Relaxation and Intermediate Asymptotics of a Rectangular Trench in a Viscous Film
Oliver Bäumchen, Michael Benzaquen, Thomas Salez, Joshua D. McGraw, Matilda Backholm, Paul Fowler, Elie Raphaël, Kari Dalnoki-Veress
RRelaxation and Intermediate Asymptotics of a Rectangular Trench in a Viscous Film
Oliver B¨aumchen, ∗ Michael Benzaquen, ∗ Thomas Salez, Joshua D. McGraw, † Matilda Backholm, Paul Fowler, Elie Rapha¨el, and Kari Dalnoki-Veress
1, 2, ‡ Department of Physics & Astronomy and the Brockhouse Institutefor Materials Research, McMaster University, Hamilton, Canada Laboratoire de Physico-Chimie Th´eorique, UMR CNRS Gulliver 7083, ESPCI, Paris, France (Dated: November 20, 2018)The surface of a thin liquid film with nonconstant curvature flattens as a result of capillary forces.While this leveling is driven by local curvature gradients, the global boundary conditions greatlyinfluence the dynamics. Here, we study the evolution of rectangular trenches in a polystyrenenanofilm. Initially, when the two sides of a trench are well separated, the asymmetric boundarycondition given by the step height controls the dynamics. In this case, the evolution results from theleveling of two noninteracting steps. As the steps broaden further and start to interact, the globalsymmetric boundary condition alters the leveling dynamics. We report on full agreement betweentheory and experiments for: the capillary-driven flow and resulting time dependent height profiles;a crossover in the power-law dependence of the viscous energy dissipation as a function of time asthe trench evolution transitions from two noninteracting to interacting steps; and the convergenceof the profiles to a universal self-similar attractor that is given by the Green’s function of the linearoperator describing the dimensionless linearized thin film equation.
PACS numbers: 47.15.gm, 47.55.nb, 47.85.mf, 66.20.-d, 83.80.Sg
Thin films are prevalent in applications such as lubri-cants, coatings for optical and electronic devices, andnanolithography to name just a few. However, it isalso known that the mobility of polymers can be alteredin thin films [1–5]. Thus, understanding the dynamicsof thin films in their liquid state is essential to gain-ing control of pattern formation and relaxation on thenanoscale [6, 7]. For example, high-density data storagein thin polymer films is possible by locally modifying asurface with a large 2-D array of atomic force microscopeprobes [8]. This application relies on control of the timescales of the flow created by a surface profile to produceor erase a given surface pattern. In contrast to this tech-nologically spectacular example is the surface of freshlyapplied paint which relies on the dynamics of leveling toprovide a lustrous surface.Much has been learned about the physics of thin filmsfrom dewetting, where an initially flat film exposes thesubstrate surface to reduce the free energy of the sys-tem [9–18]. Other approaches, for example studying theevolution of surface profiles originating from capillarywaves [19], embedding of nanoparticles [5], or those cre-ated by an external electric field [20, 21] have also beenutilized to explore mobility in thin films. Although thecapillary-driven leveling of a nonflat surface topographyin a thin liquid film has been reported in various stud-ies [22, 23], experiments with high resolution comparedto a general theory that relates time scales to spatialgeometries and properties of the liquid are still lacking.Recently, we studied the leveling of a stepped film : a newnanofluidic tool to study the properties of polymers inthin film geometries [24–28].In this work, we study the leveling of perfect trenches in thin liquid polymer films as shown in Fig. 1. A trench ah d Si PS a xy z xy ( a ) ( b ) Monday, June 24, 13
FIG. 1. (a) Schematic of the initial geometry of a polystyrenefilm on a silicon wafer (not to scale: 2 a is a few µ m’s, while h and d ∼
100 nm). (b) Atomic force microscopy image ofa typical trench (image width 30 µ m, height range 115 nm). is composed of two opposing steps separated by a dis-tance 2 a . This structure evolves because of gradientsin the Laplace pressure resulting from gradients in thecurvature. Initially, the trench levels as two noninteract-ing steps, with an asymmetric global boundary conditionset by the height h of the film at the top of the step,and the height d of the film at the bottom of the step.However, as the steps broaden they interact with one an-other, which results in an overall symmetric boundarycondition, with h on both sides. As will be shown, thecrossover from asymmetric to symmetric boundary con-ditions modifies the scaling law for the energy dissipationin the film. This striking feature reveals the fundamen-tal role played by the boundary conditions in the evo-lution of global quantities of the system. Furthermore,the trench geometry provides an ideal illustration of thetheory of intermediate asymptotics [29]: a key tool un-derlying historical examples such as the Reynolds dragforce [30], Kelvin’s nuclear explosion [31, 32], and scaling a r X i v : . [ c ond - m a t . s o f t ] S e p (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) x [ µ m] h − h [ n m ] a) initial5 min10 min20 min40 min100 min160 min (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) x [ µ m] h − h [ n m ] b) initial5 min20 min40 min80 min160 min220 min (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) x/t [ µ m/min ] ( x − x ) /t [ µ m/min ] H c) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) x/t [ µ m/min ] H d) FIG. 2. (top row) Experimental surface profiles at various times (5 ≤ t ≤
220 min). The initial rectangular trenches are shownwith dashed lines and exhibit widths of (a) a = 5.3 µ m and (b) a = 1.6 µ m. (bottom row) The normalized height data of (a)and (b) are rescaled in accordance with the scaling predictions of the theory. The right panel of (c) shows the data shifted toillustrate the collapse of the self-similar profiles, where x ( t ) is a horizontal shift for each right front. Profiles in (a) and (c)show the approach of two noninteracting steps, whereas in (b) and (d) the steps are interacting and the trench depth decreases. theory in general, in a situation that is highly relevant forindustry and fundamental nanorheology. Intermediateasymptotics theory is based on self-similarity, addressescomplex nonlinear partial differential equations, and pro-vides solutions at intermediate times – a time range farenough from the initial state so that details of the ini-tial condition can be forgotten, and far enough from thetrivial equilibrium state. The value of those intermediatesolutions is thus to bring a certain generality by offeringan alternative to the principle of superposition, whichis lacking in nonlinear physics. In the work presentedhere, the profile does not evolve self-similarly for earlytimes. However, after some transient period it becomesself-similar and converges to a universal attractor thatdepends on the boundary condition: the intermediateasymptotic solution [33].Polymer films exhibiting rectangular trench geome-tries, as illustrated in Fig. 1, were prepared as follows.Polystyrene (PS) films were spincast from a toluene(Fisher Scientific, Optima grade) solution onto freshlycleaved mica sheets (Ted Pella Inc.). The PS has molec-ular weight M w = 31 . ∼ − mbar) for 24 hrs at130 ◦ C. This temperature is well above the glass tran-sition temperature ( T g ≈ ◦ C) and ensures removalof residual solvent and relaxation of the polymer chains.Films were then floated onto the surface of ultrapure wa-ter (18.2 MΩcm, Pall, Cascada LS) and picked up with1 cm × µ m long and only a few µ m wide. Thus, the secondtransfer creates a rectangular trench such that h = 2 d ,see Fig. 1. The edges of the trench were checked withatomic force microscopy (AFM, Veeco Caliber) and op-tical microscopy to ensure that there were no defects inthe trench and the vicinity. We stress that the samplesare prepared at room temperature, well below T g , andonly flow when heated above T g .In all cases studied, the width of the trench was con-stant and much smaller than its length. The problem canthus be safely treated as invariant in the y -coordinatealong the length of the trench. Prior to each measure-ment, the initial condition was recorded using AFM, asshown in Figs. 2(a) and 2(b): the width of the trench 2 a ,as well as the depth h − d was determined from an anal-ysis of height profiles. Independently, thicknesses weremeasured with AFM from float gaps or a small scratchmade through the film to the substrate. Samples wereannealed in ambient conditions [34] at 140 ◦ C on a hotstage (Linkam) with a heating rate of 90 ◦ C/min. Heightprofiles z = h ( x, t ) representing the vertical distance be-tween the substrate-liquid and liquid-air interfaces at po-sition x of the trenches were recorded using AFM aftervarious times t , following a quench to room temperature.Figures 2(a) and 2(b) show the evolution of rectangulartrenches for h = 2 d = 206 nm, with a = 5 . µ m and a = 1 . µ m respectively, as a function of the annealingtime t . We deliberately used two different initial widthsin order to address the two different temporal regimes ina single rescaled description, as detailed below. As canbe seen in Fig. 2(a), the profiles broaden and are accom-panied by a bump at the top of the step and a dip atthe bottom of the step that are characteristic of isolatedsteps [25]. Initially, both sides of the trench in Fig. 2(a)level independently of each other until t ∼
100 min, whenboth dips merge into a single minimum. The subsequentstage is characterized by the decreasing depth of the pro-file as shown in Fig. 2(b) while the profile continues tobroaden.The relaxation process of the surface can be describedby considering capillary driving forces originating fromthe nonconstant curvature of the surface [35]: Gradientsin the Laplace pressure provide a driving force for theleveling of the surface topography. The Laplace pressureis given by p ( x, t ) ≈ − γ∂ x h , where γ is the air-liquid sur-face tension. The height scales in this study were chosento be small enough that gravitational forces can be ne-glected [36], but sufficiently large to neglect disjoiningforces [37]. Moreover, we can safely exclude phenom-ena related to the polymer chain size, e.g. confinementeffects, as the film thicknesses are much larger than char-acteristic polymer chain length scales. The timescale ofthe longest relaxation time of PS (31.8 kg/mol) at 140 ◦ Cis orders of magnitude shorter than the time scales con-sidered here [38], thus we can treat the film as a sim-ple Newtonian liquid. The theoretical description of theproblem is thus based on the Stokes equation for highlyviscous flows, and the lubrication approximation whichstates that all vertical length scales are small compared tohorizontal ones [39–42]. A no-slip boundary condition atthe liquid-substrate interface and the no-stress boundarycondition at the liquid-air interface result in the familiarPoiseuille flow along the z axis. Invoking conservation ofvolume leads to the capillary-driven thin film equation(TFE): ∂ t h + γ η ∂ x (cid:0) h ∂ x h (cid:1) = 0 , (1)where η is the viscosity of the film [39–41]. The ratio γ/η provides the typical speed of leveling and is termed the capillary velocity . Equation (1) is highly nonlinear andhas no known general analytical solution. Nevertheless,it can be solved numerically using a finite difference al-gorithm [27, 43]. Equation (1) can also be linearized andsolved analytically in the particular situation where thetrench is a perturbation: h ≈ d [26, 33]. In this linearcase, we recently showed that the solutions converge toa self-similar attractor [33]: h ( x, t ) −−−→ t →∞ h + 2 ( d − h ) G ( U, T ) , (2) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) H a)
80 minNumerical calculation (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) ! ηγh " x/t H b) FIG. 3. Normalized height profiles for experiments (solid line)annealed at 140 ◦ C and best-fit theory (dashed lines). (a) Atrench with a = 1.6 µ m, h = 206 nm, d = 103 nm after an-nealing for t = 80 min, with corresponding numerical solutionto Eq. 1. (b) A trench where the depth is a small perturbation( a = 2.5 µ m, h = 611 nm, d = 59 nm) at 140 ◦ C. The dashedline is the analytical Green’s function of the linear operatordescribing the linearised thin film equation. where we introduced the two dimensionless variables: U = xt / (cid:18) ηγh (cid:19) / (3) T = γh t ηa , (4)as well as the Green’s function of the linear operator de-scribing the dimensionless linearized thin film equation: G ( U, T ) = 12 π T / (cid:90) ∞−∞ dQ e − Q e iQU . (5)Thus, the experimental profiles should collapse at longtimes when the vertical axis is normalised by the depthof the trench, H = [ h ( x, t ) − h ] / [ h − h (0 , t )], and thehorizontal axis is rescaled as x/t / . We checked numer-ically that this statement is still true in the nonlinearcase of Eq. (1) [33]. The self-similar regime representsthe intermediate asymptotics solution of this thin filmproblem [29].We first consider the nonlinear situation, where h =2 d . In Figs. 2(c) and 2(d) we show the rescaled datacorresponding to the noninteracting (a) and interacting(b) regimes of the trench evolution. Excellent agreementwith the theoretical predictions is obtained. The nonin-teracting steps all have the same profile in their frameof reference. This is demonstrated by shifting the dataas shown in the right side of Fig. 2(c), where x ( t ) is ahorizontal shift for each right front. At long times, self-similarity in x/t / is obtained for the interacting steps. - 1/4 h t ⌘a a L / h Friday, June 21, 13
FIG. 4. Excess contour length ∆ L of rectangular trenches(half-width a , reference film height h ) in nondimensional-ized representation as a function of the nondimensionalizedtime. Experimental data (open circles, rectangles and trian-gles) and nonlinear numerical calculations (solid circles) bothshow a transition from a − / − / Thus, there are two distinct regimes: first, an initial stagewhere the steps broaden, and are self-similar in theirframe of reference, but are not yet interacting; second, acrossover to a final self-similar stage where the steps havemerged and the depth of the profile diminishes. Thesetwo distinct stages correspond to the crossover of the sys-tem from the asymmetric boundary condition of the non-interacting steps to the symmetric global boundary con-dition of the profile. As discussed, it is possible to numer-ically solve Eq. (1) for the trench geometry [27, 43]. Theexperimental data are in excellent agreement with non-linear calculations as shown in Fig. 3(a) for t = 80 min.We note that this agreement is typical and obtained forall such comparisons. In the self-similar representationof the data, only a lateral stretch is required to matchthe experimental data and the calculation. According toEq. (1), the stretching factor is directly related to thecapillary velocity and we obtain η/γ = 0 .
034 min/ µ mfor PS (31.8 kg/mol) at 140 ◦ C. This value is in excellentagreement with literature values [28, 44] and consistentwith the capillary velocity that can be determined fromthe noninteracting steps according to the technique de-scribed in [25].With the theoretical tools described above, we are ableto analyze the entire evolution from noninteracting tointeracting steps. A relevant quantity that can be ex-tracted from the data is the excess contour length ∆ L asa function of time. For small slopes, one has:∆ L ≈ (cid:90) dx
12 ( ∂ x h ) . (6)From the specific dimensional invariance of Eq. (1), one can show that: a ∆ L h = (cid:18) γh t ηa (cid:19) − / f (cid:18) γh t ηa , h − d h (cid:19) , (7)where f is a function of two variables. This intrinsicsimilarity comes from the fact that any initial profile h λ ( λx,
0) = h ( x,
0) obtained from a horizontal stretchof factor λ yields the evolution h λ ( λx, λ t ) = h ( x, t )through Eq. (1). Thus, the excess contour length evo-lution of trenches with the same vertical aspect ratio, h /d = 2, but with different widths can be rescaled in asingle master plot as shown in Fig. 4. In accordance withthe theory, our experiments demonstrate a ∆ L ∼ t − / scaling regime for early times which corresponds to twoindependent relaxing steps [25]. Afterwards, the twosteps interfere and there is a crossover to a long-time∆ L ∼ t − / scaling regime. This observation can be un-derstood by combining Eqs. (2) and (6). Having done sowe obtain: a ∆ L h −−−→ t →∞ Γ(3 / / π (cid:18) γh t ηa (cid:19) − / (cid:18) h − d h (cid:19) , (8)which is valid in the linear case. The numerical calcula-tions shown in Fig. 4 confirm this asymptotic ∆ L ∼ t − / scaling in the nonlinear case as well. This is expectedsince any profile will eventually become a perturbationat long time. The comparison of the data for threetrenches with different widths and the numerical calcu-lations shown in Fig. 4 clearly validates the theoreticalpredictions.Finally, in order to study the intermediate asymptoticsin more detail, we realized experimentally the linear caseof a small surface perturbation: h ≈ d . This geom-etry evolves faster towards the intermediate asymptoticregime and a full analytic solution was obtained for thelinearized thin film equation [33]. The sample was pre-pared by spincoating a thick PS film directly onto aclean Si wafer, while another, much thinner, PS film wasprepared on a freshly cleaved mica sheet. The sampleannealing, and floating of the top layer to prepare thetrenches were done exactly as described for the previous h = 2 d samples. In Fig. 3(b), we show the self-similarprofiles for three different annealing times, as well asthe fit of the universal self-similar attractor given by theGreen’s function of the linearized thin film equation [33].Excellent agreement is found with the only free param-eter being the ratio: η/γ = 0 .
