Dewetting of Thin Films on Periodic Physically and Chemically

Department of Chemical Engineering, Indian Institute of Technology, Kanpur, UP ..... Mechanically Modulated Dewetting by Atomic Force Microscope for M...
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Langmuir 2002, 18, 1893-1903

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Dewetting of Thin Films on Periodic Physically and Chemically Patterned Surfaces Kajari Kargupta and Ashutosh Sharma* Department of Chemical Engineering, Indian Institute of Technology, Kanpur, UP 208016, India Received March 27, 2001. In Final Form: November 5, 2001 The instability, dynamics, and morphological transitions of patterns in thin liquid films on physically and chemically heterogeneous patterned surfaces are investigated on the basis of 3D nonlinear simulations. On a chemically striped surface (consisting of alternating less and more wettable stripes) the film breakup is suppressed on some potentially destabilizing nonwettable stripes when their spacing is below a characteristic length scale of instability (λh), which is of the same order as the spinodal length scale (λl) of instability on the less wettable stripes. The thin film pattern replicates the substrate surface energy pattern closely only when (a) the periodicity of the substrate pattern lies between λh and 2λh and (b) the less wettable stripe width is within a range bounded by a lower critical length, below which no heterogeneous rupture occurs, and an upper transition length, above which complex morphological features bearing little resemblance to the substrate pattern are formed. The thin film pattern on a periodic physically heterogeneous surface shows the loss of ideal templating when the periodicity of the surface is smaller than about 0.8 times the spinodal wavelength evaluated at the minimum film thickness. On a physicochemically patterned periodic surface, the chemical heterogeneity largely controls the thin film pattern and the effect of small to moderate physical heterogeneity is minimal.

1. Introduction Self-organization during dewetting of thin films on deliberately tailored physicochemically heterogeneous substrates is of increasing promise for engineering of desired nano- and micropatterns in thin films by templating.1-19 A recent study11 reported anisotropic spinodal dewetting of polymers on a rough surface made by rubbing. In another recent study, Rockford et al.6 reported on the dewetting behavior of thin polymer films on a periodically rough Si substrate exhibiting a regular pattern of Au stripes at the peaks. The oriented anisotropic wetting behavior of thin polystyrene film on regularly grooved physically and physicochemically heterogeneous silicon substrate was recently studied also by Rehse et * To whom correspondence should be addressed. E-mail: ashutos @iitk.ac.in. (1) Lenz, P.; Lipowsky, R. Phys. Rev. Lett. 1998, 80, 1920. (2) Lenz, P.; Lipowsky, R. Eur. Phys. J. E 2000, 1, 249. (3) Lipowsky, R.; Lenz, P.; Swain, P. S. Colloid Surf., A 2000, 161, 3. (4) Gau, H.; Herminghaus, S.; Lenz, P.; Lipowsky, R. Science 1999, 283, 46. (5) Kumar, A.; Whitesides, G. M. Science 1994, 263, 60. (6) Rockford, L.; Liu, Y.; Mansky, P.; Russell, T. P. Phys. Rev. Lett. 1999, 82, 2602. (7) Boltau, M.; Walhelm, S.; Mlynek, J.; Krausch, G.; Steiner, U. Nature 1998, 391, 877. (8) Nisato, G.; Ermi, B. D.; Douglas, J. F.; Karim, A. Macromolecules 1999, 32, 2356. (9) Gleiche, Chi, L. F.; Fuchs, H. Nature 2000, 403, 173. (10) Kataoka, D. E.; Troian, S. M. Nature 1999, 402, 794. (11) Hlggins, A. M.; Jones, R. A. L. Nature 2000, 404, 476. (12) Karim, A.; Douglas, J. F.; Lee, B. P.; Glotzer, S. C.; Rogers, J. A.; Jackman, R. J.; Amis, E. J.; Whitesides, G. M. Phys. Rev. E 1998, 57, R6273. (13) Lopez, G. P.; Biebuyck, H. A.; Daniel Frisbie, C.; Whitesides, G. M. Science 1993, 260, 647. (14) Bauer, C.; Dietrich, S.; Parry, A. O. Europhys. Lett. 1999, 47, 474. (15) Bauer, C.; Dietrich, S. Phys. Rev. E 2000, 61, 1664. (16) Rascon, C.; Parry, A. O. Phys. Rev. Lett. 1998, 81, 1267. (17) Rascon, C.; Parry, A. O.; Sartori, A. Phys. Rev. E 1999, 59, 5697. (18) Rehse, N.; Wang, C.; Hund, M.; Geoghegan, M.; Magerle, R.; Krausch, G. Eur. Phys. J. E 2001, 4, 69. (19) Kargupta, K.; Sharma, A. Phys. Rev. Lett. 2001, 86, 4536.

al.18 Thus, the technique utilizing spontaneous dewetting9 appears to be a simple method to engineer patterns in soft materials on the nanometer to micrometer scales without the conventional lithographic processes.20,21 Theoretically the rupture of a thin film on a single heterogeneous patch is now well understood. On a chemically heterogeneous substrate, dewetting is driven by the spatial gradient of microscale wettability22-24 rather than from the nonwettability of the substrate itself as in the spinodal dewetting25-28 on homogeneous surfaces. The equilibrium structures of a thin film on a substrate containing single and multiple stripes have been reported on the basis of free energy considerations.1-3,14,15 Patterned substrates pack a large density of surface heterogeneities that are closely spaced. How do hydrodynamic interactions between the neighboring heterogeneities affect the pattern evolution dynamics and morphology in thin films? How faithfully are the substrate patterns reproduced in a thin film spontaneously; i.e., how effective is the templating of soft materials using dewetting and what are the conditions for ideal templating? An associated question for both the patterned and naturally occurring heterogeneous surfaces is whether all the potentially dewetting sites remain active or “live” in producing rupture when they are in close proximity. These questions were recently addresses in a letter19 for a thin film on a chemically heterogeneous substrate. Here we present a comprehensive account of the instability, dynamics, and morphology of thin films on periodic physically and chemically (20) Wang, R.; Hashimoto, K.; Fujishima, A. Nature 1997, 388, 431. (21) Calvert, J. M. J. Vac. Sci. Technol. 1993, B11, 2155. (22) Konnur, R.; Kargupta, K.; Sharma, A. Phys. Rev. Lett. 2000, 84, 931. (23) Kargupta, K.; Konnur, R.; Sharma, A. Langmuir 2000, 16, 10243. (24) Kargupta, K.; Konnur, R.; Sharma, A. Langmuir 2001, 17, 1294. (25) Reiter, G.; Khanna, R.; Sharma, A. Phys. Rev. Lett. 2000, 85, 1432. (26) Herminghaus, S.; Jacobs, K.; Mecke, K.; Bischof, J.; Fery, A.; Ibn-Elhaj, M.; Schlagowski, S. Science 1998, 282, 916. (27) Oron, A. Phys. Rev. Lett. 2000, 85, 2108. (28) Sharma, A.; Khanna, R. Phys. Rev. Lett. 1998, 81, 3463.

