Approach to universal self-similar attractor for the levelling of thin liquid films
Michael Benzaquen, Paul Fowler, Laetitia Jubin, Thomas Salez, Kari Dalnoki-Veress, Elie Raphaël
AApproach to universal self-similar attractor for the levelling of thinliquid films
Michael Benzaquen a ∗ , Paul Fowler b ∗ , Laetitia Jubin b , Thomas Salez a , Kari Dalnoki-Veress a , b andElie Rapha¨el ‡ a We compare the capillary levelling of a random surface perturbation on a thin polystyrene film with a theoretical study on thetwo-dimensional capillary-driven thin film equation. Using atomic force microscopy, we follow the time evolution of samplesprepared with different initial perturbations of the free surface. In particular, we show that the surface profiles present long termself-similarity, and furthermore, that they converge to a universal self-similar attractor that only depends on the volume of theperturbation, consistent with the theory. Finally, we look at the convergence time for the different samples and find very goodagreement with the analytical predictions.
Introduction
In the past decades, thin films have been of undeniable inter-est to scientific and industrial communities . Indeed, under-standing the dynamics and stability of thin films is essential totechnological applications such as nanolithography and thedevelopment of non-volatile memory storage devices . More-over, thin films have enabled the study of the effect of confine-ment on polymers . Several experiments have been per-formed in order to gain insights into the dynamics of thesefilms. Examples are provided by the broad class of dewet-ting experiments , as well as studies on capillary level-ling . Levelling experiments on thin polymer films in thevicinity of the glass transition temperature have recently giveninsights into the surface flow in glassy polymers . The effectof viscoelasticity related to the polymeric nature of these filmshas been addressed as well .Thin liquid films are also of great interest to the hydrody-namics and applied mathematics community, as the viscousrelaxation of a perturbed free surface is described by a non-linear partial differential equation that, to date, remains onlypartially solved. This equation is called the capillary-driventhin film equation . Several analytical and numerical studies have allowed for a deeper understanding of its math-ematical features. Recently, it was shown that the solution ofthe thin film equation for any sufficiently regular initial surfaceprofile uniformly converges in time towards a universal self-similar attractor that is given by the Green’s function of the lin-ear capillary-driven thin film equation . In the terminologyof Barenblatt , this attractor corresponds to the intermediate a Laboratoire de Physico-Chimie Th´eorique, UMR CNRS 7083 Gulliver,ESPCI ParisTech, PSL Research University. b Department of Physics & Astronomy and the Brockhouse Institute forMaterials Research, McMaster University, Hamilton, Canada ∗ These authors contributed equally to this work. ‡ E-mail: [email protected] asymptotic regime. “Intermediate” refers to time scales thatare large enough for the system to have forgotten the initialcondition, but also far enough from the generally predictablefinal equilibrium steady state; which, for capillary-driven thinfilms is a perfectly flat surface. For thin films, the question ofthe convergence time to this universal attractor has not beenaddressed so far and is the focus of this paper.Here, we report on levelling experiments on thinpolystyrene films that corroborate the theoretical predictionson the convergence of the surface profiles to a universal self-similar attractor. In the first part, we recall the main resultsof the theoretical derivation of the intermediate asymptoticregime, and address the question of the convergence time. Inthe second part, we present the experiments where we followthe time evolution of samples prepared with different randominitial perturbations of the free surface. Consistent with thetheory, we show that the surface profiles present long termself-similarity, and converge to a universal self-similar attrac-tor that only depends on the volume of the perturbation. Inparticular, the convergence times measured in the differentsamples show very good agreement with the theory.
