ?t is the simulation time
step and ?ijis a symmetric random variable, with zero mean,
and is uncorrelated for different times and for different pairs
of particles, i.e.,
1 − r/rc
for r ? rc
for r ? rc.?
?4?
??ij?t?? = 0,
?5?
??ij?t??kl?t??? = ??ik?jl+ ?il?jk???t − t??,
?6?
with i?j and k?l.
In order to achieve thermal equilibrium in the canonical
ensemble at some temperature T, the fluctuation-dissipation
theorem requires that
? = ?2/2kBT.
?7?
The equations of motion of a fluid particle i are then
dri?t? = vi?t?dt
?8?
and
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Page 3

dvi?t? =
1
mi?
j
?Fij
?C?+ Fij
?D?+ Fij
?R??dt,
?9?
where miis the mass of fluid particle i.
In the present study, a slit geometry is considered, where
the thin film is confined between two parallel walls. This is
achieved by creating a rigid wall parallel to the xy-plane
from an arrangement of DPD particles in a face-centered
cubic lattice. The wall lateral dimensions are Lx?Lyand its
thickness is W=2rc. The two walls are separated by a dis-
tance D. Due to the use of periodic boundary conditions
along the three axes, in fact, the simulation box contains only
one wall. The dimensions of the simulation box are therefore
Lx?Ly?Lz, with Lz=W+D. The fluid therefore interacts
with both top and bottom faces of the wall. This method was
recently used by Laradji et al.42–44for studies of flow of
polymer solutions and polymer brushes in nanoscale slit
pores.
The system is composed of three types of coarse-grained
particles representing fluid A, fluid B, and wall W. The inter-
action amplitudes between fluid particles are chosen as ?AA
=?BB=25?/rcand ?AB=80?/rc. The fluid number density is
?=3.0rc
and ?WB, combined with the wall particles number density, is
important to the velocity field boundary condition and the
nature of wetting. The effect of the wall boundary condition
on the velocity field was investigated through the consider-
ation of a one-component A-fluid in the slit and driving it via
a uniform pressure gradient along the x-axis with a relatively
low driving force, such as the flow is laminar and follows the
Poiseuille–Hagen law.43,44The wall acts as a nonslip bound-
ary condition if the velocity field vanishes continuously at
the wall-fluid interface. For the used wall number density,
?w=61.35rc
order to prevent wall penetration by the fluid particles, ?WA
=3?/rcis found to be sufficiently low to produce a nonslip
boundary condition, while being sufficiently high to prevent
fluid particles from penetrating the wall. On the other hand, a
?WA?10?/rc produces a slip boundary condition.43The
effect of the velocity field boundary condition on the kinetics
of phase separation will be discussed in more detail in
Sec. IV.
Other parameters used in the present study are kBT/?
=1.0 and ?=3.0??3m/rc
=0.01?, with the time scale ?=?mrc
thickness values D=5rc, 10rc, 18rc, 20rc, and 22rcare con-
sidered. The lateral dimensions of the thin film are Lx=Ly
=128rc.
The equations of motion were integrated numerically us-
ing the velocity-Verlet algorithm. The simulations were per-
formed in a massively parallel fashion using the Message
Passing Interface to handle interprocess communication on a
112 CPU IBM system x LINUX cluster at the University of
Memphis and an 800 CPU Dell LINUX cluster at the Univer-
sity of Southern Denmark. Simulations generally utilized be-
tween 8 and 16 CPUs, with a maximum of 64 nodes being
used at times. Ten independent runs were made toward av-
eraging for each set of parameters in order to reduce statis-
tical deviations.
−3. The amplitude of the wall-fluid interactions, ?WA
−3, which is much higher than the fluid density in
2?1/3. The integration time step ?t
2/??1/2. Fluid films with
Domain growth is monitored through the circularly av-
eraged structure factor along the xy-plane,
S?q?,t? = ???˜?q?,t??2?,
?10?
where q?=?qx,qy? and
?˜?q?,t? =??dxdyeiq?·r???A?r?,t? − ?B?r?,t??,
?11?
where r?=?x,y?. ?A?r?,t? and ?B?r?,t? are the local volume
fractions of components A and B in a thin slice of the film
along the xy-plane of thickness 1.0rcand thus excluding the
wetting layers close to the walls. The characteristic length
scale of the morphology is then calculated, from the second
moment of S?q?,t?, as
???t? = 2???
q?=0
q?=0
qc
S?q?,t?/?
qc
q?
