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Mesopatterning of Thin Liquid Films by Templating on Chemically Patterned Complex Substrates Kajari Kargupta*,† and Ashutosh Sharma‡ Department of Chemical Engineering, Indian Institute of Technology, Kanpur, UP 208016, India, and Department of Chemical Engineering, Jadavpur University, Kolkata, West Bengal 700032, India Received September 30, 2002. In Final Form: February 12, 2003 Surface directed instability and dewetting in thin films and resulting morphologies are studied using 3D nonlinear simulations based on the equations of motion, both for the isotropic and anisotropic 2D substrate patterns. Three different substrate (wettability) patterns are considered: (a) arrays of more (or completely) wettable rectangular blocks on a less wettable substrate, (b) arrays of less wettable blocks on a more (or completely) wettable substrate, and (c) a checkerboard pattern of alternating more and less wettable blocks. An ideal replication of the surface energy pattern produces an ordered 2D array of liquid columns (in case 1), or a matrix of holes on a flat liquid sheet (in case 2), or a checkerboard pattern of alternating liquid columns and holes/depressions (in case 3). The effects of pattern periodicity, domain widths, anisotropy, and wettability on the morphological phase transitions are presented. Regardless of the precise geometry of the substrate pattern, templating is found to be better on a completely wettable substrate containing the less wettable blocks, rather than on a less wettable substrate having more wettable blocks. Complete wettability (zero contact angle at all thicknesses), of either the blocks or their surroundings, ensures the pinning of the liquid contact line at the block boundaries. Thus, complete wettability leads to better templating compared to partially wettable substrates. Ideal templating is found possible only when the following conditions are met: (a) the periodicity (Lpx and Lpy) of the pattern is more than λh; where λh is a characteristic length scale found to be close to the spinodal length scale of the less wettable part, λm, (b) the less wettable area fraction, Af, should be less than a transition value beyond which the liquid spills over the less wettable part leading to a morphological transition from discrete columnar structure to continuous liquid ridges, (c) less wettable block/channel width should be less than a transition length scale (∼0.5λm), and (d) the aspect ratio of the periodicity intervals (Lpx/Lpy) should be close to 1. Anisotropy in the substrate periodicity leads to stripe-like liquid patterns whenever either Lpx or Lpy is less than λh. Large values of periodicity lead to the formation of novel “block mountain-rift valley” or “flower” like microstructures that do not replicate the substrate energy pattern. Interestingly, the near ideal templating of more complex substrate patterns, e.g., alphabets, is also guided by the above conditions.
1. Introduction Nano- and microscale patterned materials play an important role in photonics, biology, microfluidics, and a variety of optoelectronic devices, for example, high-density recording devices. Closely spaced rectangular arrays of cylindrical polymer posts or columns are used as sensors and synthetic media for separation processes involving DNA. Recently, techniques utilizing dewetting,1-6 condensation,7-9 and phase separation10-12 on physicochemically heterogeneous substrates have appeared as simple meth* To whom correspondence should be addressed. † Jadavpur University. ‡ Indian Institute of Technology. (1) Higgins, A. M.; Jones, R. A. L. Nature 2000, 404, 476. (2) Rehse, N.; Wang, C.; Hund, M.; Geoghegan, M.; Magerle, R.; Krausch, G. Eur. Phys. J. E 2001, 4, 69. (3) Gleiche, Chi, L. F.; Fuchs, H. Nature 2000, 403, 173. (4) Rockford, L.; Liu, Y.; Mansky, P.; Russell, T. P. Phys. Rev. Lett. 1999, 82, 2602. (5) Sehgal, A.; Ferreiro, V.; Douglas, J. F.; Amis, E. J.; Karim, A. Langmuir 2002, 18, 7041. Sehgal, A. Personal communication. (6) Han, Y. Personal communication. (7) Gau, H.; Herminghaus, S.; Lenz, P.; Lipowsky, R. Science 1999, 283, 46. (8) Kumar, A.; Whitesides, G. M. Science 1994, 263, 60. (9) Braun, H.-G.; Meyer, E. Thin Solid Films 1999, 345, 222. (10) Boltau, M.; Walhelm, S.; Mlynek, J.; Krausch, G.; Steiner, U. Nature 1998, 391, 877. (11) Nisato, G.; Ermi, B. D.; Douglas, J. F.; Karim, A. Macromolecules 1999, 32, 2356. (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: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1998, 57, R6273.
ods to engineer mesoscale patterns in soft materials without the lithographic processes. The basic idea is to modulate the instability of the free surface of a thin film by application of chemical3-32 or electrical33,34 force fields (13) Kataoka, D. E.; Troian, S. M. Nature 1999, 402, 794. (14) Lenz, P.; Lipowsky, R. Phys. Rev. Lett. 1998, 80, 1920. (15) Lenz, P.; Lipowsky, R. Eur. Phys. J. E 2000, 1, 249. (16) Lipowsky, R.; Lenz, P.; Swain, P. S. Colloid Surf., A 2000, 161, 3. (17) Kargupta, K.; Sharma, A. Phys. Rev. Lett. 2001, 86, 4536. (18) Kargupta, K.; Sharma, A. J. Chem. Phys. 2002, 116, 3042. (19) Kargupta, K.; Sharma, A. Langmuir 2002, 18, 1893. (20) Bauer, C.; Dietrich, S.; Parry, A. O. Europhys. Lett. 1999, 47, 474. (21) Bauer, C.; Dietrich, S. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2000, 61, 1664. (22) Rascon, C.; Parry, A. O. Phys. Rev. Lett. 1998, 81, 1267. (23) Rascon, C.; Parry, A. O.; Sartori, A. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1999, 59, 5697. (24) Bru¨sch, L.; Ku¨hne. H.; Thiele, U.; Ba¨r, M. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2002, 66, 011602. (25) Tsori, Y.; Andelman, D. Macromolecules 2001, 34, 2719. (26) Tsori, Y.; Andelman, D. Europhys. Lett. 2001, 53, 722. (27) Tsori, Y.; Andelman, D. J. Chem. Phys. 2001, 115, 1970. (28) Kielhorn, L.; Muthukumar, M. J. Chem. Phys. 1999, 111, 2259. (29) Pereira, G. G.; Williams, D. R. M. Phys. Rev. Lett. 1998, 80, 2849. (30) Pereira, G. G.; Williams, D. R. M. Macromolecules 1999, 32, 758; Langmuir 1999, 15, 2125. (31) Pereira, G. G.; Williams, D. R. M. Langmuir 1999, 15, 2125. (32) Nath, S.; Nealey, P. F.; de Pablo, J. J. J. Chem. Phys. 1999, 110, 7483. (33) Scha¨ffer, E.; Thurn-Albrecht, T.; Russell, T. P.; Steiner, U. Nature 2000, 403, 874. (34) Scha¨ffer, E.; Thurn-Albrecht, T.; Russell, T. P.; Steiner, U. Europhys. Lett. 2001, 53, 518.
