substrateAlejandro Guillermo González, Javier Alberto Diez, Yueying
Wu, Jason D. Fowlkes, Philip D. Rack, and Lou KondicLangmuir, Just Accepted Manuscript • DOI: 10.1021/la4009784 • Publication Date (Web): 27 Jun 2013
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Instability of liquid Cu films on a SiO2 substrate
Alejandro G. González,∗,†,‡ Javier A. Diez,†,‡ Yueying Wu,¶ Jason D. Fowlkes,§
Philip D. Rack,§,¶ and Lou Kondic‖
Instituto de Física Arroyo Seco, Universidad Nacional del Centro de la Provincia de BuenosAires, Tandil, Argentina, CIFICEN-CONICET, Tandil, Argentina, Department of Materials
Sciences and Engineering, University of Tennessee, Knoxville, TN, USA, Center for NanophaseMaterials Sciences, Oak Ridge National Laboratory, TN, USA, and Department of Mathematical
Sciences, New Jersey Institute of Technology, Newark, NJ, USA
AbstractWe study the instability of nanometric Cu thinfilms on a SiO2 substrate. The metal is melted bymeans of laser pulses for some tens of nanosec-onds, and during the liquid life time the free sur-face destabilizes, leading to the formation of holesat first and then in later stages of the instabilityto metal drops on the substrate. By analyzing theFourier transforms of the SEM (scanning electronmicroscope) images obtained at different stages ofthe metal film evolution, we determine the emerg-ing length scales at relevant stages of the instabil-ity development. The results are then discussedwithin the framework of a long-wave model. Wefind that the results may differ whether early or fi-nal stages of the instability are considered. Basedon the interpretation of the experimental results,we discuss the influence of the parameters describ-ing the interaction of the liquid metal with the solidsubstrate. By considering both the dependence ofdominant length scales on the film thickness and
∗To whom correspondence should be addressed†Instituto de Física Arroyo Seco, Universidad Nacional
del Centro de la Provincia de Buenos Aires, Tandil, Ar-gentina
‡CIFICEN-CONICET, Tandil, Argentina¶Department of Materials Sciences and Engineering,
University of Tennessee, Knoxville, TN, USA§Center for Nanophase Materials Sciences, Oak Ridge
National Laboratory, TN, USA‖Department of Mathematical Sciences, New Jersey In-
stitute of Technology, Newark, NJ, USA
the measured contact angle, we isolate a modelwhich predicts well the trends found in the experi-mental data.
IntroductionInstabilities of thin liquid films deposited on solidsubstrates have attracted a significant interest fora number of years. These instabilities are impor-tant in numerous applications, in particular in thefast growing field of nanofluidics. They lead to theformation of drops, which in the case of liquefiedmetals, solidify into particles, which find its rele-vance in the applications that range from plasmon-ics, to liquid crystal displays and solar cells;1–3 forexample, the size and distribution of metallic par-ticles is known to be related to plasmon-couplingwith incident energy, that has a huge potential ofincreasing the yield in solar cell devices.4 To makefuture progress of relevance to these and other ap-plications, it is important to understand the basicmechanism driving the instabilities.
Stability of a thin film on nanoscale have beenextensively studied in the case of polymer films,see, for instance, the recent review by Jacobs etal.5 However, metal films liquefied by laser irra-diation, have been considered to a much smallerextent. Instabilities of Cu, Au, and Ni filmswere considered experimentally,6 and more re-cently both experimental and theoretical analysesof dewetting of Co, Ag, Fe, and Ni films have
been carried out.3,7–9 In our earlier works, wehave considered instabilities of other liquid metalstructures, such as polygons, lines, and rings.10–13
While these more complex geometries lead to ad-ditional insight regarding the instability develop-ment, it is important to analyze carefully the sim-plest case (uniform films), since some of the thebasic mechanisms leading to instability in differ-ent geometries are related. Therefore, a better un-derstanding of uniform films should help us gainbetter insight into the instability of more complexstructures.
