Beading instability and spreading kinetics in grooves with convex curved sides

0.59 MB·English

by  Patrick B Warren

Beading instability and spreading kinetics in grooves with convex curved sides
Patrick B. Warren∗
Unilever R&D Port Sunlight, Quarry Road East, Bebington, Wirral, CH63 3JW, UK.
(Dated: December 30, 2015)
The coarsening kinetics for the beading instability for liquid contained in a groove with convex
curved sides (for example between a pair of parallel touching cylinders) is considered as an open
channel flow problem. In contrast to a V-shaped wedge or U-shaped microchannel, it is argued
that droplet coarsening takes place by viscous hydrodynamic transport through a stable column of
liquid that coexists with the droplets in the groove at a slightly positive Laplace pressure. With
some simplifying assumptions, this leads to a t1/7 growth law for the characteristic droplet size as
a function of time, and a t−3/7 law for the decrease in the droplet line density. Some remarks are
5 also made on the spreading kinetics of an isolated drop deposited in such a groove.
1
0 PACSnumbers: 47.55.nb,47.20.Dr
2

c
e I. INTRODUCTION kinetic spreading stages as it empties into the unfilled
D regions. This aspect will be discussed at the end.
Open channel flow problems have attracted much in-
0
3 terest not only because of possible applications in mi-
II. WICKING EQUATION
crofluidics [1–9], but also because of their relevance to

] liquids spreading on topographically patterned surfaces
t
f such as human skin [10–12]. The paradigmatic case of To start with, consider the general problem of open
o spreading in a V-shaped wedge has been analysed both channel flow in a channel of arbitrary cross section, and
s
. when liquid is supplied by a reservoir [10, 13], and in letA(x,t)bethecrosssectionoccupiedbyliquid. Alocal
at the starved (no reservoir) situation [14]. Various aspects mass conservation law holds [13, 14, 19],
m of these predictions have been confirmed experimentally
∂A ∂(Av)
- [2–4, 8]. Flows in U-shaped channels (i.e. with concave + =0. (1)
d sides) have also been considered since such channels are ∂t ∂x
n
easilymicro-machinedandarethusrelevantformicroflu-
o Herein, the mean flow rate v satisfies a Hagen-Poiseuille
idics applications [7, 11]. In the present study, I revisit
c (HP) law, v = −(k/η)∂p/∂x, in which k is the per-
[ theproblem,inthecontextofagroovewithconvex sides, meability (a quantity with units of length squared, cf.
such as that formed between a pair of parallel touching Darcy’slaw),η isviscosity,andp=p(A)isthe(loading-
2
cylinders (Fig. 1). In addition to possible microfluidics
v dependent) Laplace pressure. Combining the HP law
applications,interestinthisproblemismotivatedbycon-
0 withEq.(1)giveswhatcanperhapsbecalledthewicking
sideration of oily soil spreading along the fibres in tex-
2 equation,
2 tile yarns in woven fabrics [15]. It also has other poten-
5 tialtechnologicalrelevance,forexampletomoltensolder ∂A ∂ (cid:16)Ak dp ∂A(cid:17)
= . (2)
0 wicking in stranded copper wires and braids. ∂t ∂x η dA ∂x
1. This problem throws up some interesting aspects not
0 foundinthepreviouscases. Agroovewithconvexcurved Thisisthebasisformuchofthesubsequentdevelopment,
5 sides supports a stable uniform liquid column at low andalsocodifiesthestaticsviastabilityanalysis. Itgen-
1 loads,butathigherloadingauniformliquidcolumnmay erally has the character of a non-linear diffusion equa-
v: display a beading instability in which it breaks up into a tion. IhaveassumedthattheoccupiedcrosssectionAis
i stringofdroplets,similartoRayleigh’sobservationofthe weaklyvaryingwithx,sothecontributiontotheLaplace
X
breakup of a thread of treacle on a paper surface [16]. pressure from the interface curvature in the longitudinal
r Although the statics are by now quite well understood
a
[5, 6, 9, 15, 17, 18], I shall argue here that the loading
duality uniquely differentiates the present case from V-
shaped wedges and U-shaped microchannels, since the
droplets (beads) can coarsen by mass transport through
connecting liquid columns.
Additionally, the loading duality implies an isolated
droplet deposited into such a groove may show several
FIG. 1. Liquid in a groove between parallel touching cylin-
∗ Email: [email protected] ders.