2S?q?,t??
1/2
,
?12?
where qcis a cutoff wave vector taken to be 2.5rc
Simulations carried out by first creating a one-
component thin film, composed of A-particles only, which is
let to equilibrate at kBT/?=1. Once equilibrium is reached, a
quench is mimicked through sudden and random switching
50% of the A-particles into B-particles, which corresponds to
the case where ?¯A=?¯B. The phase separation process is typi-
cally monitored during 40 000–60 000 time steps after the
quench. The system contains between 250 000 and 1 200 000
fluid particles, depending on the thickness of the film.
−1.
III. THE SYMMETRIC CASE: ?AW=?BW
We first present results pertaining to the effect of film
thickness on domain growth in the case where the interac-
tions between the wall and A- and B-fluid particles are sym-
metric ?nonwetting regime?. In Fig. 1, the average domain
size as calculated using Eq. ?12? is shown for different values
of the film thickness ranging from D=5rcto D=22rc, and for
the case of wall-fluid interactions ?AW=?BW=3?/rc. Figure 1
indicates that the rate of domain growth increases with in-
creasing film thickness. Figure 1 also shows the presence of
a kink ?highlighted by an arrow? which occurs when the
average domain size along the xy-plane, ???D. This kink
corresponds to the crossover time from three-dimensional
phase separation kinetics to quasi-two-dimensional kinetics.
To substantiate this point, snapshot series of the thin film
with D=20rcare shown in Fig. 2. Note that the kink occurs
at about 100? for D=20rc. From Fig. 2, it is clear that when
t?100?, the domain size is smaller than the thickness of the
film, and that for t?100?, domains are larger than the film
thickness, and hence domain growth becomes effectively two
dimensional at later times.
Figure 1?b? shows that at late times, the average domain
size scales asymptotically as ???tnwith n?1/3 for the thin-
nest film ?i.e., for D=5rc?. This indicates that for the thinnest
film considered here, hydrodynamics play a minor role on
the kinetics of phase separation, and that growth is mainly
mediatedbythe long-range
mechanism in line with the Lifshitz–Slyozov theory.5In con-
trast, Fig. 1?b? shows that for thicker films ?i.e., D?10rc?,
evaporation-condensation
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Page 4

faster dynamics is clearly observed during intermediate
times. During intermediate times, when the domain size is
smaller than the film thickness, three-dimensional viscous
domain growth with ???t is expected. However, since even
in the case of D=22rc, the average domain size at interme-
diate times is not much smaller than D, as shown by Figs. 1
and 2, the system is prevented from reaching this regime. At
late times, the kinetics in systems with D?10rcis more
consistent with the growth law ???t1/2, an indication that the
systems behave as quasi-two-dimensional where domain
growth is dominated by viscous hydrodynamics.4
IV. EFFECT OF VELOCITY FIELD BOUNDARY
CONDITION
It was shown previously42,43that the nature of the
boundary condition of the velocity field depends on the
strength of the interaction between the fluid and the wall. In
particular, it was found that when the interaction is weak,
e.g., ?WA?3?/rc, a nonslip boundary condition is observed.
In contrast, for ?WA?10?/rc, a slip boundary condition is
observed. In this section, the effect of the velocity field
boundary condition on the phase separation kinetics of the
film is discussed.
In Fig. 3, the average domain size, ??, is shown versus
time for a film thickness D=5rcand for both cases of slip
and nonslip boundary conditions. This figure shows that at
relatively early times, the kinetics of the two systems are
practically identical. At late times, however, the kinetics of
the system with slip boundary condition becomes markedly
faster than that with nonslip boundary condition. In particu-
lar, the growth exponents at late times are n=1/2 and 1/3 in
the cases of slip and nonslip boundary conditions, respec-
tively. Note that in the case of D=5rc, the domain size from
the beginning is about or larger than the thickness of the film,
implying that the phase separation process in this case is
essentially two-dimensional. The exponent n=1/3, in the
case of nonslip boundary condition, is an indication that do-
main growth in this system is mainly due to the evaporation-
condensation mechanism.5This is because the drag due to
the nonslip wall on the binary fluid causes a breakdown in
the long-range correlations of the velocity field along the
xy-plane, and therefore screening of the hydrodynamic inter-
actions. In contrast, the level of screening is reduced in the
case of a slip boundary condition, leading to a growth law
that is closer to that of purely two-dimensional fluids, with a
growth exponent n=1/2, as observed previously in many
numerical studies using various approaches. The effect of
0100 200300
t/τ
0
10
20
30
40
ξ||/rc
(a)
(b)
11.522.5
ln10(t/τ)
0.8
1
1.2
1.4
1.6
log10(ξ||/rc)
n=1/3
n=1/2
FIG. 1. ?a? Average domain size along the xy-plane vs time. Solid lines from
bottom to top ?at late times? correspond to D=5rc, 10rc, 18rc, and 22rc,
respectively. ?b? Same data shown in ?a? plotted in a double logarithmic
graph. Again curves from bottom to top correspond to D=5rc, 10rc, 18rc,
and 22rc, respectively. The dashed straight lines in ?b? have slopes of 1/2
and 1/3. Arrows in ?a? indicate crossover times for respective film thickness.
FIG. 2. Time sequence of snapshots for the case of thin film with symmetric
wall interactions, nonslip boundary condition, and D=22rc. The length of
the arrow below the left-bottom snapshot is 20rc. ?a?, ?b?, ?c?, and ?d? cor-
respond to times 10?, 50?, 100?, and 200?, respectively.
0.51 1.52 2.5
log10(t/τ)
0.6
0.8
1
1.2
1.4
1.6
log10(ξ||/rc)
n=1/3
n=1/2
FIG. 3. Average domain size along the xy-plane as a function of time for the
film with thickness D=5rcwith a slip boundary condition ?top curve? and a
nonslip boundary condition ?bottom curve?. Broken straight lines have
slopes 1/2 ?top? and 1/3 ?bottom?.
024908-4M. J. A. Hore and M. LaradjiJ. Chem. Phys. 132, 024908 ?2010?
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Page 5

wall-fluid boundary condition in the correlations of the ve-
locity profile is further substantiated by the velocity autocor-
relation along the xy-plane, defined as
Gv?t? = ?Vx?t + t0?Vx?t0? + Vy?t + t0?Vy?t0??,
where Vx=?ivix?t?/N and likewise for the y-component. The
brackets in the equation above designate averages over initial
time, t0, and over independent runs. The velocity autocorre-
lation is shown in Fig. 4, where it is clear that the velocity is
much more correlated in time in the case of slip boundary
condition than in the case of nonslip boundary condition.
?13?
V. THE ASYMMETRIC CASE: ?AWÅ?BW
This section focuses on the case where the wall interacts
more favorably with the A-component than with the
B-component. The particular case where ?WA=3?/rcand
?WB=5?/rcis considered. This corresponds to the situation
where the A-fluid wets the two walls, while the walls act as a
nonslip boundary condition for the velocity field. Simula-
tions were performed on thin films with thickness D=5rc,
10rc, 18rc, and 22rc.
Figures 5 and 6 show evolution snapshots of the films
center planes for systems with thickness 5rcand 22rc, re-
spectively. Figure 6 also shows transverse cuts of the film.
Both snapshot series qualitatively show that the interplay
between wetting and spinodal decomposition leads to phase
separation kinetics markedly different from the case of non-
wetting discussed in Sec. III. The contrast becomes even
stronger for thicker films. Figures 5 and 6 show that imme-
diately after the quench, the process of phase separation
starts in a manner similar to that of the nonwetting case,
namely, the emergence of a fine bicontinuous domain mor-
phology, typical to spinodal decomposition after a critical
quench. However, a topological asymmetry in the domain
morphology of the two components sets in relatively early in
the process. At late times, in both the thinner and thicker
films, the wetting ?A? component forms, in the core of the
film, bridging tubular domains that connect the two wetting
layers. The nonwetting component forms a laterally perfo-
rated network in the core of the film, sandwiched between
the two wetting layers.