For sufficiently thin metal films, of the thick-nesses of few nanometers, there is a strong evi-dence that the developing instability is of spinodaltype, i.e. it is caused by growth of surface per-turbations due to destabilizing effect of liquefiedmetal/substrate interaction forces.7,11 The instabil-ity evolution proceeds typically by the formationof holes in the liquid metal surface, and a net-work of connected bridges. The rupture of thesebridges lead to a pattern of drops which solidifyto form isolated metal particles. The emerginglengthscales – the distances between holes and/ordrops/particles – can then compared to theoreticalmodels expected to govern the evolution of the de-scribed process. How to carry out this comparisonis not always clear - for example, one needs to de-cide which stage of the instability development isto be used. A most obvious question is whether thedistances between the holes that form in the initialstages of the instability, or the distances betweenthe drops/particles that remain at the later times areto be considered. To our knowledge, this issue hasnot been analyzed carefully in the literature so far.This is not surprising, since the evolution happenson a very fast time scale and it is difficult to cap-ture its properties. We will discuss this topic ex-tensively in the present paper and study to whichdegree the results are influenced by the temporalevolution of the available data.
This work centers on experimental and theoreti-cal analysis of Cu films on SiO2 substrate exposedto laser irradiation. By considering the parts ofthe experimental domain exposed to different ir-radiance we are able to reach some understand-ing regarding the time evolution of the instability,and therefore distinguish between different stagesof its development. The Fourier spectrum of the
film thickness allows us to define in a precise man-ner the emerging wavelengths and, in addition,their time evolution. We should note here that thedata analysis that we carry out for the purpose ofquantifying the instability development is signifi-cantly different from the one which was requiredfor polymer films, due to the difference in the typeof the available data. On one hand, the domainsizes that can be used when liquid metals are con-sidered is orders of magnitude larger than the typi-cal instability wavelength, so that very good statis-tics can be obtained. This is in contrast to poly-mer films where domain size is typically a singleorder of magnitude larger than the typical wave-length, thus reducing the quality of the informa-tion that can be reached based on Fourier spectra,and requiring additional methods for data analy-sis, e.g. based on Minkowski functionals.5 On theother hand, the evolution in the case of metal filmshappens much faster (on nanosecond time scales)due to different material properties, and partiallyfor this reason precise information about temporalevolution of instability is rarely available.
After carrying out Fourier spectra-based analysisof the experimental data, we consider the applica-tion of a long-wave (LW) model for the purposeof studying the instability development. One hasto be careful regarding the choice of informationthat is being used: in particular, we find that oneobtains different results when holes or drops areconsidered. Note that previous works have usuallyconsidered drops, since they are easier to study andcorrespond to a fully developed final stage. How-ever, there is no clear evidence whether the exper-imental data obtained for drops could be reliablyrelated to existing linear theories or not. There-fore, we discuss to which degree the LW modelwithin its linear regime can be used in conjunc-tion with experimental data to extract the materialproperties entering the governing model, such asthe Hamaker constant. Furthermore, we considerdifferent models describing liquid metal/solid sub-strate interaction and discuss to how the choice ofa model influences the results.
ExperimentsCopper thin films with thickness ranging from4.8nm to 15.5nm were sputter deposited onto 100nm SiO2 coated silicon chips14 using radio fre-quency (RF) magnetron sources. The sputteringconditions are: 5 cm diameter sputter Cu targetsand a RF power of 30 W (148 V self bias), 25 sccmAr at 5 mTorr processing pressure. The Cu filmthickness was measured via optical reflectometry(Filmetrics F20-UV thin-film analyzer).
The films were irradiated in air by either a sin-gle or multiple pulses using a normal incident laserbeam (≈ 12 mm x 12 mm) that impinges at thecenter of the sample. The KrF laser has a wave-length of 248 nm and a Gaussian temporal pro-file whose width (full-width-at-half-maximum) is≈ 18 ns. The laser energy density was chosento be 140mJ/cm2 to ensure the films reach melt-ing threshold based on the numerical simulationmethod used in our earlier work.13 The total flu-ency of energy received by the metal in a particu-lar region during its irradiation determines the liq-uid lifetime. Since the laser energy has a Gaussianspatial profile, the decreasing energy density fromthe center to the edge of the thin films leads to dif-ferent liquid lifetimes at different regions. This ef-fect allows us to capture information of differentstages of the evolution of the liquid film dewettingwithin a single pulse. Therefore, a region in thecenter of the pulse has longer liquid lifetime thanthe regions at the borders. The regions subject tolower fluency can be correlated to early dewettingstages, while those at higher fluency, as well asthose irradiated with more pulses, have longer life-times and then correspond to later stages.