10.1021/la010469n CCC: $22.00 © 2002 American Chemical Society Published on Web 02/08/2002

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patterned surfaces, based on 2D and 3D nonlinear simulations. The chemically heterogeneous substrate considered here consists of alternating less wettable and more wettable (or completely wettable) stripes that differ in their interactions with the overlying film. The periodic physical heterogeneity considered here is represented by the sinusoidal undulations of the substrate surface. There are two key geometric parameters of the substrate patterns considered here. The first is its periodicity interval, Lp, which is defined as the total width of a repeat unit consisting of one less wettable and one more wettable stripe or the distance between two consecutive peaks on a sinusoidal rough surface. The second parameter is the width, W, of the less wettable stripes for a chemically heterogeneous surface. The paper is organized as follows. First, the dynamics and evolution of the thin film morphology on a chemically heterogeneous surface are presented. In the next section, we have considered the evolution of the thin film patterns on physically heterogeneous substrates. Finally, the most general situation is studied where a combination of the physical and chemical heterogeneities occurs. 2. Theory 2.1. Thin Film Equation for Evolution on a Smooth Surface. The following nondimensional thin film equation governs the stability and spatiotemporal evolution of a Newtonian nonslipping thin film on a physically uniform substrate:22,23,28

∂H/∂T + ∇‚[H3∇(∇2H)] - ∇‚[H3∇Φ] ) 0

(1)

H(X,Y,T) is the nondimensional local film thickness scaled with the mean thickness ho, Φ ) [2πho2/|As|][∂∆G/∂H]), ∆G is the excess intermolecular interaction energy per unit area, As is the effective Hamaker constant for van der Waals interaction, X, Y are the nondimensional coordinates in the plane of the substrate, scaled with a characteristic length scale (2πγ/|As|)1/2ho2, the nondimensional time T is scaled with (12π2µγho5/As2), and γ and µ refer to the film surface tension and viscosity, respectively. The terms (from left to right) in eq 1 correspond to the accumulation, curvature (surface tension), and the intermolecular forces, respectively. The unsteady force, which includes viscous effects, merely retards the growth of instability. The net surface tension effect in a 3D geometry may be stabilizing (due to the in-plane curvature as in a 2D geometry) or destabilizing (due to the transverse curvature as in the Rayleigh instability of circular cylinders). Finally, a gradient of the conjoining pressure causes the flow of fluid in the direction of decreasing force/unit area, Φ. For homogeneous substrates, Φ ) Φ(H), and the spinodal dewetting is caused by flow from the thinner to thicker regions only when ∂Φ/∂H < 0. Linearization of eq 1 on a homogeneous surface admits the solutions of the form H ) 1 +  exp[ι(KxX + KyY) + ωT], where Kx and Ky are wavenumbers in the orthogonal X and Y directions, ω is the growth coefficient, and  is the nondimensional amplitude (scaled with the mean thickness, ho) of the initial disturbance. The resulting linear dispersion relation with K2 ) Kx2 + Ky2 and ΦH0 ) (∂Φ/∂H) evaluated at H ) 1 is28

ω ) -(Kx2 + Ky2)ΦH0 - (Kx2 + Ky2)2

(2)

Equation 2 gives the necessary condition for the spinodal instability: (Kx2 + Ky2)0.5 < Kc, where Kc ) (-ΦH0)1/2 is the nondimensional critical wavenumber. Thus, only the modes with the nondimensional wavenumbers Kx and Ky less than Kc (or length scales Λx and Λy larger than 2π/Kc) can grow. This implies, interestingly, that the spinodal dewetting can never occur on a very narrow stripe (width < Λ ) 2π/Kc) irrespective of its length, if the stripe is surrounded by a completely wettable substrate (∂Φ/∂H > 0). However, rupture by the heterogeneous mechanism engendered by the wettability contrast still remains possible in such a case. The dimensional critical length scale of the spinodal instability on a uniform surface is given by λ ) (-4π2γ/(∂2∆G/ ∂h2))1/2. Further, the dominant wavelength (Λm ) 2π/Km ) 2π-

Figure 1. Schematic diagram of the thin film on a physicochemically patterned substrate. (-0.5ΦH0)-0.5) of the fastest growing (dω/dK ) 0) linear mode is λm ) (-8π2γ/(∂2 ∆G/∂h2))1/2. On a chemically heterogeneous striped surface, Φ ) Φ(H,X,Y). At a constant film thickness, we model the variation of Φ in X direction by a periodic step function of periodicity, Lp. Gradient of force ∇Φ at the boundary of the stripes causes flow from the less wettable (higher pressure) stripes to the more wettable (lower pressure) stripes, even when the spinodal stability condition ∂Φ/ ∂H > 0 is satisfied everywhere.22,23 A chemically heterogeneous striped surface introduces one additional spinodal length scale. The critical spinodal length scales on the less wettable and more wettable stripes are denoted as λl and λw, respectively. It is known that a single stripe in the absence of its neighbors can cause rupture only if its width exceeds a critical length scale, WC , λl.22,23 2.2. Thin Film Equation for Evolution on a Rough Surface. We now consider the general evolution equation for a thin film on a substrate containing both chemical and physical heterogeneities. Figure 1 shows a thin film on a rough (physically heterogeneous) substrate. The surface is described by a function of position coordinates (Appendix A): Z ) BF(X,Y). B is the nondimensional amplitude (B ) a/ho < 1) of the roughness function, F(X,Y) ∈ [-1,1]. In the simulations F(X,Y) is considered as a sinusoidal wavelike function sin(kX) of periodicity interval Lp ) 2π/k. The following nondimensional thin film equation, derived from the Navier Stokes equations with the lubrication approximation, governs the stability and spatiotemporal evolution of such a thin film system on a rough surface subjected to the excess intermolecular interactions (nondimensional form of eq A10, Appendix A):

∂H/∂T + ∇‚[(H - BF)3∇(∇2H)] - ∇‚[(H - BF)3∇Φ] ) 0 (3) H(X,Y,T) is the nondimensional local height of the film, measured from a datum Z ) 0, scaled with the mean thickness ho. The nondimensional conjoining pressure, Φ, is a function of local nondimensional thickness of the film, χ ) (H - BF). Thus, Φ ) Φ(χ) ) ∂∆G/∂χ. Φ is related to the dimensional conjoining pressure, φ (φ ) ∂∆G/∂η, where η ) (h - af) is the dimensional local thickness), by Φ ) (2πho3/|As|)(∂∆G/∂η). Thus on a physically heterogeneous surface, a spinodally stable film may become unstable whenever the condition ∂∆G/∂χ < 0 is satisfied at any local thickness. Clearly, the peaks of the hills on the surface shown in Figure 1 are the potentially dewetting sites for dewetting due to the reduced local film thickness. Equation 3 reduces to the well-known thin film eq 1 for a geometrically homogeneous surface, for which a ) B ) 0 (η ) h) and the surface is described by Z ) 0. Also when a rough periodic surface contains chemical heterogeneity, the thin film system is still described by the same eq 3, with Φ ) Φ((H - F),X,Y). In the simulations reported here, the periodicity interval for the chemical heterogeneity is kept identical to that of the physical heterogeneity, 2π/k (Figure 1). 2.3. Excess Intermolecular Interactions. We consider a fairly general representation of the antagonistic (attractive/ repulsive) short-range and long-range intermolecular interactions applicable to the aqueous films (eq 4)28-30 and to the polymer (29) Thiele, U.; Mertig, M.; Pompe, W. Phys. Rev. Lett. 1998, 80, 2869. (30) Padmakar, A. S.; Kargupta, K.; Sharma, A. J. Chem. Phys. 1999, 110, 1735.

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Langmuir, Vol. 18, No. 5, 2002 1895 An analytical representation of combined antagonistic (attractive/repulsive) short- and long-range intermolecular interactions for a polymer-like film on a coated (e.g. oxide covered) substrate is23

-12π∆G ) [(As - Ac1)/(η + dc1 + dc2)2 + (Ac1 - Ac2)/(η + dc2)2 + Ac2/η2] (5)

Figure 2. Stationary state thin film profiles on a striped surface. The potential given by eq 4 is used for the specified values of the parameters as given in the text with W ) 0.16 and λl ) 0.107 µm. The less wettable stripes are denoted by the black rectangles. (a)-(c) are for a 6 nm film on the substrates of increasing periodicity (λl ) 0.67 µm, nondimensional Λl ) 3.265). (d) shows the profile for a 5.5 nm thin film on the same substrate as shown in (b) (λl ) 0.42 µm, nondimensional Λl ) 2.43).