The characterization of the kinetics of the phase separa-
tion of the thin film will be carried through investigation of
the volume fraction profile, the thickness of the wetting
layer, the number density of lateral domains in the center
plane, the lateral characteristic length scales as calculated
from the structure factor and the clusters distribution, and the
area fraction of component A in the center plane.
The evolution of the laterally averaged profile of the
difference in volume fractions of the two components along
the z-axis, which is defined as
LxLy??dxdy??A?r?,z? − ?B?r?,z??,
???z? =
1
?14?
is shown in Figs. 7 and 8 for systems with D=5rcand 22rc,
respectively. z=0 and z=D correspond to the loci of the in-
terfaces between the walls and fluid mixture. Both figures
show a rapid migration of the A-component toward the sub-
strates at early times. In the thicker film ?see Fig. 8? oscilla-
tions in the volume fraction profile are clearly identified dur-
ing the interval 0?t?50?. These waves are the result of
wetting of the A-component on the substrates which leads to
an anisotropy in the domain structure. This is reminiscent of
SDSD. In Fig. 9, the thickness of the wetting layer is de-
picted as a function of time for all systems investigated. Fig-
ures 7–9 clearly indicate that the time dependence of the
thickness of the wetting layer is nonmonotonic. Namely, the
thickness of the wetting layers grows at early times, reaches
a maximum, and eventually decays with time. The time, at
which the maximum in the thickness of the wetting layer is
reached, increases with increasing film thickness. It is inter-
0100 200300
t/τ
0
2×10-6
4×10-6
6×10-6
Gv(t)
0100200
t/τ
-0.006
-0.003
0
0.003
0.006
vx(t)/N
FIG. 4. The velocity correlation function as calculated using Eq. ?13? for the
two systems shown in Fig. 3. Black and red curves correspond to nonslip
and slip boundary conditions, respectively. The inset figure shows the aver-
age momentum per particle along the x-axis as a function of time for the two
systems.
FIG. 5. Snapshot series for the thin film case with D=5rcwith asymmetric
interactions between the walls and the two fluid components. Blue corre-
sponds to the A-component ?component that wets that walls? and red corre-
sponds to the B-component. The views are cuts along the xy-plane taken at
z=2.5rc.
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Page 6

esting to note that the maximum value of the wetting layer
thickness is proportional to the film thickness. The growth of
the wetting layers is qualitatively different from that of
SDSD of a fluid mixture with macroscopic thickness and in
contact with a single substrate, where the wetting layer
grows monotonically with time in the case of complete
wetting.33The nonmonotonic growth of the wetting layers is
therefore due to confinement of the binary fluid. Neverthe-
less, the growth of the wetting layer at early times is found to
be linear in time, implying that this early growth is due to the
advective transport of the preferred ?A? component from the
film core to the substrate. This is in agreement with the ex-
perimental study of Wang and Composto.20These results are
also in qualitative agreement with the recent computational
studies of Das et al. based on both a time-dependent
Ginzburg–Landau model29and molecular dynamics.35Due
to the finite amount of A-component per unit of area in a thin
film, growth of the wetting layer cannot be sustained indefi-
nitely, leading to its saturation at intermediate times. It is
important to note that the subsequent decay in the thickness
of the wetting layer coincides with a tubular domain mor-
FIG. 6. Snapshot series for the asymmetric thin film with D=22rc. For each
time, the top ?rectangular? snapshot corresponds to a cut of the film across
the channel taken at y=0 and the square view corresponds to a cut along the
xy-plane taken at z=11rc. Color coding is the same as in Fig. 5.
01.25 2.5
z / rc
3.755
-1
-0.5
0
0.5
1
∆φ(z)
FIG. 7. Volume fraction profiles of system with D=5rcshown in Fig. 5.
Dashed curve corresponds to t=?. Solid lines from bottom to top at z/rc
=2/5 correspond to t=10? and 20?, 100?, 200?, 300?, and 400?. Note that
the profiles for t?100? are practically indistinguishable.
0 5.51116.5 22
-1
1
-0.5
0
0.5
1
∆φ(z)
0 5.5 11 16.5 22
z / rc
-1
-0.5
0
0.5
∆φ(z)
(a)
(b)
FIG. 8. Volume fraction profile of system with D=22rcshown in Fig. 6. ?a?
corresponds to the profiles at earlier times. Dashed curve in ?a? corresponds
to t=?. The solid lines in ?a? from top to bottom and at z=11rccorrespond
to t=20?, 50?, 70?, 80?, 100?, and 130?, respectively. ?b? corresponds to the
profiles of the same system at later times. Curves from bottom to top and at
z=11rccorrespond to t=130?, 200?, 300?, and 400?, respectively.