In contrast to the dewetting process of poly-meric films where the dynamic evolution is read-ily accessible, since it takes place in time scales ofthe order of seconds, minutes, or even hours, theliquid metal thin films typically require nanosec-onds and hence capturing experimentally differentdewetting stages is a complex challenge. To thisend, nano-second laser irradiation with a gradi-ent energy distribution provides an effective wayto freeze different stages corresponding to differ-ent liquid life times. Therefore, it is meaningful toexplore the change in experimental length scalesduring morphology evolution considering differ-
ent regions, and d etermine which stage corre-sponds more closely to existing theoretical modelsand yields a better estimate of the Hamaker con-stant.
The morphologies of different stages of the evo-lution are captured by taking scanning electron mi-croscopy (SEM) images after each pulse of theresolidified sample. In order to have a roughlyconsistent localization of the energy profile on thethin film after each laser pulse, we align the sub-strate at practically the same laser position. Inthis way, we are able to capture different dewet-ting stages as ones moves from the center to theedge.
Figure 1 shows a typical case of the series ofSEM pictures obtained in experiments for Cu filmswith a thickness of h = 8.0 nm on SiO2. Fig-ures 1a–d correspond to pictures taken at the bor-der of the central region of the irradiated sample ina single laser pulse. In Fig. 1a, we observe a rathernon uniform distribution of holes which becomesmore uniform in Fig. 1b. As one considers a regioncloser to the center of the pulse, the holes becomebigger and tend to coalesce, forming a net of rims.In Fig. 1d, the rims break up and retract leadingto a new configuration which after a second pulseends up in a distribution of drops (Fig. 1e). In sum-mary, this type of analysis of the SEM’s allowsto correlate both the position of a picture with re-spect to the center of the pulse (and/or the numberof pulses) with a time sequence that describes theevolution of the film instability.
We have repeated the procedure in a series of 26experiments with thickness varying from 4.8 nmto as high as 17 nm. We took an average of fourpictures for each experiment corresponding to dif-ferent stages. These images were used to computethe results given in the following section.
Measurement of contact anglesWe now proceed to measure the static contact an-gles of the drops resulting after a large number ofpulses. We have used two methods. The first oneconsists in applying a sufficient number of laserpulses to assure fully developed drops shape. Thefluence is controlled and limited in order to min-imize evaporation. Examples of these drops areshown in Fig. 2a, where the approximating spheres
(a) First stage: border, one pulse. (b) Second stage: outer intermedi-ate, one pulse.
(c) Third stage: inner intermediate,one pulse.
(d) Fourth stage: center, one pulse. (e) Last stage: center, two pulses.
Figure 1: SEM images of various regions of a 8.0nm thick Cu film on a SiO2 substrate. The positionof each picture with respect to the center of the laser beam decreases from (a) to (d), and thus the corre-sponding liquid lifetime increases. Picture (e) is also central as (d), but the film has been irradiated withan extra pulse, and therefore it corresponds to the largest lifetime.
and the calculated contact angles are also shown.The distribution of these angles with respect todrop volumes is shown in Fig. 2b for drops ob-tained from films of different thicknesses. We ob-serve a significant variation of θ for smaller drops(V < 3× 10−3 µm3), while the mean value 〈θ〉vary from 69◦ to 80◦. We note that a possible de-pendence of contact angle on the drop size is an in-teresting question, which we leave for future work.For the present purposes, the mean values are suf-ficient.
(a)
0 0.002 0.004 0.006 0.008 0.01
V (µm3)
50
60
70
80
90
100
110
θ (
°)
h= 8.6 nmh= 9.1 nmh=12.0 nmh=16.8 nm
(b)
Figure 2: (a) Measurement of contact angles ina sample of solidified drops (particles) resultingfrom a film of thickness h = 8.7nm. (b) Distribu-tion of contact angles from films of different thick-ness, h, versus the drops volume.
A drawback of the method described above isthat it is difficult to determine the points of con-tact at the base of a solidified drop, particularly thesmaller ones. Therefore, we have also explored analternative approach that enhances the contrast ofthe surface of a particle. To do so, we sputtereda thick layer of Ni (≈ 150nm) on top of Cu drops,and then we milled them by means of a focused ion
beam (FIB). This milling process was done withan FEI Nova 600 dual beam scanning electron mi-croscope (SEM) with a gallium ion source underan accelerating voltage of 30KeV. The ion millingarea was defined to show the whole drop cross sec-tion (≈ 800nm×800nm area in this case) and wascarefully aligned through the center of the particle.After ion milling of each drop, cross section SEMimages were taken to determine the contact angleas shown in Fig. 3. As a result, the average contactangle of a total of 15 drops is 65◦ with a standarddeviation of 6.9◦. This result is consistent with theone given above, suggesting that despite relativelylarge standard deviations, contact angle is a welldefined quantity on the scales considered.