016 min/ µ m. This valueis again in very good agreement with expected literaturevalues [28, 44] demonstrating the validity of the interme-diate asymptotics.To conclude, we provided new insights in the relaxationof a trench at the free surface of a viscous film. The sur-face relaxes due to the nonconstant curvature of the freeinterfaces which drives flow. We found that the tran-sition from the asymmetric boundary condition of twononinteracting steps, to the symmetric boundary condi-tion of the two interacting steps greatly influences thescaling of global properties such as the energy dissipa-tion in the film. Specifically, the excess contour lengthcrosses over from a − / − / universal and should not depend on the initial profile, oron the viscous material used.The authors thank NSERC of Canada, the Ger-man Research Foundation (DFG) under grant numbersBA3406/2 and SFB 1027, the ´Ecole Normale Sup´erieureof Paris, the Fondation Langlois and the ESPCI JoliotChair for financial support. ∗ These authors contributed equally to this work. † Present address: Department of Experimental Physics,Saarland University, D-66041 Saarbr¨ucken, Germany ‡ [email protected][1] P. A. O’Connell, and G. B. McKenna, Science , 1760(2005).[2] L. Si, M. V. Massa, K. Dalnoki-Veress, H. R. Brown, andR. A. L. Jones, Phys. Rev. Lett. , 127801 (2005).[3] H. Bodiguel and C. Fretigny, Phys. Rev. Lett. , 266105(2006).[4] K. Shin, S. Obukhov, J.-T. Chen, J. Huh, Y. Hwang,S. Mok, P. Dobriyal, P. Thiyagarajan, and T. Russell,Nat Mater. , 961 (2007).[5] Z. Fakhraai and J. A. Forrest, Science , 600 (2008).[6] T. Leveder, S. Landis, and L. Davoust, Appl. Phys. Lett. , 013107 (2008).[7] J. Teisseire, A. Revaux, M. Foresti, and E. Barthel, Appl.Phys. Lett. , 013106 (2011).[8] P. Vettiger, G. Cross, M. Despont, U. Drechsler,U. D¨urig, B. Gotsmann, W. H¨aberle, M. A. Lantz, H.E. Rothuizen, R. Stutz, and G. K. Binnig, IEEE Trans-actions on Nanotechnology , 39 (2002).[9] D. J. Srolovitz and S. A. Safran, J. Appl. Phys., , 255(1986).[10] F. Brochard Wyart and J. Daillant, Can. J. of Phys. ,1084 (1990).[11] G. Reiter, Phys. Rev. Lett. , 75 (1992).[12] R. Seemann, S. Herminghaus, and K. Jacobs, Phys. Rev.Lett. , 196101 (2001).[13] R. Fetzer, K. Jacobs, A. M¨unch, B. A. Wagner, and T. P. Witelski, Phys. Rev. Lett. , 127801 (2005).[14] G. Reiter, M. Hamieh, P. Damman, S. Sclavons,S. Gabriele, T. Vilmin, and E. Rapha¨el, Nat. Mater. ,754 (2005).