Table Of ContentInflueInnflcueeonfceSloipf SolinptohnetPhelaPtelaatue-aRua-RylaeyilgehighInIsntsatbaibliitliytyoonnaaFFiibbrree
SabrinaSaHbrainefaneHra,1e,f2neMr,1i,c2haMeilchBaeenlzBaeqnuzeanq,u3enO,3livOelrivBer¨auBm¨aucmhecnh,e2n,,42,T4hTohmomasasSaSlaelze,z3,3RRoobbeerrtt
Peters,P2eJteorssh,2uaJoDsh.uMa cDG.rMawcG,1raKwa,1rinKaJraincoJbasc,o1bEs,l1ieERlieapRhaap¨ehla,3¨el,a3nadndKaKrairDi Dalanlonkoki-iV-Veerreessss22,,33,,∗⇤
1Depart1mDeepnatrtomfeEnxtpoefriEmxepenrtiamlePnhtaylsiPchsy,sSicasa,rSlaanardlaUndniUvenrisvietrys,itDy,-6D6-0646104S1aSaarabrrbu¨rcu¨kceknen,,GGeerrmmaannyy
2Depar2tDmeepnatrtomfenPthyosfiPcshy&sicAss&troAnsotrmony,omMyc,MMacsMtearstUerniUvneirvseirtysi,tyH,aHmaimltioltno,n,CCananaaddaa
3PCT Lab, UMR CNRS 7083 Gulliver, ESPCI ParisTech, PSL Research University, Paris, France
3PCT Lab, UMR CNRS 7083 Gulliver, ESPCI ParisTech, PSL Research University, Paris, France
4Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Go¨ttingen, Germany
4Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Go¨ttingen, Germany
The Plateau-Rayleigh instability (PRI) of a liquid column underlies a variety of
ThePlateau-Rayleighinstabilityofaliquidcolumnunderliesavarietyoffascinatingphenomena
hydrodynamic phenomena that can be observed in everyday life. In the classical case
that caonfbaefroebeselirqvuedidicnoeluvemrynd,aliyneliafer.pIenrtcuornbtaratisotntothtehoerycapsreeodficatsfrcehealriaqcutidericsytliicndriesre,-dtiemscersibainngd
the evowluatvioenlenofgtahsliqoufisdurlafayecreounndauslaotliidonfisb.rIenrceoqnutirreasstcotnosaidferreaetiloinquoifdtchoelusmolind,-ltihqeuiddeisnctreirpftaicoen.
5
In thisoafrtaicllieq,uwidelraeyveisritonthae sPollaitdeafiub-RreayrleeqiguhireInssctaobnisliidtyeroaftiaonliqoufidthceoasotilnidg-laiqfiubidreinbtyervfaarcyeinign
1
the hydardoddiytnioanmitcobtohuendfraereyicnotnedrfitaicoen. aHtetrhee, fiwberer-elivqiusiidt tinhteerPfaRceI,ofrfoamlinqou-isdliplatyoersloipn. aWsohliilde
0
the wavfieblernegtbhyivsanroytinsgentshiteivheytdorothdeynsoalmidi-cliqbuoiudnidnaterryfaccoe,ndwietiofinndatthtahtethfiebrger-olwiqtuhidratineteorfftahcee
2
undulatfiroonms sntoro-snlgiplytdoespleipn.dsWoenfitnhde hthyadtrotdhyengamroiwctbhourantdearoyf tchoenduitnidonu.latTiohnesesxtpreornimgleyntdsepareendins
y excellenotnatghreeemsyesnttemwitgheoamneetwrytahsinwfiellml asthoenorythienchoyrdprooradtyinngamsliicp,btohuunsdparroyvicdoinngditainono,riwgihniaclh,
a quantitiastiivne aenxdcerlolebnutstatgoroelemtoemnteawsuitrhe stlhipeolernyg.thWs.hile the wavelength is not sensitive to the
M boundary condition, the rise-time is strongly a↵ected and provides an opportunity to
quantify the slip-length.
4 Glistening pearls of water on a spider’s web [1], or the
2 breakupoGfalisctyenliinndgrpiceaalrljsetofowfwataetreorninatospdidreorp’slewtse,ba[r1e],foar-the (a) (b)
] miliar mbarenaifkeusptaotfioancsyloinfdtrhiecaPlljaetteoafuw-Rataeyrlienitgohdirnosptlaebtsi,liatryefa- h0
n (PRI) [2m,i3li]a.rBmyaneivfeosltvaitniognisnotof tdhreoPpllaettesa,ut-hReasyulerifgahceinastraebaility e0
a
x
y oftheliq(PuRidI,)a[2n,d3c].onBsyeqeuveonlvtilnygthinetosudrrfoapcleetesn,etrhgey,suarrfeacree-area
d duced. Tofhtihseinliqstuaidb,ilaitnydaclosnoseaqcutsenftolryathleiqsuuirdfacfielmenecrogayt,inargere- t=0
u- a solid fidbucreed[.4–T1h2is],inthstoaubgilhitythaelsflooawctsbofournadalirqyuicdofinldmitcioonating ⇣ h(�x⇤,t)
l at the soalisdo-lildiqfiubidrein[4t–e1rf2a],cethporuogvhidtehseafldodwitbioonuanldcaorymcpolnexd-ition e0
.f ity to thaettshyestseomlid.-lWiquhiidleinttheerfabcreeapkruovpidoefsaadhdoimtioongaelnceoomupslex- a xx
s filmintoitydrtooptlehtessoynstaemfi.brWemhilaeytbheeabrneuakisuapncoefianhcoomatoignegnous t=20min
c
filmintodropletsonafibremaybeanuisanceincoating h(x,t)
i technologies of e.g. wires and optical fibres, this funda- ⇣ �⇤
s technologies of e.g. wires and optical fibres, this funda-
mental instability turns out to be very useful: take for
y mental instability turns out to be very useful: take for e0
h exampleexwaamtperlecwolalteecrticoonlletchtrioonugthhrfoougghhaforgvehsatrinvegst[1in3g,[1143],,14], a xx
p abiomimeticapproachthatisperfectedinnaturebythe
abiomimeticapproachthatisperfectedinnaturebythe t=3900min
[ spider’s web [1]. Hence, taking advantage of the PRI on
spider’sweb[1]. Here,wetakeadvantageofthePRIona
2 a fibre,fiebnraebtloesgaginaipnhinygsicpahlyinssicigahltiinnstioghthteisnotloid-tlhiqeuisdolbido-und- FIG. 1. Plateau-Rayleigh instability on a fibre.
v liquid baoruyncdoanrdyiticoonn.dIintdioened., uInnddeeresdt,anudnindgertshteanbdreinakg-dtohwen of (a) Schematic and (b) optical micrographs illustrating the
4 break-dtohwennoo-fsltiphebonuon-dslairpybcoonudnidtiaorny[1c5o]nidsiotfiomnaj[o1r5]initseroefsttoFIGP.R1I.for aPlliaquteidaupo-Rlysatyylreenigehfilminosntaabgillaitsys fiborne. Aat tfi=br0e,.
9 major inthteersecsitenttoifitchaensdciienndtuifistcriaalndcominmduusntirtiiaesl acsomitmhuasnip-rac-(a)tShcehepmolaytsitcyraennedfi(lbm) oopntitchael mnoi-csrliopgrfiabprheshilalsusatrathtiinckgnetshse
1 ties as ittichaalsimprpalicctaitciaolnsiminplaicraeatisotnhsaitninavroelavsetshmaatlli-nsvcaollevefluidPRIe0fo=r1a3.l2iq±ui1dµpmolaynsdtytrheengelafislsmfibornearadgiluasssisfiabr=e.9.A6±t t1=µm0.,
2 small-scsaylseteflmusi,dssuycshteamssl,abs-uocnh-aa-cshilpabd-oevni-cae-sc,hflipowdsevinicpeso,roustheTphoelywsitdytrhenoefthfielmoptoincaltihmeagneos-sisli5p60fiµbmre. has a thickness
1.0 flaofwews.inmpeoTdrhioaeuascslmawsesedilcliaaalsacbsaiswoeleolgolifcaatslhbeflioobwlrosegatikocuanplaflmoofewaasftleaowmn.ianmarefreeTe0h=e w13id.2th±o1fµthmeaonpdtictahleigmlaasgsesfibisre56ra0dµimus.isa=9.6±1µm.
0 The cflloawssinicgalliqcuaisdesotrfeatmheisbrdeeapkeunpdenotf oanlmamatienraiarlfpreroeper- the growth dynamics of the PRI in experiments and an-
5 ties and geometry [16]. While there are some studies of alytical theory involving hydrodynamic slip has thus far
flowing liquid stream is dependent on material proper-
1 ties andthgiesoimnsettarbyili[t1y6]i.nWgeohmileettrhieesreotahreer sthomanefrseteudciyelsinodfricalnotnboetebneeanchaicehvieedve.d.
: liquid flows [17–23], the role of the boundary condition
v this instability in geometries other than free cylindrical ChaCrhaacrtaecrtisearitziaotnionofofslisplipbobuounnddaarryyccoonnddiittiioonnss aatt iinn--
at the solid-liquid interface on the evolution of the PRI
Xi liquid flioswlesss[1w7e–l2l3u]n,dtehrestrooolde[o2f4,th25e].boItunhdasarbyeecnonshdoitwionnthatterftaecrefasceasndantdhethireicrocnotnrtorlolilnligngppaararammeetteerrss hhaass bbeeeennaacc--
tivelyinvestigatedintheliterature[27–31]. Theclassical
at the solid-liquid interface on the evolution of the PRI tivelyinvestigatedintheliterature[27–31]. Theclassical
r the appearance of a dewetting liquid rim undergoing a boundaryconditionassumesnoslipatthesolid-liquidin-
a is less wPellaltueanud-eRrasytoleoigdh-[2ty4p,e2i5n]s.taIbtilhitaysisbieneflnuesnhcoewdnbyththatehy-boundaryconditionassumesnoslipatthesolid-liquidin-
terface. That is, for simple shear flow in the x direction
the appderaordaynncaemoifcbaoduenwdaertytincogndliiqtiuoindartimtheusnodliedr-glioqiunigdianter-terface. That is, for simple shear flow in the x direction
along a solid interface placed at z = 0, the tangential
Plateauf-aRcaey[l2e6i]g;hh-otwypeveeirn,satagbenileitryaliqsuinanfltuiteantciveedmbyattchhebheytw-eenalong a solid interface placed at z = 0, the tangential
flowvelocity,v (z),vanishesatthesolid-liquidinterface:
x
drodynamicboundaryconditionatthesolid-liquidinter- flowvv(e0l)oc=ity0,.vxH(zow),evvaern,isahsesalaretatdhyensooltiedd-libqyuiNdaivniteerr[f3a2c]e,:
x
face [26]; however, a general quantitative match between vx(0th)er=e i0s.noHofuwnedvaemr,enatsalaplrreinadciyplenorteeqduirbinygNvav(0ie)r=[320],,
x
the growth dynamics of the PRI in experiments and an- theraendisonneocfaunnadlasomheanvtealhypdrrinodciypnlaemriecqsuliipripnaggevaxt(0t)his=in0-,
alytical⇤[email protected] slip has thus far andteornfaecec,anasadlseofinheadvbeyhythderosldipynleanmgitch:slbip=pa[vgxe/a@tzvtxh]izs=i0n,-
|
terface, as defined by the slip length: b = [v /∂ v ] ,
x z x z=0
|
where b=0 corresponds to the classical no-slip case.
Inthisarticle,weexplicitlyaddresstheeffectofavary-
∗ [email protected] ing boundary condition, from no-slip to slip at the solid-
2
liquid interface, on the PRI of a viscous liquid layer on a hydrodynamic boundary condition.
solidfibre. Wefindthatthegrowthrateoftheinstability
is strongly affected by the solid-liquid interface, with a The theoretical approach. The experimental find-
faster breakup into droplets for a slip boundary condi- ings can be understood within the lubrication approxi-
tion compared to an equivalent sample with no-slip. In mation, from a thin-film model based upon the Laplace
contrast, the wavelength λ of the fastest growing mode pressure-drivenStokesequation. Weassumeincompress-
∗
is not sensitive to the solid-liquid interface. The linear ible flow of a viscous Newtonian liquid film, of thickness
stabilityanalysisofanewlydevelopedthinfilmequation varying from 5 to 93 um which is well above the film
incorporatingslipisinexcellentagreementwithourdata. thickness where disjoining pressure plays a role (a few
The theory is valid for all Newtonian liquids and enables tens of nanometers) [34]. Gravitational effects can be
the precise determination of the capillary velocity γ/η, neglected since all length scales involved in the problem
and most importantly of the slip length b. arewellbelowthecapillarylengthlc 1.73mm. Finally,
’
thevelocitiesoftheliquidfilmsaresmall(e.g. thefastest
observed rate of change in the amplitude is 25nm/s).
∼
Thus, theReynoldsandWeissenbergnumbersareorders
ofmagnitudesmallerthan1andinertialandviscoelastic
RESULTS
effects can be ignored.
Wenon-dimensionalisetheproblem(seeFig.1forvari-
The experimental approach. An entangled able definitions) through:
polystyrene (PS, 78kg/mol) film with homogeneous
h h x λ b γ t
0 ∗
thickness e0 (5 to 93 µm), is coated onto a fibre with ra- H0 = a , H = a, X = a, Λ∗ = a , B = a, T = ηa,
dius a (10 to 25 µm), resulting in a PS coated fibre with
(1)
radius h = a+e , as schematically shown in Fig. 1(a).
0 0 wherethecapillaryvelocityγ/η istheratiooftheliquid-
Glass fibres provide a simple no-slip boundary condition
air surface tension to the viscosity of PS. By assuming
[36]. In contrast, a slip interface results from coating the
volumeconservation, nostressattheliquid-airinterface,
entangled PS film onto a glass fibre pre-coated with a
and the Navier slip condition at the solid-liquid bound-
nanometric thin amorphous fluoropolymer (AF2400, 14
ary, one obtains (see Supplementary Methods) the gov-
1 nm) [33]. The fluoropolymer coating on glass was
± erning equation for the dimensionless profile H(X,T):
used because it is well established that PS, above a crit-
ical molecular weight (Mc 35kg/mol), exhibits signifi- 1 H0+H2H000 0
∼ ∂ H + (H,B) =0 , (2)
cant hydrodynamic slip at this solid-liquid interface [33]. T H 16 M
Henceforth, we will refer to these as the “no-slip” and (cid:20) (cid:21)
where: (H,B)=
“slip” fibres.
M
All samples were prepared and stored at room tem- 4B 1
4H2logH +(4B 3)H2+4 8B+ − , (3)
perature, well below the PS glass transition temperature − − H2
(T 100 C), thereby ensuring that there is no flow in
g ◦ and where the prime denotes the partial derivative with
∼
the PS film prior to the start of the experiment. Prior
respect to X. Note that Eq. (2) is a composite equation
to each experiment, the PS coated fibre was measured
inthesensethatwehavekeptasecondorderlubrication
with optical microscopy as shown in the t = 0 image
terminthepressurecontribution: theaxialcurvature. It
of Fig. 1(b). To initiate the experiments, samples were
is thelowest orderterm counterbalancing thedriving ra-
annealed in ambient atmosphere at 180 C – well above
◦ dial curvature, and it is thus crucial to obtain the actual
T – which causes the PRI to develop. The evolution of
g threshold of the instability. The dynamical aspect of the
the surface profile was recorded with optical microscopy.
flow is however well described at the lowest lubrication
Figure 1(b) shows a typical evolution for a PS film on a
order. An interesting discussion on this matter can be
no-slip fibre.
found in Ref. [9].
Above T , the liquid PS film becomes unstable, which Performing linear stability analysis, namely letting
g
causes variations in the local axisymmetric surface pro- H(X,T) = H + ε(T)eiQX, where ε(T) 1, yields
0
(cid:28)
file, h(x,t) = h + ζ(x,t), over the axial coordinate x an exponential growth of the perturbation of the form
0
and time t. The amplitude ζ(x,t) of the undulations ε(T) eT/τ(Q), where the rate function is given by:
∝
grows with time and finally results in a droplet pattern
1 (H ,B)
dspisaptliaaylivnagriaatuionnifoofrmζ(xw,atv)eilnentghtehi,nλit∗ia.lBdyevmeloeapsmuernintg,atnhde τ(Q) =Q2 1−H02Q2 M16H00 . (4)
locatingthemaxima,thePRIwavelengthofeachsample We define the fastest(cid:0)growing m(cid:1)ode Q corresponding
∗
was determined (see Fig. 1(a)). Typically, four or five to the smallest time constant τ = τ(Q ), and obtain
∗ ∗
wavelengths were averaged per sample. In addition, by Q =1/(H √2). Writing Q in terms of the dimension-
∗ 0 ∗
measuring the temporal change in radius of an individ-
less wavelength leads to:
ual bulge, we gained information about the growth rate
of the instability, and consequently the influence of the Λ =2π√2H , (5)
∗ 0
3
1,200 1
Glass (no-slip) (a) (b)
AF2400(slip)
900 Eq. (5)
0
m]
µ 600
[
∗ −1
λ
300 )
ζa
(
n −2
0 l
0 20 40 60 80 100 120
h [µm]
0
−3
e =21.1±1µm e =33.5±1µm
0 0
FIG.2. Influenceofthegeometryonthewavelengthof −4 a =19.0±1µm a =18.7±1µm
the instability. Wavelength of the fastest growing mode as
afunctionofthetotalinitialradius(seeFig.1(a)). Theblack
0 1,000 2,000 0 1,000 2,000
dashed line represents Eq. (5). The error bars are calculated t [s] t [s]
from the error in the geometry and the inaccuracy given by
the wavelength measurement.
FIG. 3. Temporal growth of the perturbation. Semi-
whichissimilartotheclassicalPRIdominantwavelength logarithmic plot of the evolution of the perturbation on (a)
for inviscid jets with no solid core. We note that the a no-slip fibre (glass), and (b) a slip fibre (glass coated with
AF2400). The radius of the fibre a and initial thickness of
correspondencebetweenRayleigh’sresultandoursisde-
the polymer film e (see Fig. 1 (a)) are indicated. The solid
pendent on both the assumption of Stokes flow and on 0
line is the best linear fit in the initial regime.
the presence of a solid core in our system.
The dimensionless growth rate of the fastest growing
mode is given by:
substrates [26]. The spatial morphology of the instabil-
1 ity at short times is thus unaffected by the solid-liquid
=αB+β , (6)
τ boundarycondition. WhileitisclearfromFig.2thatthe
∗
wavelength is the same on slip and no-slip fibres, there
where B is the dimensionless slip length, defined in
is a small systematic deviation from the theory. This
Eq. (1), and where α and β are parameters that depend
slight deviation could perhaps be related to the lowest
on geometry only:
lubrication order of the present model [9], but could also
1 1 2 be attributed to the experimental contribution of sev-
α = H (7a)
16H3 0− H eral modes and the asymmetry of the rate function (see
0 (cid:18) 0(cid:19) Eq. (4)) in the vicinity of the fastest growing mode.
1 1
β = 4H2logH 3H2 +4 .(7b)
64H03 (cid:18) 0 0− 0 − H02 (cid:19) The temporal evolution of the instability. We
As one sees, the wavelength of the fastest growing mode now turn from the spatial morphology of the instability
depends exclusively on the initial total radius, while the to the temporal evolution. From the experimental im-
corresponding growth rate is a linear function of the slip ages (see Fig. 1 (b)), we extract the maximal radius of
length. Note that in the case of shear-thickening liq- an individual bulge as it develops, in order to obtain the
uids, the PRI should be significantly slowed down as a amplitude ζ as a function of time. The linear stability
result of the increasing viscosity associated with an in- analysispresentedpredictsaperturbationthatgrowsex-
creasing strain rate. Conversely, a shear-thinning liquid ponentiallywithadimensionlessgrowthrate1/τ∗ forthe
isexpectedtoacceleratetheriseoftheinstability. Inthe fastestgrowingmode(seeEq.(6)). Figure3displaystyp-
case of a viscoelastic material, Maxwell-like rheological icaldataforthelogarithmoftheperturbationamplitude
models[35]canbeimplementedandmayrevealinterest- normalised by the radius of the fibre, ζ/a, as a function
ing physics, beyond the scope of the present study. of t, for both no-slip and slip fibres. The data for both
boundary conditions is consistent with the expected ex-
The spatial evolution of the instability. Figure 2 ponential growth in the early regime. Thus, the initial
displays the wavelength λ of the fastest growing modes, slopes of these curves provide reliable measurements of
∗
measured on no-slip and slip fibres, as a function of the the growth rates.
initial total radius h . As expected from Eq. (5), the The dimensionless growth rates 1/τ are shown in
0 ∗
wavelength λ grows linearly with increasing radius of Fig. 4 for both slip and no-slip fibres, as a function of
∗
thefibre-polymersystem,andisidenticalonslipandno- the dimensionless initial total radius H . We see that
0
slipfibres. Thisisconsistentwithexperimentsandathe- for both the slip and no-slip boundary conditions, the
oretical framework for retracting liquid ridges on planar growth rates show a similar geometry dependence. The
4
0.02 the no-slip data is consistent with the theoretical pre-
Glass(no-slip) diction of 1/τ = β for all geometries. In contrast the
AF2400(slip) ∗
Eq.(6),B=0 ratio1/(τ∗β)revealsthattheamplificationduetoslipon
0.016 Eq.(6),b=4.0±0.4µm the rise of the instability is more pronounced as α/(βa)
Eq.(6),B=0.3
increases. For the fibre radii used here, large α/(βa)
corresponds to small H (see Eq. (7)). The stronger in-
0.012 0
fluence of slip observed for smaller values of H is due
∗ 0
τ to a non-zero velocity of polymer molecules at the solid-
/
1 liquid interface [31, 33]. For the smallest value of H ,
0.008 0
α/(βa)[1/µm]
80 0.2 0.4 0.6 0.8 1 corresponding to e0/a ∼ 0.4, the slip-induced amplifica-
Eq.(6),b=4.0±0.4µm tion factor of the growth rate is as large as 4. On the
0.004 β)6 contrary, in thick polymer films, the impact∼of slip is di-
∗τ/(4 minished. From a best linear fit to the slip-data shown
12
in the inset of Fig. (4), the slip length is found to be
0 0 b = 4.0 0.4µm. Obtaining a slip length in the range
1 2 3 4 5 6 7 8 9 10 ±
of micrometers is in accordance with former studies on
H
0
thedewettingofentangledpolymerfilmsfromsubstrates
with a fluoropolymer coating [33].
FIG. 4. Influence of slip on the growth rate. The inset
shows the dimensionless growth rate 1/τ∗ normalised by the Having determined the value of b and knowing the fi-
no-slip case β, as a function of α/(βa), see Eqs. (6) and (7). bre radii, we obtain B for each of the experimental ge-
The slip length b is obtained from a best linear fit (dash- ometries. Basedonthetheoreticalmodel(Eq.(6))corre-
dotted) to the slip data. The error bars are calculated from spondinggrowthratescanbecalculatedandareshownto
the error in the geometry and the inaccuracy given by the beinexcellentagreementwiththeexperimentaldata(see
growthratemeasurement. Themaincurveshowsthedimen- Fig. (4)). The dimensionless slip length B = b/a ranges
sionless growth rate of the fastest growing mode on no-slip
from0.25to0.37inourexperiments. Toguidetheeye,a
(glass) and slip (AF2400) fibres, as a function of the dimen-
typicalcurvecalculatedwithB =0.3isshowninFig.(4).
sionless initial total radius H = 1+e /a (see Fig. 1 (a)).
0 0 We see that for the fibres with the slip boundary condi-
OpensymbolsrepresentgrowthratescalculatedfromEq.(6)
tion the growth rate is larger than in the no-slip case for
usingb=4.0 0.4µmandtherespectiveexperimentalgeome-
± a given geometry, and the maximum of the growth rate
tries. Alsoshownisthetheoreticalcurveforno-slip(Eq.(6),
with B = 0). Furthermore, the theoretical curve for slip 1/τ∗ corresponds to a smaller value of H0. As expected,
(Eq. (6), with B =0.3) is plotted as a guide to the eye. aslipperysurfacefacilitatesahighermobility,andhence
a faster growth of the instability. This enhanced mass-
transport also explains the horizontal shift of the maxi-
maxima for the slip and no-slip data can be easily un- mumofthegrowthrate: foragivengeometry–andthus
derstood: a decreasing growth rate as H converges to curvature – the mobility of the slip case is increased in
0
1 is due to the diminishing thickness of the liquid film, comparison to no-slip. Therefore, the maximum of the
and thus to the reduced mobility, while the decreasing growth rate is shifted to lower values of H0.
growthrateforlargeH isduetosmallercurvaturesand
0
thus a smaller driving force of the instability.
According to Eq. (6) with B = 0 (no slip), a maxi- DISCUSSION
mum in the growth rate is expected for H =1+e /a=
0 0
5.15. Thispredictionisentirelyconsistentwiththedata. WereportonthePlateau-Rayleighinstabilityofavis-
Thus, a coated fibre of a given diameter is maximally cous liquid polystyrene layer on a solid fibre of radius a,
unstable when the ratio of the film thickness to the fi- whentheboundaryconditionatthesolid-liquidinterface
bre radius is about e0/a 4, consistent with an earlier isvariedbetweentheclassicalcaseofno-slipandtherel-
∼
theoretical study [5]. The capillary velocity is the only evantsituationofslippage. Thewavelengthofthefastest
adjustable parameter in the no-slip case. We obtained growing mode on a slip fibre shows a linear dependence
γ/η =294 43µm/min, which is in agreement with pre- on the initial total polymer-fibre radius, h = a + e ,
± 0 0
vious data [36] adjusted to the temperature used here and is not affected by the boundary condition, consis-
through the William-Landel-Ferry equation [37, 38]. For tentwiththelubricationtheorydeveloped. Forbothslip
theslipcase,therearenowtwofreeparameters: thecap- and no-slip fibres, we observe an exponential temporal
illary velocity γ/η and the dimensionless slip length B. growth of the instability at short times, and the respec-
If we take the value of the capillary velocity to be that tive growth rates show a qualitatively similar geometry
of the no-slip case, we are left with only one true fitting dependence. Inthecaseofaslipfibre, thegeometrycor-
parameter. responding to the maximum of the growth rate 1/τ is
∗
To quantify the slip length in our system the growth shifted to a smaller value of h /a, and the rise of the in-
0
ratesnormalisedbyβ areplottedasafunctionofα/(βa) stability is faster due to the added mobility at the solid-
(see inset Fig. (4) and Eqs. (6) and (7)). As expected, liquid interface. The slip induced amplification of the
5
growth rate is significant in a parameter range that is of viscous polymer solution was placed between two glass
paramount technological relevance. The linear stability slides, forming a meniscus at the edge of the glass
analysisofthethinfilmequationdevelopedhereisinex- slides. We note that the chloroform does not dissolve
cellent agreement with the data, valid for all Newtonian the underlying AF2400 coating on the slip fibres. A
liquids, andprovidesarobustmeasureoftwofundamen- slip or no-slip fibre could then be placed in the middle
tal quantities: the capillary velocity γ/η, and the slip of the gap between the slides and pulled out of the
length b. droplet with a constant speed using a motorised linear
translation stage. By varying the pulling speed v in the
0
range 80<v <150mm/s, we obtained film thicknesses
0
that ranged from e = 5 to 93µm after the solvent had
METHODS 0
evaporated.
Preparation of fibres
Glassfibreswerepreparedbypullingheatedglasscap- Experimental setup
illary tubes to final radius in the range 10 < a < 25 µm
using a pipette puller (Narishige, PN30). To prepare
The as-prepared samples were placed into a heated
the slip boundary condition, the glass fibres were
sample cell to initiate the PRI. With two 0.5 mm
hydrophobised by dip-coating in a 0.5 wt% solution ∼
thick spacers, the coated fibres were suspended above a
of AF2400 (Poly[4,5-difluoro-2,2-bis(trifluoromethyl)-
reflective Si wafer (to improve contrast), and placed on
1,3-dioxole-co-tetrafluoroethylene]) (Aldrich) in a
a microscope hot stage (Linkam). A metal ring in di-
perfluoro-compound solvent (FC72TM, Fisher Sci-
rect contact with the hot stage supported a glass cover
entific), to form an amorphous fluoropolymer layer
over the sample and Si wafer (see Supplementary Figure
(T 240 C). A dip-coating speed of 1mm/s resulted
g ∼ ◦ 1 and Supplementary Methods), thereby ensuring good
in an AF2400 layer with a thickness of 14 1nm.
± thermal contact and temperature control to within 1◦C.
The hydrophobised fibres were annealed in a vacuum
The surface profiles were analysed from the optical mi-
chamberat80 Cfor90minutestoremoveexcesssolvent.
◦ crographs taken at various times using a custom-made
edge detection software written in MATLAB.
Preparation of homogeneous PS films ACKNOWLEDGEMENTS
To prepare homogeneous PS films, a concentrated The authors wish to thank Martin Brinkmann for
solution (35 wt%) of atactic polystyrene (Polymer insightful discussions. The authors thank the gradu-
Source Inc.) with a molecular weight of 78kg/mol and ate school GRK 1276, NSERC of Canada, the Ger-
low polydispersity (M /M = 1.05) was dissolved in man Research Foundation (DFG) under grant numbers
w n
chloroform (Fisher Scientific). A droplet of the highly BA3406/2 and SFB 1027 for financial support.
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1
Supplementary Figures
(a) (b)
F
E D
C
B
5 A
1
0
Supplementary Figure 1: Schematic of the sample chamber. Representation of (a) the side view and (b) the top view
2
of the experimental setup. On top of a hot stage (A), a Si wafer (B) with two teflon spacers (C) is placed. Freely suspended
y filaments are realized by depositing the fibre (D) across the spacers. The fibre is surrounded by a metal ring (E) with a glass
a cover (F).
M
4
2
]
n
y
d
-
u
l
f
.
s
c
i
s
y
h
p
[
2
v
4
9
1
2
0
.
1
0
5
1
:
v
i
X
r
a
2
Supplementary Method
Fibre thin film equation with slip:
Here,weprovidethemainstepsofthederivationofthefibrethinfilmequation(seeEqs.(2)and(3)inthearticle).
Let us consider a viscous liquid film coating a cylindrical fibre of constant radius a (see Fig. 1 in the article). The
axis of the fibre, along the x direction, is chosen as a reference for the radial coordinate r. We assume cylindrical
invariance of the free-surface profile described by r = h(x,t), at position x and time t. Furthermore, we assume
an incompressible flow of the viscous Newtonian fluid, where gravity, disjoining pressure and inertia are negligible.
Definingthelocalexcesspressurep(r,x,t)andthevelocityfieldv(r,x,t),andassumingthelubricationapproximation
to first order, namely that the slopes of the profiles remain small and the velocity is mainly oriented along the fibre
axis, v = ve , leads to the two projections of the Stokes equation:
x
∂ v
r
∂ p = η ∂ v+ (1a)
x rr
r
(cid:18) (cid:19)
∂ p = 0 , (1b)
r
where η is the dynamical shear viscosity that we assume to be homogeneous and constant. The lubrication approx-
imation is valid since the profile slopes remain small in comparison to 1 at early times – the regime in which the
linearanalysisisperformed–andsincethetypicalhorizontallengthscaleoftheflowisexpectedtobelargerthanthe
minimalwavelengthofthePlateau-Rayleighinstability, thatisgivenbyλ =2πh e . Thisisindeedalwaysthe
min 0 0
(cid:29)
case as even in the worst case geometry, e 5a, one has e /λ 0.13 1. According to Supplementary Eq. (1b),
0 0 min
≈ ≈ (cid:28)
theexcesspressureisinvariantintheradialdirection,tofirstorderinthelubricationapproximation. Thus,theexcess
pressure is set by the Laplace boundary condition at the free surface which, within the small slope approximation,
reads:
1
p = γ h , (2)
00
h −
(cid:18) (cid:19)
where γ denotes the air-liquid surface tension, taken to be homogeneous and constant, and where the prime denotes
the derivative with respect to x. The two terms on the right hand side of Supplementary Eq. (2) correspond to the
radial and axial curvatures of the free surface. Note that, as already mentioned in the body of the article, we have
kept a second order lubrication term in the pressure contribution: the axial curvature h . This is because it is
00
−
the lowest order term counterbalancing the driving radial curvature 1/h, and it is thus crucial to obtain the actual
threshold of the instability [9]. Regarding the boundary conditions, we assume no shear at the free surface and the
Navier slip condition at the solid-liquid interface, namely:
∂ v = 0 (3a)
r r=h
|
v
r=a
∂ v = | , (3b)
r r=a
| b
where b denotes the slip length. Integrating Supplementary Eq. (1), together with Supplementary Eqs. (2) and (3),
yields:
v = γ h0+h2h000 2h2log r r2+ 2bh2+a2 2ab , (4)
4ηh2 a − a −
(cid:0) (cid:1)(cid:20) (cid:16) (cid:17) (cid:21)
for all r [a,h]. Volume conservation requires that:
∈
Q
0
∂ h+ = 0 , (5)
t
h
where we introduced the volume flux per radian:
h
Q = drrv . (6)
Za
3
Combining Supplementary Eq. (5) together with Supplementary Eqs. (4) and (6) yields the fibre thin film equation:
γ h 4b a2 0
∂ h+ h +h2h 4h2log + 3 h2+4a2 8ab+ 4ab a2 = 0 . (7)
t 16ηh 0 000 a a − − − h2
(cid:20) (cid:18) (cid:18) (cid:19) (cid:18) (cid:19) (cid:19)(cid:21)
(cid:0) (cid:1) (cid:0) (cid:1)
Finally, introducing the dimensionless variables (see Eq. (1) in the article) into Supplementary Eq. (7) yields the
dimensionless fibre thin film equation (see Eqs. (2) and (3) in the article).
Experimental setup and details:
The as-prepared samples were annealed in ambient atmosphere at 180 C - well above T - which causes the
◦ g
polystyrene films to melt and the PRI to develop. The evolution of the PRI was recorded with an inverted optical
microscope (Olympus BX51). At various times, images were taken using a 5x magnification objective and a camera
with a resolution of 1392 x 1040 pixels (QImaging, QIClick), resulting in a pixel size of 1.28µm. To ensure free
standing filaments, the coated fibres (D) were placed along two 0.5mm thick teflon spacers (C) located on a Si
∼
wafer (B) (see Supplementary Figure 1). The Si wafer was used to improve the optical contrast. A metal ring (E)
in direct contact with the hot stage supported a glass cover (F) over the sample and Si wafer. The temperature
within the sample cell is uniform (within 1 C) and convection does not affect the measurements. By using spacers
◦
(C) made of teflon, we ensured that there is no drainage of the liquid PS film towards the edges (PS does not wet
teflon). The surface profiles were analysed from the optical micrographs taken at various times, using a custom-made
edge detection software written in MATLAB.