(a)
100 200 300 400 500Drop diameter (nm)
50
55
60
65
70
75
80
θ (
° )
(b)
Figure 3: (a) Image of a solidified Cu drop coveredwith nickel and viewed with a 52◦ tilt. (b) Distri-bution of contact angles for different drops versustheir projected diameters.
Evolution of the instabilityIn order to study the characteristic length scalesof the patterns formed by the thin films instabil-ity, we have computed the fast Fourier transforms(FFT) of pictures corresponding to different stagesof the evolution. For the reasons described in thenext paragraph, we consider four sub-regions of512× 512 pixels at the corners of each picture(whose size is 2048×1764 pixels), and obtain thecorresponding FFT’s. First, we verify that there isno preferred direction in the film pattern. Figure 4shows the density of the transform for a typicalsub-region. The annular white zone confirms thatthe distribution of length scales is isotropic. Wehave verified that isotropic distribution holds forall considered experimental images.
Figure 4: Density plot of the FFT of Fig.1c.
Once confirmed that the FFT’s are isotropic, wecompute (for each of the four sub-regions) the ra-dial spectral distribution of wavenumbers, k, byaveraging the FFT’s over all directions for annulibetween k and k+ δk. A good overlapping of thespectra for all four sub-regions guarantees that thewhole picture corresponds to the same stage. Fig-ure 5 shows that this is indeed the case, and there-fore we conclude that each of the pictures shownin Fig. 1 corresponds to a fairly uniform laser irra-diation, and have endured the same liquid lifetime.The peaks of the spectra can be correlated to typ-ical lengths, ` = 2π/k, which, depending on thestage under study, characterize the sizes and dis-tances between holes or drops.
In order to describe the evolution of the film
instability, we must correlate the average radialspectra of a picture with a time sequence. Forinstance, in Fig. 1a there is a rather random pat-tern of holes, and so its average spectra has a weakand wide peak at kmax ≈ 100 µm−1 (`max ≈ 63nm)as shown in Fig. 5a. This indicates that there isno clear dominant length in this very early stage.The apparent peak around k = 0 is an artifactof the method due to the finite size of the sam-ple, and it is more pronounced in the early stageswhere there are rather extensive connected flat ar-eas. Fig. 5b shows instead a spectra with a clearpeak at kmax ≈ 63 µm−1 (`max ≈ 100nm) in corre-spondence with the more uniform distribution ofholes in Fig. 1b. This peak is due to the fact thatthere is a better defined spacing between holes. Inthe next stage, the holes are growing and coalesc-ing thus leading to an increase in the characteris-tic spacing as seen in the peak at kmax ≈ 50 µm−1
(`max ≈ 125nm) in Fig. 5c. At the fourth stage inFig. 1d, the bridges and rims around the holes havebroken up and have given place to a filament-likestructure whose spectra is shown in Fig. 5d. Thisfigure shows a narrower peak at a lower wavenum-ber kmax ≈ 38 µm−1 (`max ≈ 166nm). As theseelongated structures further contract and break up,a rather uniform distribution of drops shows up(see Fig. 1e). The spectrum shown in Fig. 5e has apeak at kmax ≈ 25 µm−1 (`max ≈ 250nm), thus in-dicating the final drop spacing. The shape of thesedrops is very similar to spherical caps, and theiraverage radius contributes to the formation of anincipient secondary peak in the spectra, as seen inFig. 5e for k ≈ 69 µm−1 (`≈ 90nm).
A typical evolution of these spacings is shownin Fig. 6a. Since the total liquid life time is notexactly known, we sort the lengths according tothe stages, shown in Fig. 1, providing therefore arough idea of the time evolution. The tendencyof the pattern to increase its typical length scale isclearly visible. The error in the determination ofthe distances is calculated from the dispersion ofthe corresponding k’s obtained for each of the foursub-regions.
We find it useful to separate the evolution intofour main stages:
1. a preliminary early stage, where no well-defined peak is observed in its spectrum;
Figure 5: FFT spectra of the experimental images shown in Fig. 1. Each part of the figure shows four FFTspectra computed from four sub-domains of the experimental images. All the spectra are plotted with thesame symbol. The overlap shows that the patterns in each experimental figure belong to the same stage ofthe evolution. The exposure time increases from (a) to (e).
2. a developed early stage, where the holes dis-tribution is characterized by a clear peak;
3. a series of intermediate stages, where bridgebreakups and coalescence are produced; and
4. a final stage, where drops are clearly visibleand characterized by a narrow peak in thespectrum.
The first length scale, `max, which appears inthe evolution corresponds to the developed earlystage, and it characterizes the distance between thecenters of the holes formed in the film. We de-note such length as `holes. Similarly, when the in-stability saturates to a final pattern of drops, `maxcorresponds to the average distance between dropcenters, which we call `drops (see Fig. 6b).
While analyzing the information from more than450 spectra, we have found consistently a mono-tonic increase of `max as we progress from theearly second stage to the advanced fourth oneabove. Therefore, by studying the early secondstage (when the thin film linear instability is fully
developed, but not affected by nonlinear effectsor other types of instability) and the fourth stage(when the drops are completely formed), we coverthe full range of emerging length scales. Further-more, intermediate stages are conceptually com-plicated to consider since they involve coupling oflinear and nonlinear effects. For these reasons, weconcentrate on the developed early stage and onthe final stage in what follows.
Modeling of the instabilityThe typical length scales obtained from the spec-tra discussed in the previous section are analyzedwithin the framework of the linear thin film in-stability under the LW model. In this approach,it is assumed that the flow is driven by both sur-face tension and van der Waals forces. The lat-ter are described by the so-called disjoining pres-sure, Π, whose dependence on thickness h in-volves the competition between long- and short-range forces.15 While the exact function form ofΠ is not known, in particular for liquid metals11,16
Figure 6: (a) Evolution of the characteristic spac-ing of the patterns, `max for the experimental im-ages shown in Fig. 1. The horizontal axis indicatesthe numbering of the stage. (b) Average distancebetween holes (`holes) at the developed early stageand drops (`drops) at the final stage for differentfilm thickness, h.
(see these references for further discussion), acommonly used expression is17–19
Π(h) = κ
[(h∗h
)n
−(
h∗h
)m], (1)
where h is the fluid thickness, n > m are posi-tive exponents, and h∗ is the equilibrium thick-ness where repulsive and attractive forces balance.Within the present context, the second term in-cludes the electronic component.20 The pressurescale, κ , is related to the Hamaker constant, A, as
A = 6πκh3∗, (2)
Only a small subset of exponents (n,m) has beenconsidered in previous works. It is difficult to ob-
tain them from very first principles, except usingadditional simplifying assumptions that cannot bealways taken for granted in liquid metal films sub-ject to laser irradiation. Typical examples foundin the literature21 considering polymeric films are:(9,3), (4,3)and (3,2). For liquid metals, some ofthese exponents have been used, and both simpli-fied versions with attractive forces only,3,7,8 andmore complex models with additional exponentialterm22 have been used.
Since it is unclear from the literature which ex-ponents should be used, in the present work wewill explore whether a simple model, specifiedby Eq. (1), with (commonly used) sets of expo-nents provides consistent results, and/or whetherone could identify the pair which provides a goodagreement with the experimental results for Cu,while remembering that the functional form spec-ified by Eq. (1) is an approximation itself. An-other approximation - the use of the LW modelfor a problem where equilibrium contact angles arelarge - should be recalled as well. We expect thatthis approach is applicable in particular when con-sidering initial stages of instability (leading to holeformation) since during most of the instability de-velopment the slopes of the film surface are not toolarge.
Using this disjoining pressure within the LWmodel, the wavelength of the mode of maximumgrowth rate obtained from the linear stability anal-ysis (LSA) is given by23
λm = 2π/
√A
12πγhh3∗
[m(
h∗h
)m
−n(
h∗h
)n],
(3)where γ is the liquid surface tension. Since thisformula refers to the linear regime, it is appropri-ate to assume that λm should be related to `holes,although it is frequently found in the literature thatit is related to `drops (as well). We also considerthis possibility and discuss its implications in anappropriate section below. Here, we proceed todiscuss the basic framework which can be usedto compare the experimental results to the predic-tions of the model leading to Eq. (3). We note thattypical values of h∗ are estimated in the theoriesthat describe the intermolecular interactions, suchas those using Lennard-Jones potentials, and they
To proceed, we first study in some detail the be-havior of Eq. (3) with respect to the exponents(n,m). It is convenient to rewrite this equation as
λm =
√√√√48π2γ
mAhm+1h3−m
∗
[1− n
m
(h∗h
)n−m]−1
.