Here we recall the main theoretical results from our previouswork , and derive an expression for the convergence time asa function of the volume of the perturbation. The levelling of a supported thin liquid film can be describedwithin the lubrication approximation. Assuming incompress-ible viscous flow, together with a no-slip boundary conditionat the substrate and a no-stress boundary condition at the free1 a r X i v : . [ c ond - m a t . s o f t ] A ug urface, yields the so-called capillary-driven thin film equa-tion : ∂ t h + γ η ∂ x (cid:0) h ∂ x h (cid:1) = , (1)where h ( x , t ) is the thickness of the film at position x and time t , γ is the surface tension, and η is the viscosity. Equation(1) can be nondimensionalised through h = h H , x = h X and t = ( η h / γ ) T , where h is the equilibrium thickness of thefilm infinitely far from the perturbation. This leads to: ∂ T H + ∂ X (cid:0) H ∂ X H (cid:1) = . (2)The height of the film can be written as h ( x , t ) = h + δ ( x , t ) ,where δ ( x , t ) is the perturbation that levels with the passingof time. For the case of small perturbations compared to theoverall thickness of the film, Eq. (2) can be linearised by let-ting H ( X , T ) = + ∆ ( X , T ) where ∆ ( X , T ) (cid:28)
1. This yieldsthe linear thin film equation: ∂ T ∆ + ∂ X ∆ = . (3)For a given sufficiently regular initial condition ∆ ( X , ) = ∆ ( X ) , the solution of Eq. (3) is given by: ∆ ( X , T ) = (cid:90) d X (cid:48) G ( X − X (cid:48) , T ) ∆ ( X (cid:48) ) , (4)where G is the Green’s function of Eq. (3), and reads : G ( X , T ) = π (cid:90) d K e − K T e iKX . (5)By ’sufficiently regular’, we mean in particular that the ini-tial perturbation of the profile is summable, with a non-zeroalgebraic volume, and that this perturbation vanishes when X → ± ∞ . The Green’s function is obtained by taking the spa-cial Fourier transform of Eq. (3). Equations (4) and (5) arecentral to the problem as, for a given initial condition, theygive the profile at any time. Guided by the mathematical structure of Eq. (3), we intro-duce the self-similar change of variables: U = XT − / , and Q = KT / , together with ˘ ∆ ( U , T ) = ∆ ( X , T ) . These vari-ables, together with Eqs. (4) and (5), yield:˘ ∆ ( U , T ) = (cid:90) d X (cid:48) ˘ G ( U − X (cid:48) T − / , T ) ∆ ( X (cid:48) ) , (6)where ˘ G ( U , T ) = T − / φ ( U ) , and: φ ( U ) = π (cid:90) d Q e − Q e iQU . (7) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) Figure 1
Schematic illustrating the convergence of any given initialprofile to the universal intermediate asymptotic solution.
Note that the integral in Eq. (7) can be expressed in termsof hypergeometric functions (see appendix). The main resultfrom our previous work was that, for any given initial con-dition ∆ ( X ) the rescaled solution T / ˘ ∆ ( U , T ) / M , where M = (cid:82) d X ∆ ( X ) (cid:54) = φ ( U ) (see Fig. 1):lim T → ∞ T / ˘ ∆ ( U , T ) M = φ ( U ) . (8)According to Barenblatt’s theory , this is the intermediateasymptotic solution. The solution is universal in the sense thatit does not depend on the shape of the initial condition. Notethat in the particular case of a zero volume perturbation, theattractor is given by the derivatives of the function φ ( U ) . Thequestion of the time needed to reach this fundamental solutionis important as it quantifies how long one has to wait to forgetthe initial condition. In order to study the approach to the self similar attractor, welook at the surface displacement at x = ∆ ∞ ( X , T ) be the perturbation profile in the intermedi-ate asymptotic regime, then according to Eq. (8) at U = ∆ ∞ ( , T ) = M φ ( ) / T / . (9)We then define the convergence time T c as being the intersec-tion of the initial central height and the central height in theintermediate asymptotic regime: ∆ ( ) = ∆ ∞ ( , T c ) , (10)which leads to: T c = (cid:18) Γ ( / ) π M ∆ ( ) (cid:19) . (11)2ote that the choice of origin, x =
0, is arbitrary and will bediscussed in the experimental section.
Samples were prepared using polystyrene (PS) with weightaveraged molecular weight M w = . = .
06 (Polymer Source Inc.). Solutions of PS intoluene (Fisher Scientific, Optima grade) were prepared withvarious weight fractions, 1 < φ <
10 wt%. Films with thick-ness h Si were spincast onto clean 10 mm ×
10 mm Si wafers(University Wafer) and films with thickness h Mi were spincastonto freshly cleaved 25 mm ×
25 mm mica substrates (TedPella Inc.).To prepare samples with various surface geometries the fol-lowing procedure was used. First, ∼
10 mm ×
10 mm sec-tions of the films prepared on mica were floated onto thesurface of an ultrapure water bath (18.2 M Ω cm, Pall, Cas-cada, LS). These pieces of film were then picked up using thepreviously prepared films with thickness h Si on the Si sub-strate. During this transfer, the floating films were intention-ally folded back on themselves to create random non-uniformsurface geometries. We emphasize that samples were preparedat room temperature, well below the glass transition tempera-ture T g ≈ ◦ C. Two types of samples were prepared: • small perturbations : Films with a relatively small thick-ness perturbation, where the linear thin film equationis expected to be valid. Such films were preparedwith thicknesses h Mi (cid:28) h Si to create surface perturba-tions with max [ δ ( x , )] / h (cid:28)
1. We used film thick-ness combinations { h Si , h Mi } ≈ {
600 nm, 80 nm } and {
200 nm, 25 nm } . • large perturbations : Films with large thickness per-turbations relative to h . Varying geometries wereprepared with thicknesses h Mi ≈ h Si to create sur-face perturbations with max [ δ ( x , )] / h ∼
1. Sam-ples were prepared using film thickness combinations { h Si , h Mi } ≈ {
100 nm, 100 nm } , {
150 nm, 150 nm } , and {
200 nm, 200 nm } .The shapes of the non-uniform perturbations were not pre-pared by design, rather, during the preparation process manyprofiles are found on a single sample. Regions of interest werethen located and chosen such that, while the height is varyingin one direction, it is sufficiently invariant in the orthogonalhorizontal direction, i.e. h can be taken to be a function of x and t alone. Ensuring that the profiles were invariant in one di-rection was crucial for the comparison to the two-dimensionaltheory discussed above. Having prepared non-uniform surfaceperturbations, a second piece of film with thickness h Mi wasfloated onto a portion of the sample with thickness h Si to create a stepped bilayer geometry, the details of which are fully ex-plained elsewhere . Briefly, the initial height profile of sucha step is well described by a Heaviside step function. Whena stepped film profile is annealed above T g the step levels dueto capillary forces. For this well defined and well studied ge-ometry, measuring the evolution of the film height profile overtime gives an in situ measurement of the capillary velocity, γ / η . We emphasize that each sample has both the perturba-tion of interest as well as a region where there is a stepped bi-layer. By obtaining the capillary velocity γ / η from the bilayerportion of the sample while also probing the perturbation onthe same sample, we reduce measurement error (for exampledue to small sample-to-sample variations in annealing temper-ature). The final stage in the preparation of the samples is a1 min anneal at 130 ◦ C on a hot stage (Linkam Scientific In-struments Inc.) to ensure that the floated films were in goodcontact with the substrate film and to remove any water fromthe system. Note that although there is some evolution of thegeometry during this short initial annealing stage, as will be-come clear below, t = ◦ C using a heating rate of 90 ◦ C/min.Above T g , capillary forces drive the non-uniform surface ge-ometries to level. After some time the samples were rapidlyquenched to room temperature and both the perturbation andbilayer film profiles were measured using AFM. From theAFM scans of the stepped bilayer (not shown), we use thetechnique described previously to extract the capillary ve-locity. For all samples we measure the capillary velocity γ / η ≈ µ m / min, which is in excellent agreement with pre-vious measurements . In Fig. 2(a), (b) and (c) are shown the evolution of three ex-amples of small perturbations, with the highest profiles corre-sponding to the initial t = x = x < x >
0. In the initial stages of annealing,the perturbations quickly lose any asymmetry in their shape.With additional annealing, the symmetric profiles broaden andtheir maximal heights decrease. Since the heights of the lin-ear profiles are small compared to the equilibrium film thick-nesses h , we expect their evolution to be governed by Eq. (3)(the linearized thin film equation). In particular, at long timeswe expect the profiles to converge to the universal self-similarattractor described in Section 1.2.3
10 −5 0 5 10020406080 x [ µ m ] δ ( x , t ) [ n m ] (a) −10 −5 0 5 1000.20.40.60.81 U δ ( x , t ) / δ ( , t ) (d) φ ( U ) /φ (0) −30 −20 −10 0 10 20 30050100150 x [ µ m ] δ ( x , t ) [ n m ] (b) −10 −5 0 5 1000.20.40.60.81 U δ ( x , t ) / δ ( , t ) (e) φ ( U ) /φ (0) −30 −20 −10 0 10 20 30020406080100120 x [ µ m ] δ ( x , t ) [ n m ] (c) −10 −5 0 5 1000.20.40.60.81 U δ ( x , t ) / δ ( , t ) (f ) φ ( U ) /φ (0) Figure 2