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to form ordered mesostructures. The morphology of such self-organized liquid mesostructures is largely controlled by the length scales of the substrate pattern. This method of pattern transfer appears to be of increasing promise for engineering of desired nano- and micropatterns in thin films by templating.1-9 There are now several experimental1-6 and theoretical14-24 studies of dewetting of thin films on simple 1D structures composed of parallel nonwettable and wettable stripes. Although these studies have resulted in significant advances in our understanding of some basic mechanisms of rupture and morphological organization on 1D patterns, most of the patterns envisaged to be useful in actual applications are fully 2D and anisotropic and are thus much more complex. In this paper, we consider the problem of self-organization and conditions for ideal templating of thin films on 2D, isotropic, and anisotropic complex patterns. In addition to exploring the mechanisms that are common both to the 1D and to the 2D dewetting, we establish the novel self-organizational features related to the 2D complex patterns. Toward this end, a brief overview of the previous work on the uniform and 1D substrates is presented first. While spinodal dewetting on a homogeneous surface occurs by a somewhat randomly placed collection of holes and droplets,35-38 recent studies predict the strong influence of chemical heterogeneity of the substrate in determining the dewetting behavior and in orienting the morphology of microstructures.2,4-9,14-24,39,40 For example, Rockford et al.4 reported the pattern directed dewetting of thin polymer films on a laterally periodic Si substrate decorated by regular Au stripes, which pointed to a relation between surface interactions and macromolecular ordering. Gau et al.7 reported the formation of liquid microchannels and bridges on structured surfaces consisting of hydrophilic and hydrophobic stripes. They reported a new “bulging instability” of liquid channels where the fluid spills over to the nonwettable part of the substrate as the liquid volume is increased beyond a critical value. Substrate-induced ordering in block copolymers on chemically patterned substrate has been extensively studied.11,12,25-32 For example, Nisato et al.11 predicted a coupling between phase separation and the surface deformation modes, when the scale of phase separation becomes commensurate with the period of striped surface pattern. Karim et al.12 suggested the control of phaseseparation morphology through a surface pattern that modulates the polymer-surface interactions. They also observed an upper limit on the scale of striped surface pattern for good templating, as theoretically predicted.17-19 These studies have revealed the importance of commensurability of the substrate pattern and the intrinsic spinodal thin film pattern for ideal templating. It is now known that on a chemically heterogeneous substrate, dewetting is driven by a spatial gradient of microscale wettablility, rather than by the nonwettability of the substrate itself as in the spinodal dewetting.17-19,39,40 The equilibrium structures of liquid domains on periodic, chemically heterogeneous substrate have been investigated on the basis of minimization of the free (35) Reiter, G.; Khanna, R.; Sharma, A. Phys. Rev. Lett. 2000, 85, 1432. (36) Herminghaus, S.; Jacobs, K.; Mecke, K.; Bischof, J.; Fery, A.; Ibn-Elhaj, M.; Schlagowski, S. Science 1998, 282, 916. (37) Oron, A. Phys. Rev. Lett. 2000, 85, 2108. (38) Sharma, A.; Khanna, R. Phys. Rev. Lett. 1998, 81, 3463. (39) Konnur, R.; Kargupta, K.; Sharma, A. Phys. Rev. Lett. 2000, 84, 931. (40) Kargupta, K.; Konnur, R.; Sharma, A. Langmuir 2000, 16, 10243.
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Figure 1. Substrate energy patterns: (a) arrays of more (or completely) wettable blocks on a less wettable matrix; (b) arrays of less wettable blocks on a more (or completely) wettable substrate; (c) checkerboard pattern of more and less wettable blocks.
energy.14-16,20-23 On the basis of nonlinear simulations,17-19 we have recently reported induced dewetting and conditions for ideal templating of a thin film on a substrate consisting of alternating more and less wettable stripes. It was found that the simulated thin film pattern on chemical stripes ideally replicates the surface pattern only when the pattern periodicity is greater than a characteristic length scale (λh), which is close to (but less than) the spinodal length scale of instability on the nonwettable stripes. In addition, good templating required that the width of stripes should be less than an upper cutoff width beyond which dewetting is initiated at the stripe boundaries.17-19 Recent experiments by Sehgal et al.5 have indeed observed similar conditions for ideal templating of polystyrene thin films on micro-contact-printed thiol substrates with stripes of hydrophobic (-CH3) and hydrophilic (-COOH) end groups.5 They have reported morphological transitions from droplet doublets on each stripe to ideally templated confined channel states and finally droplet bridging and transitions to a disordered morphology as the stripe periodicity (less wettable stripe width) decreases. This study thus confirms the existence of the theoretically proposed lower and upper cutoff length scales for efficient pattern recognition. Recently, Han