(4)Since h∗� h for the considered experiments, thesquare bracket is approximately equal to unity, andconsequently λm becomes independent of the ex-ponent n, which accounts for the repulsive forceterm in Eq. (1). Moreover, for m = 3, the depen-dence on h∗ also disappears, and we have
λm =
√16π3γ
Ah2, for m = 3, (5)
which is an expression frequently used in the liter-ature,7,8,25 albeit for different metals. For m 6= 3,the functional dependence of λm on both h and h∗is modified. In particular, for m = 2 as, for in-stance, in the commonly used pair (3,2), we have
λm =
√24π3γ
Ah3/2h1/2
∗ , for m = 2. (6)
Regarding the time scales involved in the prob-lem, the LSA gives
τ =3µ
16γπ4λ 4
mh3 , (7)
where µ is the fluid viscosity. Thus, we have
τ =48π2γµ
A2 h5, for m = 3, (8)
and
τ =108π2γµ
A2 h2∗h
3, for m = 2. (9)
Note that m = 2 case retains the dependence onh∗ unlike the m = 3 case, and this implies that theinstability develops much faster for the pair (3,2)than for pairs such as (9,3) and (4,3), since h∗�h.
The static contact angle, θ , of the drops resultingfrom the instability process can be related to theparameters κ and h∗ of the disjoining pressure by
23
κ =γ
2Mh∗tan2
θ , (10)
where M = (n−m)/((m− 1)(n− 1)). Note thathere we are assuming the dependence on θ in theform of (tan2 θ)/2 instead of (1− cosθ), sincethe former dependence23,26 results from the lin-earized form of the free surface curvature,27 con-sistently with the hypothesis of small slope in theLW model, while the latter is derived from thecomplete (nonlinear) form. By using Eq. (2), wecan now express θ in terms of the Hamaker con-stant, A, and the equilibrium thickness, h∗,
tanθ =
√MA
3πγh2∗. (11)
In the next two sections we proceed to discussthe utility of the outlined expressions for λm andθ for the purpose of explaining the experimentalresults.
Analysis based on the distancesbetween holesThe early stages of the thin film instability, forwhich the LSA is appropriate, yields as immedi-ate consequence to the formation of holes. Sinceduring a considerable part of this hole formationprocess the slopes of the film surface are not large,the LW model should accurately describe the dy-namics. The process leading to drop formation ismuch more complex. Therefore, we consider thedistance between holes first.
Since one expects that `holes is basically given byλm as predicted by the LSA within the LW model,we attempt to fit the experimental values of `holeswith Eq. (3). To do so, we start by choosing twopairs of exponents (9,3) and (4,3), and a value ofh∗ = 0.1nm that is consistent with the underlyingtheories. Figure 7 shows that the fittings with bothpairs are very similar, which according to Eq (2)yield A = 1.36× 10−16 J. This value is signifi-cantly larger than the reported by Eichenlaub etal.,28,29 which vary between a theoretically calcu-lated value of Acalc = 1.67×10−19 J and an exper-imentally measured one of Aexp = 1.42×10−19 J.One may wonder whether the assumed value of h∗
is too small; however, even for h∗ as large as 2nmor so, too high values of A are obtained. Even ifEichenlaub’s deductions are disregarded, the val-ues of the Hamaker constant obtained using thepairs of exponents (9,3) and (4,3) are too largeto be acceptable in comparison to other results re-ported in the literature.3,9,30 Note also h∗ of theorder of a few nanometers is inconsistent with theunderlying molecular models. These results sug-gest that m = 3 for any usual n > m does not leadto acceptable values of A for the considered exper-imental case of Cu on SiO2 substrate; it is possi-ble that different metals and/or different substratesmay lead to a different conclusion.
10 15
100
200
500
h HnmL
Λm
Hnm
L
Figure 7: Fitting of `holes (symbols) with λm forh∗ = 0.1nm. Both exponents (9,3) (dotted line)and (4,3) (dashed line) yield A = 1.36×10−16 J.
Next, we consider m = 2 and, in particular, thepair (3,2) which has also been suggested in the lit-erature. Figure 8 shows the results for h∗ = 0.1nm(solid line) which leads to A= 2.58×10−18 J. Thisvalue, although still larger than those reported byEichenlaub et al.,28,29 is two orders of magnitudesmaller than the one obtained for m = 3, and fur-thermore it is comparable to other results reportedin the literature.3,9,30 Figure 8 also shows that, asexpected, the exponents (9,3) and (4,3) for thisvalue of A cannot be used to fit the experimentaldata (dashed and dotted lines in Fig. 8).