2
direction can be neglected. Note that the permeability
1.0
also depends on the loading, so that k =k(A).
Insight can be gained by linearising about the uni- (a)
P Q
formlyloadedstaticsolution,viz.A=A andv =0. Let 0.5
0
us write A/A =1+(cid:15)(x,t). Then ∂(cid:15)/∂t=D ∂2(cid:15)/∂x2
0 eff
where D = (Ak/η)dp/dA is an effective diffusion co-
eff 0.0
efficient, evaluated at A = A . If this is positive, then
0
perturbations will die away and the liquid will be self-
-0.5
levelling. If it is negative, then perturbations will grow
indicative of the aforementioned beading instability. It
is clear that the behavior depends, not on the sign of -1.0
20 60 100 140 180
Laplace pressure p, but rather on the sign of dp/dA.
Physically, if dp/dA>0, an overfilled region will have a
higher Laplace pressure than an underfilled region, and
theliquidwillflowtoeventhingsout. Ontheotherhand, (b)
ifdp/dA<0,liquidwillflowfromunderfilledregionsinto
overfilled regions, magnifying the initial imbalance. The
interestingandunusualpropertyofagroovewithconvex
curved sides is that both situations occur, depending on
the loading.
III. STATICS
Aspecificexampleoftheloadingdualityisprovidedby FIG.2. (a)ThedimensionlessLaplacepressureasafunction
the groove between a pair of parallel touching cylinders, ofthefillingangle,withliquid-solidcontactangleθ=30◦. (b)
shown in Figs. 1 and 2. Since I neglect the interface cur- Fillingstagesinthisproblem: thethicksolidlineisthemax-
vatureinthelongitudinaldirection,thetransverseprofile imum Laplace pressure at α = 120◦; uniform loading above
this (dashed lines) is predicted to be unstable.
of the free surface is characterised by an arc of a circle
with radius R. Taking R > 0 to indicate the surface is
convexoutwards,onehasp=γ/Rwhereγ issurfaceten-
sion. In this problem it is convenient [9] to parametrise IV. KINETICS
the loading by the wrapping angle α, shown in Fig. 1.
Notethattheareaisamonotonicallyincreasingfunction A. Beading instability
of α, so dA/dα>0.
Elementary trigonometric arguments, first presented
Aswehaveseenforthecaseofparalleltouchingcylin-
to my knowledge by Princen [15], show that
ders, the Laplace pressure in a groove with convex sides
a cos(θ+α) may show a maximum as a function of the loading. Be-
=− (3)
yond the maximum, dp/dA<0, and therefore a beading
R 1−cosα
instability arises in which the uniformly loaded state is
whereθistheliquid-solidcontactangle. Thisresultholds
unstable towards the growth of perturbations. But first,
for both convex and concave interfaces, and the sign has
whatcouldbethefinalstateinsuchasituation? Inprin-
been inserted in accord with the above convention that
cipleonecanhavetwodifferentstatesofloadingatsame
R is positive if the interface is convex.
Laplace pressure, which can therefore be in coexistence,
From this we find (after a little rearrangement)
for example points P and Q in in Fig. 2b. However the
dR R2cos(1α+θ) higherloadedstateisalwaysintheunstableregion. The
=− 2 . (4) logicalconclusionisthattheexcessliquidisexpelledinto
dα 2asin3(1α)
2 alargedropletthatsitssomewhereonthetwocylinders,
Since p ∝ a/R, Eqs. (3) and (4) show that p increases coexisting with a stable column of liquid in the groove.
through zero at α+θ = 1π to reach a weak maximum In the case of parallel touching cylinders, the wrapping
2
at 1α+θ = 1π (i.e. α = π −2θ). A specific example angle for this stable liquid column would have to satisfy
is s2hown in F2ig. 2a, for a contact angle θ = 30◦. The 1π−θ <α<π−2θ. The first inequality arises because
2
zero crossing is at α = 60◦ and the Laplace pressure the liquid column coexists with a large drop at a (weak)
maximumisatα=120◦. Fig.2bshowsaselectionofthe positive Laplace pressure [21]. The second inequality is
corresponding filling states. The stability requirement the column stability condition.
that dp/dA ∝ dp/dα > 0 indicates that filling states I turn now to the coarsening kinetics. A uniformly
with α > 120◦ are unstable with respect to the above- overloaded state will certainly break up into a string of
mentioned beading instability [20]. droplets, but in a groove with convex curved sides these
3
5
10
2
4
10
1.0
3
10 0
12
1 10
2
10
10
1
3 6 9 12 0.0
1 10 10 10 10 0.0 1.0 2.0
FIG.3. Meandropletvolumeasafunctionoftime. Droplets
FIG. 4. Scale invariance of droplet size distribution. His-
coarsen by viscous hydrodynamic transport along the con-
tograms are computed when there are 10000, 5000, 1000,
nectingliquidcolumns(diagrammaticinset),accordingtothe
500,300and100dropletsremainingoutofaninitial5×104,
rules prescribed in the Appendix. Results from 10 indepen-
combining data from 100 independent simulation runs. Sim-
dent simulation runs are shown. Each simulation was ini-
ulation parameters as for Fig. 3.
tialised with 5×104 equispaced droplets, with random sizes
taken from a uniform distribution V < V < 2V . Droplets
0 0
are removed when V < 0.9V , and their liberated contents
0 particular geometry though.
are added to the liquid column (β = 1 in the model). The
Themassfluxisincorporatedintoalocal massconser-
time step parameter was (cid:15)=0.02.
vation law, cf. Eq. (1), which governs the growth of the
mean droplet volume,
droplets are always connected by liquid columns with
dV
a finite filling depth according to the above argument. ∼J. (6)
dt
Thereforethelargerdropletscaneatthesmallerones,by
transporting liquid along the connecting liquid columns. At the same time there is a global mass conservation law
ThisstandsincontrasttotheV-shapedwedgeforexam- which relates the mean dropletsizeto the mean spacing,
ple, where spatially separated droplets are disconnected
V
[4, 22, 23] and the droplet population has to coarsen by
∼ω ∼const, (7)
some other mechanism, such as a prewetting film [24– L
26], or transport through the vapor phase in the case of
where ω is the mass per unit length [30].
a volatile liquid.
Combining Eqs. (5)–(7), together with V ≡ R3 and
WithasimplifyingassumptionabouthowtheLaplace
∆p∼γ/R, shows that
pressure depends on the droplet size, aspects of the
Lifshitz-Slyozov-Wagner (LSW) theory of droplet coars- dR γa4ω
R2 ∼ . (8)
ening [27] can be adapted to the present case. The sim-
dt ηR4
plifyingassumptionisthattheLaplacepressurescalesin-
versely with droplet size as p∝V−1/3, where V ≡R3 is This integrates to
thedropletvolume. Thisislikelytobetrueonlyasymp-
(cid:16)γa3ωt(cid:17)1/7
totically [28, 29]. Nonetheless, let us suppose that this R∼ . (9)
is true, and the situation has evolved so that there is η
a string of droplets sitting in the groove. The follow-
Thus the prediction is that the mean droplet size should
ing mean-field scaling ansatz predicts how the character-
growast1/7andthedropletlinedensity(i.e.1/L∼R−3)
istic droplet size R, and characteristic spacing between
should diminish as t−3/7.
droplets L, evolve with time.
The mean-field assumption is questionable given the
First, the mass flux between adjacent droplets will be
one-dimensional nature of the coarsening problem. To
a4 ∆P investigate specifically just this aspect, I undertook nu-
J ∼ . (5) mericalsimulationsusingthemodeldescribedintheAp-
η L
pendix. This confirms that the mean-field scaling ansatz
This just expresses the HP law in scaling form. Shown doesindeedpredicthowthemeandropletsizegrowswith
here is the prefactor for the case of parallel touching time (Fig. 3), and also demonstrates LSW-like scale in-
cylinders,wherethefourthpowerofthecylinderradiusa variance for the droplet size distribution (Fig. 4).
arisesfromtheproductoftheDarcypermeabilityk ∼a2 The origin of the mean-field behaviour is apparent if
and the cross sectional area of the connecting liquid col- oneexaminestheequal-timedropletsizecorrelationfunc-
umn A ∼ a2. The analysis does not depend on this tion, C(|i − j|), shown in Fig. 5. As can be seen the
4
Washburn (BLCS) type flow from what is in effect a
1.0
0.0 shrinking droplet reservoir [10, 13, 19]. This should per-
sist all the way until the loading falls below the Laplace
-0.1 pressure maximum. Past this point, by analogy to the
0.5 V-shaped wedge [14], one expects the spreading rate to
-0.2
slow down since the reservoir has been exhausted.
-0.3 For the case of parallel touching cylinders, a second
102 106
power law appears at a very late stage where every-
0.0 where α (cid:28) 1. In this limit, the wetted portion of the
groove has shrunk to a narrow fissure with a width of
the order R ∼ α2a (see Eq. (3) in the limit α → 0)
and a depth of the order αa. One therefore expects
-0.5
0 5 10 A∼α3a2,andpresumablyk ∼α4a2sincethepermeabil-
ity should largely be determined by the width. Similar
to the coarsening kinetics problem, a scaling analogue
FIG.5. Equal-timesizecorrelationfunction,C(|i−j|),com-
of the HP law can be introduced. In the present case
puted for the same set of simulations used for Fig. 4. The
inset shows the time dependence of the depth of the nearest- this is dL/dt ∼ (k/η) × ∆p/L where L is the length
neighbourminimum. Errorbarsarefromblockaveraging(10 of the wetted portion of the groove and ∆p ∼ γ/R is
blocks × 10 runs). the Laplace pressure. Substituting the above scaling ex-
pressions gives dL/dt ∼ γα2a/(ηL). An additional con-
straint comes from the analogue of the global mass con-
only significant correlation appears to be between near- servation law in Eq. (7), namely that the total droplet
estneighbours,at|i−j|=1. Thisindicatesthatdroplets volume Ω ∼ α3a2L should be conserved. Eliminating α
which are larger than average tend to be adjacent to between this volume constraint and the HP scaling law
droplets which are smaller than average, but apart from yields dL/dt ∼ γΩ2/3/(ηa1/3L5/3). This integrates to
this no significant long-range correlations develop. the final rather esoteric result
To see the origin of this nearest-neighbour correla-
L∼(γt/η)3/8Ω1/4a−1/8. (10)
tion hole, consider an artificial situation in which one
large droplet sits in a uniform string of equisized smaller Inotherwords,theinitialL∼t1/2spreadinglaw(BLCS)
droplets. Away from the large droplet, the ambient
should weaken when the droplet reservoir vanishes, and
Laplace pressure is uniform and no coarsening takes eventually enter an L ∼ t3/8 power law in the final
place. However the large droplet has a sub-ambient
starved state.
Laplace pressure and so starts to draw in material from
its immediate neighbours. This causes the immediate
neighbours to shrink, and increases their Laplace pres-
V. DISCUSSION
sure relative to the ambient background. The next-
nearest neighbours then see that these droplets have
Ihavearguedthatthekineticaspectsofspreadingand
started to shrink, and so they in turn start to grow. The
(de)wetting for a liquid contained in a groove with con-
process continues, and it is easy to see that it will gener-
vexcurvedsidespresentssomeuniqueaspectswhencom-
ateastaggeredarrayofdropletsizes,withaconcomitant
pared, for example, to a V-shaped wedge or a U-shaped
nearest-neighbour negative size correlation.
microchannel. The novel aspects arise from an underly-
The nearest-neighbour correlation hole diminishes
ing loading duality, wherein a liquid column is stable at
somewhat as time progresses, but eventually appears to
low loading, but becomes unstable at higher loading.
settle down to a value C(1)≈−0.15 (Fig. 5 inset). One
Forthecaseofuniformloadingabovethecriticalload-
should point out that the coarsening dynamics in the
ing threshold, a beading instability should be observed
model is quite subtle, for instance, the largest droplet at
in which the liquid column breaks up into a string of
timetmaynotnecessarilybethelargestdropletatsome
droplets, which subsequently coarsen by mass transport
later time t(cid:48) > t. This is because the growth rate of a
along connecting liquid columns. With a simplifying as-
droplet depends not only on its size but also on the sizes
sumption about the dependence of the Laplace pressure
of its neighbours, and how far away they are.
on the droplet volume, a mean-field scaling ansatz in-
dicates that the droplet size and line density scale with
non-trivialpowerlawsintime,andsimulationsshowthat
B. Droplet spreading thisisnotdestroyedbytheone-dimensionalnatureofthe
problem.
The remaining point of kinetic interest concerns the Of course, the simplifying assumption about the
fate of a droplet of liquid deposited onto an initially Laplace pressure scaling does not hold in reality, and in
empty groove. The droplet will start to empty into general I would expect the clean power law behaviour to
the groove, presumably driving a Bell-Cameron-Lucas- be modified by finite-droplet-size effects, which may be
5
quite persistent. Some general predictions should be ro- Eqs. (A1) and (A2) are the required set of non-
bust however, such as the relatively slow droplet growth linearkineticequations. Sincetheypredictthatdroplets
via transport along connecting liquid columns, and the shrink,aswellasgrow,weneedarulewhichgovernshow
negative correlation between nearest neighbour droplet shrinking droplets can disappear. At this point it is con-
sizes shown in Fig. 5. These could perhaps be tested in venient to introduce a fiducial volume V ∼ R3, where
0 0
anelectrowettingexperiment[3,4]. Ishouldcautionthat the fiducial length R is set by the height of the con-
0
the predicted slow coarsening kinetics may be overtaken necting liquid column (in the simulations, V =R =1).
0 0
by other, ultimately faster, mechanisms. For example A simple rule for shrinking droplets is that they vanish
transport through the vapour phase for a volatile liquid when V < αV . If this happens, the droplet is removed
i 0
may ultimately lead to a R ∼ t1/3 coarsening law, as in and the distance between the remaining droplets is set
LSW theory [27]. equal to L +L +βV1/3, where the third term is an
i−1 i i
Another prediction arising from the loading duality is ad hoc correction for the length contributed by the van-
that a droplet deposited in the groove should show a ished droplet (taking β as a free parameter).
staged spreading kinetics, starting with the classic Bell-
As an initial condition I set V = r V where r is a
Cameron-Lucas-Washburn law as the droplet initially i i 0 i
random number chosen from a uniform distribution, 1≤
actslikeareservoir,slowingwhentheloadingfallsevery-
r <r . Thedropletsareinitiallyequispaced,withL =
where below the critical value, and possibly ending with i m i
R . Periodic boundary conditions are imposed.
a new power law in the final starved state. Again, this 0
may perhaps be probed experimentally. The droplet volumes are evolved according to
Eqs. (A1) and (A2), alongside the above rule for remov-
ingdropletswhichbecometoosmall. Eqs.(A1)and(A2)
Appendix A: Simulation of coarsening kinetics are integrated using a simple, adaptive, Euler-type for-
wardfinitedifferencescheme,withatimestep∆tchosen
I introduce a simplified model of the coarsening kinet- suchdV /dt×∆t/V ≤(cid:15), inotherwordssothatthefrac-
i i
ics to test specifically the mean-field ansatz presented tionalchangeinanydropletsizedoesnotexceed(cid:15)inany
in the main text. In the model, I consider a one- time step. For the reported simulations I used α = 0.9
dimensional string of i = 1...N droplets, connected by and β = 1 for the vanishing rule, r = 2 for the max-
m
liquid columns as indicated in the lower inset in Fig. 3. imum initial drop size relative to V , and (cid:15) = 0.02 for
0
Big droplets grow, and small droplets shrink, under the the choice of time step. I have checked the results are
influenceofviscoushydrodynamictransportthroughthe insensitive to these choices.
liquid columns. This is driven by differences in the
As time evolves, the larger droplets grow at the ex-
Laplace pressure between neighbouring droplets. To es-
pense of the smaller droplets, and the smallest droplets
tablishasystemofkineticequationsforthedropletsizes,
shrinkandvanish. Eventuallythesimulationstopswhen
I suppose that the Laplace pressure in the i-th droplet
there is one large final droplet (N = 1). I monitor the
is proportional to Vi−1/3, where Vi is the droplet volume mean droplet volume (cid:104)V(cid:105) = (1/N)(cid:80)N V as a func-
(thisisthesimplifyingassumptionmentionedinthemain i=1 i
tionoftime(notethatN changesasdropletsdisappear),
text). AccordingtotheHPlaw,theLaplacepressuredif-
and at selected time points record the drop size distri-
ference drives a mass flux through the connecting liquid
bution. Typical results, aggregated from multiple sim-
column as, cf. Eq. (5),
ulation runs from independent starting points, are sum-
marised in Figs. 3 and 4.
V−1/3−V−1/3
Ji = i L i+1 . (A1) I also calculate periodically the equal-time correlation
i function
In this L is the distance between the i-th and (i+1)-th
i
droplets,andallothermaterialpropertiesintheproblem
(cid:104)∆V ∆V (cid:105)
havebeensubsumedintothedefinitionsoflength,volume C(|i−j|)= i j , (A3)
(cid:104)∆V2(cid:105)
andtime. NotethatV >V impliesJ >0,sothatliq-
i+1 i i
uid flows from smaller droplets to larger droplets. Given
the fluxes, mass conservation dictates that, cf. Eq. (6),
where ∆V = V −(cid:104)V(cid:105) is the deviation from the mean
i i
droplet volume, at time t. This function is shown in
dV
i =J −J . (A2) Fig. 5, evaluated at various points in the simulation.
dt i−1 i
[1] T.M.SquiresandS.R.Quake,Rev.Mod.Phys.77,977 mann, Langmuir 23, 5200 (2007).
(2005). [3] K. Khare, et al. Langmuir 23, 12997 (2007).
[2] J.-C.Baret,M.M.J.Decr´e,S.Herminghaus, andR.See- [4] K. Khare, et al. Eur. Phys. J. Special Topics 166, 151
6
(2009). than60◦.Thebehaviorstandsinmarkedcontrasttothe
[5] X.-F. Wu, A. Bedarkar, and K. A. Vaynberg, J. Colloid influence of contact angle on the height of a droplet sit-
Interface Sci. 341, 326 (2010). ting as a spherical cap on a flat surface. This considera-
[6] A. Bedarkar, X.-F. Wu, and A. Vaynberg, Appl. Surf. tionpotentiallyinfluencesthewickingofliquidsinwoven
Sci. 256, 7260 (2010). fibrous networks, bearing in mind the applications men-
[7] D. Yang, et al. J. Phys. Chem. C 115, 18761 (2011). tioned in the introduction.
[8] J. Barman, et al. Langmuir 31, 1231 (2015). [21] Aconsequenceofthisisthatasmoreliquidisadded,the
[9] A. Sauret, et al. Soft Matter 11, 4034 (2015). amount contained in the groove should go down, since
[10] A.D.Dussaud,P.M.Adler, andA.Lips,Langmuir19, adding liquid increases the size of the large droplet and
7341 (2003). reduces its Laplace pressure (towards zero), thereby in
[11] Y. Chena, et al. Micro. Engn. 86, 1317 (2009). paralleldiminishingtheloadingintheliquidcolumncon-
[12] R.Seemann,M.Brinkmann,S.Herminghaus,K.Khare, tained in the groove.
B. M. Law, S. McBride, K. Kostourou, E. Gurevich, [22] P.ConcusandR.Finn,Proc.Natl.Acad.Sci.(USA)63,
S. Bommer, C. Herrmann, and D. Michler, J. Phys.: 292 (1969).
Condens. Mat. 23, 184108 (2011). [23] K.Rejmer,S.Dietrich, andM.Napio´rkowski,Phys.Rev.
[13] L. A. Romero and F. G. Yost, J. Fluid Mech. 322, 109 E 60, 4027 (1999).
(1996). [24] A. O. Parry, C. Rasco´n, and A. J. Wood, Phys. Rev.
[14] P. B. Warren, Phys. Rev. E 69, 041601 (2004). Lett. 85, 345 (2000).
[15] H. M. Princen, J. Colloid Interface Sci. 34, 171 (1970). [25] J.M.Romero-EnriqueandA.O.Parry,NewJ.Phys.9,
[16] Lord Rayleigh, Phil. Mag. 34, 145 (1892). 167 (2007).
[17] A.E.Sa´ezandR.G.Carbonell,J.FluidMech.176,357 [26] A. Malijevsky´ and A. O. Parry, Phys. Rev. Lett. 110,
(1987). 166101 (2013).
[18] C. Duprat, A. D. Bick, P. B. Warren, and H. A. Stone, [27] A. J. Bray, Adv. Phys. 43, 357 (1994).
Langmuir 29, 7857 (2013). [28] B. J. Carroll, J. Colloid Interface Sci. 57, 488 (1976).
[19] M. Reyssat, L. Courbin, E. Reyssat, and H. A. Stone, [29] P. G. de Gennes, F. Brochard-Wyart, and D. Qu´er´e,
J. Fluid Mech. 615, 335 (2008). Capillarity and Wetting Phenomena (Springer, New
[20] Aninterestingcorollaryisthatthemaximumstablefilled York, 2004).
state may protrude above the cylinders (cf. the heavy [30] Itseemslikelythatω≡R3/L∼a2 sincethisistheonly
filledlineinFig.2b).Forparalleltouchingcylindersone length scale. This would correspond to an initial droplet
can show that this occurs if the contact angle is smaller volume of order a3 and droplet spacing of order a.