Dynamical density functional theory for the dewetting of evaporating thin films of nanoparticle suspensions exhibiting pattern formation

1.4 MB·English

by  A. J. Archer

Dynamicaldensityfunctionaltheoryforthedewettingofevaporatingthinfilmsofnanoparticle
suspensionsexhibitingpatternformation
A.J. Archer, M.J. Robbins, and U. Thiele
DepartmentofMathematicalSciences, LoughboroughUniversity, LeicestershireLE113TU,UK
Recentexperimentshaveshownthatthestrikingstructureformationindewettingfilmsofevaporatingcol-
loidal nanoparticle suspensions occurs in an ultrathin ‘postcursor’ layer that is left behind by a mesoscopic
dewettingfront. Variousphasechangeandtransportprocessesoccurinthepostcursorlayer,thatmayleadto
nanoparticledepositsintheformoflabyrinthine,networkorstronglybranched‘finger’structures. Wedevelop
aversatiledynamicaldensityfunctionaltheorytomodelthissystemwhichcapturesallthesestructuresandmay
beemployedtoinvestigatetheinfluenceofevaporation/condensation,nanoparticletransportandsolutetrans-
portinadifferentiatedway.Wehighlight,inparticular,theinfluenceofthesubtleinterplayofdecompositionin
0 thelayerandcontactlinemotionontheobservedparticle-inducedtransverseinstabilityofthedewettingfront.
1
0
2 How surface patterns and structures evolve over time is of tion/condensation of the solvent. As a consequence, one
great interest for a wide range of scientific fields. Striking must distinguish between ‘normal’ convective dewetting and
n
examples include river network patterns [1], the growth of evaporativedewetting. Experimentalstudiesperformedwith
a
J rocksaroundgeothermalsprings[2]evaporation-causedcof- volatile solutions/suspensions of polymers [8, 9], macro-
5 fee stain patterns [3], and the patterns in the distribution of molecules[10–12]andcolloidsafewnanometersinsize(re-
1 living organisms [4]. Many such structures are generated by ferred to as ‘nanoparticles’) [5–7, 25] describe a variety of
theinteractionoffluidmotionoverthesurfaceanddeposition richly structured deposits of the solutes. One may observe

] and/or abrasion of material. A particular process of high re- labyrinthine and polygonal network structures similar to the
t
f cent interest that concerns us in this Letter is the formation structures observed following ‘classical’ dewetting. As the
o
ofstructuresduringthe(evaporative)dewettingofnanoparti- solvent evaporates, the solute remains dried onto the sub-
s
. cle suspensions on solid substrates [5–7]. The patterning is strateandtherefore‘conserves’thetransientdewettingpattern
t
a generictoawideclassofdewettingevaporatingsuspensions [6,10]. However,thesoluteisnotjustapassivetracer: itmay
m
andsolutions[8–12]anddependscruciallyontheinterplayof influence the thresholds and the rates of the initial film rup-
- severalcompetingphase-changeandtransportprocesses. ture processes. Most importantly, it may also destabilise the
d
straight dewetting fronts and trigger the creation of strongly
n The rapidly expanding study of such systems currently re-
ramifiedstructures–aprocessobservedinmanydifferentsys-
o ceives strong impetus from research in two distinct areas.
tems[7–9,26].
c On the one hand, studies from the last couple of decades of
[
thedynamicsofdewettingofsurfacesbynon-volatileliquids For instance, the dewetting of films of a suspension of thiol-

1 [13, 14] have been extended to investigate the interplay be- coated gold nanoparticles in toluene may result in the depo-
v tween(de)wettingandevaporationofvolatileliquids[15,16]. sition of the nanoparticles in branched finger patterns. The
1 On the other hand, there is interest in the non-equilibrium precisepropertiesofthesedependonthestrengthoftheattrac-
6 thermodynamics and rheology of the respective phase and tionbetweenthecolloidalparticles[7]. Employingcontrast-
6
flow behaviour of bulk suspensions and solutions – see e.g. enhancedvideomicroscopytostudythedynamicsofthesys-
2
Ref. [17] and references therein. A thin film of pure non- tem, initially the receding of a mesoscopic dewetting front
.
1 volatileliquidthatisdepositeduponasmoothsubstrate(e.g., (equivalent to the receding three phase contact line) is ob-
0
apolystyrenefilmhavingathicknessofafewtensofnanome- served,leavingbehindanunstableultrathin‘postcursor’film,
0
ters, deposited on silicon oxide [13]) may rupture due to ef- havingathicknesssimilartothediameterofthenanoparticles.
1
fectivemolecularinteractionsbetweenthefilmsurfaceandthe Subsequently,thepostcursorfilmbreaks,formingapatternof
:
v solid substrate. The rupture mechanism can be (i) via a sur- holes that themselves grow in an unstable manner, resulting
i
X face instability (often called spinodal dewetting) that occurs inanarrayofbranchedstructures. Notethatthemesoscopic
spontaneouslyandresultsinpatternsofacertaincharacteristic frontmayalsobeunstable–aneffectthatdoesnotconcernus
r
a wavelength,or(ii)vianucleationatrandomlydistributedde- here.
fects[18,19]. Theresultingholesthengrowtoformapolyg-
Theoriesformodellingsuchprocessesareratherlimitedatthe
onalnetworkofliquidrimsthatmaysubsequentlydecayinto
present time. Hydrodynamical models of dewetting by thin
drops. All stages of this process are intensively studied: the
filmsbasedonalong-waveapproximation[24,27]providea
rupture mechanisms [13, 14, 18], the hole growth [20], the
mesoscaledescriptionoftheliquidfilm, andcanaccountfor
morphologiesandevolutionoftheresultingpatterns[21],and
evaporation of volatile liquids [28] and for the presence of a
thestabilityofrecedingliquidfronts[22,23]. Forreviewsof
solute [29]. However, they are not able to describe the pro-
thisbodyofwork,seeRef.[24].
cessesoccurringinthepostcursorfilmastheydonotaccount
The dewetting processes of solutions and suspensions are for the interactions between the solute particles and between
more involved than those of a pure liquid because they in- thesoluteparticlesandthesolvent. Alternatively,todescribe
volve several interdependent dynamical processes: transport theseprocessesonemayemploytwo-dimensional(2d)kinetic
of solute or colloids, transport of the solvent and evapora- Monte-Carlo (KMC) lattice models that focus solely on the
Preprint–[email protected]–[email protected]–January15,2010
2
FIG. 1: (Color online) (a) Phase diagram of the pure liquid in the
planespannedbythedensityρ andtemperatureT. (b)Liquidand
l
nanoparticledensitiesρ andρ atcoexistenceasafunctionofthe
l n
averagenanoparticledensityρ¯ ,forε /k T = 1.25,ε /k T = FIG. 2: (Color online) Results for spinodal evaporative dewetting
n ll B nl B
0.6,ε =0. of a nanoparticle suspension. (a) and (b) are typical nanoparticle
nn
densityprofiles,fortimest/t = 6and20,and(d)and(e)arethe
l
correspondingliquiddensityprofiles,forMc =0. Thedomainsize
l
is200σ×200σ. (c)and(f)givethecorrespondingtimeevolution
dynamics of the solute diffusion and the solvent evaporation
ofthemeanliquiddensity(cid:104)ρ(cid:105)andthestructurefactor(cid:104)ρ (k)2(cid:105)for
[30–32]. The neglect of convective solvent transport can be l n
various values of Mc. The remaining parameters are ε /k T =
justified based on estimates comparing its influence with the 1.25,ε /k T =0.l6,ε =0,α=0.4Mncσ4,µ/k Tll =B−3.4,
nl B nn l B
oneofevaporativesolventtransport[32]. However,sofarthe and(cid:104)ρ (cid:105)=0.3.
n
2dKMCmodelshavenotincorporateddiffusivesolventtrans-
portthatmightbeimportantinthepostcursorlayer.
In this Letter we present an alternative description for the agradientexpansionofthe(non-local)freeenergyfunctional
structure formation in the postcursor film that does not have of a continuous system [37]. The rate of evaporation of the
the limitations of the previous approaches. We develop a 2d liquidfromthesubstrateisdeterminedbythechemicalpoten-
dynamical density functional theory (DDFT) [33–36] to de- tialµinthereservoir(i.e.inthevapourabovethesubstrate).
scribe the coupled dynamics of the density fields of the liq- When the temperature is sufficiently low, and the chemical
uid ρl(r,t) and the nanoparticles ρn(r,t). In this approach, potentialµ = µcoex,weobservecoexistencebetweenathick
diffusive liquid transport can be incorporated in a straight- (highdensity)andathin(lowdensity)liquidfilm. InFig.1(a)
forwardmannerenablingustogobeyondprevious2dKMC wedisplaythelimitoflinearstability(spinodal)andtheequi-
studies and examine its influence. To construct the DDFT librium coexistence curve (binodal) for the pure liquid (i.e.
model, we (i) develop via coarse-graining an approximation withρ = 0). Toindicatetheinfluenceofthesoluteweplot
n
for the underlying free energy functional of the system and inFig.1(b)thedensitiesρ andρ atcoexistenceforafixed
l n
(ii) form equations governing the dynamics of the two den- temperaturek T/ε = 0.8asafunctionoftheaveragecon-
B ll
sity fields. These are able to account for the non-conserved centrationρ¯ = 1(ρa+ρb),whereρa andρb arethedensities
n 2 n n n n
andconservedaspectsofthedynamics,i.e.,phasechangeand ofthenanoparticlesinthecoexistingaandbphases. Forfur-
diffusivetransportprocesses,respectively. therdetailsconcerningthephasediagramanditstopologysee
InordertocompareresultswiththeestablishedKMClattice Ref.[39].
model [30–32], we start from the lattice Hamiltonian to de- Atequilibrium,thederivativeµ ≡ δF[ρ ,ρ ]/δρ isacon-
n n l n
velop a mean-field (Bragg-Williams) approximation for the stant,correspondingtothechemicalpotentialofthenanopar-
free energy functional [35, 37]. Expressing the interaction ticles. However, when the system is out of equilibrium, µ
n
terms(sumsoverneighbouringlatticesites)asgradientopera- may vary along the substrate. We assume that the thermo-
tors[35],thefollowingsemi-grand[38]freeenergyfunctional dynamicforce∇µ drivesthedynamicsofthenanoparticles
n
isobtained: andthatthenanoparticlecurrentisj = −M ρ ∇µ , where
n n n
(cid:90) (cid:20) ε ε Mn(ρl)isamobilitycoefficient. Thisexpressionforthecur-
F[ρ ,ρ ]= dr f(ρ ,ρ )+ ll(∇ρ )2+ nn(∇ρ )2 rent, together with the continuity equation, yields the time
l n l n 2 l 2 n
evolutionequationforthenanoparticledensityprofile:
(cid:21)
+εnl(∇ρn)·(∇ρl)−µρl ,(1) ∂ρ (cid:20) δF[ρ ,ρ ](cid:21)
n =∇· M ρ ∇ n l . (2)
∂t n n δρ
n
where f(ρ ,ρ ) = k T[ρ lnρ + (1 − ρ )ln(1 − ρ )] +
l n B l l l l
k T[ρ lnρ +(1−ρ )ln(1−ρ )]−2ε ρ2 −2ε ρ2 − Thisequationmayalsobeobtainedbyassumingover-damped
B n n n n ll l nn n
4ε ρ ρ , includes entropic contributions and various inter- stochasticequationsofmotionforthenanoparticles[33,34].
nl n l
action terms – the parameters ε , where i,j = n,l, are the To model the fact that nanoparticles do not diffuse over the
ij
energies for having neighbouring pairs of lattice sites occu- drysubstrate(whenρ issmall)wesetthemobilityM (ρ )to
l n l
piedbyspeciesiandj,respectively,T isthetemperature,k switchatρ = 0.5(smoothly)fromzeroforthedrysubstrate
B l
is Boltzmann’s constant and we have set the lattice spacing (low ρ ) to α for the wet substrate (high ρ ). Note that our
l l
σ =1. NotethatF inEq.(1)canalsobeobtainedbymaking resultsarenotsensitivetothepreciseformofM (ρ ).
n l
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FIG.5: (Coloronline)Nanoparticledensityprofilesfromtheevolu-
tionofanunstabledewettingfront, forthetimet/t = 20000, (a)
l
forMc =0and(b)Mc =5,forα=0.5Mncσ4,µ/k T =−3.28
l l l B
anddomainsizeis800σ×800σ; remainingparametersandinitial
densityprofilesareasinFig.4. In(c)weshowthedependenceof
the mean finger number (cid:104)f(cid:105) on the mobility coefficient for liquid
FIG.3: (Coloronline)Densityprofilesforevaporativedewettingin diffusion,Mc.
l
thenucleationregime.(a)-(c)arethenanoparticledensityprofilesat
timest/t = 20,80and4000,(d)and(e)aretheliquidprofilesfor
l
t/tl = 20and80, forMlc = 2andµ/kBT = −3.33; remaining liquiddensityprofile:
parametersareasinFig.2. In(f)weplottheaveragedensityofthe
(cid:20) (cid:21)
liquidonthesubstrate(cid:104)ρl(cid:105), asafunctionoftimeforthiscaseand ∂ρl =∇· Mcρ ∇δF[ρn,ρl] −MncδF[ρn,ρl]. (3)
thecaseMlc = 0. Thesystemwasinitialisedwiththe(discretised) ∂t l l δρl l δρl
densityprofiles: ρ(x,y,t=0)=0.9+0.05χ,ρ (x,y,t=0)=
l n
0.3+0.27χ,whereχisarandomnumberuniformlydistributedon WeassumethatthetwomobilitycoefficientsMcandMncare
l l
theinterval[−1,1]. Itisduetothisrandomnoisethattheholesare constants. In what follows we set k T = 1 and Mnc = 1.
B l
nucleatedinsomeplacesandnotinothers. Notethatinthelowdensitylimit,whenρ → 0andρ → 0,
n l
theconservedpartinbothEq.(2)andEq.(3)correspondsto
Fickiandiffusion.
Beforediscussingresultsfromourtheory,wefirstmakeacou-
pleofcommentsaboutthestatusofthetheory.Firstly,wenote
thatEq.(1)constitutesasimple‘zeroth-order’mean-fieldap-
proximation for the free energy of the system and omits (for
example) terms such as ln(1−ρ −ρ ) which describe the
n l
excluded area correlations between the liquid and the nano-
particles. Second, due to the fact that we derive the theory
from the (already) coarse-grained lattice Hamiltonian rather
thanbyintegratingoverdegreesoffreedom(coarse-graining)
inthefullDDFTtheoryforthethree-dimensionalliquidfilm,
thetheorycannotberegardedasa‘fully’microscopictheory.
Thereasonthatwehavemodelledthesystemusingthissim-
FIG. 4: (Color online) Density profiles from the evolution of an pletheoryisbecauseourinterestisthebasicquestionofwhat
unstable dewetting front. (a)–(c) are the nanoparticle density pro- physicsdrivesthebehaviourdisplayedintheexperiments. In
files at times t/tl = 2000, 20000 and 40000, (d) and (e) are the thisworkwechoosetostartfromthelatticetheory,inorderto
liquid profiles for t/tl = 2000 and 20000, for α = 0.2Mlncσ4, comparewiththeKMC.However,asfutureworkweplanto
µ/kBT =−3.28anddomainsize800σ×800σ;remainingparam- gobeyondthelatticetheory. Ourgoalhereisnottoconstruct
etersareasinFig.2.Theinitialdensityprofilesarethesameasthose
a model describing every detail of these systems; instead we
inFig.3,exceptfory<0wesetρ =ρ =10−10,tocreateanini-
l n seek to examine what physics is involved in determining the
tiallystraightdewettingfrontaty=0.In(f)weplotthemeanfinger
observedpatternformation. Nonetheless,thisDDFTdoesal-
number(cid:104)f(cid:105)asafunctionofthemobilitycoefficientfornanoparticle
diffusionα,forthecasewhenMc =0. lowustoinvestigatethetimeevolutionofthepostcursorfilm
l
ofanevaporatingnanoparticlesuspensionunderfewerrestric-
tionsthantheKMCmodel. Todiscusstheimportanceofliq-
uidtransportwithinthelayerwecompareresultsobtainedfor
For the liquid, the density may change either by evapora- differentliquidmobilities(Mc >0)andresultswithoutliquid
l
tion/condensation from/to the substrate (non-conserved dy- transport(i.e.,settingMc =0). Wefocusonthreeexamples:
l
namics)ormaydiffuseoverthesubstrate(conserveddynam- (i) spinodal dewetting (see Fig. 2), (ii) dewetting via nucle-
ics). The latter dynamics is treated in a manner analogous ationofholesinaninitiallyflatfilm(seeFig.3),and(iii)the
to that for the nanoparticles. For the non-conserved dynam- unstablerecedingofanevaporativedewettingfront,whichex-
ics, we assume a standard form [40], i.e., that the change hibitsbranchedfingering(seeFigs.4and5).
of the density over time is proportional to −(µ − µ) = Fig. 2 shows snapshots from a purely evaporative spinodal
surf
−δF[ρ ,ρ ]/δρ , where µ (r,t) is the local chemical po- dewettingprocess. Panel(c)givestheevolutionofmeanliq-
n l l surf
tential of the liquid on the substrate. Combining these two uid density with time, (cid:104)ρ (cid:105) ≡ A−1(cid:82) drρ (r,t), where A is
l l
contributions, we obtain the time evolution equation of the theareaofthesubstrate, andpanel(f)givesthenanoparticle
Preprint–[email protected]–[email protected]–January15,2010
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structurefactorS(k) ≡ (cid:104)ρ (k)2(cid:105),whereρ (k)istheFourier the present continuum model (DDFT) is similar to results of
n n
transform of ρ (r). For small times, the unstable film de- theKMC[32], andprovesthatjammingofdiscreteparticles
n
velops a typical spinodal labyrinthine pattern with a typical is not a necessary mechanism for causing the instability. In
wavelength2π/k (notethatthesymmetrybreakinginthe Fig.4(f)weshowhowtheaveragenumberoffingersperunit
max
densityprofilesinallourcalculationsisduetotheadditionof length,(cid:104)f(cid:105),variesasafunctionofα,themobilitycoefficient
asmallamplituderandomnoisetothedensityprofilesattime of the nanoparticles on the wet substrate, for the case when
t=0andnonoiseisaddedatlatertimes). Thenanoparticles Mc = 0. We see that as α is decreased, the number of fin-
l
concentratewheretheremainingliquidissituated. However, gersincreases. Thisincreasein(cid:104)f(cid:105)occursbecausewhenthe
they are ‘slow’ in their reaction: when ρ already takes val- mobilityofthenanoparticlesisdecreasedthefront‘collects’
l
ues in the range 0.08 to 0.83, the nanoparticle concentration more particles (less of them diffuse further from the front).
has only deviated by about 25% from its mean value. The Theresultingregionofhighconcentrationsolutionatthefront
film thins rapidly forming many small holes. The competi- maybe‘dynamicallyunstable’: Asthefrontvelocitydepends
tion for space results in a fine-meshed network of nanoparti- non-linearlyontheamountofparticlescollected,anyfluctua-
cle deposits with a much higher concentration of particles at tionalongthefrontmaytriggeratransverseinstability.
thenetworknodes–aneffectthatcannotbeseenwithinthe Fig. 5 shows that the finger number (cid:104)f(cid:105) depends non-
KMCmodel. Becausetheliquidwetsthenanoparticles,some monotonicallyonthemobilitycoefficientforliquiddiffusion,
liquid always remains on the substrate. Accounting for sol- Mc. Althoughtheoveralltrendisanincreaseof(cid:104)f(cid:105)within-
l
vent diffusion, the rate of the dewetting process is increased creasingMc,thereexistsanintermediateregion(5 (cid:46) Mc (cid:46)
l l
[see Fig. 2(c)] and leads to a more strongly modulated final 7)where(cid:104)f(cid:105)slightlydecreases.Theoveralltrendresultsfrom
pMatlcte=rn0–[i2.e(.f)t]h.epeaksinS(k)arehigherforMlc >0thanfor afinxeidncpraeratsiceleindiffrfounstivvietylo.cHitoyw(edvueer,towethceuirnrecnretlaysehaivneMnolc)exa-t
Fig. 3 shows snapshots from a dewetting process triggered planationfortheintermediateslightdecrease.
by nucleation events. The holes nucleate at several arbitrary
Notealsothatinallcasestheinstabilitymaybestronglyam-
places due tothe random initial noise inthe density profiles,
plified when the particle interactions favour the clustering of
and grow to form a random polygonal network of rims of
thenanoparticles(thisisnotthecasefortheparametersused
highly concentrated solution. On a very long time scale the
here), where the higher concentration at the receding front
networkcoarsensintoanarrayofdrops. Theinfluenceofliq-
leadstoalocaldemixingofnanoparticlesandliquid, thatit-
uid transport can be seen in the final panel of Fig. 3, where
self enforces the deposition of a highly branched finger pat-
we display a plot of the average density of the liquid on the
tern. In this ‘demixing regime’ the instability is determined
substrateasafunctionof(log)time. Weseethattheliquidis
bythedynamicsandtheenergeticsofthesystemwhereasfor
able to evaporate from the substrate faster and that for times
thecasestudiedinFig.4itmainlydependsonthedynamics.
t/t ∼102−104,thetotalamountofliquidonthesubstrateis
l
We note finally, that the fingering process can be seen as a
lesswhenMc = 2,thanwhenMc = 0. However,oververy
l l self-optimisationofthefrontmotion,sothattheaveragefront
long times, the drops on the substrate slowly move and ‘eat-
velocity is kept constant by expelling particles into the fin-
up’ the network pattern. This process is faster with solvent
gers. Asimilareffectexistsfordewettingpolymerfilms[22],
diffusion, leading to a faster increase of (cid:104)ρ (cid:105) at long times.
l
wheresurplusliquidisexpelledfromthegrowingmovingrim
Since the liquid wets the nanoparticles, some liquid also re-
whichcollectsthedewettedpolymer. However,frontinstabil-
condensesbackontothesubstrate.
itiesfoundfordewettingpolymersonlyresultinfingerswith-
ThefinalexampleinFigs.4and5istheevolutionofthefin-
outside-branches[41]orfieldsofdropletsleftbehind[22].
geringinstabilityforarecedingdewettingfront.Thefingering
instability is caused by a build up of the nanoparticles at the InthisLetterwehavedevelopedaversatileDDFTthatisca-
receding front, which collects the nanoparticles due to their pable of describing the pattern formation observed in evapo-
attractiontotheliquid. InFig.4(a)weseethatatearlytimes rating dewetting thin films of suspensions. Since our DDFT
theinitiallystraightfrontshowsarathershort-waveinstabil- takesaccountofallthebasicsolventandsolutetransportand
ity; about 20 short ‘fingers’ can be seen. However, the fin- phase change processes in a consistent manner, we believe
gerpatterncoarsensrapidlytoastationarypatterncontaining that it will be the basis for many successful future studies of
onlyabouthalftheinitialnumberoffingers. Intriguingly,the thebehaviourofsuspensionsandsolutionsatinterfaces.
mean finger number remains constant although at the mov- AJA and MJR gratefully acknowledge financial support by
ing contact line new branches are created and old branches RCUKandEPSRC,respectively.Weacknowledgesupportby
merge continuously. The occurrence of this phenomenon in theEUviagrantPITN-GA-2008-214919(MULTIFLOW).
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