[15] T. Vilmin and E. Rapha¨el, Eur. Phys. J. E , 161(2006).[16] O. B¨aumchen, R. Fetzer, and K. Jacobs, Phys. Rev. Lett. , 247801 (2009).[17] J. H. Snoeijer and J. Eggers, Phys. Rev. E , 056314(2010).[18] O. B¨aumchen, R. Fetzer, M. Klos, M. Lessel, L. Mar-quant, H. H¨ahl, and K. Jacobs, J. Phys.-Condens. Mat. , 325102 (2012).[19] Z. Yang, Y. Fujii, F. K. Lee, C.-H. Lam, and O. K.C. Tsui, Science , 1676 (2010).[20] D. R. Barbero and U. Steiner, Phys. Rev. Lett. ,248303 (2009).[21] F. Closa, F. Ziebert, and E. Rapha¨el, Phys. Rev. E ,051603 (2011).[22] E. Rognin, S. Landis, and L. Davoust, Phys. Rev. E ,041805 (2011).[23] E. Rognin, S. Landis, and L. Davoust, J. Vac. Sci. Tech-nol., B , 011602 (2012).[24] J. D. McGraw, N. M. Jago, and K. Dalnoki-Veress, SoftMatter , 7832 (2011).[25] J. D. McGraw, T. Salez, O. B¨aumchen, E. Rapha¨el, andK. Dalnoki-Veress, Phys. Rev. Lett. , 128303 (2012).[26] T. Salez, J. D. McGraw, O. B¨aumchen, K. Dalnoki-Veress, and E. Rapha¨el, Phys. Fluids , 102111 (2012).[27] T. Salez, J. D. McGraw, S. L. Cormier, O. B¨aumchen,K. Dalnoki-Veress, and E. Rapha¨el, Eur. Phys. J. E ,114 (2012).[28] J. D. McGraw, T. Salez, O. B¨aumchen, E. Rapha¨el, andK. Dalnoki-Veress, Soft Matter , 8297 (2013).[29] G. I. Barenblatt Scaling, Self-Similarity, and Interme-diate Asymptotics (Cambridge University Press, Cam-bridge, 1996).[30] O. Reynolds, Phil. Trans. Roy. Soc. London,
159 (1950).[32] G. I. Taylor, Proc. Roy. Soc.,
175 (1950).[33] M. Benzaquen, T. Salez, and E. Rapha¨el, Eur. Phys. J.E , 82 (2013).[34] We have previously verified that for these experimentsthere is no difference in annealing in an ambient or inertenvironment.[35] P.-G. de Gennes, F. Brochard-Wyart, and D. Qu´er´e, Cap-illarity and Wetting Phenomena: Drops, Bubbles, Pearls,Waves (Springer, 2003).[36] H. Huppert, J. Fluid Mech. , 43 (1982).[37] R. Seemann, S. Herminghaus, and K. Jacobs, Phys. Rev.Lett. , 5534 (2001).[38] A. Bach, K. Almdal, H. Rasmussen, and O. Hassager,Macromolecules , 5174 (2003).[39] L. G. Stillwagon and R. Larson, Journal of AppliedPhysics , 5251 (1988).[40] A. Oron, S. Davis, and S. Bankoff, Rev. Mod. Phys. ,931 (1997).[41] R. Craster and O. Matar, Rev. Mod. Phys. , 1131(2009).[42] R. Blossey, Thin Liquid Films . ISBN 9789400744547,Springer, Dordrecht, (2012).[43] A. Bertozzi, Notices Amer. Math. Soc. , 689 (1998).[44] S. Wu, Polymer Handbook , vol. 4 Wiley-Interscience,, vol. 4 Wiley-Interscience,