024908-6M. J. A. Hore and M. LaradjiJ. Chem. Phys. 132, 024908 ?2010?
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Page 7

phology of the A-component in the core of the film. This will
later be discussed in more detail.
The areal fraction of A-component in the center plane,
ac, is calculated from the difference in volume fraction at the
center plane as ac=?1+???D/2??/2, and is depicted as a
function of time for all systems in Fig. 10. This figure shows
that at the center plane, the two components alternate be-
tween being the minority and majority components at early
times. This is associated with the propagation of the spinodal
waves during the early stages of the phase separation pro-
cess. At intermediate times, the center plane becomes de-
pleted in the wetting ?A? component with a minimum value
about 30% for D=5rc, 7% for D=10rc, 2% for D=18rc, and
1% for D=22rc. The degree of depletion in the wetting com-
ponent at the center plane is therefore enhanced as the film
thickness is increased. The depletion in the wetting compo-
nent is due to the wetting of the substrates with the
A-component. Examination of Fig. 10 in conjunction with
Fig. 6 leads us to note that the enrichment of the center plane
in the nonwetting component coincides with the breakup of
the interconnected morphology of the A-component in the
core of the film. At later times, Figs. 6 and 10 show that the
growth of actakes place during the lateral growth of the
bridging A-domains in the core of the film.
The kinetics of the phase separation of the thin film fluid
mixture is further elucidated through the growth of average
domain size, as calculated form the lateral structure factor at
the center plane, Eq. ?12?. This is shown for all systems in
Fig. 11. The kinetics is also investigated through the number
density of the A-domains in the center plane, shown for all
systems in Fig. 12. Figure 11 shows that the time dependence
of the average domain size in the case of wetting is qualita-
tively different from that in the nonwetting case, particularly
when the thickness of the film exceeds 10rc, where the aver-
age domain size clearly exhibits a nonmonotonic time depen-
dence. The number of A-clusters in the center plane, shown
in Fig. 12, decays monotonically in the case of D=5rc. This
implies that in the case of very thin films, a broken symmetry
in the morphology of the two components sets in very early
in time. This is supported by the snapshots of Fig. 5, where
the A-component clearly exhibits isolated compact domains
in the center plane, while the B-component forms essentially
a single laterally percolating domain. This broken symmetry
in morphology in the case of D=5rcis associated with the
fact that in this case, the average domain size since early
times is comparable to the thickness of the film, leading to
quasi-two-dimensional growth. Since the thickness of the
wetting layers is constant at late times in the case of D
=5rc, the growth of the domains is mainly driven by coales-
cence.
Figure 12 shows that the number of clusters is very small
FIG. 9. Thickness of the wetting layer for films with asymmetric interac-
tions. Curves from bottom to top at t=200? correspond to D=5rc, 10rc,
18rc, and 22rc, respectively. The inset shows the maximum value of the
wetting layer thickness vs film thickness. Notice the linear dependence of
lmaxon D.
0100 200
t/τ
300400
0.0
0.2
0.4
0.6
ac
FIG. 10. Area fraction along the center plane of A-domains. Curves from
top to bottom at t=200? correspond to D=5rc, 10rc, 18rc, and 22rc,
respectively.
0100200
t/τ
300400
0
10
20
30
40
50
ξ||/rc
FIG. 11. Average domain size vs time for films with asymmetric interac-
tions. Curves from bottom to top at t=400? correspond to D=5rc, 10rc,
18rc, and 22rc, respectively.
0 100200
t/τ
300400
0.000
0.004
0.008
0100200
t/τ
300400
0.0
0.1
0.2
0.3
0.4
ρcAc
ρc[rc
-2]
FIG. 12. Clusters number density, ?c, vs time for films with asymmetric
interactions. Dashed curve corresponds to D=5rc. Curves from left to right
at ?c=0.002rc
inset shows the product ?cAc, where Acis the average domain cross-
sectional area. Curves from top to bottom at t=200? correspond to D
=10rc, 18rc, and 22rc, respectively.