Figure 3. Variation of the ratio of characteristic to spinodal length scale with the ratio ∆Φ/(-∂Φ/∂H), obtained from simulations using a different set of parameters. The ratio ∆Φ/ (-∂Φ/∂H) is varied by changing the value of Sp from -0.61 to 550 mJ/m2 on the more wettable hydrophilic stripes, keeping other parameters the same as used in Figure 2. films on high-energy surfaces such as silicon (eq 5).6,23,31 In both cases, the long-range van der Waals force exerts a stabilizing influence and, thus, only the films thinner than a critical thickness become unstable due to the shorter range destabilizing (attractive) interactions. Films thicker than a critical thickness are metastable. Very thick (∼100 µm) films that are stabilized because of gravity are not considered here.

∆G ) -(As/12πη2) + Sp exp(-η/lp)

(4)

η ) (h - af) is the local film thickness. The excess interaction free energy/unit area (∆G) and the corresponding potential or conjoining pressure (φ ) ∂∆G/∂η) represent aqueous films when the effective Hamaker constant, As, and Sp are both negative (long-range van der Waals repulsion combined with a shorterrange attraction, e.g., hydrophobic attraction).32 The potential of eq 4 is used for simulations shown in Figures 2 and 3. As an example of aqueous films, a film of thickness less than 7.4 nm would be spinodally unstable (∂Φ/∂H < 0) on a physically smooth, less wettable hydrophobic stripe characterized by the parameters Ahs ) -1.41 × 10-20 J, Shp ) -65 mJ/m2, and lp ) 0.6 nm. The equilibrium contact angle obtained from the extended YoungDupre equation, cos θ ) 1 + ∆G(he)/γ, in this case is 61°, where he is the equilibrium thickness and where φ ) 0. On a more wettable hydrophilic stripe characterized by As ) -1.41 × 10-20 J, Sp ) -0.61 mJ/m2, and lp ) 0.6 nm, the film of any thickness is spinodally stable (∂Φ/∂H > 0) and therefore completely wets the surface. The surface tension, γ, and the viscosity, µ, of the water film used for the simulations are 72.8 mJ/m2 and 10-3 kg/ms, respectively. Accordingly, the nondimensional coordinate, X, and time, T, are obtained by the above-mentioned scalings using the characteristic length scale and time scale that are 5.7ho2 nm and 4.33 × 10-8ho5 s, respectively, and Φ ) 4.454 × 10-7ho3[∂∆G/∂η], where ho is in nm. (31) Kim, H. I.; Mate, C. M.; Hannibal, K. A.; Perry, S. S. Phys. Rev. Lett. 1999, 82, 3496. (32) Van Oss, C. J.; Chudhury, M. K.; Good, R. J. Chem. Rev. 1988, 88, 927.

Negative value of the effective Hamaker constant on the substrate (As ) Ahs ) -1.88 × 10-20 J) signifies a long-range repulsion, whereas a positive value on coating (Ac1 ) Ahc1 ) 1.13 × 10-20 J) represents an intermediate-range attraction.22,23 The potential, eq 5, is used in simulations shown in Figures 4-18. The nonwettable coating (e.g., oxide) thickness (dc1 ) 2.5 nm, dhc1 ) 4 nm) is increased on alternating stripes which causes the macroscopic contact angle to increase from 0.58° on the more wettable stripes to 1.7° on the less wettable stripes. A still shorter range repulsion may rise due to a chemically adsorbed or grafted layer of the polymer (Ac2 ) -0.188 × 10-20 and dc2 ) 1 nm are the film Hamaker constant and the thickness of the adsorbed layer, respectively).23 In this example, the films are spinodally unstable below the critical thicknesses (hc) of 6.82 and 12.4 nm on the less wettable and more wettable stripes, respectively. The surface tension, γ, and the viscosity, µ, of the polymer film used for the simulations are 38 mJ/m2 and 1 kg/ms, respectively. Accordingly, the nondimensional length, X, and time, T, are obtained by the above-mentioned scaling by using the characteristic lengthscale and time scale that are 3.56ho2 nm and 1.27 × 10-5ho5 s, respectively, and Φ ) 3.34 × 10-7ho3[∂∆G/∂η], where ho is in nm. In both the potentials, eqs 4 and 5, the near-surface cutoff is provided by an extremely short-range repulsion which gives a primary minimum in the free energy, ∆G, near h ) 0. Thus, the minimum film thickness of the “adsorbed” layer after dewetting equals the equilibrium thickness, where (∂∆G/∂h) ) 0. The contact line is thus defined as the line where a liquid domain meets the flat adsorbed film of equilibrium thickness. The contact corner is very sharp due to the steep near-substrate repulsion. The analytical representations, eqs 4 and 5, are chosen here for illustration without affecting the general underlying physics of the thin films subjected to a short-range attraction combined with a long-range repulsion. Qualitatively aspects are not expected to differ for these models since both the models show the same qualitative variation of the excess interaction energy/ unit area and its second derivative with the film thickness.1-3,6,18,19 2.4. Numerical Methods. 1D and 2D forms of eqs 1 and 3 were numerically solved using a central difference scheme in space combined with the periodic boundary conditions, and the Gears algorithm for stiff equations was employed for the time marching. An initial small amplitude, volume preserving random perturbation was used. The excess nondimensional potential, Φ, is approximated as a continuous function across the stripe boundary by the finite difference scheme. For uniform grids with width ∆X, the function Φ was interpolated as a quadratic function while calculating the first derivative, (∂Φ/∂X), by the central difference scheme. The convergence of results was verified by increasing the grid density and also by the use of nonuniformly spaced grids. In the latter case, the finer grids of constant grid width (a multiple of coarser grids, ∆X/M, M ∼ 10) were used on the potentially destabilizing less wettable stripes and near the boundaries where most of the “action” occurs. These finer grids were surrounded by the coarser grids of gradually decreasing grid density (∆Xsmooth ) ∆X (P)Q, where P is a grid coefficient between 0.8 and 0.96 and Q is a natural number between 1 and M). The nonuniform grids were in turn surrounded by the coarsest grids of a constant grid width, ∆X. The spatial derivatives of the functions (H, Φ) were written in terms of the central difference scheme using variable coefficients. The coefficients for each nonuniform grid were calculated by solving a set of linear equations of the form AC ) B, where C is the unknown coefficient (variable) matrix. The constant matrix A depends on the choice of the grid. A three points difference rule for the first derivative expresses the variable as a quadratic function of X.

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Chemically patterned substrates were numerically simulated by considering at least four alternating less wettable stripes and four more wettable stripes. Periodic boundary conditions were applied. Qualitative aspects (for example, morphology) could be well resolved by about 80 times 80 grids in 3D simulations, but quantitative convergence (for example, kinetics) usually required more grids and about 400 grids were found to be adequate in the 2D simulations for this purpose.

