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Pattern Formation and Dewetting in Thin Films of Liquids Showing Complete Macroscale Wetting: From “Pancakes” to “Swiss Cheese” Ashutosh Sharma* and Ruhi Verma Department of Chemical Engineering, Indian Institute of Technology, Kanpur, Kanpur 208 016, India Received May 29, 2004. In Final Form: July 31, 2004 Based on the complete 3D numerical solutions of the nonlinear thin film equation, we address the problems of surface instability, dynamics, morphological diversity and evolution in unstable thin films of the liquids that display complete macroscale wetting. The twin constraints of complete macroscale wettability and nanoscale instability produce a variety of microscopic morphological phases approximating sharp crystal surfaces with flat tops resembling a mesa or a micro “pancake” or a slice of Swiss cheese. While the maximum thickness of flat regions is found to be independent of the initial film thickness, the precise lateral morphology of microdomains formed depends on the film thickness. As the film thickness is increased, the initial pathway of evolution changes from the formation of small spherical droplets, to long mesas (parapets) and islands, to circular holes, all of which eventually resolve by ripening into a collection of round pancakes at equilibrium. However, beyond a certain transition thickness, a novel metastable honeycombed morphology, resembling a membrane or a slice of Swiss cheese, is uncovered, which is produced by an abrupt “freezing” of the evolution during hole growth. In contrast, the spinodal dewetting in thin films of partially wettable systems always engenders spherical droplets at equilibrium. The equilibrium dewetted area from simulations, as well as from simple mass balance, is shown to decline linearly with the initial film thickness.
Introduction The problem of spontaneous pattern formation in thin (∼10-100 nm) liquid films due to the intermolecular interactions has attracted much recent attention, both theoretically1-17 and experimentally.6,16-31 In addition to their many technological applications (e.g., coatings), thin * To whom correspondence should be addressed. E-mail:
[email protected]. (1) Vrij, A. Discuss Faraday Soc. 1966, 42, 23. (2) Ruckenstein, E.; Jain, R. K. J. Chem. Soc., Faraday Trans. 2 1974, 70, 132. (3) Brochard-Wyart, F.; Daillant, J. Can. J. Phys. 1991, 68, 1984. (4) Sharma, A. Langmuir 1993, 9, 861. (5) Sharma, A.; Jameel, A. T. J. Colloid Interface Sci. 1993, 161, 190. (6) Sharma, A.; Reiter, G. J. Colloid Interface Sci. 1996, 178, 383. (7) Khanna, R.; Sharma, A. J. Colloid Interface Sci. 1997, 195, 42. (8) Sharma, A.; Khanna, R. Phys. Rev. Lett. 1998, 81, 3463. (9) Sharma, A.; Khanna, R. J. Chem. Phys. 1999, 110, 4929. (10) Sharma, A.; Singh, J. J. Adhes. Sci. Technol. 2000, 14, 145. (11) Oron, A. Phys. Rev. Lett. 2000, 85, 2108. (12) Thiele, U.; Velarde, M. G.; Neuffer, K. Phys. Rev. Lett. 2001, 87, 016104. (13) Besterhorn, M.; Neuffer, K. Phys. Rev. Lett. 2001, 87, 046101. (14) Thiele, U.; Neuffer, K.; Pomeau, Y.; Velarde, M. G. Colloids Surf., A 2001, 206, 135. (15) Warner, M. R. E.; Craster, R. V.; Matar, O. K. Phys. Fluids 2002, 14, 4040. (16) Sharma, A. Eur. Phys. J. E 2003, 12, 397. (17) Warner, M. R. E.; Craster, R. V.; Matar, O. K. J. Colloid Interface Sci. 2003, 268, 448. (18) Reiter, G. Phys. Rev. Lett. 1992, 68, 75. (19) Reiter, G. Langmuir 1993, 9, 1344. (20) Zhao, W. M.; Sokolov, H. J.; Fetters, L. J.; Plano, R.; Sanyal, M. K.; Sinha, S. K.; Sauer, B. B. Phys. Rev. Lett. 1993, 70, 1453. (21) Bischof, J.; Scherer, D.; Herminghaus, S.; Leiderer, P. Phys. Rev. Lett. 1996, 77, 1536. (22) Jacobs, K.; Herminghaus, S.; Mecke, K. R. Langmuir 1998, 14, 965. (23) Xie, R.; Karim, A.; Douglas, J. F.; Han, C. C.; Weiss, R. A. Phys. Rev. Lett. 1998, 81, 1251. (24) Thiele, U.; Mertig, M.; Pompe, W. Phys. Rev. Lett. 1998, 80, 2869. (25) Herminghaus, S.; Jacobs, K.; Mecke, K.; Bischof, J.; Fery, A.; Ibn-Elhaj, M.; Schlagowski, S. Science 1998, 282, 916.
films are intimately involved in a spectrum of nanoscale phenomena, for example, wetting, phase separation and mobility in confined geometries, biomembrane instability, drying, heterogeneous nucleation, and so forth. Further, unstable thin films are (a) a convenient prototype for the understanding of patterns in nonlinear spinodal processes in general, (b) a tool for determination of intermolecular forces based on facile observations of patterns, and (c) of potential use in engineering of regular nanoscale structures and thin membranes/coatings of controlled porosity. In particular, recent soft-lithographic techniques such as directed dewetting on physically or chemically patterned templates32-41 and application of electric-field42,43 and capillary-flow or confined film lithographies42-46 all make (26) Reiter, G.; Sharma, A.; David, M.-O.; Casoli, A.; Khanna, R.; Auroy, P. Langmuir 1999, 15, 2551. (27) Reiter, G.; Khanna, R.; Sharma, A. Phys. Rev. Lett. 2000, 85, 1432. (28) Higgins, A. M.; Jones, R. A. L. Nature 2000, 404, 477. (29) Khanna, R.; Sharma, A.; Reiter, G. Eur. Phys. J. direct E 2000, 2, 1. (30) Seemann, S.; Herminghaus, S.; Jacobs, K. Phys. Rev. Lett. 2001, 86, 5534. (31) Becker, J.; Gru¨n, G.; Seemann, R.; Mantz, H.; Jacobs, K.; Mecke, K. R.; Blossey, R. Nat. Mater. 2003, 2, 59. (32) Konnur, R.; Kargupta, K.; Sharma, A. Phys. Rev. Lett. 2000, 84, 931. (33) Kargupta, K.; Sharma, A. Phys. Rev. Lett. 2001, 86, 4536. (34) Sehgal, A.; Ferreiro, V.; Douglas, J. F.; Amis, E. J.; Karim, A. Langmuir 2002, 18, 7041. (35) Kargupta, K.; Sharma, A. J. Colloid Interface Sci. 2002, 245, 99. (36) Bru¨sch, L.; Ku¨hne, H.; Thiele, U.; Ba¨r, M. Phys. Rev. E 2002, 66, 011602. (37) Kargupta, K.; Sharma, A. Langmuir 2002, 18, 1893. (38) Zhang, Z.; Wang, Z.; Xing, R.; Han, Y. Polymer 2003, 44, 3737. (39) Kargupta, K.; Sharma, A. Langmuir 2003, 19, 5153. (40) Luo, C. X.; Xing, R. B.; Zhang, Z. X.; et al. J. Colloid Interface Sci. 2004, 269, 158. (41) Luo, C. X.; Xing, R. B.; Han, Y. C. Surf. Sci. 2004, 552, 139. (42) Schaffer, E.; Thurn-Albrecht, T.; Russell, T. P.; Steiner, U. Nature 2000, 403, 874. (43) Morariu, M. D.; Voicu, N. E.; Schaffer, E.; Lin, Z.; Russell, T. P.; Steiner, U. Nat. Mater. 2003, 2, 48.