−2correspond to D=10rc, 18rc, and 22rc, respectively. The
024908-7 Spinodal decomposition and wettingJ. Chem. Phys. 132, 024908 ?2010?
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Page 8

during early times of the phase separation process in the case
of D?10rc. This is due to the fact that the domain morphol-
ogy during this stage is bicontinuous. Figure 11 shows that
the characteristic length scale, ??, calculated from the second
moment of the structure factor, momentarily decays during
intermediate time for D=18 and 22rc. The decay in the av-
erage domain size for D?18rcat intermediate times coin-
cides in fact with the rapid rise in the number of domains, as
shown by Fig. 12, and the breakup of the bicontinuous do-
main morphology of component A in the center plane, as
demonstrated by Fig. 6. The number of clusters reaches a
maximum and then decays with time at late times. It is noted
that the maximum number of clusters decreases with increas-
ing film thickness. This was also noted previously by
Tanaka.13
Stages of Phase Separation of Thin Film Fluid Mixtures
with Wetting: The time dependence of the various physical
quantities presented above suggests that the kinetics of phase
separation with wetting of a thin film binary fluid that is
sandwiched between two substrates is divided into three
stages. To substantiate this, various measured quantities are
plotted together in Fig. 13 for the film with thickness D
=22rc, where the three stages are also delineated. Systems
with D=10rcand 18rcexhibit similar behavior.
During stage I, which in the case of D=22rcoccurs dur-
ing the time interval 0?t?50?, the domain morphology is
bicontinuous and domain growth is essentially three dimen-
sional, with the average domain size being smaller than the
thickness of the film. During stage I, domain growth pro-
ceeds simultaneously with a linear growth of the wetting
layers. The end of stage I is characterized by a marked slow-
ing down in the growth of the average domain size and a
slight slowing down in the growth of the wetting layer. The
kinetics of stage I is reminiscent of SDSD observed in mac-
roscopically thick fluid mixtures undergoing both spinodal
decomposition and wetting kinetics.
Stage II is characterized by the breakup of the biconti-
nous domain morphology and formation of tubular domains.
The onset of stage II takes place when the average domain
size reaches a value close to the thickness of the film. Stage
II, which occurs during the interval 50??t?130?, can itself
be subdivided into two substages: substage II?a? which oc-
curs during 50??t?80? and substage II?b? which occurs
during 80??t?130?. Figure 13 shows that the first substage
of stage II is characterized by a rapid increase in the number
of clusters, a decay in the characteristic length, ??, a rapid
growth of the wetting layer, and a rapid enrichment of the
center plane with the nonwetting ?B? component. As demon-
strated by Fig. 6, during substage II?a?, the bicontinuous do-
main morphology breaks up into domains with compact
cross section along the center plane. The domains are how-
ever tubular along the z-axis and are connected to the two
wetting layers. Most of these domains shrink in size as dem-
onstrated by the momentary decrease ??and qualitatively by
Fig. 6. The domain shrinkage of substage II?a? is mediated
by the rapid migration of material of the A-component from
the core of the film toward the two substrates, thereby con-
tributing to a rapid increase in the thickness of the two wet-
ting layers. It is interesting to note that the slope of the linear
growth of the wetting layers in substage II?a? seems to be
equal to that in stage I. Substages II?a? and II?b? are delin-
eated by the time at which the number of clusters reaches a
maximum. During substage II?b?, the average domain size is
essentially constant, while the number of clusters rapidly de-
cays and the thickness of the wetting layer slowly grows.
Indeed, inspection of snapshots at t=100? and 130? in Fig. 6
leads to the conclusion that during substage II?b? many of the
domains shrink and disappear, contributing to the decay in
the number of A-clusters in the center plane and slow growth
of the wetting layers. On the other hand several domains
remain unchanged with a somewhat fixed diameter during
substage II?b?. Inspection of the diameters of these domains
leads us to conclude that the shrinking domains have diam-
eters smaller than the thickness of the film, while the surviv-
ing domains are those with diameters slightly larger than the
film thickness.
