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.

THE JOURNAL OF CHEMICAL PHYSICS 132, 024908 ?2010?

0021-9606/2010/132?2?/024908/10/$30.00© 2010 American Institute of Physics

132, 024908-1

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

Page 2

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

024908-2M. J. A. Hore and M. Laradji J. Chem. Phys. 132, 024908 ?2010?

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

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

024908-3Spinodal decomposition and wettingJ. Chem. Phys. 132, 024908 ?2010?

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

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?

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

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.

024908-5Spinodal decomposition and wettingJ. Chem. Phys. 132, 024908 ?2010?

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

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?

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

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?

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

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?

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

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?

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

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.

1R. W. Balluffi, S. M. Allen, and W. C. Carter, Kinetics of Materials

?Wiley-Interscience, Hoboken, New Jersey, 2005?.

2A. Onuki, Phase Transition Kinetics ?Cambridge University Press, Cam-

bridge, 2002?.

3Dynamics of Ordering Processes in Condensed Matter, edited by S. Ko-

mura and H. Furukawa ?Plenum, New York, 1988?.

4A. Bray, Adv. Phys. 43, 357 ?1994?.

5I. L. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids 19, 35 ?1961?.

6V. S. Nikolayev, D. Beysens, and P. Guenoun, Phys. Rev. Lett. 76, 3144

?1996?.

7H. Chen and A. Chakrabarti, J. Chem. Phys. 108, 6006 ?1998?.

8E. D. Siggia, Phys. Rev. A 20, 595 ?1979?.

9P. Guenoun, D. Beysens, and M. Robert, Phys. Rev. Lett. 65, 2406

?1990?.

10R. A. L. Jones, L. J. Norton, E. J. Kramer, F. S. Bates, and P. Wilziusm,

Phys. Rev. Lett. 66, 1326 ?1991?.

11P. Wiltzius and A. Cumming, Phys. Rev. Lett. 66, 3000 ?1991?.

12F. Bruder and R. Brenn, Phys. Rev. Lett. 69, 624 ?1992?.

13H. Tanaka, Phys. Rev. Lett. 70, 53 ?1993?; 70, 2770 ?1993?.

14G. Krausch, C. Dai, E. J. Kramer, and F. K. Bates, Phys. Rev. Lett. 71,

3669 ?1993?.

15G. Krausch E. J. Kramer, F. K. Bates, J. F. Marko, G. Brown, and A.

Chakrabarti, Macromolecules 27, 6768 ?1994?.

16G. Krausch, Mater. Sci. Eng. R. 14, 5 ?1995?.

17W. Straub, F. Bruder, R. Brenn, G. Krausch, H. Bielefeldt, A. Kirsch, O.

Marti, J. L. Mlynek, and J. F. Marko, Europhys. Lett. 29, 353 ?1995?.

18H. Tanaka, Phys. Rev. E 54, 1709 ?1996?.

19L. Sung, A. Karim, J. F. Douglas, and C. C. Han, Phys. Rev. Lett. 76,

4368 ?1996?.

20H. Wang and R. J. Composto, J. Chem. Phys. 113, 10386 ?2000?.

21H. Wang and R. J. Composto, Phys. Rev. E 61, 1659 ?2000?.

22H.-J. Chung and R. J. Composto, Phys. Rev. Lett. 92, 185704 ?2004?.

23G. Brown and A. Chakrabarti, Phys. Rev. A 46, 4829 ?1992?.

24S. Puri and K. Binder, J. Stat. Phys. 77, 145 ?1994?.

25S. Puri and H. L. Frisch, J. Phys.: Condens. Matter 9, 2109 ?1997?.

26S. Puri and K. Binder, Phys. Rev. Lett. 86, 1797 ?2001?.

27S. Bastea, S. Puri, and J. L. Lebowitz, Phys. Rev. E 63, 041513 ?2001?.

28M. Müller and G. D. Smith, J. Polym. Sci., Part B: Polym. Phys. 43, 934

?2005?.

29S. K. Das, S. Puri, J. Horbach, and K. Binder, Phys. Rev. E 72, 061603

?2005?.

30L. Ó. Náraigh and J. L. Thiffeault, Phys. Rev. E 76, 035303 ?2007?.

31S. K. Das, J. Horbach, and K. Binder, Phys. Rev. E 79, 021602 ?2009?.

32J. M. Geoghegan and G. Krausch, Prog. Polym. Sci. 28, 261 ?2003?.

33S. Puri, J. Phys.: Condens. Matter 17, R101 ?2005?.

34H. Chen and A. Chakrabarti, Phys. Rev. E 55, 5680 ?1997?.

35S. K. Das, S. Puri, J. Horbach, and K. Binder, Phys. Rev. Lett. 96,

016107 ?2006?.

36S. K. Das, S. Puri, J. Horbach, and K. Binder, Phys. Rev. E 73, 031604

?2006?.

37K. Bucior, L. Yelash, and K. Binder, Phys. Rev. E 77, 051602 ?2008?.

38P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, 435 ?1977?.

39P. J. Hoogerbrugge and J. M. V. A. Koelman, Europhys. Lett. 19, 155

?1992?.

40P. Español and P. Warren, Europhys. Lett. 30, 191 ?1995?.

41P. Español, Europhys. Lett. 40, 631 ?1997?.

42J. Huang, Y. Wang, and M. Laradji, Macromolecules 39, 5546 ?2006?.

43J. A. Millan, W. Jiang, M. Laradji, and Y. Wang, J. Chem. Phys. 126,

124905 ?2007?.

44J. A. Millan and M. Laradji, Macromolecules 42, 803 ?2009?.

024908-10M. J. A. Hore and M. LaradjiJ. Chem. Phys. 132, 024908 ?2010?

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp