New Hydrodynamic Mechanism for Drop Coarsening

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by  Vadim Nikolayev

New hydrodynamic mechanism for drop coarsening
∗
Vadim S. Nikolayev, Daniel Beysens, and Patrick Guenoun
Service de Physique de l’Etat Condens´e, CE Saclay, F-91191 Gif-sur-Yvette Cedex, France
(Dated: 9 March 1995)
Wediscussanewmechanismofdropcoarseningduetocoalescenceonly,whichdescribesthelate
stages of phase separation in fluids. Depending on the volume fraction of the minority phase, we
identifytwodifferentregimesofgrowth,wherethedropsareinterconnectedandtheircharacteristic
sizegrowslinearlywithtimeandwherethesphericaldropsaredisconnectedandthegrowthfollows
(time)1/3. The transition between the two regimes is sharp and occurs at a well defined volume
6
fraction of order 30%.
1
0
2
InthisLetterweconcentrateonthekineticsofthelate coefficient of the drops of the same viscosity

n stagesofthephaseseparation. Thissubjecthasreceived
a considerable attention recently [1]–[4]. Most of the ex- D = kBT . (2)
J
5πηR
periments on growth kinetics have been performed near
6
thecriticalpointofbinaryliquidmixtures(orsimpleflu-
2 The factor f represents the correction which takes into
ids) because there the critical slowing down allows the accountthehydrodynamicinteractionbetweenthedrops.
] phenomenon to be observed during a reasonable time.
n It depends on the ratio of the viscosities of the liquid
After a thermalquenchfromthe one-phaseregionto the
y inside and outside the drops and the average distance
d two-phase region of the phase diagram (Fig. 1) the do- betweenthedropsand,therefore,onthe volumefraction
- mains of the new phases nucleate andgrow. It turns out
φ. This correctionhasbeencalculatedinthedilute limit
u
that two alternative regimes of coarsening are possible.
l (φ 0)byZhangandDavis[10]. Intheproximityofthe
f The first can be observed when the volume fraction φ →
. criticalpoint we can assume the viscosities of the phases
s of the minority phase is lower than some threshold [5]
c to be equal and
and the domains of the characteristic size R grow ac-
i
s cording to the law R t1/3 (t is the time elapsed after f(0)=0.56. (3)
y ∝
h the quench) as sphericaldrops. The secondregime man-
p ifestsitselfwhenthequenchisperformedathighvolume According to the model, the drops coalesce immediately
[ fraction, the coarsening law is R t1 and the domains after the collision. Coalescenceis the only reasonfor the
∝
grow as a complicated interconnected structure. Recent decrease of the total number of the drops with the rate
1
v experiments [2] show that when 0.1 < φ < 0.3 the t1/3
dn
4 growth can be explained by a mechanism of Brownian = N . (4)
B
0 drop motion and coalescence rather than the Lifshitz- dt −
0
Slyozov mechanism [6] which holds for φ 0 and which With the relation
7 →
we will not discuss here. We are interested in the late
0 4
. stagesofgrowthwhenphase boundariesarealreadywell φ= πR3n=const, (5)
1 3
developed and the concentrations of the phases are very
0
6 closetotheequilibriumvaluesatgiventemperatureT as Eq.(4) yields a R t1/3 law. The further improvements
1 definedbythecoexistencecurve(Fig.1). Thenthedrops (see[11]andrefs. ∝therein)ofthismodelinfluencemainly
v: growjustbecausethesystemtendstominimizethetotal thenumericalfactorin(1)whichisnotimportantforthe
i surface separating the phases (i.e. due to coalescence) present considerations.
X
and φ no longer depends on time. Hydrodynamic approaches. The originof the R
ar Brownian coalescence. The Brownian mechanism t1 growth law, observed at high volume fraction wher∝e
was considered first by Smoluchowski [7] for coagulation domainsareinterconnected,ismuchlessclear. Bymeans
of colloids and was then applied to phase separation by ofadimensionalanalysisSiggia[9]hasshownthathydro-
Binder and Stauffer [8] and Siggia [9]. According to this dynamics is needed to explain the kinetics. It assumed
mechanism, the rate of collisions per unit volume due to the growth to be ruled by the Taylor instability of the
the Brownian motion of spherical drops in the liquid of long tube of fluid, which breaks into separate drops and
shear viscosity η is associatedthe growthrate with the rate ofthe evolution
of the unstable fluctuations. This idea has been devel-
oped by San Miguel et al. in [12]. However, it was not
N =16πDRn2f(φ), (1)
B clear how this process was related to the growth.
Another approach has been considered by Kawasaki
where n is an average number of drops per volume, R and Ohta [3] who used a model of coupled equations of
is the average radius of the drops and D is the diffusion hydrodynamics and diffusion. It was assumed that the
2
X'
0 --1.5 -1 -0.5 ψ0 0.5 1 1.5
2
)
(K cee Initial position
T-T c-0.02 CoexistencurvDits1c/o3n- Intte1r- Dits1c/o3n- -0.9063 d0 d0
nected connected nected
-0.04
5 6 5 0 R
0.1 0.2 0.3 .18 R
7 d
5
0.66 c+ 0.7 0.72 c- 0.76 0
ψ d
Concentration 0
1
FIG. 1: Coexistence curve for a model two-phase system .28 R
1
(density-matched cyclohexane–methanol) from Ref. [1]. The
dotted curve is the calculated boundary (see text) between
thet1/3 andt1 growthregionswhichcorrespondstoφ=0.26. d0
The triangles are the experimental data from Ref. 1. The 2.3 "Coalescence"
curves corresponding to φ=0.15 (random percolation limit) 75 position
andφ=0.35 (thevaluewhichgivesthebestfittotheexper- X
imental data from Ref. 1) are presented also for comparison.
Thevolumefraction oftheminorityphaseatthepoint(c,T) FIG. 2: The positions of the drop surfaces at the beginning
can be calculated as φ = 1/2−|c−c |/(c− −c+) where c (dotted line) and at the end (solid line) of the simulation. ψ
c c
(=0.707 for thiscase) is thecritical concentration. is the coalescence distance, d0 is the initial distance between
′
thedrops, X –X is theaxis of cylindrical symmetry.
growth is controlled by diffusion and the hydrodynamic
correction to this process was calculated. However, the Browniandiffusion but rather flows induced by previous
translational movement of the drops due to the pressure coalescence. We use the conceptof “coalescence-induced
gradient was not taken into account. The motion of the coalescence” as introduced by Tanaka [4] who, however,
liquid was supposed to be induced by the concentration thought that induction by the hydrodynamic flow was
gradient only. At the same time, it is well known that not relevant, having stated that coalescence takes place
the concentration variation does not enter the equations after the decay of the flow. We consider here a coales-
ofhydrodynamicsofthe liquidmixture inafirstapprox- cence process between two drops and study numerically
imation[13]. Moreover,it is evidentthatathighvolume thegeneratedflowanditsinfluenceonthethirdneighbor-
fraction the coalescence process induced by the transla- ingdrop. Thefullydeterministichydrodynamicproblem
tional motion of the drops becomes very important. within the creeping flow approximation (which is well
Recently, several groups [14–18] performed large scale justified near the critical point [9]) was solved. The free
direct numerical simulations by using different ap- boundary conditions were applied on all drop interfaces
proaches to solve coupled equations of diffusion and hy- whose motion is driven by surface tension. At each time
drodynamics. Some recovered the linear growth law stepthevelocityofeachmeshpointontheinterfacecon-
[15, 17], whilst the others were notable to reachthe late tours was computed using a boundary integralapproach
stages of separation and measured the transient values [19]. When the new positions ofthe interfaces havebeen
of the growth exponent (between 1/3 and 1). In spite calculated, the procedure was continued iteratively. The
of these efforts, the physical mechanism for the linear details of the solution will be presented elsewhere [20].
growth has not been clarified. To our knowledge, the Webeginthesimulationwhencoalescencestartsbetween
simulationsnevershowedtwoasymptoticallawsdepend- two drops of size R (Fig. 2), i.e. when the drops of the
ing on the volume fraction: the exponent is either larger minority phase approach to within a distance of coales-
than 1/3 when accounting for hydrodynamics or 1/3 for cence ψ which corresponds to the interface thickness of
pure diffusion. Thus the precise threshold in φ separat- the drop [9]. We choose for the simulation R = 10ψ.
ingthe t1 andt1/3 regimesisnotpredictedeitherbyany Then we place another drop at the distance d from the
0
theory or by simulation. composite drop (defined as the aggregateof two coalesc-
Simulation of coalescence. We show here that a ing drops) and envelope these two drops by a spherical
t1 growth can originate from a coalescence mechanism shell to mimic the surrounding pattern of tightly packed
whose limiting process between two coalescences is not drops. Thusthedistancebetweenthedropsandtheshell
3
has been chosen to be d also. The surface tension σ is always repulsive due to the lubrication force. Thus they
0
supposed to be the same for all the interfaces. Unfor- tend to be as far from each other as possible. More-
tunately, due to the prohibitively long computing times, over, experiment shows a liquid-like order for the drop
we could not simulate the process of coalescence of two positions. Such a correlation explains why the drops do
spherical drops. Instead, we had to use a configuration not percolate [1] when the volume fractionφ reaches the
with cylindrical symmetry with respect to the axis X′ random percolation limit (φ 0.15).
≈
– X (see Fig. 2) which is expected to retain the main Sincenoquantitativeinformationisavailabletodeter-
features associated with the spherical shape. In the be- mineb,wecalculateitsupperandlowerbounds. Ideally,
ginning of the simulation the composite droplooks more we can assume that the drops are arranged into a reg-
like a torus. The spherical shell approximation can be ular lattice, with the vertices as far from each other as
justified by the fact that the main effect of the assembly possible. This is the face centered cubic lattice where
of surrounding drops (as well as of the sphericalshell) is b=π/3√2 0.74andwhichcorrespondstothe fullyor-
≈
to confine the motion of the neighboring drops – see [20] dered structure. We can also consider as a lower bound
foranadvanceddiscussion. Asetupwith onlytwodrops the random close packing arrangementfor spheres of ra-
without either a shell or surrounding drops would not dius R + d /2. This corresponds to the absence of a
0
enable a new coalescence even for the smallest interdrop short-range order [22] and implies b 0.64. We note
≈
distances. Though we cannot control quantitatively this that the value of b is not very sensitive to the particular
approximation,it is the simplest one which captures the spacearrangement. Inthefollowingweadoptthemedian
main features of the process. value b=0.69.
Afirstimportantresultfromthesimulationisthatthe Generalization of the hydrodynamic model.
flow generated by the first coalescence is able to gener- Now we can generalize the above hydrodynamic mech-
atea secondcoalescencebetweenthecompositedropand anism for an arbitrary shape of the drops. The self-
the neighboring drop (Fig. 2). This means that the lu- similarity of the growth implies the following relation
brication interaction with the surrounding drops (with for the characteristicsizes of the drops between i-th and
the shell in our model) make attract the composite and (i+1)-thcoalescences: R(i+1) =βR(i), where β is a uni-
the neighboring drops (note that the second coalescence versal shape factor, which depends on φ only. We can
does not take place between this neighboring drop and rewrite also the Eq. (6) for the time between the coales-
the shell). This leads to the formation of a new elon- cencesinthe formt(i) =α(φ)ηR(i)/σ, whereα(φ) isalso
c
gated droplet. When the drops are close enough to each the universal function. The last expression conforms to
other the composite drop will have no time to relax to a the scaling assumption which implies the independence
spherical shape since a new coalescence takes place be- of t(i) on the secondlength scale ψ. Then after n coales-
c
fore relaxation. An interconnected pattern naturally fol-
cences
lows. Incontrast,whenthedropsarefarfromeachother
(d /R 1),thesecondcoalescencewillnevertakeplace. n−1
0
The dr≫oplets take a spherical shape and the liquid mo- R=βnR(0), t= Xtc(i)
tion stops. It is also clear that if d /R < l , which we i=0
0 G
call “geometric coalescence limit”, coalescence necessary
and
occursduetogeometricalconstraint. Wefindl 0.484
G
≈
[20], [21]. β 1 σ
R=R(0)+ − t, (8)
The second important result is that the coalescence α · η
takes place also for l < d /R < l where l is a value
G 0 H H
which we call “hydrodynamic coalescence limit”. It is where R(0) is the initial size of the drop. Noting that
defined as a reduced initial distance where the time be- β = 21/3 for the spheres and β > 1 for the long tubes,
tweentwocoalescences(tc)becomesinfinite. Sincethere we can take β 1.1 for the estim∼ate. Since α 10 for
∼ ∼
is only one length scale (R) in the problem, tc can be φ = 0.5, as it follows from Fig. 3 and Eq.(7), we obtain
written in the scaled form R 0.01σ/ηt,whichcompareswellwiththe experiment
∼
[23], which gives 0.03 for the numerical factor.
t =αηR/σ, (6)
c Competition between two mechanisms. Using
where α is a reduced coalescence time which depends on now Eq. (7) we can relate l to a volume fraction φ .
H H
d /R only (Fig. 3). It is clear that the described hydrodynamic mechanism
0
The quantity d /R is relatedto the volume fractionof worksonlywhenφ>φ whiletheBrowniancoalescence
0 H
the drops (minority phase) φ: takes place in the whole range of φ. Below we shall con-
φ=b[1+d /(2R)]−3. (7) sider the regime for whichφ>φH inorderto obtainthe
0 position of the boundary between t1/3 and t1 regions on
The constantb depends onthe spacearrangementofthe the phase diagram.
drops. The hydrodynamic interaction between them is Taking into account the competition between the two
4
60
Now we aim to estimate the function G(φ) by using for
α(φ) the calculated function in Fig. 3 along with Eq.(7).
50 It turns out that the function G(φ) well fits the power
law
40 G(φ) (φ φ )−δ (12)
H
∝ −
forφ>φ whereφ 0.26,δ 0.33andthedivergency
H H
α 30 comes from α(φ). Fr≈om (12),≈it is easy to deduce that
(11) is valid when 0 < φ φ < 10−6. In practice, this
H
20 meansthatforallφ>φ −,theh∼ydrodynamicmechanism
H
only will determine the growth from the very beginning
10 of the drop coarsening. Alternatively, for φ < φH, the
l l drops will grow according to the Brownian mechanism
g H
only. This explains the sharp transition in the kinetics
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 (t1 t1/3) which is controlled by the volume fraction
→
d /R of the minority phase as observed in [1] and [2]. The
0
curve which corresponds to the threshold value φ=0.26
is plotted in Fig. 1. It shows a reasonable agreement
FIG. 3: The simulation data on the dependence of the re-
duced coalescence time α on d0/R. The vertical lines show withtheexperimentaldatainspiteofourveryparticular
thegeometric and hydrodynamiccoalescence limits. choice of the form and arrangementof the drops.
It should be mentioned that our model can be applied
toanysystemwherethegrowthisduetothecoalescence
mechanisms, we consider the relation
ofliquiddropsinsideanotherfluid(phaseseparation,co-
dn agulation, etc.).
= (N +N ), (9)
dt − B H Oneoftheauthors(V.N.)wouldliketothankthecol-
laboratorsof SPEC Saclay for their kind hospitality and
instead of (4), where N is the rate of the coalescences
H Minist`eredel’EnseignementSup´erieuretdelaRecherche
due to the hydrodynamics which can be calculated by
of France for the financial support.
using Eq.(8) and the relationbetween n and R (Eq.(5)).
The latter, however, depends on the shape of the drop.
Weassumethatintheearlystagesthedropsarespherical
and we use Eq.(5). It should be noted that the Eq.(9)
rewritten for the scaled wavenumber exactly coincides ∗ On leave from: Bogolyubov Institute for Theoreti-
withthe semi-empiricalequationsuggestedby Furukawa cal Physics, National Ukrainian Academy of Sciences,
[24]. 252143, Kiev,Ukraine; e-mail: [email protected]
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5
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c
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G
[17] A.ShinozakiandY.Oono,Phys.Rev.E48,2622(1993). conditionD =D +2R+3ψwhichmeansthattwodrops
s c
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s