3. Results and Discussion The major objective of this paper is to investigate the morphological patterns that develop spontaneously in a thin film on a periodic heterogeneous substrate and thereby explore the conditions for the ideal or nearly ideal templating. The term ideal templating can reasonably be defined as follows: (a) All less wettable stripes are dewetted fully without any liquid remnants of the film remaining there, (b) liquid cylinders of constant crosssectional curvature form on the more wettable stripes, (c) the base of the liquid cylinders formed on the more wettable stripes cover them completely (i.e., the contact lines are parallel and reside close to the stripe boundaries), and (d) the liquid cylinders formed on the more wettable stripes remain stable or quasi-stable for a long time in the transverse direction. In short, the contours of the resulting structures closely replicate the surface energy pattern. Similar considerations apply also in the case of physically heterogeneous substrates. As discussed below, good templating under some conditions can also occur as an intermediate structure during the evolution rather than as a very long time stable or quasi-stable structure. This is also useful in the most practical applications where the structure can be “frozen” at this stage for further processing in the solid state. 3.1. Chemically Heterogeneous Substrates. In this section, we first address the question of whether all the potentially dewetting sites can affect rupture when they are in close proximity to each other. Clearly, a necessary condition for good templating requires that dewetting should occur on every less wettable site (number of dewetted stripes ) nD ) number of less wettable stripes ) nS). As shown below, such a condition is however not always met. On a chemically heterogeneous periodic substrate consisting of alternating completely wettable hydrophilic and nonwettable hydrophobic stripes, rupture occurs on the hydrophobic sites due to the potential gradient caused by the heterogeneity. The excess interaction energy of water film on the hydrophilic and hydrophobic stripes is represented by eq 4. The contact line cannot cross the boundary of the nonwettable sites when the neighboring stripes are completely wettable (e.g., Figure 2, based on 2D simulations). Thus, the effect of periodicity interval is easily understood from the number of holes or liquid domains produced. The critical length, WC, of a single heterogeneity (hydrophobic patch) that engenders rupture on a large hydrophilic substrate in this case was found to be 0.08λl. The hydrophobic stripe width was considered to be larger than WC to ensure heterogeneous rupture. Ideal templating requires that the number of dewetted sites (nD) should equal the total number of less wettable sites (nS); i.e., dewetting should occur on every less wettable site of the patterned substrate. Figure 2a shows a single rupture on a substrate containing four potentially dewetting sites (nS ) 4) when the periodicity interval is small, Lp ) 0.45λl. The resulting lone drop spans across the remaining “inactive” hydrophobic sites. Figure 2a-c shows increased number of dewetted stripes (nD ) 1, 2, 4) and resulting liquid domains as Lp is increased from 0.45λl f 0.68λl f

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0.9λl. This, as well as a large number of simulations not displayed here, showed that, on a substrate containing many potentially destabilizing sites, only the sites (randomly picked) separated by a characteristic length scale of the instability, λh, which is of the order of the spinodal scale, λl, remain “live” or effective in causing the film rupture. Dewetting on the remaining intervening sites is suppressed since rupture on each site would require surface deformations on smaller scales resulting in high surface energy penalty. The ratio of this characteristic length scale, λh, to the spinodal length scale, λl, obtained from simulations, decreases as the potential difference across the stripe boundary (∆Φ) increases (Figure 3), where ∆Φ ) (Φl - Φm) is evaluated at the initial mean thickness. Subscripts l and m denote the less and more wettable stripes. For a large value of spinodal parameter (e.g. for thinner films), the ratio approaches very close to 1 unless ∆Φ is very large (Figure 3). A simple scale analysis of eq 122,23 leads to the following characteristic length scale for the growth of instability: λh R [∆Φ/c - ∂Φ/∂H] -0.5. In the above analysis, H ) 1 +  and ∇2Φ ∼ [∆Φ/cL2 (∂Φ/∂H)/L2], where L is the length scale of instability and Lc1/2 is the lengths cale of potential gradient due to the heterogeneity (c,  < 1). L is optimized to give the maximum growth rate of instability. Thus, the ratio of heterogeneous to spinodal length scale λh/λl R (1 + ∆Φ/(-c∂Φ/∂H))-0.5. For very large potential difference, λh/λl R (∆Φ)-0.5. However, λh ∼ λl, when the spinodal term (-∂Φ/∂H) is strong compared to the potential difference created by the heterogeneity. An interesting implication of the above result is that even when the rupture occurs rapidly by the heterogeneous mechanism on a substrate containing a large density of heterogeneities, the length scale (and number density of holes) of the resulting pattern can give the illusion of spinodal dewetting by mimicking the characteristic length scale of the latter. Differentiating true spinodal dewetting from heterogeneous dewetting therefore requires a careful consideration of their distinct time scales and morphological features.22,23 Further, the contact line always remains pinned at the boundary of the stripes whenever alternate stripes are completely wettable (Figure 2a-d). Decreased mean film thickness increases both the terms ∆Φ and -∂Φ/∂H and thus decreases λh nonlinearly. Thus, a greater number of nonwettable heterogeneities become active in causing the film breakup for thinner films. Comparison of Figure 2b (thickness ) 6 nm, λl ) 0.67 µm) and Figure 2d (thickness ) 5.5 nm, λl ) 0.42 µm) clearly shows this for an identical substrate pattern. This morphological transition uncovered based on the dynamical simulations is also in conformity with the earlier equilibrium energy considerations1,3 that are related however only to the equilibrium structures. The equilibrium structures reported by Lenz et al.1 show the suppression of dewetted regions on some of the hydrophobic stripes with an increase in the volume of fluid (i.e. the increase of the mean thickness of the film on the same substrate area). The “film state” reported by them1,3 has also been obtained in our simulation for film thickness higher than a critical thickness, hc (where ∂Φ/∂H > |∆Φ|/ c), at which the characteristic length scale, λh, tends to infinity. For a fixed thickness, the instability is also suppressed and the “film state” results when the width of nonwettable stripes is less than a critical width, WC. From the above discussion, a necessary condition for good templating is that Lp g λh which ensures dewetting on every less wettable site (nD ) nS). As shown below, the condition Lp g λh, for nD ) nS, also remains valid when both the stripes are nonwettable. However, in this case,

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Figure 5. Morphological evolution in a 2.5 nm thick film on a striped surface (W ) 0.3 µm ∼ 0.6λl , Γ ) 13.5, and Λl ) 22.5). In (A)-(C), Lp ) 0.8, 1.5, and 3 µm, respectively (Χp ) 36, 67.4, 135). Gray scale images from left to right in (A) correspond to T ) 124, 383, 753, and 3045, respectively. Images from left to right in (B) correspond to T ) 1127, 6510, 8766, and 115 140, respectively. Images in (C) are at T ) 1281, 16 157, 62 876, and 235 456, respectively.

Figure 4. Morphological evolution in a 5 nm thick film on a striped surface (W ) 0.8 µm ) 0.53λl, nondimensional stripe width Γ ) 8.9, and Λl ) 17). The potential of eq 5 is used for the specified values of parameters as given in the text for the simulations shown in this figure, as well as in Figures 5-9 and 13-18. In (A)-(D) the periodicity of the substrate pattern, Lp, is 1, 1.2, 1.45, and 3 µm, respectively (nondimensional periodicity Xp ) 11.23, 13.5, 16.28, and 33.7). The first image in these figures as well as in the subsequent figures represents the substrate surface energy pattern; black and white represent the more wettable part and the less wettable part, respectively. For other images describing the film morphology, a continuous gray scale between the minimum and the maximum thickness in each picture has been used. Gray scale images from left to right in (A) correspond to T (nondimensional time) ) 528, 731, and 1049, respectively. Images from left to right in (B) correspond to T ) 113, 226, and 464, respectively. Images in (C) correspond to T ) 69, 133, and 363, respectively. Images in (D) are at T ) 139, 226, and 518, respectively.