10.1021/la048669x CCC: $27.50 © 2004 American Chemical Society Published on Web 10/07/2004
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use of controlled instability of thin liquid films for patterning. It is now well understood1-17,47 that the free surface of a thin film is unstable and deforms spontaneously whenever the spinodal parameter, defined as the second derivative of the excess free energy (per unit area) due to the sum of intermolecular interactions, is negative. A competition between the antagonistic (attractive-repulsive) long-range (e.g., van der Waals) and the short-range (e.g., entropic effects due to adsorption and grafting) forces creates at least one window of instability, over a range of film thickness, where the spinodal parameter is negative.1-31,47 The theory of nonlinear spinodal patterns in the unstable range has largely been resolved5,7-17,31 for a class of thin films made of fluids which show partial macroscopic wetting on homogeneous substrates; that is, macroscopic droplets of such fluids do not spread out completely on the substrate but show a finite macroscale equilibrium contact angle. Nonlinear 3D simulations for such systems have shown7-14 two distinct pathways of morphological evolution for the surface instability leading to dewetting and, eventually, to the formation of disjointed spherical droplets. Relatively thin films dewet by the formation of a bicontinuous pattern which directly transforms into droplets, whereas relatively thick films first dewet by the formation of isolated circular holes which grow and coalesce to form liquid ridges, which then further fragment/retract to form droplets. A slow ripening/ coalescence leads to fewer and bigger droplets. Most importantly, the equilibrium morphology of partially wetting systems is always a set of spherical droplets.7-14 The objective of this paper is to address the problem of morphological evolution in an equally fascinating but distinct class of unstable thin film systems which display perfect macroscale wetting due to repulsive long-range van der Waals force, but their thin films are nonetheless rendered unstable by the short-range attractive forces.48-63 Put in a different jargon, the total spreading coefficient and its apolar van der Waals component are positive, but the non-van der Waals polar component is slightly negative. The total free energy (per unit area) maintains its positive definite character at all thicknesses, implying (44) Chabinyc, M. L.; Wong, W. S.; Salleo, A.; Paul, K. E.; Street, R. A. Appl. Phys. Lett. 2002, 81, 4260. (45) Kim, Y. H.; Lee, H. H. Adv. Mater. 2003, 15, 332. (46) Harkema, S.; Schaffer, E.; Morariu, M. D.; Steiner, U. Langmuir 2003, 19, 9714. (47) Sharma, A.; Mittal, J. Phys. Rev. Lett. 2002, 89, 186101. (48) Joanny, J. F.; De Gennes, P. G. C. R. Acad. Sci., Ser. II 1984, 299, 279. (49) De Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (50) Brochard-Wyart, F.; Dimeglio, J.-M.; Quere, D.; De Gennes, P. G. Langmuir 1991, 7, 335. (51) Leger, L.; Joanny, J. F. Rep. Prog. Phys. 1992, 55, 431. (52) Sharma, A. Langmuir 1993, 9, 3580. (53) Ruckenstein, E. J. Colloid Interface Sci. 1996, 179, 136. (54) Riegler, H.; Asmussen, A.; Christoph, M.; Davydov, A. In Proceedings of 30th Rencontres de Moriond, Short and Long Chains at Interfaces; Daillant, J., Guenoun, P., Marques, C., Muller, P., Van, J. T. T., Eds.; Editions Frontieres: Gif-sur-Yvette, France, 1995; p 307. (55) Samid-Merzel, N.; Lipson, S. G.; Tannhauser, D. S. Physica A 1998, 257, 413. (56) Samid-Merzel, N.; Lipson, S. G.; Tannhauser, D. S. Phys. Rev. E 1998, 57, 2906. (57) Sharma, A. Langmuir 1998, 14, 4915. (58) Padmaker, A. S.; Kargupta, K.; Sharma, A. J. Chem. Phys. 1999, 110, 1735. (59) Kim, H. I.; Mate, C. M.; Hannibal, K. A.; Perry, S. S. Phys. Rev. Lett. 1999, 82, 3496. (60) Scarpulla, M. A.; Mate, C. M.; Carter, M. D. J. Chem. Phys. 2003, 118, 3368. (61) Mate, C. M. Appl. Phys. Lett. 2004, 84, 532. (62) Leizerson, I.; Lipson, S. G.; Lyushnin, A. V. Langmuir 2004, 20, 291. (63) Asmussen, A.; Riegler, H. J. Chem. Phys. 1996, 104, 8151.