The results of three experiments on small perturbations. The top panel shows the height of the perturbation, δ ( x , t ) = h ( x , t ) − h , asa function of position for annealing times 0 ≤ t ≤
60 min for samples with (a) h =
221 nm, (b) h =
681 nm, and (c) h =
681 nm. Thebottom row shows the height of the perturbation scaled by the height at x = U = XT − / = x ( η / h γ t ) / . For comparison,we also plot the rescaled self-similar attractor (see Eq. (7)) which is shown as a black dashed line in the bottom row. To test this prediction we plot the normalized height ofthe perturbation as a function of the variable U = XT − / = x ( η / h γ t ) / , as shown in Fig. 2 (d), (e) and (f). We observethat at late times the profiles converge to the rescaled self-similar attractor φ ( U ) / φ ( ) regardless of the initial condition,as predicted in Section 1.2. Here, we emphasize that since wehave determined the capillary velocity in situ by measuringthe evolution of a stepped bilayer geometry near the perturba-tion on each sample, there is no free parameter in the aboverescaling and comparison to the theoretical prediction (shownas a dashed black line in Fig. 2 (d), (e) and (f)). Furthermore,at late times, the error between the experimentally measuredprofiles and the attractor is less than 1%. Measurement of the samples with large perturbations (see ex-ample in Fig. 3) were more challenging because at long an-nealing times ( t >
100 min) the lateral extent of the heightprofiles exceeds the accessible range of the AFM ( ∼ µ m).Here, we resort to imaging ellipsometry (Accurion, EP3) to record height profiles. Imaging ellipsometry (IE) has ∼ nmheight resolution with lateral resolution comparable to an op-tical microscope: ∼ µ m. Thus, IE and AFM are complimen-tary techniques. For the example in Fig. 3, data was acquiredwith AFM for t ≤
63 min, while IE was used for the threelongest annealing times. With IE there is one caveat: in certainranges of thickness there is a loss of sensitivity depending onthe wavelength of laser light and the angle of incidence used(658 nm, 42-50 deg). † For the IE data the regions where theIE was insensitive were interpolated with a quadratic spline asindicated by dashed lines to guide the eye.The evolution of a large perturbation is shown in Fig. 3.In this case, the perturbation does not obey the condition δ ( , ) / h (cid:28)
1. As can be seen in Fig. 3(a), with sufficientannealing, the large perturbations become symmetric. Similarto the evolution observed for the small perturbations, once theprofiles are symmetric, the maximal height δ ( , t ) decreaseswith further annealing and the profiles broaden. † This issue can be circumvented by varying the angle of incidence. How-ever this was not possible for the experiments presented here because chang-ing angle of incidence also shifts the region of interest slightly.
50 0 50050100150200250300350400 x [ µ m ] δ ( x , t ) [ n m ] (a) −30 −20 −10 0 10 20 3000.511.5 U δ ( x , t ) / δ ( , t ) (b) φ ( U ) /φ (0) Figure 3
An example data set with large perturbation. (a) Height ofthe perturbation as a function of position and annealing time with h =
216 nm; (b) the normalized profiles. For times t ≥
303 min,profiles were measured using imaging ellipsometry (IE). Regionswhere IE is insensitive have been interpolated with quadratic splinesas indicated by the dashed lines. The black dashed line correspondsto the rescaled self-similar attractor (see Eq. (7)).
The normalized profiles are shown in Fig. 3(b). Althoughthe perturbations are initially large, upon long enough anneal-ing the condition δ ( , t ) (cid:28) h can be reached. In particular,the final state of a large perturbation is still expected to be theself-similar attractor. For the data shown in Fig. 3, even after11022 min of annealing, the profile has not reached the con-dition that δ ( , t ) (cid:28) h . While the height profiles are clearlysymmetric at long times, and they are converging towards self-similarity, the final profile is not yet equivalent to the final at-tractor of Fig. 2. The fact that the sample has not yet fullyreached the self-similar attractor is simply because the start-ing profile was so tall, that the long annealing times requiredand the width of the profile (while still requiring good heightresolution) place this outside our experimental window. −1 −0.4 −0.3 −0.2 −0.1 t [min] δ ( , t ) / δ ( , ) -1/4 t c Figure 4