Stage III, which in the case of D=22rcoccurs when t
?130?, is marked by the decrease in the number of A bridg-
ing domains and the thickness of the wetting layer, and an
increase in the volume fraction of the A-component in the
film core as well as an increase in the volume fraction of the
A-component in the center plane. The decrease in l?t? and the
increase in ac?t? implies a backflow of the A-component
from the wetting layers toward the film core. It is noted that
during the latter part of stage III, the number of clusters
becomes practically constant. This is substantiated by Fig. 6,
where the number of clusters is constant for t?200?.
We now focus on discussing the growth mechanisms
during stage III. In order to answer this question, we will
look at the characteristic length determined from the struc-
ture factor ???t?, shown in Fig. 14, the areal cluster diameter
d??t?, shown in Fig. 15, and the number cluster density ?c?t?,
shown earlier in Fig. 12. We note that the length scale ??
measures the average distance between lateral domains in the
center plane. Figure 14 shows that except for D=5rcand
0
0.001
0.002
ρc(t)
0100 200300
t/τ
0
1
2
3
l(t) /rc
0
0.2
0.4
0.6
ac(t)
0
10
20
30
40
ξ||(t)
I
II
III
FIG. 13. Cluster number density, characteristic length scale from the struc-
ture factor, thickness of the wetting layer, and difference in volume fraction
at the center plane for the film with asymmetric interactions and D=22rc.
This figure shows the three main stages of the phase separation process
labeled as I, II, and III. The three stages are separated by vertical dashed
lines. The dotted-dashed line within stage II separates substages II?a? and
II?b? discussed in the text.
024908-8M. J. A. Hore and M. Laradji J. Chem. Phys. 132, 024908 ?2010?
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Page 9

10rc, no power law is observed over an appreciable amount
of time during late stages, an indication of a lack of dynami-
cal scaling. However, it is interesting to note that asymptoti-
cally, the growth of ?? in systems with D?10rcseems to
approach a t1/3law, maybe indicating that growth becomes
mediated by lateral diffusion and collision, i.e., coalescence,
of the tubular domains. However, while in systems with D
=18rcand 22rc, the number of clusters during the later part
of stage III becomes practically constant implying that colli-
sion events during stage III are rare and cannot contribute to
domain growth. Furthermore, if diffusion and collision are
the main growth mechanisms, the product ?cd?
constant, which as demonstrated by the inset in Fig. 12, is
not the case for D=18rcand 22rc. In the case of D=5rcand
10rc, on the other hand, growth is mediated by coalescence
of the lateral domains. We note that these results are quali-
tatively different from those obtained by Das et al.36using
molecular dynamics, where a growth law t2/3was reported. It
is interesting to note that Wang and Composto also found
that ???t1/3but they observed an appreciable amount of coa-
lescence of domains.
In Fig. 16, the ratio between the two length scales, ??/d?,
is shown for all systems. This figure shows that ??/d? be-
comes practically constant for systems with D=5rcand 10rc
during later times, an implication that these systems ap-
proach a dynamical scaling regime. This is expected since
2should be
for these two systems, the thickness of the wetting layer
reaches saturation, implying that backflow of A-material can-
not drive domain growth in these systems at late times. Fur-
thermore, we found that for the thinnest films and during late
times, the number of clusters scale as t−2/3, implying that
growth must be the result of domains diffusion and their
coalescence. In systems with D=18rcand 22rc, the ratio
??/d?is strongly time dependent during late times, implying a
lack of scaling during stage III in these systems. In these two
systems, the thickness of the wetting layer is very time de-
pendent during late stages, which implies that growth is es-
sentially the result of backflow of the A-material. We note
that both Tanaka13and Wang and Composto20found that
during late stages, the average domain diameter grows lin-
early with time, but then slows down to a logarithmic growth
at very late times. Our results shown in Fig. 15 do indeed
corroborate their findings.