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a zero macroscopic contact angle from the generalized Young-Dupre equation.50-52 However, it is the nonmonotonic decay of the free energy with thickness (due to the short-range attraction) which creates the unstable spinodal regions where the curvature of the free energy becomes positive. Clearly, macroscopic droplets of arbitrary heights cannot result from the instability-driven fragmentation of the film and subsequent ripening, since this would invalidate the macroscale considerations of perfect wettability. A clue is provided by the behavior of droplet spreading in such systems, which results in the formation of a microscopically thin circular “pancake” with a flat top, predicted first by Joanny, de Gennes, and coworkers48-51 and later interpreted in terms of the antagonistic nature of the long- and short-range apolar and polar components of the spreading pressure.4,52 Here, we address the inverse problem of the morphological evolution starting with a nonequilibrium thin flat film rather than a macroscopic drop. In particular, we explore whether an equilibrium membrane-like morphology can be engineered by the spinodal dewetting in completely wettable systems, which is not possible in partially wettable systems. Seminal experiments of Riegler et al.54,63 showed that the thin films of organic multilayers of behenic acid on silicon dioxide surfaces are unstable in a range of thickness and evolve into long flat-topped ridges or “parapets”, the (nanoscale) thickness of which is independent of the initial film thickness. Experiments of Mate and co-workers59-61 on perfluoropolyether films supported on silicon wafers show formation of islands with flat summits, rather than round droplets, in a range of film thickness. Further, measurements of Lipson and co-workers55,56,62 on evaporation of thin unstable water films on hydrophilic mica show structures and dynamics that are consistent with a potential representing perfect bulk wetting but spinodal instability in a range of film thickness.56-58 Thus, it may be proposed that these systems have the same type of potential which governs the behavior of the Joanny-de Gennes pancakes. Limitations of the linear analysis and 2D nonlinear simulations for uncovering of true 3D morphologies are already known in the context of partially wettable systems.7-14 Interestingly, as will be shown here, these limitations are even more profound for the perfectly wetting systems considered here. In what follows, we address the problem of morphological diversity and evolution for the first time based directly on the complete 3D numerical solutions of the nonlinear thin film equation in which the excess intermolecular interactions in the film are incorporated as body forces. We show that simultaneous macro- and microscale constraints produce a variety of microscopic morphological phases approximating sharp crystal surfaces. All of these structures share a common characteristic of having a flat top or resembling a mesa. Thus, such systems may be especially suitable for patterning applications requiring sharp features that are otherwise not readily obtained by processing in liquid form. A novel metastable honeycombed morphology, resembling a slice of Swiss cheese, is also uncovered, which is produced by an abrupt “freezing” of the evolution during hole growth. Among other things, we seek relationships between the morphology and the interaction potential, which is the most important structure-property correlation for thin films. This will also be an aid to rational design and interpretation of experiments. Theory We consider a thin fluid film of mean thickness h0 on a solid substrate. The free surface of such a film is unstable
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when the spinodal parameter, defined as ∂2∆G/∂h2, is negative, where h is the local film thickness. The excess free energy per unit area, ∆G, due to the sum of intermolecular interactions is composed of antagonistic (repulsive and attractive) long- and short-range components,4-10,24,52,56,64,65 which can be represented in a fairly general form as follows.
∆G ) -A/12πh2 + SP exp[(d0 - h)/lp]
(1)
When the Hamaker constant for the substrate (van der Waals component of the substrate surface tension) exceeds that of the film material, the effective Hamaker constant A is negative, signifying a long-range van der Waals repulsion that promotes film stability and spreading. This is almost always the case for aqueous films and for polymers on a majority of surfaces, for example, silicon wafers, mica, and glass, due to a relatively low surface tension of polymers. The shorter range non-van der Waals attraction (SP < 0) is engendered by a variety of effects depending on the film-substrate interactions, for example, entropic confinement effects for polymers due to adsorption/grafting at the solid-fluid interface, hydrophobic attraction for aqueous films, the presence of a nonwettable solid coating on the substrate (e.g., thin oxide layer on otherwise high energy silicon), acid-base interactions, and so forth. lp is the decay length of the non-van der Waals interaction, which is usually of the order of the radius of gyration for the adsorbed polymer, and d0 is a molecular cutoff. The generalized Young-Dupre equation50-52 gives the macroscopic contact angle, cos θ ) 1 + ∆G(he)/γ, where γ is the film surface tension and ∆G(he) is the free energy at equilibrium thickness for which ∂∆G/∂he ) 0 and ∂2∆G/ ∂he2 > 0. Clearly, materials for which the minimum of ∆G is negative display finite contact angles (cos θ < 1; partial wetting), whereas systems with positive definite ∆G show perfect macroscale wetting. In the latter case, if ∆G decays monotonically with h, spreading eventually results in a monomolecular film, which is the trivial case.28 Qualitative variation of ∆G for liquids that show perfect macroscale wetting but nanoscale nonwettability resembles either of the two nonmonotonic curves shown in Figure 1. Both of these allow a window of instability where the spinodal parameter ∂2∆G/∂h2 is negative (Figure 1C), but at the same time, ∆G is always positive (Figure 1A). The latter implies perfect macroscale wetting (zero contact angle) from the Young-Dupre equation. The only difference between the systems represented by curves 1 and 2 of Figure 1 is that the force (energy) per unit area (volume) (∂∆G/∂h) has two zero intercepts for curve 2 and none for curve 1 (Figure 1B). For the analytical representation of the potential, eq 1, curve 1 is obtained when Rp ) -SLW/SP lies between [27lp2 exp(d0/lp - 3)/d02] and [128lp2 exp(d0/lp - 4)/3d02], and curve 2 is obtained for Rp values between [4lp2 exp(d0/lp - 2)/d02] and [27lp2 exp(d0/lp - 3)/d02].4,5 The van der Waals spreading coefficient, SLW, is defined as A ) -12πd02SLW.4-9,52,65 While a reasonable analytical representation for the potential, eq 1, and the numerical values of parameters are chosen in Figure 1 for simulations and illustrations, all of the morphological features obtained here are independent of these details. Thus, a thin film governed by any potential ∆G that is always positive but qualita(64) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1992. (65) Van Oss, C. J.; Chaudhury, M. K.; Good, R. J. Chem. Rev. 1988, 88, 927.
Figure 1. Variations of the (a) free energy per unit area, ∆G (×105 for curve 1, ×104 for curve 2), (b) conjoining pressure, ∂∆G/∂h (×102 for curve 1, ×104 for curve 2), and (c) spinodal parameter, ∂2∆G/∂h2 (×10-10 for curve 1, ×10-13 for curve 2). Curves labeled 1 correspond to SLW ) 5 mJ/m2, SP ) -0.024 969 mJ/m2, and lp ) 2.5 nm, and curves labeled 2 correspond to SLW ) 5 mJ/m2, SP ) -0.415 8508 mJ/m2, and lp ) 0.6 nm. Instability occurs in the spinodal region where ∂2∆G/∂h2 < 0.
tively resembles the nonmonotonic decay shown in Figure 1A would show the same behavior. The only relevant characteristics are the qualitative shapes described by Figure 1, which result from a long-range van der Waals repulsion combined with a sufficiently weak shorter range attraction, so as to produce a positive definite nonmonotonic variation of ∆G. The following simple analysis describes why a mesa of microscopic thickness can result under these conditions. The following generalized equation of capillarity, which can also be directly obtained from eq 6 below at steadystate,52,57 describes the 2D morphology of a thin fluid domain at equilibrium.48-52,56-58
hxx(1 + hx2)-3/2 - (φ(h)/γ) + K ) 0
(2)
where γ is the surface tension of the fluid, φ(h) ) ∂∆G/∂h, x is a space coordinate parallel to the substrate surface, and subscripts are a shorthand notation for differentiation, that is, hx ) ∂h/∂x, hxx ) ∂2h/∂x2. Integration of eq 2 produces (K and C are constants of integration)
(1 + hx2)-1/2 + (∆G(h)/γ) - Kh + C ) 0
(3)
We seek a particular, but physically illuminating, solution of the type shown in Figure 2 in an unbounded domain which satisfies the following far-field conditions.
x f -∞, h ) hmin, hx ) 0, hxx ) 0 and x f ∞, h ) hmax, hx ) 0, hxx ) 0 Thus,
γK ) φ(hmax) ) φ(hmin)
(4)
and
C ) -1 - ∆G(hmax)/γ + Khmax ) -1 - ∆G(hmin)/γ + Khmin (5)
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viscosity, and ΦH ) [2πh04/|A|][∂2∆G/∂h2]). A renormalized time (in units of nm5),
tN ) t(A2/12π2µγ) ) Th05
Figure 2. Schematic representation of a 1D nanoscopic, laterally unbounded pancake with far-field boundary conditions.
The above conditions ensure the mechanical equilibrium between the two morphological phases of thicknesses hmin and hmax. Equations 4 and 5 are the equations for the unknowns hmax and hmin. Equations 4 and 5 are satisfied at the highest and the lowest points of intersection of one particular horizontal line with the φ curve in Figure 1B, in the range where three intersections are produced. The middle root, hm, corresponds to the point of maximum slope (Figure 2) where the curvature hxx vanishes (see eq 2). Thus, positive definite character of ∆G rules out formation of an equilibrium macroscopic drop (which would imply a finite contact angle), and the nonmonotonic decay of ∆G allows the formation of a nanoscopic mesa or island in which two flat films are joined by a sharp edge. Thus, hmax and hmin are easily obtained for an infinite island for any choice of the positive definite nonmonotonic potential. The complete equilibrium profile (Figure 2) is obtained by numerical integration of eq 2. While this Joanny-de Gennes pancake is only a particular solution, it does provide a simple physical meaning and estimates of the twin “morphological phases”, hmax and hmin. If one defines an equilibrium pseudo spreading coefficient, Se ) [∆G(hmin) - φ(hmin)hmin], eq 5 can be recast in a more familiar form:48-52
∆G(hmax) - φ(hmax)hmax - Se ) 0 However, Se cannot be determined independent of the potential, since it is defined by eqs 4 and 5. We can now address the fate of thin films of perfectly wetting systems in the spinodally unstable range where ∂2∆G/∂h2 is negative (Figure 1C). The morphological pathways of evolution of the surface instability and their dynamics are governed by the following thin film equation5-10 derived from the Navier-Stokes equations in which intermolecular potential is incorporated as a body force.
3µht + [h3{γ (hxx + hyy) - φ }x]x + [h3{γ (hxx + hyy) - φ}y]y ) 0 (6) where subscripts are shorthand notation for differentiation, h(x,y,t) is the local film thickness, the x-y plane coincides with the substrate, t is time, µ is viscosity, and γ is the surface tension of the film. The following nondimensional thin film equation5-10 facilitates the economy of computations and their compact presentation.
∂H/∂T + ∇‚[H3∇(∇2H)] - ∇‚[H3ΦH∇H] ) 0
(7)
where H(X,Y,T) is the nondimensional local film thickness scaled with the mean thickness h0; X,Y coordinates in the plane of the substrate are scaled with the characteristic length scale for the van der Waals case, (2πγ/|A|)1/2h02; and nondimensional time, T, is scaled with (12π2µγh05/ A2), where γ and µ refer to film surface tension and
(8)
is also defined to remove the influence of mean film thickness, when comparing results for different thickness films at the same real time. The nondimensional representation, eq 7, greatly facilitates the computations and a compact representation of results despite a large number of parameters. To calculate hmax and hmin from eqs 4 and 5, γ ) 30.8 mJ/m2 has been used. Dimensional time (or renormalized time from eq 8) is reported for µ ) 1 kg/m s. The linear stability analysis of the thin film eq 7 predicts a dominant nondimensional characteristic length scale, L, of the surface instability.1-17
L2 ) -8π2/ΦH0
(9)
where ΦH0 ) ΦH evaluated at the mean thickness, H ) 1. The most probable “unit cell” therefore has dimensions of L × L during the initial evolution of the surface instability. Numerical Methods For most of the simulations reported here, the nonlinear thin film equation, eq 7, was numerically solved over an area of (64L2) with periodic boundary conditions, starting with a small nondimensional amplitude (≈1% of initial film thickness), volume preserving random perturbation. A central finite difference scheme was used for spatial discretization, and Gear’s algorithm was used for time marching. An 80 × 80 grid was found sufficient to track the morphology when the difference between hmin and hmax was not too large (h < 10.5 nm for curve 1 of Figure 1 and h < 3 nm for curve 2 of Figure 1). Simulations with higher thickness could be resolved by a 200 × 200 grid and by reducing the solution domain (in Figure 11 shown later). The resulting set of equations were solved with the help of a banded matrix solver. Some of the simulations were repeated with higher grid densities and also with higher amplitude initial random perturbations to rule out the possible effects of these parameters on the resulting morphologies. In addition, an initially periodic perturbation also produced similar morphological results as the random perturbations. These additional simulations regarding the robustness of the results are presented in the Supporting Information. Also, the initial to intermediate time morphological evolution was found to be unaffected by the domain size, since this size is much larger than the unit cell size, L2. The long-term ripening merely induces lateral growth of flat-topped structures without the introduction of any new morphological features. Results and Discussion The free surface of a thin film governed by the potentials of the kind shown in Figure 1 is unstable when the mean film thickness is in the spinodal range which extends between two critical thicknesses (hc1 < h < hc2, where the spinodal parameter, ∂2∆G/∂h2 ) φh, is negative; Figure 1C). In addition to the zone of instability (hc1 < h < hc2), simultaneous eqs 4 and 5 predict the minimum and maximum equilibrium thickness (hmin and hmax), which are independent of the initial film thickness. The existence of an unstable zone and a maximum height of the resulting structures are precisely the behavior witnessed in the
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Figure 3. Variations of the maximum thickness (hmax), minimum thickness (hmin), and upper critical thickness (hc2) with Rp ) -SLW/SP for lp ) 2.5 nm. The vertical line separates the regions where the qualitative variation of the potential resembles curves 1 and 2 in Figure 1.
experiments where dewetting produces flat-topped islands.54-63 Based on the qualitative variations of ∆G and ∂∆G/∂h (Figure 1A,B), there are two subclasses of completely wettable but nanoscopically unstable systems. These are exemplified by curves 1 and 2 of Figure 1A,B, where ∆G for curve 1 shows monotonic decay, and consequently, ∂∆G/∂h for curve 1 does not change sign. For curve 2, ∂∆G/∂h shows two zero intercepts (not counting the one at infinite thickness). Figure 3 shows the regions of existence where the behavior exemplified by curves 1 and 2 is found. This figure also shows the variation of hmin and hmax and the upper critical thickness, hc2, beyond which a thin film becomes stable (∂2∆G/∂h2 > 0). Results for both types of systems, an extremely weak short-range attraction (Figure 1, curve 1) and the moderate attraction (Figure 1, curve 2), are presented here. While the qualitative variation of the spinodal parameter is identical in both cases, the ∆G and ∂∆G/∂h curves are qualitatively different. Interestingly, qualitative variation of the spinodal parameter for partially wettable systems is also identical; the difference lies solely in the behavior of ∆G.5,52 For the potential of curve 1 in Figure 1, the lower and the upper critical thicknesses are 8.65 nm (hc1) and 11.48 nm (hc2), respectively, and hm ) 9.63 nm. For a laterally unbounded structure, theoretical estimates of the maximum (hmax) and minimum (hmin) equilibrium thicknesses are 13.03 and 7.90 nm, respectively. For the potential of curve 2, the lower and the upper critical thicknesses are 1.72 nm (hc1) and 3.23 nm (hc2), respectively. The maximum (hmax) and the minimum (hmin) equilibrium thicknesses are 5.03 and 1.44 nm, respectively. Regardless of the parameter values, it can be shown that hmax > hc2 and hmin < hc1. We found that the selection of the initial nonlinear pattern depends crucially on the distance between the mean film thickness and the location of the minimum in the spinodal parameter (hm), as has also been observed earlier for partially wettable systems.7-9,11-14 In particular, the onset of dewetting in partially wettable systems on homogeneous substrates occurs by one of the following two morphological pathways:7-9,11-14 (a) for all films thinner than the minimum in the spinodal parameter and for slightly thicker films, dewetting occurs by the formation and retraction of isolated droplets, and (b) for relatively thicker films, dewetting occurs by the formation and expansion of isolated holes. The observations reported below for perfectly wettable systems also confirm this picture for the early phases of evolution. For example, Figure 4 depicts the morphology at the instant of film rupture for different mean thickness films
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Figure 4. Morphologies at the instant of (pseudo-)dewetting, when the minimum thickness declines to hmin at at least one grid point. Pictures from left to right are for h (in nm) ) 9, 9.75, 10, 10.5, 11, and 11.35 nm, respectively. The area of each picture is 64L2. Parameters correspond to the potential of curve 1 in Figure 1 (SLW ) 5 mJ/m2, SP ) -0.024 969 mJ/m2, lp ) 2.5 nm, µ ) 1 Pa‚s, γ ) 30.8 mJ/m2). The maximum thicknesses (in nm) are 12.86, 12.19, 11.93, 11.2, 11.44, and 12.05, respectively. A continuous linear gray scale between the minimum and the maximum thicknesses is employed in each image.
of curve 1 in Figure 1. Rupture is defined when the minimum film thickness declines to hmin at some point. Films to the left of the spinodal minimum (e.g., picture 1 in Figure 4) dewet by the formation of round isolated droplets. Increasingly thicker films to the right of the spinodal minimum (pictures 2-6) dewet by the formation of an undulating bicontinuous structure composed of long hills and valleys (pictures 2-4), which resolves into increasingly circular isolated holes for still thicker films (pictures 5 and 6). The initial morphology until the film breakup can thus be modulated either by changing the potential (or hm) or by a change of the film thickness. However, unlike the case of partially wettable systems (pictures 2-4), the morphology after the onset of dewetting in completely wettable systems is profoundly affected by the location of hmax. As is shown below, novel morphologies not witnessed in partially wettable systems are thus produced. Formation of Parapets and Pancakes. Figure 5 (pictures 1-6; for curve 1 of Figure 1) summarizes the major events in the time evolution of patterns in a 9 nm thick film (h f hc1) which is slightly thinner than the location of the minimum in the spinodal parameter (hm). Figure 6 shows the evolution of a typical 2D cross-sectional profile. Initial random disturbances are first organized into a bicontinuous undulating structure, the ridges of which directly fragment into droplets (Figure 5, pictures 1 and 2). The minimum thickness declines to the equilibrium thickness (hmin) to form a largely flat thin film surrounding the droplets, on which pseudo-dewetting occurs by retraction of droplets. The droplet height increases by retraction and later by ripening of the structure in which bigger drops grow at the expense of smaller ones (picture 3). Ripening is engendered by the Laplace pressure and disjoining pressure induced flow from the smaller to bigger drops. The droplet height however does not increase indefinitely (as in partially wettable systems), but the top flattens out to produce a mesa when the maximum thickness reaches hmax. Further ripening produces fewer flat-topped islands of greater base area, but the height remains constant (pictures 4 and 5). Eventually a truly thermodynamic, globally stable state is reached, which is represented by the coexistence of a single round pancake with its surrounding equilibrium
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Figure 5. Different stages of the evolution of a 9 nm thick film, which is thinner than the location of the minimum in the spinodal parameter vs thickness curve 1 of Figure 1. The area of each picture is 64L2 (L ) 52 µm). Parameters correspond to the potential of curve 1 in Figure 1 (SLW ) 5 mJ/m2, SP ) -0.024 969 mJ/m2, lp ) 2.5 nm, µ ) 1 Pa‚s, γ ) 30.8 mJ/m2). Nondimensional times (from left to right) are T ) 1.6 × 104, 4.8 × 104, 5.1 × 106, 1.1 × 107, 1.4 × 107, and 7.8 × 107, respectively. The corresponding dimensional maximum and minimum thicknesses (in nm) are (10.89, 8.56), (12.87, 8.01), (13.16, 7.95), (13.14, 7.93), (13.11, 7.92), and (13.05, 7.92), respectively.