Central height of the small perturbation shown in Fig. 2(a)and (d) normalized by its initial value as a function of time. Thehorizontal dashed line represents the initial value. A power law of t / is fit to the late time data. In accordance with Eq. (10), theconvergence time is defined as the intersection of these two regimes,as indicated by the vertical dashed line. One of the main predictions of the theory outlined in Sec-tion 1.3 is that the time taken to converge to the attractor de-pends on the algebraic volume of the perturbation accordingto Eq. (11). The convergence time is determined in accor-dance with Eq. (10) as the crossover from an initial regime,which is highly dependent on δ ( x , ) , to a universal interme-diate asymptotic regime. In Fig. 4, we plot the normalizedcentral height of the perturbation, δ ( , t ) / δ ( , ) , for the smallperturbation shown in Fig. 2(a) and (d). The initial state canbe characterized by the central height of the perturbation at t = δ ( , t ) / δ ( , ) decreases in time following the t − / power law. Note that the t − / line is fit to the last three data points which correspond tothe latest profiles shown in Fig. 2(d). These three profiles arein excellent agreement with the calculated asymptotic profile(see black dashed line in Fig. 2(d)). The crossover from theinitial regime to the intermediate asymptotic regime shown inFig. 4 gives the experimentally determined convergence time, t c . From t c , the non-dimensionalized convergence time, T c ,can be obtained.The theory predicts a very clear dependence of the dimen-sionless convergence time, T c , on M / ∆ ( ) , a measure of thedimensionless width of the initial profile (see Eq. (11)). InFig. 5 is plotted the dimensionless convergence time obtainedas in Fig. 4 as a function of M / ∆ ( ) , for seven small per-turbations, as well as four large perturbations. For small per-turbations, we observe excellent agreement between experi-5 −1 M / ∆ (0) T c Eq. (11)small perturbationslarge peturbations Figure 5
Non-dimensionalized convergence time as a function ofnon-dimensionalized width. Here the square data points representdata from small perturbation samples which are in excellentagreement with the dashed black line. The dashed black line is thetheoretical prediction of Eq. 11. In the inset both the smallperturbation results, which have reached the self-similar regime, andthe large perturbation data (circles and arrows) which are not yetself-similar are shown. The large perturbation data provides only alower-bound for T c , which is why that data falls below the predictedline. ments and the theoretical prediction of Eq. (11) with no fittingparameters. We also show the convergence time for the largeperturbation data (see inset of Fig. 5). However, since the largeperturbations have not fully reached the intermediate asymp-totic regime, the T c one obtains by forcing a t − / power lawthrough the latest data point corresponds to a lower bound. Forthis reason, the data points provided are shown with verticalarrows. Conclusions
We have studied, both with theory and experiment, thecapillary-driven levelling of an arbitrary surface perturbationon a thin liquid film. Using atomic force microscopy andimaging ellipsometry we follow the evolution of the perturba-tions and compare the results to the theoretical predictions ofthe two-dimensional capillary-driven thin film equation. Wehave shown that regardless of the initial condition, the per-turbations converge to a universal self-similar attractor that isgiven by the Green’s function of the linear thin film equation.Furthermore, we have shown that the time taken to convergeto the attractor depends on the volume of the perturbation. Wemeasured the convergence time for both small and large per-turbations and found good agreement between theory and ex- periment. Specifically, the experimental results are consistentwith the theory over two orders of magnitude in the dimen-sionless typical width of the initial profile and six orders ofmagnitude in dimensionless convergence time, with no freeparameter.
Acknowledgements
The financial support by NSERC of Canada and ´Ecole Nor-male Sup´erieure of Paris, are gratefully acknowledged. Theauthors also thank M. Ilton, J. D. McGraw, M. Backholm, O.B¨aumchen, and H. A. Stone for fruitful discussions, as well asEtienne Rapha¨el for the cover artwork.
Appendix
We here wish to calculate the integral in Eq. (7) in terms ofHypergeometric functions. Performing a Taylor expansion ofthe integrand yields: φ ( U ) = π ∞ ∑ k = ( iU ) k k ! (cid:18) (cid:90) d Q Q k e − Q (cid:19) . (12)At this stage one can see that all terms corresponding to anodd k = p + f is real.Furthermore, changing the variables through S = Q leads to: φ ( U ) = π ∞ ∑ p = ( iU ) p ( p ) ! (cid:90) ∞ d S S ( + p ) / − e − S , (13)where we recognise a Γ function: φ ( U ) = π ∞ ∑ p = ( iU ) p ( p ) ! Γ (cid:18) + p (cid:19) . (14)Then, separating the sum over p in even p = m and odd p = m + φ ( U ) = π ∞ ∑ m = U m ( m ) ! Γ (cid:18) m + (cid:19) − U π ∞ ∑ m = U m ( m + ) ! Γ (cid:18) m + (cid:19) . (15)Developing the Γ functions in terms of Pochhammer risingfactorials Γ ( m + α ) = Γ ( α )( α ) m where ( α ) m = α × ( α + ) × ... × ( α + m − ) , and using the relation Γ ( α + ) = α Γ ( α ) yields: φ ( U ) = π Γ (cid:18) (cid:19) ∞ ∑ m = U m ( m ) ! (cid:18) (cid:19) m − U π Γ (cid:18) (cid:19) ∞ ∑ m = U m ( m + ) ! (cid:18) (cid:19) m . (16)6eveloping the factorials and rising factorials and proving bymathematical induction that:4 m × × ... × ( + ( m − )) m × ( m − ) × ... × = m ! 1 (cid:0) (cid:1) m (cid:0) (cid:1) m , (17)and that:4 m × × ... × ( + ( m − ))( m + )( m + ) × ... × =
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