We note that the fitting of `holes versus h withEq. (6) for λm is obtained for a given value of theratio A/h∗, provided h∗ � h. In fact, we find anapproximate linear relationship between the valuesof A obtained by varying h∗ and using the full ex-pression for λm, Eq. (4) (symbols in Fig. 9a). Thesmall departure with respect to the average straight
line A= 2.89×10−17h∗ J, with h∗ in nm, is due theadditional term with n = 3. This result implies thatin order to obtain A as small as 10−19 J, one needsto consider h∗ ≈ 0.01nm, which is unacceptablysmall from the physical point of view.
10 15
100
500
200
300
150
700
h HnmL
Λm
Hnm
L
Figure 8: Fitting of `holes (symbols) with λm forh∗= 0.1nm. The exponents (3,2) (solid line) yieldA = 2.58×10−18 J, while (4,3) (dashed line) and(9,3) (dotted line) are unable to fit the data.
Finally, by using a linear approximation for therelation between A and h∗ for (3,2) in Eq. (11),we can relate θ with h∗ as shown in Fig. 9b. Forh∗ = 0.1nm, Eq. (11) yields θ = 72.8◦, which isin reasonable agreement with the measured con-tact angles (see Figures 2 and 3). This can be in-terpreted as a further confirmation of the selectionof exponents (3,2), since the theory gives theoret-ical angles that are in correspondence with inde-pendently measured ones.
We note that several factors may contribute toyield a Hamaker constant larger than that reportedby Eichenlaub et al.29 The studies based on Lif-shitz theory,24 have demonstrated that Van derWaals or Casimir forces can be modified due toa change of the optical properties of the materi-als.31–38 This change can be produced by the gen-eration of extra free electrons due to the laser irra-diation, which may take place in our case in bothmetal and Si under the 100nm SiO2 layer. For in-stance, under an irradiation of a 1.7ns (full widthhalf maximum) laser with an effective energy of30µJ and a photon energy of 3.68eV, Inui37 re-ported an increase of Casimir force between two100nm thick Si plates with a separation of 1µmof about 1.5 times the force without laser irra-diation. In our case, the free carrier generation
Figure 9: (a) Hamaker constant, A, versus h∗ fromthe fitting of `holes with λm. The solid line showsan average linear relationship. (b) Contact angle,θ , versus h∗ for (3,2). The symbol on the curvecorresponds to the predicted contact angle for h∗=0.1nm (θ = 72.8◦).
rate under the laser pulse energy of 0.14J/cm2 canreach above 1016−1017cm−3 per nanosecond, andhence a modified dielectric constant is expected.As for the metal film, modified dielectric constantand nonlinear optical response under ultrafast laserirradiation have been widely explored.39–41 Bigotet al.42 used an 80fs pump laser with a maximumpump energy density of 0.5mJ/cm2 to investigatethe effect of interband and intraband transition onoptical properties of Cu nanoparticles. With amodified energy status distribution due to an in-crease of electron temperature39 and enhancementof Fermi energy as a result of interband transi-tions, as well as a different inverse collision timeof the collective mode, it is showed that the cor-responding dielectric constant can be modified bythe laser excitation. Comparing the laser energydensity used by Bigot et al. ( 6× 10−6J/fs cm2)to the laser energy density applied in our experi-
ment (≈ 8×10−6J/fs cm2), a similar phenomenonmay take place in our Cu film while the laser dura-tion is much longer than a femtosecond laser. Be-sides the laser effect, an increased temperature thatmay smear the electronic distribution around theFermi energy can also modify the liquid metal op-tical constant. This brief discussion provides onlysome of the possibilities that may lead to an in-crease of the actual Hamaker constant. Furtherwork is needed to understand how relevant any ofthese effects actually is.
Analysis based on the distancesbetween dropsThe distance between drops, `drops, has been alsoconsidered in determining the characteristic lengthscale of the instability, despite the fact that dropsare the outcome of complex dynamics which isnot necessarily well described (if at all) by thelinear stability analysis. Still, we consider dropsas well in order to illustrate the differences inthe results when considering drops versus holes.Figure 10 shows the distance between the drops,and corresponding fits obtained using Eq. (3) forh∗ = 0.1nm, and the same pairs of exponents asin previous section. Due to the larger dispersionof the experimental results, the linear regressionin the log-log plot in the figure is not so goodas for `holes; for instance, the standard deviationof the data with respect to the fit is now almosttwice as large. Nevertheless, the pair (3,2) stillyields a reasonable fit for `drops, while pairs suchas (9,3) and (4,3) lead to fitting lines which donot describe the experimental results well. The(3,2) pair leads to the best fit for a Hamaker con-stant A = 7.18×10−19 J, which is smaller than theone obtained with `holes, and it is coincidentallycloser to the reported values.28 However, follow-ing a similar procedure to the one described in theprevious section to calculate the contact angle, wefind θ = 59.7◦, which is out of the range of themeasured angles (see Fig. 2b and Fig. 3b). As dis-cussed previously, this finding is not surprising dueto the complexity of nonlinear effects which areresponsible for the drop formation.
The failure to get a good fit for drops, and thereasonable results for holes, is not so surprising if
Figure 10: Fitting of `drops (symbols) with λm forh∗= 0.1nm. The exponents (3,2) (solid line) yieldA = 2.58×10−18 J, while (4,3) (dashed line) and(9,3) (dotted line) do not provide good fits for thisvalue of A.
one takes into account that the considered model isbased on the linear stability analysis, and thereforesmall perturbations from a base state (flat film).Furthermore, one also cannot expect a good agree-ment with the theory by considering intermediatestages of instability, since during these stages non-linear effects already become dominant. The inter-mediate stages can provide interesting informationabout growth rates, mixing of instabilities or evo-lution to a nonlinear regime, but these studies willrequire further extensive analysis that is out of thescope of the present work.
ConclusionsIn this paper, we have considered the evolutionof the instability of Cu films on SiO2 substrates,where the metal is liquefied by exposure to laser ir-radiation. One distinguishing feature of this studyis that, by analyzing regions of the film exposedto different levels of irradiation, we are able to de-fine several stages of the instability development.When considering initial stages, we find a clearsignature of the spinodal type of instability man-ifesting itself in a well defined peak in the Fourierspectrum, and thus suggesting the existence of alength scale that characterizes the emerging fea-tures. The experimental results are then analyzedusing a long-wave (LW) model. The wavelengthof maximum growth rate as given by the linearstability analysis performed within this model al-
lows to extract the values of the Hamaker constantdescribing the liquid metal/solid substrate interac-tion. The values thus obtained are found to belarger than expected based on the (limited) avail-able literature, but consistent with other worksconsidering similar systems. We find that the func-tional form of the disjoining pressure model de-scribing liquid/solid interaction has a strong influ-ence on the comparison between the experimentand the theory; only one of the usual exponentpairs defining the disjoining pressure can providea reasonable fit to the experimental results. Fur-thermore, significantly different results are foundwhen the model is applied to the drops/particles(which are formed in the last stages of the insta-bility development and do not change their shapewith additional laser pulses) instead of holes thatform during early stages. This suggests that muchcare is required when applying the tools of LW(or any other) theory for the purpose of under-standing the experimental data: the conclusionsmay be sensitive to the properties of the consid-ered model, as well as to the choice of the timeat which the experimental data are extracted. Forexample, the standard results regarding scaling ofthe emerging length and time scales with the filmthickness, h, do depend on the parameters enter-ing the model. For the present study of Cu filmson SiO2 substrates we find that the exponent pair(3,2) provides a good agreement for the scaling ofthe wavelength, λ , with h, (λ ∼ h3/2), while theother pairs of exponents, leading to λ ∼ h2, do notprovide accurate description of the experimentaldata, at least for the system considered here.
The results given in this paper show that lin-ear stability analysis of the underlying LW model,when used carefully, can be utilized for developingan accurate description of the experimental data.We expect that these results will bring us a stepcloser in developing better understanding of theinstability evolution in other, more complex liq-uid metal geometries. This will be a subject of ourfuture work.
Acknowledgement J.D.F. and Y.W. acknowl-edge support from the U.S.Department of En-ergy, Basic Energy Sciences, Materials Sciencesand Engineering Division. P.D.R. and L.K. ac-knowledge partial support by the NSF Grant No.
CBET 1235710. A portion of this work was con-ducted at the Center for Nanophase Materials Sci-ences, which is sponsored at Oak Ridge NationalLaboratory by the Office of Basic Energy Sci-ences, U.S. Department of Energy. A.G.G. andJ.A.D. acknowledge support from Consejo Na-cional de Investigaciones Científicas y Técnicasde la República Argentina (CONICET, Argentina)with grant PIP 844/2011 and visits to ORNL witha joint CONICET-NSF international cooperationproject.
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