Figure 6. Typical cross-sectional views of the 3D morphologies shown in Figure 5. Nondimensional times for curves 1-4 are 321, 42 471, 567 453, and 1.2 × 1016, respectively.
flat film (picture 6). The same sequence of evolution, film f circular droplets f round pancake, was also obtained in simulations (not shown) for the potential represented by curve 2 of Figure 1 for h < hm. The minimum and maximum thicknesses in these simulations were found to be always within about 2% of the predictions of the infinite 2D pancake solution, that is, hmax and hmin. Simulations in larger domains (with larger liquid volume) produce a larger number of initial droplets (droplets per area remaining approximately constant), and therefore, a larger round mesa is finally produced by the ripening. However, the morphological pathways of evolution remain unaltered. Figure 7 shows another pathway of evolution for slightly thicker films close to the right of the minimum in the spinodal curve, which also leads finally to the equilibrium structure of a round pancake. In Figure 7, the (pseudo-)dewetting is initiated by the fragmentation of the initial bicontinuous structure into uneven ridges of varying shapes, sizes, and heights. Due to a larger volume of these fragments, hmax is attained before a significant ripening and rounding of the ridges has occurred. Thus, long parapets or rectangular mesas result, which gradually ripen and transform into circular pancakes due to the surface tension effect. A cross section of Figure 7 shows the flat-topped structures (figure given in the Supporting Information). The same type of evolution was also seen in curve 2 of Figure 1 immediately to the right of hm (results not shown). As the film thickness is further increased, dewetting is initiated by the formation of circular holes, rather than
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Figure 7. Evolution in a 10 nm thick film of the same system as in Figure 5. The area of each picture is 64L2 (L ) 43 µm). Nondimensional times (from left to right) are T ) 125, 6.1 × 103, 2.4 × 104, 3.6 × 105, 1.01 × 106, 2.2 × 106, 7.4 × 106, and 8.6 × 106, respectively. The corresponding dimensional maximum and minimum thicknesses (in nm) are (10.76, 9.2), (11.75, 8.2), (13.0, 7.9), (13.0, 7.9), (13.0, 7.92), (13.0, 7.92), (13.1, 7.9), and (13.14, 7.92), respectively.
Figure 8. Different stages of the evolution of a 10.25 nm thick film, which is thicker than the location of the minimum in the spinodal parameter vs thickness curve 1 of Figure 1. The area of each picture is 64L2 (L ) 46 µm). Nondimensional times (from left to right) are 1679, 4.6 × 103, 6.6 × 103, 3.6 × 104, 2.4 × 105, 4.9 × 105, 4.0 × 106, and 5.3 × 107, respectively. The corresponding dimensional maximum and minimum thicknesses (in nm) are (10.93, 9.43), (11.44, 8.4), (11.89, 8.1), (12.8, 7.89), (13.02, 7.89), (13.1, 7.89), (13.2, 7.89), and (13.2, 7.89), respectively.
by droplets and bicontinuous structure. Figure 8 (for curve 1 of Figure 1) shows this yet another pathway of evolution leading eventually to a circular pancake for intermediate thickness films where dewetting is initiated by the formation of largely circular holes. Holes grow and coalesce to produce long liquid threads, which in turn fragment due to the Rayleigh instability (flow from high to low crosssectional curvatures). Ripening of the fragments engenders fewer and thicker ridges, which eventually flatten at the top and slowly transform into a round pancake. The same sequence of evolution, film f circular holes f hole coalescence f ridges f pancake, was also seen in the simulations for the intermediate thickness films of the potential curve 2 of Figure 1 (results not shown). Thus, even dewetting initiated by holes can eventually lead to the pancake morphology for sufficiently thin films, whenever the liquid mass produced by hole formation, growth, and coalescence can fragment before its height reaches the maximum allowed thickness, hmax. The equilibrium round pancakes are the only equilibrium structures at long times, regardless of whether dewetting is initiated by isolated liquid droplets, ridges, or holes. In many cases, intermediate metastable structures such as curved-top droplets and irregular islands with maximum height < hmax can form by the evolution of instability in relatively thin films. Of course, very long term ripening of fragmented (isolated) liquid domains always engenders round mesas. However, ripening is a
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Figure 9. Different stages of evolution in a 11.35 nm thick film, which is close to hc2. Parameters correspond to the potential of curve 1 in Figure 1 (SLW ) 5 mJ/m2, SP ) -0.024 969 mJ/m2, lp ) 2.5 nm, µ ) 1 Pa‚s, γ ) 30.8 mJ/m2). The area of each cell is 64L2 (L ) 140 µm). The onset of dewetting occurs by the formation of circular holes as seen in picture 2. Nondimensional times are 1046, 1.9 × 104, 4.5 × 104, 1.0 × 105, 3.7 × 105, 1.9 × 106, 5.5 × 106, and 1.3 × 107, respectively. The corresponding maximum and minimum thicknesses (in nm) are (11.73, 10.7), (12.03, 7.85), (13.05, 7.84), (12.82, 7.83), (12.93, 7.85), (12.93, 7.83), (12.94, 7.83), and (12.94, 7.83), respectively.
Figure 10. Typical cross-sectional views of 3D morphologies shown in Figure 9. Nondimensional times for curves 1-4 are 2334, 69 610, 1.4 × 105, and 3.2 × 1011, respectively.
much slower process, and therefore, the intermediate, kinetically stable structures such as long mesas and droplets may appear to be the “final” morphologies in some experiments. Formation of “Swiss Cheese” or Membrane Morphology. A more interesting scenario arises for relatively thicker films closer to the upper spinodal boundary, hc2. Figure 9 (11.35 nm film; for curve 1 of Figure 1) shows that dewetting occurs by the formation of circular holes, which develop rims during their expansion. Repeated formation, growth, and coalescence of holes leads to thickening of the surrounding liquid, which however cannot rise above the maximum permissible thickness, hmax. Once most of the liquid surface levels off close to hmax, hole expansion slows down tremendously and eventually almost stops. This produces a honeycombed morphology resembling a slice of Swiss cheese or a craterriddled mesa. A cross-sectional view of the structure is shown in Figure 10. At later times, the structure coarsens by the formation of fewer and bigger holes, which afterward remain frozen since the maximum height of the continuous liquid phase reaches hmax. A similar sequence of evolution is illustrated in Figure 11 for another system (for a 3.2 nm thick film of the potential represented by curve 2 of Figure 1). The same behavior was confirmed in numerous other simulations (not shown) starting with some threshold thickness to the right of the spinodal minimum all the way to the upper spinodal boundary. The final structures remained unchanged even after continued integration to about 3 orders of magnitude in time. That these structures are indeed at equilibrium was also directly confirmed by the observation that the final profiles satisfied the steady-state version of
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Figure 11. Different stages of evolution in a 3.2 nm thick film, which is close to hc2. Parameters correspond to the potential of curve 2 in Figure 1 (SLW ) 5 mJ/m2, SP ) -0.415 8508 mJ/m2, lp ) 0.6 nm, µ ) 1 Pa‚s, γ ) 30.8 mJ/m2). The area of each cell is 16L2 (L ) 7.1 µm). Nondimensional times are 74, 4.2 × 104, 4.6 × 104, 4.7 × 104, 5.0 × 104, and 9.5 × 105, respectively.
eq 7 with an accuracy of 10-11 or better at all grid points. This curious phenomenon of the evolution frozen midway to produce a metastable or a quasi-metastable phase (dispersion of air in liquid) is not possible in partially wettable systems, since there is no constraint on the maximum thickness. The evolution in partially wettable systems therefore always concludes with the formation of spherical macroscopic droplets (dispersion of liquid in air), which is the only and globally stable state in partially wettable systems. Thus, a thin membrane-like morphology is produced in perfectly wetting systems whenever dewetting is initiated by the formation of holes and the maximum cutoff thickness is reached before a possible fragmentation of the continuous liquid phase can produce isolated liquid ridges or droplets. The tendency for fragmentation due to the Rayleigh-like instability becomes weaker for thicker films, since attainment of hmax during early stages of evolution (hole growth and coalescence) eliminates strong gradients of cross-sectional curvature which occur in relatively thinner films of perfectly wetting systems and for all thicknesses in (unconstrained) partially wettable systems. Another question which may now be addressed is whether the metastable Swiss cheese geometry is the only stable equilibrium attained at large times for relatively thick films. In other words, would one still get the same membrane morphology if one started not with a continuous thin film but from other initial morphologies? Figure 12 shows the evolution starting with two different initial conditions: the first is a set of disjointed droplets, and the other is a single big droplet. Both of these initial structures are chosen with the same volume per unit area (mean thickness) and the potential as in Figure 9. Both of these initial states end up in the formation of disjointed mesas or the pancake structures, rather than a porous membrane structure as in Figure 9. Clearly, it is only the evolution starting with a film morphology which is trapped into the metastable state of dispersion of holes in the liquid, even though the round pancake solution also continues to be a stable equilibrium. However, this solution can be realized only if the continuous liquid structure surrounding the holes is initially broken by external means to produce isolated liquid fragments or droplets. Thus, while the spreading of a droplet always results in a pancake, dewetting or retraction of a thin film can lead either to a membrane morphology (for relatively thick films) or to
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Figure 12. Equilibrium morphologies (second picture in each row) corresponding to two different initial morphologies (first picture in each row). The second picture in the first row is an intermediate time, pseudo-equilibrium morphology which also resolves into a pancake structure by ripening at very long times. The mean thickness ()volume/area) of both structures is 11.35 nm. Parameters correspond to the potential of curve 1 in Figure 1 (SLW ) 5 mJ/m2, SP ) -0.024 969 mJ/m2, lp ) 2.5 nm, γ ) 30.8 mJ/m2).
Figure 13. Number of droplets (curves 1-3 for h0 ) 9, 9.75, and 10.25 nm) and number of holes (curves 4 and 5 for h0 ) 10.5 and 11 nm, respectively) within an area of 64L2. Parameters correspond to the potential of curve 1 in Figure 1 (SLW ) 5 mJ/m2, SP ) -0.024 969 mJ/m2, lp ) 2.5 nm, µ ) 1 Pa‚s, γ ) 30.8 mJ/m2). A renormalized time, tN in units of nm5 (as defined by eq 8, tN ) T(h0)5), is used to remove the influence of the film thickness from the definition of the nondimensional time, T.
a large number of microscopic pancakes at intermediate times. A morphological hysteresis therefore results depending on different initial conditions, which can engender different morphological pathways of evolution. Other Morphological Characteristics. It also becomes clear that 2D simulations can neither differentiate between the pancake and Swiss cheese geometries (the appearance of the 2D cross sections for both is similar, as in Figures 6 and 10) nor uncover the metastable membrane-like phase. Since in two dimensions every hole is flanked by two mesas (and vice versa), ripening of broader mesas continues unhindered by the concurrent shrinking of smaller mesas in the 2D simulations. However, such a pathway of ripening is available only for disjointed ridges but not for the 3D dispersion of holes in the liquid. Although the linear stability analysis cannot predict the actual 3D structure, how does it fare in prediction of the length scale of the instability before a significant ripening or hole coalescence has occurred? The linear theory predicts, on average, formation of one hole or one droplet per L2 of the substrate area initially. Figure 13 shows the number of droplets or holes as a function of time in a typical 64L2 (8L × 8L) simulation in which about 64 structures are initially anticipated before a significant coalescence or ripening has occurred.
Sharma and Verma
Figure 14. Kinetics of increase in the dewetted area. Curves 1-5 correspond to mean film thicknesses of 9, 9.5, 10.25, 10.5, and 11 nm, respectively. Renormalized time is as defined in Figure 13. As expected, curves 1-5 show progressively slower dewetting (and smaller equilibrium fractional dewetted area) as the film thickness increases. However, at early times, a 9 nm film (curve 1) shows an apparently anomalous behavior. This is explained by the fact that the minimum of the spinodal parameter for this case (curve 1 of Figure 1) occurs at 9.63 nm, so that the spinodal parameter for a 9 nm film is similar to that of a 11 nm film (curve 5). This causes the onset of dewetting in these two cases to coincide.
Simulations show a maximum of 54 holes for a relatively thick film to a low of 40 droplets for a thin film, before the coarsening of the structure diminishes these numbers. For the thinner films to the left of the spinodal minimum, this discrepancy from the linear theory is expected, since the linear theory overestimates the magnitude of the spinodal parameter since more less negative values of ΦH are obtained at the thinner spots during the course of evolution. Thus, the linear theory, L2 ) -8π2/ΦH0, underestimates the characteristic length scale or overestimates the density of droplets. In addition, droplet formation and ripening are concurrent processes and cannot be as strictly separated as hole formation and coalescence. This factor (drop ripening) also leads to a somewhat smaller drop density. However, as a first estimate, the linear theory appears to be satisfactory for prediction of the length scale of the structures prior to coarsening. Figure 14 for the potential of curve 1 in Figure 1 shows a typical kinetics of dewetting for different thickness films. A cutoff of 1.05hmin was chosen to define the extent of the dewetted area. Dewetting is faster for thinner films, and the equilibrium dewetted area is more. For all thicknesses to the right of the spinodal minimum (curves 3-5), the onset of dewetting occurs at increasing times as the film thickness increases. This is expected1-12 since the characteristic time scale for the instability increases with decreased magnitude of the spinodal parameter (-φh). Complex events such as the drop ripening (for relatively thinner films), hole coalescence (for relatively thicker films), and fragmentation of ridges engender somewhat abrupt changes in the kinetics of dewetting in Figure 14. Interestingly, the fractional dewetted area at equilibrium is always less for thicker films. This conclusion is summarized in Figure 15, where the solid line is the best fit to the simulations. Equilibrium dewetted areas from simulations show an almost linear decline with the film thickness regardless of the final equilibrium morphology obtained (pancake or Swiss cheese). This observation may be understood based on the following simple argument of mass conservation. To simplify the matter, the final equilibrium structure is idealized as made up of flat mesas of area Am and thickness hmax and flat dewetted areas of area AD and thickness hmin, with perfectly vertical edges separating the two regions. Thus the total area AT ) AD + Am, and the conservation of mass for a film of initial thickness h0 gives Amhmax + ADhmin ) ATh0. A rearrange-
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droplets at equilibrium. A proper selection/tailoring of the substrate is therefore essential for obtaining the novel morphologies discussed here. A high-energy substrate would ensure the long-range repulsive van der Waals force, and a shorter range weak attraction could be induced, for example, by chain adsorption/grafting leading to nanoscale autophobicity.
Figure 15. Variation of the fractional dewetted area at equilibrium with the mean film thickness for two different systems, the potentials of which are characterized as follows: (curve 1) SLW ) 5 mJ/m2, SP ) -0.415 8508 mJ/m2, lp ) 0.6 nm, hc1 ) 1.72 nm, hc2 ) 3.23 nm, hmax ) 5.03 nm, hmin ) 1.44 nm; (curve 2) SLW ) 5 mJ/m2, SP ) -0.024 969 mJ/m2, lp ) 2.5 nm, hc1 ) 8.65 nm, hc2 ) 11.48 nm, hmax ) 13.03 nm, hmin ) 7.91 nm. Symbols represent results from simulations, solid lines are the best linear fit to the simulations, and the broken line represents a simplified mass balance, eq 10.
ment gives the equilibrium fractional dewetted area,
aD ) AD/AT ) 1 - [(h0 - hmin)/(hmax - hmin)]
(10)
The broken line in Figure 15 shows the above theoretical correlation, which has about the same slope as the best fit to the simulations but a slightly different intercept since the finite volume of the edges is not accounted for in the simple mass balance. Figure 15 summarizes all of our results for the dewetted area from simulations for the two different potentials considered. Predictions of eq 10 are given as solid lines in the same figure, indicating a good correlation. Interestingly, transition from pancake to Swiss cheese equilibrium morphology occurs close to a film thickness for which the dewetted area equals the coverage by liquid mesas, that is, the fractional dewetted area is close to 0.5. For example, simulations showed the onset of membranelike morphology at about 10.3 nm for curve 2 of Figure 15. Thus, this morphological transition is a close analogue of an ideal phase inversion close to equal volume fractions: liquid-in-air dispersion (pancakes) f air-in-liquid dispersion (Swiss cheese). Finally, the equilibrium island area covered by the flat mesas with perfectly vertical edges may be defined as am ) (1 - aD) ) (h0 - hmin)/(hmax - hmin), which increases linearly with the initial film thickness. Indeed, dewetted islands of behenic acid on silicon surfaces show such a linear increase.54,63 Experiments54-63 have also confirmed that resulting structures have unique heights (hmax and hmin) irrespective of the original film thickness and that films beyond a certain critical thickness remain perfectly stable and wetting. All of these observations are in accord with the theory discussed above. Interestingly, any flat-topped structure in general, and an equilibrium membrane-like morphology in particular, can never be produced by the spinodal dewetting in the thin films of systems showing partial macroscale wetting. The latter always leads to the formation of spherical
Conclusions The instability, dynamics, and morphologies and their pathways of evolution are studied in thin films of liquids that show perfect macroscale wetting; that is, the YoungDupre equation predicts zero contact angle due to a positive spreading coefficient. Spreading of a liquid droplet in such systems results in the de Gennes-Joanny pancake of nanoscopic thickness due to the influence of short-range non-van der Waals attractive forces, which also open a window of spinodal instability in a range of film thickness (hc1 < h < hc2). We have shown that the surface instability of such films in the spinodal range eventually produces either a collection of equilibrium round pancakes (for relatively thin films) or a novel membrane- or Swisscheese-like honeycombed equilibrium morphology. The latter morphology is obtained for relatively thick films beyond a certain transition thickness to the right of the spinodal minimum. The transition thickness for the onset of membrane-like morphology is close to where the equilibrium area coverage of liquid equals the dewetted area, that is, close to the fractional dewetted area of 0.5. Regardless of the equilibrium morphology, the fractional dewetted area at equilibrium decreases linearly with the film thickness. The maximum thickness in the flat part of any structure is independent of the initial film thickness and the precise morphology but depends only on the surface properties. Long mesas, flat-topped parapets, or even curved droplets with round tops can also result during the evolution, but these are merely kinetically stable (long lasting) structures which must eventually resolve into round pancakes by retraction of edges and ripening. It is hoped that the theory and simulations reported here will aid the design and interpretations of future experiments and may also help development of membranes and coatings with specific architecture, for example, perforated optical coatings with increased transmission and reduced refraction of light. Acknowledgment. The assistance of Dinni Lingraj in the initial stages of this work is gratefully acknowledged. This work was supported by a grant from the Department of Science and Technology, India, Nanosciences program. Supporting Information Available: Figure A: Crosssectional profile of the structure in Figure 7. Figure B: Simulation of Figure 5 redone with higher grid density showing grid convergence. Figure C: Simulation of Figure 5 redone with higher initial amplitude showing insensitivity to amplitude. Figure D: Dewetting kinetics for different initial amplitudes. Figure E: Simulation of Figure 5 redone with a periodic initial condition. This material is available free of charge via the Internet at http://pubs.acs.org. LA048669X