VI. SUMMARY AND DISCUSSION
In summary, in this paper results on the kinetics of phase
separation of thin film binary fluids using DPD are pre-
sented. Both cases of wetting and nonwetting were investi-
gated for films with varying values of the film thickness. The
effect of velocity field boundary condition on the kinetics is
also investigated. The thin films are found to exhibit dynam-
ics far richer than that in bulk binary fluids. In the case of
equal interactions between the two fluid components and the
walls, corresponding to a nonwetting case, the system exhib-
its three-dimensional-like kinetics at intermediate times
when the average domain size is smaller than the film thick-
ness, followed by quasi-two-dimensional kinetics at late
times. For the thinnest films, a growth exponent n=1/3 is
observed at late times, an indication that growth is mediated
by the evaporation-condensation mechanism. However, for
the thicker films, the growth exponent at late times is closer
to n=1/2, presumably indicating that at late times, growth is
mediated by domain coalescence. The investigation of the
velocity field boundary condition indicate that as expected, a
slip boundary condition leads to faster dynamics than in the
case of nonslip boundary condition.
The present investigation of asymmetric interactions be-
tween the substrates and the two fluid components, leading
2.02.22.42.6
log10(t/τ)
1.2
1.4
1.6
log10(ξ||/rc)
n=1/3
n=1/3
n=1
n=1
FIG. 14. The average characteristic length ??from the structure factor at late
times for all systems with asymmetric interactions between the wall and the
two fluid components. Solid curves from bottom to top at log10?t/??=2.6
correspond to D=5rc, 10rc, 18rc, and 22rc, respectively. The dashed lines
have a slope n=1/3 and the dotted-dashed line has a slope n=1.
100200
t/τ
300 400
0
10
20
30
40
50
d||/rc
FIG. 15. The average domain diameter d?along the xy-plane. Curves from
left to right at d?/rc=20 correspond to D=10rc, 18rc, and 22rc, respectively.
Dashed lines are fit to a logarithmic form, d?=a+b ln?t/??, during stage III.
0100200
t/τ
300 400
0
2
4
6
8
ξ||/d||
FIG. 16. Ratio between length scale ??and domains lateral diameter d?as a
function of time. Solid lines from bottom to top at t=300? correspond to
D=5rc, 10rc, 18rc, and 22rc, respectively. The dashed line corresponds to a
constant function of time.
024908-9Spinodal decomposition and wettingJ. Chem. Phys. 132, 024908 ?2010?
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Page 10

to preferential wetting of component A on the walls, reveals
more complicated dynamics, and can be divided into three
main stages. Stage I, is characterized by three-dimensional
bicontinuous domain morphology in the core of the film and
SDSD with growth of both the average domain size and the
wetting layers. This is followed by stage II which occurs
when the average domain size becomes comparable to the
film thickness. Stage II is characterized by the breakup of the
bicontinuous domain structure into tubular domains connect-
ing the two wetting layers. Most of these domains shrink
during the early part of stage II, leading to a rapid increase in
the thickness of the wetting layer and a depletion of the core
in the wetting component. The latter part of stage II is char-
acterized by a slowing down in the growth of the wetting
layer, disappearance of the small tubular domains, and
growth of tubular domains with diameter larger than the film
thickness. Stage III is characterized by a decay in the thick-
ness of the wetting layer, indicating a backflow of the
A-material from the wetting layers to the core, leading ini-
tially to a rapid growth of the diameter of the tubular do-
mains with a growth exponent close to one. As the amount of
material in the wetting layers becomes increasingly depleted,
a logarithmic growth is observed.
It is important to note that as the thickness approaches its
equilibrium value, the amount of material backflow from the
wetting layers is reduced, and thus backflow induced growth
is not expected to be the dominant driving force at very late
times. Eventually, the amount of A-component in the film
core then becomes conserved, and growth becomes domi-
nated by diffusion and collision ?that is coalescence? of do-
mains. During this stage, dynamical scaling is expected with
a growth law ???d??t1/3. This regime is indeed observed
for the thinnest systems, with D=5rcand 10rc. For the
thicker films, simulations on larger systems are needed to
indeed to validate this proposal. We should also note that the
systems we investigated are different from those studied by
Wang and Composto, which possess one free surface. The
added free surface complicates the kinetics even further at
very late times due to the interplay between spinodal decom-
position and capillary fluctuations. We plan to investigate
these systems computationally in the near future.
ACKNOWLEDGMENTS
This work was supported by a grant from the Petroleum
Research Fund and a grant from the University of Memphis
Faculty Research Fund. Simulations were performed at the
High Performance Computing Facility at the University of
Memphis and the Danish Center for Scientific Computing at
the University of Southern Denmark.
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Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp