Dynamics of Structure Formation in Thin Liquid Films

20 Pages·2000·0.61 MB·English

Dynamics of Structure Formation in Thin Liquid
Films: A Special Spatial Analysis
Karin Jacobs1, Ralf Seemann1, and Klaus Mecke2
1 Abteilung Angewandte Physik, Universit¨at Ulm
D-89069 Ulm, Germany
2 Fachbereich Physik, Bergische Universit¨at Wuppertal
D-42097 Wuppertal, Germany
Abstract. Thecharacterizationofmorphologiesthatarenotperfectlyregularisavery
difficult task, since there is no simple ”measure” for imperfections and asymmetries.
We faced this problem by trying to describe the pattern that evolves in the course of
the dewetting of a thin polymer film as compared with the scenario that takes place
in a thin gold film. With the help of the Minkowski functionals we found significant
differences in the pattern of the two systems: We were able to distinguish a spinodal
dewetting mechanism for the gold film from heterogeneous nucleation for the polymer
film.Moreover,weshowhowatemporaldevelopmentofapatterncanbecharacterized
by Minkowski functionals.
1 Introduction
What catches our eye in Fig. 1 on page 80? The form of the artishokes and the
clover, their color or the way of stacking? Where are the guiding lines for the
eye? Some of the photograph’s fascination might be explained by the symmetry
ofthearrangementoftheartishokes,whichisnearlyahexagonalclosestpacking,
and some might arise from the contrast of the different morphologies: radially
arrangedleavesfortheartishokesandalinearlylayeredstructurefortheclover.
The patterns are not perfect, yet clearly visible by eye. How can one describe
such patterns scientifically?
Researchers in crystallography and in solid state physics found a nomencla-
ture which is able to characterize symmetrical, recurring patterns by sorting
them into 32 possible point groups. An example for such a regular pattern is
showninFig.2.Still,thereisnoroomforcharacterizingimperfectionsorasym-
metries in the pattern.
In nature, symmetry and symmetry-breaking lie close together. Symmetry-
breakingoccursforinstancewhenatreetrunkramifiesoracelldividesintwo.In
our experiments, we probe such a symmetry-breaking process by characterizing
itsmorphology.Theyareperformedinaneasilycontrollablemodelsystem,athin
liquid film lying uniformly and smoothly on a solid substrate. Here, symmetry-
breaking takes place when the film does not like to wet the solid surface and
beads off. This is actually an everyday phenomenon, it comprises - just to name
afew-thedewettingofawaterfilmfromafreshlywaxedcar,theprintingofink
onpaperortransparenciesorthetricklingofwaterdropletsdownwater-repellent
textiles or leaves (”the lotus effect” [1]).
K.R.MeckeandD.Stoyan(Eds.):LNP554,pp.72–91,2000.
(cid:1)c Springer-VerlagBerlinHeidelberg2000
Dynamics of Structure Formation 73
Thinliquidfilmsonsolidsurfacesareofenormousimportanceinmanyfields
of modern technology. A vast amount of research has thus been dedicated to
these systems, and the interest in this field is still growing. However, most of
the fundamental processes are not yet fully understood. Among them are the
symmetry-breakingmechanismsintheearlyaswellasinthelatestageofdewet-
ting. In the present study, we investigate two systems, a thin metal film and a
thin polymer film, which give insight into the morphology and the dynamics of
liquid structures.
2 Results and Discussion
2.1 Early Stage of Dewetting: Formation of Holes
Inmanyexperimentsinvolvingdifferentfilmsandsubstrates,ithasbeeninvari-
ably observed that rupture proceeds by the formation of circular holes whose
radius grows in time until they finally merge and lead to dewetting of the entire
film [9,10,11,20,21,22,29,30,33,36] as shown in Fig. 3. The mechanism of initial
hole formation is still under discussion. Let us first concentrate on this early
stage of dewetting.
It is generally accepted that there are two possible mechanisms that may
give rise to the formation of dry patches initiating the dewetting process. Ei-
ther there is nucleation, e. g. from dust particles or impurities in the film, or,
if long-range interactions (such as van der Waals forces) between the liquid film
material and the substrate disfavor wetting, fluctuations in film thickness ex-
perience a driving force. (Fluctuations by thermal motion are always present
on liquid surfaces.) Their amplitudes then grow exponentially with time, finally
leading to dewetting when their size becomes equal to the film thickness. From
the apparent analogy with symmetry-breaking mechanisms involved in decom-
position processes [7,13], the scenario has been termed spinodal dewetting. In
thiskindofdynamicalinstability,thereisusuallyacertainwavelengthλ the
max
amplitudeofwhichgrowsfastest,andthusdeterminesthedominantlengthscale
oftheemergingstructure.Thiswavelengthscalesasthesquareofthefilmthick-
ness under certain, quite general conditions, as was shown experimentally [2,35]
and theoretically [3,24]. Since the number density of holes appearing during the
rupture process was found in several studies to depend on the film thickness
according to this scaling law, it was widely believed that spinodal dewetting is
thestandardmechanismleadingtotheruptureofpolymerfilms.Incontrast,we
show in the present paper that nucleation by defects is in fact the dominating
mechanism leading to the generally observed rupture scenario of polymer films.
As a convenient model system, we chose polystyrene (PS) films on silicon
wafers, which is frequently used as a standard system, in particular for most of
theworkdedicatedtothestudyofrupturemechanisms.Thepolymerfilmswere
prepared by spin casting PS from a toluene solution onto freshly cleaved mica
sheets. From there they were floated onto a clean deionized water surface and
picked up with silanized Si wafers. More experimental details can be found in
[11]. Figure 3 shows the temporal evolution of the symmetry-breaking process
74 Karin Jacobs et al.
and Fig. 4a displays details of the typical rupture scenario of a thin PS film on
a non-wettable substrate. Here, a 47 nm PS film (molecular weight 600 kg/mol)
on a silanized Si wafer is shown, annealed for 7 min at 130 oC. The average
diameter of the holes, whose shape was found invariably to be circular, is 8.4
µm, the width of the distribution 0.2 µm. The narrow size distribution reflects
thefactthatalmostalloftheholesappearwithinasharptimewindow.Theareal
density of the holes was found to scale in accordance with the above mentioned
results obtained by other authors. Similarly, the (apparently) random spatial
distributionofholesisqualitativelyidenticaltowhathasbeengenerallyobserved
before[21,22].Itisthisspatialdistributionontowhichwewillconcentratenext.
Radial Pair Correlation Analysis Ifhydrodynamicallyunstablesurfacerip-
ples (spinodal dewetting) were responsible for the formation of the holes, corre-
lations were to be expected in the geometry of the hole arrangement reflecting
the presence of a critical wavelength λ . In contrast, a distribution of posi-
max
tionsofholesfollowingaPoissonpointprocesswouldbeaclearcounterevidence
of a spinodal process. We thus determined the two-point correlation function
g(r) of the point set represented by the positions of the centers of the holes and
plotted it in Fig. 5b. Obviously, no feature indicative of a dominant wavelength
is found. Let us consider for comparison the dewetting pattern of a liquid gold
film on a quartz glass, as shown in Fig. 4b and determine g(r), too (Fig. 5b).
Again, as Fig. 5b demonstrates, no modulation ing(r) can be detected. For this
system,however,unstablesurfacerippleswithadominantwavelengthλ had
max
beenclearlyidentifiedasthedewettingmechanism[2].Thesesurfaceripplescan
even be seen with a light microscope (Fig. 6a). We have analyzed the positions
of the valleys of the observed undulations (data taken from [2]), as depicted in
Fig. 4b. They have been used as a point set onto which the same calculation of
the correlation function g(r) was applied. The result is shown in Fig. 6b. The
modulation in g(r) with a wavelength λ corresponding to the mean distance
max
of the valleys in the gold film indicates correlated sites. Moreover, two valleys
are at least 0.6 mean distances separated. (Experimentally, two valleys could be
resolved for distances larger than about 0.1 mean distances.) The existence of
such a minimum distance and of the modulation=20in g(r) thus demonstrate
that the dewetting structure of a thin liquid gold film on a quartz substrate
is indeed strongly different from a Poisson point process. Further experiments
showed that the preferred wavelength λ (also observable ’by eye’ in Fig. 6a)
max
scales as the square of the film thickness [2], as is theoretically expected for a
spinodal rupture scenario.
At this point, there are two open questions: i) Can we rule out a spinodal
dewetting mechanism for the PS films by looking only at g(r)? ii) Why is no
modulation visible in g(r) for the holes in the gold film?
The answer to i) and ii) is that g(r) is only a two-point correlation function
andthereforenotsensitiveforhigher-ordercorrelations.Sofromthepointofview
of g(r), we can neither exclude nor corroborate spinodal dewetting as rupture
mechanism for any of the systems. For the gold film it is not clear yet, whether
Dynamics of Structure Formation 75
the holes stem from valleys that reached the substrate surface or if nucleation
led to holes.
Characterization by Means of Minkowski Functionals The answers to i)
andii)urgeustoanalyzethespatialdistributionofthesitesofholes(orvalleys,
resp.)ingreaterdetail.Wecouldeitherperformstandardstatisticaltestslikethe
L-test[32],oralternatively,characterizethesampleswiththehelpofMinkowski
functionals.
We decided to use the latter, since the Minkowski functionals are connected
tophysicallyusefulparameterslikethethresholdofpercolation.Moreover,they
are very handy to describe the spatial structure of the pattern directly and to
record-forinstance-structuralchangesasafunctionoftime,aswewilldemon-
strate later. Besides, characteristics based on Minkowski functionals are known
to yield stable results with small statistical errors even for small samples, which
is of particular importance for the analysis of experimental data. They provide
statistical descriptors which contain features of n-point correlation functions at
any order n and are efficient in discriminating theoretical models [15,23,27].
Let us consider the centers of the holes in a film as an ensemble of points in
the plane. On each point, we put a circular disk, each with the same radius r,
as depicted in Fig. 7. The set is now defined as the set union of all disks, the
”coverage”.Thescale-dependentmorphologicalfeaturesofthiscoveragearethen
explored by varying the disk radius and calculating the Minkowski functionals
of the coverage as functions of r. In two dimensions, the Minkowski functionals
are the area F, the boundary length U, and the Euler characteristic χ of the
coverage. The latter is defined as the integral over the boundary curvature,
extended along the entire boundary [16]. Moreover, the Euler characteristic is a
measure of the connectivity of a structure.
For testing the Poisson process, we used the normalized length x := 3Dr/L
(L denoting the average hole distance) and the normalized functions
(cid:25) (cid:26)
F∗(x) =3D− |Ω| ln 1− F(x)
πr2N |Ω|
U∗(x)=3D U(x) (1)
2πrN(1−F(X)/|Ω|)
χ∗(x) =3D χ(x)
N(1−F(X)/|Ω|)
where|Ω|denotestheareaofthesamplingwindowandN thenumberofholes.In
thecaseofaPoissonprocessitisF∗ =3D1,U∗ =3D1,andχ∗(x)=3D1−πx2.
InFig.8,thebehaviorofthethreefunctionsisshownasopensquaresforthe
point set given by the centers of the holes displayed in Fig. 5a (left) for the PS
film. The theoretical values for a Poisson point process are represented as solid
lines.Excellentagreementisfoundforallthreefunctionals;thesmalldeviations
for larger radii being solely due to finite size effects which occur when r comes
within the range of the system size. This suggests that no lateral correlations
are present in the distribution of the holes in the polymer film. We employed
76 Karin Jacobs et al.
the same analysis also to published data of hole distributions [21,28] and found,
again, no significant deviation from a Poisson point process.
Let us consider for comparison the spinodal-dewetting pattern of the liquid
gold film, the positions of the holes are shown in Fig. 5a (right). The measured
functions F∗, U∗, and χ∗ are shown as solid circles in Fig. 8. The deviation
from a Poisson process is evident; here, correlations between the sites of the
holes are present. Also classical tests of point process statistics (the L-test [32]
and an analogous test using Ripley’s K-function [25]) reject the Poisson process
hypothesis [26,31]. For the L-test, the Poisson process hypothesis was rejected
for significance level α=3D0.01 [31].
Correlations in the distance of holes could surely also be detected with the
help of a Voronoi construction [34], but this costs a little larger computational
effort and is not as versatile as the Minkowski measures. Using a Fourier trans-
formation to find out a preferred wavelength requires large point sets and a
thorough consideration of the boundaries. Besides, a Fourier transformation is
unsuitable to check for a Poisson distribution.
It is thus demonstrated that not only the valleys of the undulation in the
goldfilmarecorrelatedintheirdistance,butalsotheemergingholes.Therefore,
we can conclude that the holes in the gold film stem from unstable hydrody-
namic surface ripples. An involvement of a spinodal process in the rupture of
the polymer films shown here, however, is very unlikely.
2.2 Late Stage of Dewetting: Growth of Holes
Up to this point, we analyzed the static features of the morphology, namely the
statistical distribution of the holes. To gain further insight in the two dewetting
modes, we investigated the dynamics of the rupture process, too. Here, we enter
the late stage of dewetting, the growth of holes.
Characterizing the Growth of Single Holes According to the theory of
spinodal dewetting, the rupture time, τ, which denotes the time after which
the formation of the holes begin, should be equal for all holes and scale as
τ ∝ h5 [3]. For the gold films, τ is experimentally not accessible due to the
specificannealingmethodusingshortlaserpulses.Inordertoobtaintherupture
time of the polystyrene films, the diameters of a number of single holes were
recorded as a function of time and extrapolated to zero radius [12]. This gave
a reasonable estimate of τ with a statistical scattering of less than 10%. The
rupture times obtained this way are only weakly dependent on film thickness
(a fit yields τ ∝ h0.6±0.3). Even films with thicknesses of more than 300 nm,
which is beyond the range of the van der Waals-interaction, break up within
some hundred seconds. It is thus demonstrated that there is neither a lateral
correlationintheholedistributionnoratemporalbehaviorevenclosetowhatis
predictedforspinodaldewetting.Thisrulesoutcompletelyaspinodaldewetting
process to be responsible for the hole formation.
Dynamics of Structure Formation 77
It should be noted here that a thermally activated hole formation process
would proceed independently at different places on the sample, and hole forma-
tion would necessarily continue to take place as long as the sample is not fully
dewetted. This contradicts observation, since almost all of the holes are present
immediately after the start of dewetting, as it is clearly seen in Fig. 3. We have
thus ruled out any dynamical instability as well as thermally activated hole for-
mation as processes responsible for the generic polymer film rupture scenario.
The only remaining possibility is heterogeneous nucleation from defects in the
filmoratthesubstratesurface.Thisisourknowledgeatthemomentandfurther
experiments should clarify the kind of defect leading to rupture.
Characterizing the Entire Structure Formation Process Deeper explo-
ration of the temporal behavior as well as the dynamics of the structure forma-
tion process by recording the radii of the holes in the case of the polymer film
is limited to the roundness of these holes. As soon as the holes coalesce, or the
surrounding rim of the hole gets unstable, this method is at its limits. Here, the
Minkowski functionals again come into play.
By determining the three Minkowski functionals F, U and χ for every single
snapshot of the time sequence, we can characterize the structure formation pro-
cess from the very beginning to the final state, where the droplets of material
are lying on the substrate, as shown in Fig. 3. Here, two thresholds are set such
that the gray scale of the usually bright substrate is above the first threshold
andthetypicallydarkrimsarebelowthesecondone.SoF,U andχofonlythe
uniform film are analyzed, as illustrated in Fig. 9. (Note that now F character-
izestheareaoftheuniformfilm,andnottheareaoftheholes,incontrasttothe
analysis of the point distribution depicted in Fig. 7.) The Euler characteristic
χ(t) is determined by calculating the mean curvature of the boundary line. The
Minkowski functionals F(t), U(t) and χ(t) for the entire sequence are shown in
Fig. 10. On one hand, F(t), U(t) and χ(t) serve as a kind of ”fingerprint” for
the temporal development of the specific morphology and, on the other hand,
simultaneously, they characterize the morphology itself.
Here, the interesting question arises whether or not the growth behavior of
the holes changes as soon as the holes come close to each other. Does a hole feel
the neighborhood of another hole? Nearby the small liquid neck between two
largeholes,doesthedrainingoftheliquidchange?Byfittingatheoreticalcurve
to the data in Fig. 10, we can compare the growth rate of the holes with the
growth rate of the morphology. In one model, the radius of the holes, r, grows
linearly with time, in the other, r grows as r ∝ t2/3. The former is applicable
tosimpleliquidsflowingwithviscousenergydissipationonly[4,5],thelatterfor
complex fluids where energy is dissipated only as sliding friction between the
liquid and the solid (’slippage’) [6]. We expect slippage to play a role since the
chain length of the polystyrene is exceeding the so-called ’entanglement length’,
which means that by pulling one chain, a hole bunch of chains will be moved.
From the fits shown in Fig. 10 we learn that for the growth of the morphology,
78 Karin Jacobs et al.
slippage does not play a prominent role and that purely viscous flow can be
assumed to be the main mechanism of energy dissipation.
This is at variance with the results for the growth law obtained by recording
theradiusofsingleholesdescribedabovewherebothmechanismhadtobetaken
into account to fit the data [12]. From these experiments we know that slippage
isespeciallyimportantintheearlyregimeofholegrowth.Whydon’tweseethis
with the Minkowski functionals? The explanation is the optical resolution that
we have for characterizing the r(t)-diagram of single holes, 1 µm, in comparison
with that of the dozens of holes, as shown in Fig. 3, where features smaller than
roughly 10 µm cannot be resolved. This means that the magnification is too
small to see deviations from a linear growth mode in the early times of hole
growth with the help of the Minkowski functionals.
Another access to the growth law of the pattern is to combine the results of
the ”simulated” growth of holes as shown in Fig. 7, where we get the functions
F(r), U(r) and χ(r), with the measured functionals in the form of F(t), U(t)
andχ(t).Thisway,wegetthreeindependentgrowlawsr(t),Fig.11.Forguiding
the eye, a linear growth law is plotted. Substantial deviations from this law can
befoundforverysmallandverylargeannealingtimes.Here,thescatterofdata
is quite high indicating that at these times (or for the corresponding pictures of
the holes, resp.) the Minkowski functionals are extremely sensitive to the height
ofthethresholdwithinthepictures.Inotherwords,theresolutionoftheimages
for very small and very large annealing times is too low to characterize the
small features correctly with the help of the Minkowski functionals. However,
from the correspondence of all three r(t)-curves in Fig. 11, we can derive that
the holes indeed grow like the overlapping disks shown in Fig. 7 and that the
growth of a hole is not changed by the vicinity of other holes. This behavior
is unexpected, but is characteristic for our system and particularly convincing
since it is obtained for all three functions. Further experiments should clarify
whether or not this behavior changes in other systems, e.g. polystyrene films of
very short chain lengths (below the entanglement length) or of films on top of
other substrates.
3 Conclusions and Outlook
Before the analysis with Minkowski functionals, we knew that the surface of
dewetting gold films oscillates with a certain wavelength λ that also scales
max
with the film thickness as predicted for a spinodal dewetting scenario [2]. How-
ever,itwasnotclearwhethertheemergingholesinthegoldfilmstemfromthese
hydrodynamicallyunstablesurfacewavesorwerenucleatedbydefects.Withthe
help of the Minkowski functionals we found out that the sites of the holes are
correlated.Thisisacounterevidenceforheterogeneousnucleation.Wetherefore
conclude that in the case of gold films, spinodal dewetting is the dominating
rupture mechanism.
For the polystyrene films it was known that the areal density of holes scales
with film thickness in accordance with spinodal dewetting theories [21,22]. The
Dynamics of Structure Formation 79
rupture time τ, though, did not. In our experiment it is only weakly dependent
onfilmthickness.OuranalysisbyMinkowskifunctionalsshowsthattheholesare
most likely randomly distributed. We conclude that in the experiments shown
here,heterogeneousnucleationisresponsibleforsymmetry-breaking.Besides,by
characterizing the complete structure formation process of a polystyrene film,
wefoundthatneighboringholesdonotinfluenceeachother,theyratherbehave
like overlapping disks.
ItcanbesummarizedthatMinkowskifunctionalsarepowerfultoolstoquan-
tify morphological properties of patterns. Moreover, structure formation pro-
cessescanbefollowedandcharacterized.Inthiswaythe”fingerprints”ofdiffer-
entprocessescanbecompared,independentlyofanystatisticalassumptionsand
without any limitation of boundary conditions or minimum number of objects.
Our vision for the future is that functions based on Minkowski functionals will
become standard image analysis tools in real space, like Fourier transformation
is in k-space.
4 Acknowledgements
ItisapleasuretothankS.Herminghausforinspiringdiscussions.Wealsoappre-
ciate a critical reading of the manuscript by D. Stoyan. Support from Deutsche
Forschungsgemeinschaft DFG Grant No. Ja 905/1-2 is gratefully acknowledged.
We also acknowledge generous supply of Si wafers by Wacker Chemitronics,
Burghausen, Germany.
80 Karin Jacobs et al.
Fig.1. Artichokes and clover, the load of an Egyptian farmer’s tractor [18].
Dynamics of Structure Formation 81
Fig.2. Example for a geometric pattern: a ceramic tile of the Alhambra Palace
(Granada, Spain).