λh is close to (slightly less than) the spinodal length scale on the less wettable stripe where rupture occurs. If both the stripes are partially wettable, the contact line can move across the boundary of the stripes to the more wettable part. The complete 3D morphology including the features parallel to the stripes can only be studied by the 3D simulations reported below for a polymer-like film on an oxide (low-energy) covered silicon (high-energy) substrate made up of two different partially wetting alternating stripes. The excess energy of interaction for the system is represented by eq 5. In all of the gray scale images based on 3D simulations shown here, we have used a continuous gray scale to denote thickness with increasingly darker shade for thicker regions. The continuous gray scale between the minimum and the maximum thickness is used in each image. Figure 4A-D depicts the effect of periodicity interval, Lp, on the self-organization of a 5 nm thin film that is spinodally unstable on both types of stripes. In every case, the evolution starts with local depressions on the lesswettable stripes (Figure 4A-D). For Lp sufficiently smaller than λl (Lp ) 1 µm, λl ) 1.51 µm (Λl ) 17), based on γ ) 38mJ/m2 and µ ) 1 kg/m.s), the isolated holes or depressions that form along the less wettable stripes grow onto the more wettable regions and coalesce with each other rapidly (Figure 4A) leading to a disordered structure and

very poor templating. Increased periodicity (Lp ) 1.2 µm; Figure 4B) leads to more ordered dewetting, but the number of dewetted regions remains less than the number of less wettable stripes, and defects evolve at late times (e.g., holes in the liquid ridge; image 4 of Figure 4B). For Lp g λl (Figure 4C,D), the number of dewetted stripes equal the number of less wettable stripes. However, templating in the form of liquid ridges with straight edges is best at an intermediate periodicity, Lp ∼ λh (Figure 4C). A further increase in Lp makes the width of dewetted region bigger than the width of the less wettable stripe (image 4 of Figure 4D); i.e., the contact line resides in the interior of the more wettable stripe rather than close to the boundary. Dewetting on the less wettable stripes causes accumulation of material on the more wettable stripes and thus increases the mean thickness of the cylindrical liquid domains (how). For example, the mean thickness of the liquid domains, how, in Figure 4C is 10 nm, which is close to the critical thickness on the more wettable part (hc ) 12.4 nm). Since an increased thickness decreases the spinodal parameter, the length and time scales of the spinodal instability also increase on the more wettable stripes. In addition, the spinodal parameter is already weak for the more wettable stripes compared to the less wettable stripes even at the same thickness. Thus, the liquid cylinders on the more wettable stripes are quasistable and persist for very long periods of time compared to the time for the formation of cylinders. Figure 5A-C depicts the morphological evolution in a spinodally unstable relatively thin (2.5 nm) film for a fixed width of the less wettable stripes as the periodicity interval, Lp, is increased (width of the more wettable stripes is increased). The spinodal instability in such films evolves by the formation of drops rather than holes. For Lp close to λl (Figure 5A), the number and width of dewetted stripes are equal to the number and width of the less wettable stripes (image 5 of Figure 5A). The liquid morphology thus closely replicates the surface energy pattern. Figure 5A does not show spinodal instability in the transverse direction over the chosen simulation domain length (∼4.5λw). However, for a larger simulation domain lengths (∼9λw), the liquid regions disintegrate into droplets separated by an average distance slightly larger than λmw (simulation not shown). The distance λmw is close to the dominant spinodal length scale on the basis of the mean thickness of the cylindrical liquid domain (how). In

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Figure 6. 3D morphologies at a late stage of evolution for films of thickness (A) 2.5 nm, (B) 6.9 nm, and (C) 8 nm on a striped surface with W ) 1.2 µm and Lp ) 3 µm.

contrast to the evolution of thicker films, liquid cylinders formed by dewetting of thinner films are more likely to remain spinodally unstable due to their lower mean thickness. In addition to the spinodal instability in the transverse direction, the Rayleigh instability derived from the cross-sectional curvature of the cylinder also acts synergistically. Thus, disintegration of liquid ridges can occur even when the stripe width is somewhat less than the spinodal length scale, λw. In all cases where the transverse instability of the liquid ridges can occur leading to droplets, the ideal templating in the form of unbroken cylinders is obtained at an intermediate transient stage of evolution. At a larger periodicity interval (Lp ∼ 3λ, Wm ∼ 2.5λmw in Figure 5B), the contact line moves across the boundary of the stripes to reside in the interior of the more wettable part (image 3 of Figure 5B). The liquid cylinders again disintegrate into a matrix of droplets (image 5 of Figure 4B). As mentioned above, the average distance between two droplets (∼2 µm) is close to the spinodal length scale evaluated at the mean thickness (how ) 3.5 nm) of the liquid cylinder (1.65 µm, Λmw ) 37.8), which is larger than the spinodal length scale (λmw ) 1.07 µm, Λmw ) 48) at initial mean film thickness, ho. For very large periodicity intervals (e.g., Lp ∼ 6λl and Wm ∼ λmw in Figure 5C), the width of more wettable stripe can support several cycles of the spinodal growth in the lateral direction, resulting in the appearance of secondary spinodal structures on the more wettable stripes. For example, the formation of droplets arranged both in the lateral and the transverse directions is evidenced in the images 4 and 5 of Figure 5C. This leads to a poor templating. The formation of secondary structures also appears to slow the growth of the contact line into the more wettable stripes (images 2 and 3), in contrast to the case in Figure 5B. The condition for ideal templating (Lp g λh) can be achieved either by changing the substrate pattern, as in the above examples, or by changing λl, for example, by a change in the initial film thickness. The transition of surface morphology can also be clearly understood from Figure 6, which shows that ideal templating occurs for an intermediate thickness (image 4B of Figure 6) for which the spinodal wavelength, λl ()3 µm), is indeed equal to Lp. Thinner films (image A) evolve by the formation of droplets on the more wettable stripes due to the spinodal mechanism assisted by the Rayleigh instability. For thicker films (λl > Lp; image C), dewetting is suppressed on some stripes as discussed earlier and occurs partially on some others due to the formation of holes in the broad liquid ridges that result. Besides Lp and Wm, the other parameter that governs the thin film pattern is the width of the less wettable stripe (W). It is already known that, for a single heterogeneity on a large substrate, increase in the width of the heterogeneity shifts the onset of dewetting from the center of the heterogeneity to the boundary of the heterogeneity.22,23 An example for the system considered here is shown in the 2D simulation of Figure 7. For relatively small widths of the less wettable stripes, the rupture occurs at the center of stripes (Figure 7a). Beyond a certain

Figure 7. Evolution of instability in a 5 nm thick film on striped surface (Lp ) 3 µm ∼ 2λl, Χp ) 33.73). W ) 0.2λl and 1.3λl for (A) and (B), respectively. Profiles 1-3 in (A) correspond to T ) 180, 303, and 729, respectively. Profiles 1-3 in (B) correspond T ) 117, 192, and 291, respectively.

transition width, the film breakup shifts toward the boundaries of the stripe (Figure 7b). The off-center rupture traps a liquid cylinder at the center of the stripe, and the number of liquid domains at this stage of evolution equals twice the number of less wettable stripes. Figure 8A-D shows the transition of patterning with the increase of the less wettable stripe width (W), keeping the periodicity, Lp, fixed (>λl). The example illustrates the effect of W in improving or worsening some aspects of templating. For small stripe widths (W ∼ 0.4λl; Figure 8A), rupture is initiated at the center of the less wettable stripes by the formation of depressions that coalesce to form rectangular dewetted regions that equal the number of less wettable stripes. However, dewetted regions have widths greater than the stripe width as the contact line comes to rest in the interior of more wettable stripes (image 4 of Figure 8A). For a larger stripe width (W ∼ 0.8λl), dewetting is initiated by a layer of holes at each of the two boundaries of the stripe (image 2 of Figure 8B). Coalescence of these two layers of holes leads to the dewetted regions containing some residual droplets (image 4 of Figure 8B). Further increase in the stripe width leads to the formation of two layers of holes at the boundaries separated by an elevated liquid cylinder around the center of the less wettable stripes (image 2 of Figure 8C). Thus, at an intermediate stage of evolution, the number of cylindrical liquid ridges becomes twice the number of more wettable stripes on the substrate (image 3 of Figure 8C). Eventually, liquid ridges on the less wettable regions disintegrate into droplets (not shown). Laplace pressure gradients cause ripening of the structure leading to the merging of the droplets with the liquid ridges (image 4 of Figure 8C), leading to an uneven appearance of the contact line. Further increase in the stripe width (Figure 8D) breaks down the whole order of the substrate pattern due to the formation and repeated coalescence of several layers of holes on each stripe that evolve into arrays of irregular droplets by coalescence, fragmentation, and ripening. Thus, a good synchronization of the thin film morphology with the substrate pattern also requires an upper limit on the width of the less wettable stripes, referred to as the transition width, Wt, here (Wt < 0.8λl for this case). Similar considerations also apply to the systems where the spinodal evolution occurs by the formation of droplets

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Figure 10. Morphology diagram for the potential of eq 5 for ∆Φ/(-∂Φ/∂H) ) 0.31. The broken lines 1-4 denote the boundaries between different regimes at the onset of dewetting as shown in the figure by three symbols. The shaded region corresponds to good templating such that nD ) nS, WC < W < Wt , and the contact line is located at the stripe boundaries.

Figure 8. Morphological evolution in a 5 nm thick film on a striped surface (Lp ) 3 µm ∼ 2λl, Χp ) 33.73). W ) 0.6, 1.2, 2.1, and 2.7 µm (Γ ) 6.7, 13.5, 23,6, 30.20) respectively for (A)-(D). Gray scale images from left to right in (A) correspond to T ) 154, 186, and 615, respectively. Images from left to right in (B) correspond to T ) 274, 366, and 489, respectively. Images in (C) are at T ) 250, 363, and 6906, respectively. Images in (D) are at T ) 343, 397, and 2519, respectively.

Figure 9. Morphological evolution in a 2.5 nm thick film on a striped surface (Lp ) 1.2 µm ∼ 2λl, Xp ) 54). W ) 0.3, 0.8, and 1.02 µm (Γ ) 13.5, 36, 45.84) respectively for (A)-(C). Gray scale images from left to right in (A) correspond to T ) 1049, 73 653, and 108 525, respectively. Images from left to right in (B) correspond to T ) 1056, 8712, and 125 640, respectively. Images in (C) correspond to T ) 1685, 8885, and 24 289, respectively.

rather than by holes. Figure 9A-C depicts the sequence of evolution of a relatively thin (2.5 nm) film for different widths of the less wettable stripes and Lp ) 2λl. For small widths (Figure 9A), rupture is initiated at the center of the less wettable stripes and the number of dewetted stripes equals the number of less wettable stripes (nD ) nS). However, the liquid cylinders formed on the more wettable regions disintegrate rapidly. An increase in the width leads to off-center ruptures satisfying the condition

nD )2nS at an intermediate stage of evolution (images 2 of Figure 9B,C). The intermediate liquid cylinders on the less wettable stripes break up into droplets (image 3) that are eventually absorbed into the liquid cylinders on the more wettable stripes due to Laplace pressure gradients. Finally, the primary liquid ridges on the more wettable stripes can also break up (e.g., image 4 of Figure 9C). A good intermediate stage templating in such cases is possible only when the rate of absorption of the secondary features, formed on the less wettable stripes, into the primary ridges occurs before the disintegration of the primary ridges (as is the case in Figure 9B,C). It is clear from the above discussions that the ideal templating and different morphologies on a patterned substrate depend on the parameters Lp and W vis-a-vis the spinodal length scale. The conclusions regarding ideal templating can be quantitatively summarized on a morphology diagram displayed in Figure 10. Both the periodicity interval, Lp, and less wettable stripe width, W, are normalized with respect to the spinodal length scale, λl. Different symbols denote different types of quasiequilibrium morphologies formed before the long time processes of lateral ripening and transverse instability of liquid cylinders set in. Figure 10 shows four boundaries between different morphological “phases”. Below the horizontal boundary labeled 1, where W ) WC, the films do not rupture by the heterogeneous mechanism and therefore the underlying order dictated by the substrate is lost. The vertical boundary (2) at Lp ) λh ∼ λl indicates the shift in the number of dewetted stripes, nD, from less than nS to equal to nS. The horizontal boundary (3) indicates the transition width Wt, where initial rupture shifts from the center to the stripe boundary. Interestingly, for a periodicity interval, Lp, less than 2λl, the condition nD ) 2nS is never obtained and the morphology denoted by the solid squares is always absent regardless of the value of W. For Lp > 2λl (boundary 4) and W > Wt, the number of liquid cylinders formed are greater than the total number of stripes. Shifts in the morphology also occur (not shown) when the periodicity interval is very large (>(λmw + W)) and secondary spinodal structures can appear in the lateral direction on the more wettable part (Figure 5C). Also for a very large stripe width W, growth of spinodal structures on the less wettable part of the substrate leads to complete disorder (Figure 8D). These boundaries are not shown in the figure since these (undesirable) transitions are more obvious. Clearly, ideal templating is possible only in the parameter space λl < Lp < 2λl and Wt > W > WC. However, a part of this parameter space also does not lead to ideal templating because of the reasons discussed below.

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Figure 11. Variation of characteristic length scale of periodicity, λh, of substrate pattern (boundary 2 in Figure 10) and transition width of the less wettable stripe, Wt (boundary 3 in Figure 10), with the ratio ∆Φ/(-∂Φ/∂H) for the potential of eq 5. The ratio ∆Φ/(-∂Φ/∂H) is varied by changing the coating thickness dc1 on the more wettable stripes from 0 to 2.5 nm, keeping the other parameters the same as those given in the text for eq 5.

Figure 13. Initial evolution of a 6 nm thin film on a periodic physically heterogeneous substrate with B ) 0.2. The dotted line represents the substrate. Periodicity intervals for (A) and (B) are 0.45λmt and λmt, respectively. Profiles 1-4 in (A) correspond to T ) 2091, 2522, 2893, and 33 309, respectively. Profiles 1-3 in (B) are at T ) 429, 603, and 864 031, respectively. Figure 12. Position of contact line on the morphology diagram for the potential of eq 5. The region bounded by two solid lines denotes the morphology for which the contact line is located at the stripe boundaries.

As discussed earlier, increase in the potential difference across the stripe boundary reduces the characteristic lengthscale, λh. Figure 11 shows that Wt ≈ λh, and an increase in the potential difference reduces equally both the characteristic length scale, λh, and the transition width, Wt. Thus at a larger value of ∆Φ/(-∂Φ/∂H), the vertical boundary (2) shifts toward the left and the horizontal boundary (3) shifts down. Figure 12 shows the position of contact lines in the quasistable morphologies shown in Figure 10, for different combinations of the stripe width, W, and periodicity interval, Lp. The contact line comes to rest in the interior of the more wettable stripes when the less wettable stripes are very narrow or the ratio of the nonwettable stripe width to the periodicity interval (W/Lp) is small. A large value of W/Lp on the other hand causes the contact line to remain in the interior of the less wettable stripes. Figure 12 shows that the contact line remains close to the stripe boundaries only in a range of stripe widths. As discussed earlier, for the substrate having alternating completely wettable and partially wettable stripes, the contact line cannot move across the boundary. For such systems, the lower boundary of Figure 12 coincides with the horizontal boundary (1) of Figure 10. In summary, on a striped surface, ideal templating in the form of cylindrical liquid ridges that nearly cover the more wettable stripes (without secondary features on the less wettable stripes) occurs when (shaded region in Figure 10) (a) periodicity of the substrate pattern lies between λh and 2λh (boundaries 2 and 4 in Figure 10), (b) the stripe width is larger than a critical width, WC, which is effective in causing rupture by the heterogeneous mechanism, but smaller than a transition width, Wt, that ensures initiation of dewetting at the stripe center (boundaries 1 and 3 in Figure 10), and (c) the contact line remains close to the

stripe boundary. The shaded region in Figure 10 meets all of the above conditions for good templating. It can also be inferred that, for a system having alternating completely wettable stripes, the ideal templating can be obtained for a wider range of periodicity interval (λh < Lp < (λmw + W)). Also, in the films considered here that experience a long-range repulsion, ideal templating is more robust for thicker films due to the absence or weakness of the spinodal instability in the transverse direction. 3.2. Physically Heterogeneous Surfaces. Figure 13a,b, which is based on the 2D simulations, shows the initial evolution of a 6 nm polymer film on a periodic physically heterogeneous nonwettable substrate for two different periodicity intervals. Every elevation on the substrate is a potentially dewetting site (nS ) 6) since the film thickness is locally reduced there. Figure 13a shows the evolution, rupture, and dewetting on a substrate with relatively closely packed heterogeneities (Lp ∼ 0.2λm ∼ 0.45λmt, where λm is the dominant spinodal length scales based on the mean film thickness, h0). For a physically rough substrate, it is useful to define another spinodal lengthscale, λmt, on the basis of the minimum initial thickness, h0 - a, over the substrate peaks. For the potentials chosen here, λmt < λm. The film breakup occurs only on two spots (nD ) 2). Dewetting continues by retraction of the contact line on the substrate contours. Increase in the periodicity interval close to 0.45λm (∼λmt) results in the film rupture on all the elevated regions (nD ) nS ) 6). Large numbers of such simulations indeed showed that, on a substrate having a high density of potentially dewetting sites, a spinodally unstable film ruptures only on the sites (randomly picked) separated by the distances of the order of the local spinodal scale, λmt. Dewetting on the remaining sites is suppressed since the rupture on each site would require surface deformations on smaller scales resulting in a high energy penalty. On a physically heterogeneous substrate, a spinodally stable film (λm ) ∞, based on the mean thickness) can also rupture whenever the local thickness over the elevated

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Figure 14. Initial evolution of a 8 nm thin film on a periodic physically heterogeneous substrate with B ) 0.3. The periodicity interval is 2.5λmt. Profiles 1-4 correspond to T ) 882, 2.76 × 1013, 4.55 × 1014, and 2 × 1015, respectively.

Figure 15. Morphological evolution of a 5 nm thin film on a sinusoidal rough surface. Lp ) 0.5λm (∼0.67λmt) and 0.75λm (∼λmt) for (A) and (B), respectively. The first image shows the surface pattern. Gray scale images from left to right in (A) correspond to T ) 875, 1129, 1440, and 2358, respectively. Images from left to right in (B) correspond to T ) 794, 966, 1050, and 1375, respectively.

regions is reduced to the spinodally unstable region. Figure 14 shows the evolution of a spinodally stable film on a physically heterogeneous substrate with a periodicity interval that ensures a single rupture on every elevated region. Unlike the case of a spinodally unstable film, the periodicity interval in this case is about 2.5λmt (larger than λmt). This is due to the greater stabilizing influence of the long-range repulsion since a larger fraction of the film surface stays in the spinodally stable region. Thus, on a physically heterogeneous substrate, the film profile shows the reverse representation of the substrate profile; i.e., liquid collects in the substrate valleys, and holes form and grow on the substrate elevations. 3D simulations provide additional morphological features and conditions of ideal patterning. Figure 15A,B depicts the effect of periodicity interval on the self-organization of a 5 nm thick film on a physically heterogeneous substrate with B ) 0.1. In both the cases, depressions begin on the elevated regions. For Lp smaller than λmt, the isolated holes that initially line up along the elevated regions undergo coalescence both in the lateral and transverse directions. Coalescence among the holes belonging to different parallel layers result in a disordered structure (Figure 15A). However, for Lp ∼ λmt, the interlayer coalescence is suppressed, and the number of liquid cylinders formed equals the number of elevated regions (last image of Figure 15B). Anisotropy of the pattern originates from the difference in the length scales in the lateral and transverse direction of the substrate (x-y plane). The length scale in the lateral direction is directed by the surface pattern, whereas that in the transverse direction is dictated by the spinodal length scale, λm. In summary, the ideal reversing of substrate pattern in the film is possible only if the periodicity interval, Lp, is between the two spinodal length scales λm and λmt.

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Figure 16. Morphology diagram for a spinodally unstable film on a physically heterogeneous substrate. The broken line denotes the boundaries between different regimes at the onset of dewetting as shown in the figure by two symbols.

The strongly anisotropic orientated patterns shown in Figure 15B are in conformity with a recent experimental observation11 which shows continuous lines of polymer running across the rubbed substrate, where the length scale is close to the spinodal length scale. The morphologies shown in Figures 13b (curve 2) and 15B closely resemble the experimental images of a thin PS film on corrugated silicon substrate, reported recently by Rhese et al.18 It is clear from the above discussion that the ideal reversing of the substrate profile and the different morphologies on a physically patterned substrate depend on the parameters Lp and B vis-a-vis the spinodal length scale. The conclusions regarding the reversal of pattern can be quantitatively summarized on a morphology diagram displayed in Figure 16 for a spinodally unstable, 5 nm thick film. The y axis (1 - B) denotes the nondimensional initial minimum film thickness measured on the substrate elevation. The two distinct morphologies are denoted by two different symbols, and the broken line is the boundary between these two morphological phases. For small amplitudes (B ∼ 0.05) of the heterogeneity, the morphological transition occurs for the periodicity close to the spinodal wavelength. At higher surface roughness (at lower 1 - B), the morphological transition shifts to smaller values of the substrate periodicity compared to the mean spinodal length. A better understanding of the transition results when the transition periodicity is evaluated in terms of the spinodal length scale, λmt, on the basis of the minimum initial film thickness rather than the mean film thickness as in Figure 16. The ratio (Lp/λmt) at transition for B ) 0.03, 0.05, 0.1, 0.25, and 0.35 equals 0.85, 0.89, 0.79, 0.80, and 0.75, respectively. Thus, as mentioned earlier, the loss of ideal templating occurs when the separation distance, Lp, becomes smaller than about 0.8λmt. Ideal reversal of surface pattern demands rupture on each substrate peak without any secondary ruptures in between. For very large periodicity intervals (Lp . λmw + λml), secondary ruptures can however occur between the consecutive substrate peaks, which also destroys ideal templating. This boundary is not shown in Figure 16, since it is trivial. 3.3. Combined Physical and Chemical Heterogeneity. Finally, evolution of a thin film on a chemically heterogeneous, rough surface is considered. Although many spatial combinations of chemical and physical heterogeneities are obviously possible, we consider here the most interesting case where the chemical and physical heterogeneities exert opposing influences on the film stability. In such a system, the more wettable and less wettable regions are confined to the hills and valleys of the substrate, respectively. The arrangement is similar to an experimental system studied recently.6 Figure 17A,B

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Figure 17. Morphological evolution in a 5 nm thick film on a sinusoidal surface having more wettable stripes at the peaks (B ) 0.1; for (A) and (B), Lp ) 4 and 24 µm, Xp ) 45 and 269.5, W ) 0.8 and 2.4 µm, and Γ ) 9 and 26.95, respectively). The first image shows the surface pattern. The gray stripes at the center of the black region (peak of the hills) denote the more wettable stripes. Images from left to right in (A) correspond to T ) 108, 175, 241, and 325, respectively. Images from left to right in (B) are at T ) 124, 461, 753, and 3199, respectively.

Figure 18. Cut-sectional views of 3D images shown in Figure 17A. The more wettable part of the sinusoidal substrate is shown by the dotted line. Curves 1-3 show the evolution at nondimensional times T ) 108, 241, and 325, respectively.

shows the transition of patterning with an increase in the physical periodicity interval (Lp) on such a substrate. A cut-sectional 2D view (Figure 18) taken from the 3D simulations of Figure 17A shows this process of dewetting with greater clarity. The initial rupture is always induced by the gradient of chemical potential at the boundary of the regions of two different wettabilities (curve 2 of Figure 18, image 3 of Figure 17A, and image 2 of Figure 17B). As mentioned earlier, dewetting on a physically heterogeneous substrate is always initiated at the top of the hills, where the local thickness is the minimum. In contrast, liquid collects on the more wettable part of the substrate in this case (top of the hills), and the contact line moves toward the less wettable part of the substrate (valleys) (curve 3 of Figure 18). This behavior indicates a dominant role of chemical heterogeneity. The resulting pattern shares characteristics of dewetting on both chemically heterogeneous and physically heterogeneous substrates. As in chemical heterogeneity, liquid ridges form on the more wettable stripes, but they also form on valleys as in the case of physical heterogeneity. Thus, the number of liquid ridges equals the number of more wettable stripes plus the number of valleys. In this way, parallel ordered stripes of dewetted regions separated by liquid cylinders are formed in the case of a small periodicity interval (Lp) (Figure 17A). Of course, it is possible to decrease the volume collected on the more wettable part near the top of the hill by making the influence of physical heterogeneity dominate, e.g., by increasing the substrate roughness and by making the chemical potential contrast very weak (results not shown). However, for most realistic combinations of parameters, we found the effect of chemical heterogeneity to be more significant in causing the initial rupture. A recent study

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of Rockford et al.6 also shows the greater importance of chemical heterogeneity in determining the pattern formation by a PS film on a physicochemically heterogeneous surface, consisting of alternating Au (at the peak) and silicon stripes. No tendency of rupture on the substrate peak was evident due to the presence of more wettable Au there. Dewetting occurred on the oxide stripes and the polymer collected on the gold stripes.6 An increase in the periodicity interval (Lp) leads to the formation of several ordered layers of spinodally formed holes adjacent to the heterogeneously dewetted region at the later stage of evolution (images 2 and 3 of Figure 17B).22,23 Figure 17B also shows that the growth of the heterogeneously formed dewetted stripes is stopped as soon as ordering of satellite holes starts. These phenomena have been discussed in detail for a single stripe elsewhere.22,23 Eventually, the film dewets the less wettable part of the substrate by the coalescence of spinodally formed holes and subsequent disintegration of the liquid ridges into droplets. Thus, the pattern at late times consists of liquid cylinders separated by spaces filled with liquid fragments of varying shapes and sizes. This is however not the final equilibrium situation, and at very long times, the liquid fragments have the tendency to slowly merge with the liquid cylinders due to ripening (results not shown). In summary, the presence of chemical heterogeneity largely controls the thin film pattern and usually dominates over the effect of physical heterogeneity. 4. Conclusions The 3D thin film morphology on a patterned substrate during dewetting can be modulated profoundly by a competition among the time scales (spinodal and heterogeneous times scales for rupture and time scale of dewetting) and length scales (spinodal, stripe width, periodicity, thickness) of the problem. On a chemically patterned substrate, ideal templating occurs when (a) the periodicity of the substrate is between λh and 2λh, where λh, the characteristic length scale of instability, is close to but slightly less than the spinodal length scale (λl) of the less wettable part, (b) the less wettable stripe width is larger than WC but smaller than a transition width, Wt, which ensures the initiation of dewetting at the stripe center, and (c) the contact line remains close to the stripe boundary. Even a very weak wettability contrast (e.g. Figures 4 and 5) is sufficient to align the thin film pattern with the substrate template under the above conditions. The above predictions are found to be valid for both the models used here for expressing the excess interaction energy of long-range repulsion combined with short-range attraction. However, for the system where one of the alternating stripes is completely wettable, the widow of parameters for the ideal templating is widened. Predictions of our simulations show close resemblance to the recently reported experimental observations on dewetting of polymers on patterned surfaces that reveal (a) best templating occurs for an intermediate thickness film8,12 and (b) a correspondence between the natural length scale and periodicity of the substrate pattern for good templating.6,7,8 On a substrate containing many potentially destabilizing sites, only the sites (randomly picked) separated by a characteristic length scale (of the order of the spinodal scale), λh, remain “live” or effective in causing the film rupture. This study also reveals that periodic rough surface can lead to a good templating of liquid films only when the substrate periodicity is higher than about 0.8 times the spinodal length evaluated for the minimum initial film

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thickness, h0 - a. In this case, the thin film profile shows the ideal reverse representation of the substrate profile; i.e., liquid domains form on the valleys and holes appear on the hills (Figures 13B and 15B). The resulting morphologies obtained from simulations show close resemblance to the recently reported experimental observations by Rhese et al.18 On a physicochemically patterned substrate, the presence of chemical heterogeneity largely controls the points of initial rupture. A much richer variety of thin film patterns can be synthesized on a periodically rough, chemically heterogeneous substrate since the resulting pattern contains signatures of both types of heterogeneities. For example, we have considered a substrate containing more wettable regions on the elevated parts of the substrate. The resulting morphology under the conditions of good templating consists of liquid ridges of unequal width on top of the more wettable hills and also on the less wettable valleys. The dominant role of chemical heterogeneity on weakly rough surfaces in determining the pattern is also supported by the recently reported experimental observations by Rockford et al.6 It is hoped that this study will help in the design, interpretation, creation, and rational manipulation of selforganized microstructures in thin films by templating.

related to the excess free energy/unit area, ∆G, by the following relation: φ ) ∂∆G/∂h. The continuity equation is

∂u/∂x + ∂v/∂y + ∂w/∂z ) 0

The normal (Vn) and the tangential (Vt) components of the velocity are zero everywhere on the rough solid surface due to the conditions of impermeability and no-slip, respectively. Thus, for any arbitrary shape of the surface, all the three components of the velocity vanish at the solid-liquid boundary.

µ∂2u/∂z2 ) ∂p/∂x + ∂φ/∂x

(A1)

µ∂2v/∂z2 ) ∂p/∂y + ∂φ/∂y

(A2)

∂p/∂z + ∂φ/∂z ) 0

(A3)

u ) 0, v ) 0 at z ) af(x,y)

(A5)

w ) 0 at z ) af(x,y)

(A6)

Using the lubrication approximation, the zero shear stress condition and the pressure jump condition at the thin film surface become

∂u/∂z ) ∂v/∂z ) 0 at z ) h(x,y)

(A7)

p - p0 ) -γ(∂2h/∂x2 + ∂2h/∂y2)

(A8)

and

Appendix A Figure 1 shows the thin film system on a rigid physically heterogeneous surface. The relation z ) af(x,y) describes the surface from a datum coordinate. f(x,y) is a function of the spatial coordinates x and y, and a is the maximum amplitude calculated from z ) 0. The thin film evolution equation on a physically heterogeneous substrate is derived using the same formalism as that used in the derivation of the well-known thin film equation on a homogeneous substrate.22,23,27,28 Using the lubrication approximation,33,34 the x, y, and z momentum balance equations with V ) (u,v,w) are written respectively as follows:

(A4)

where p0 is the constant pressure outside the film. The kinematic equation is

∂h/∂t + u∂h/∂x + v∂h/∂y ) w(h)

(A9)

Solution of the above set of equations describes the evolution of a thin film on a physically heterogeneous surface with time:

3µ∂h/∂t + ∂/∂x[(h - af(x,y))3∂/∂x{γ(∂2h/∂x2 + ∂2h/∂y2) - φ}] + ∂/∂y[(h - af(x,y))3∂/∂y{γ (∂2h/∂x2 + ∂2h/∂y2) - φ}] ) 0 (A10) The thin film potential, φ(η,x,y), on a physically and chemically heterogeneous substrate is a function of the local film thickness, η ) [h(x,y) - af(x,y)], and of the spatial coordinates, x and y.

where φ is the excess intermolecular interaction potential

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(33) Ruckenstein, E.; Jain, R. K. J. Chem. Soc., Faraday Trans. 1974, 70, 132.

(34) Williams, M. B.; Davis, S. H. J. Colloid Interface Sci. 1982, 90, 220.