and co-workers6 created ordered microscale patterns by surface-directed dewetting using regularly patterned self-assembly monolayer (SAM). Regular 1D stripes, as well as isotropic block patterns were considered. The latter showed some novel 2D morphologies. Here, we explore whether spontaneous dewetting of liquid thin films on a chemically patterned surface can be a feasible route for the creation of general complex patterns by templating. Toward this end, we present the morphological transitions, variety of patterns, and conditions for the ideal templating of thin films on complex chemically patterned substrates. In particular, we have considered a number of different isotropic (square block) as well as anisotropic (rectangular block), 2D substrate patterns that involve two or more characteristic length scales in both the directions, X and Y (Figure 1), instead of a simple stripe pattern that involves characteristic length scale only in one direction. This leads to more complex hydrodynamic interactions between the chemical boundaries and to several novel patterns and dewetting scenarios not possible on 1D periodic substrates. On the basis of 3D nonlinear simulations, we study the effects of (i) periodicity (Lpx, Lpy) and width (Wx, Wy) of substrate patterns, (ii) aspect ratio of pattern periodicities (Lpx/Lpy), and (iii) wettabilities of surface phases on the morphological phase transitions. On the basis of these, it is possible to predict the general set of rules that ensure near ideal templating on both isotropically and anisotropically patterned 2D substrates. Some of the novel interesting features include (i) conditions for the formation of liquid bridges across the
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nonwettable domains and transitions from columnar to striped-like patterns, (ii) unique morphological features on a block patterned substrate due to enhanced 2D hydrodynamic interactions, (iii) morphologies and conditions for templating on a completely wettable substrate, (iv) ideal templating by surface deformation without the film rupture, and (v) effect of inversion of the substrate pattern (from less wettable blocks on a more wettable substrate to more wettable blocks on a less wettable substrate) on the final thin film pattern and templating. The above aspects are addressed for various combinations of intermolecular (attractive/repulsive) forces. This study, for the first time, also explores patterning on a substrate that is completely wettable (zero contact angle) by the liquid everywhere, but the strength of interactions (purely repulsive or island forming attractive/repulsive) is position dependent. The three different substrate patterns considered are shown in Figure 1: (a) arrays of more wettable or completely wettable blocks on a less wettable substrate, (b) arrays of less wettable blocks on a more wettable or completely wettable substrate, and (c) a checkerboard pattern of more wettable (or completely wettable) and less wettable blocks. For these three substrate patterns, an ideal replication of the substrate pattern in the film results in the formation of arrays of liquid columns, arrays of holes within a flat liquid sheet, and a checkerboard pattern of alternating liquid columns and holes, respectively. Thus, we first generalize the conditions of ideal templating on 2D periodic substrates, and finally, an alphabet-like complex pattern is considered to verify the general rules of ideal templating. 2. Theory 2A. Thin Film Equation. The following nondimensional equation derived based on the Navier-Stokes equation using the long-wave approximation governs the stability and spatiotemporal evolution of a Newtonian nonslipping thin film system on a substrate subjected to excess intermolecular interactions:17-19,39,40
∂H/∂T + ∇‚[H3∇(∇2H)] - ∇‚[H3∇Φ] ) 0
(1)
H(X,Y,T) is 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 and As is the effective Hamaker constant of the thin film on the substrate. X and Y are the nondimensional coordinates in the plane of the substrate, scaled with a characteristic length scale for the van der Waals case (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 surface tension effect in a 3D geometry may be stabilizing (due to “in-plane curvature as in 2D geometry”) or destabilizing (due to “transverse curvature as in Rayleigh instability of circular cylinders”). Finally, a gradient in conjoining pressure causes flow of fluid in the direction of decreasing force per unit area, Φ. For homogeneous substrates, Φ ) Φ(H), and the spinodal dewetting is caused by flow from 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 ho) of the initial disturbance. The resulting linear dispersion relation is (K2 ) (Kx2 + Ky2); ΦH0 ) ∂Φ/∂H evaluated at H ) 1)38
ω ) -(Kx2 + Ky2)ΦH0 - (Kx2 + Ky2)2
(2)
Thus, the nondimensional critical wavenumber, Kc ) (-ΦH0)1/2, i.e., only the modes with nondimensional wavenumbers Kx and Ky less than Kc (or length scales Λx and Λy larger than 2π/Kc) can grow. An interesting implication of the above relation is that even a very large length scale in any one of the two orthogonal directions cannot lead to the spinodal instability, when the length scale in the other direction is less than 2π/Kc. The corresponding dimensional critical length scale of the spinodal instability on a uniform surface is given by, λC ) (-4π2γ/(∂2∆G/∂h2))1/2. Further, the dominant wavelength (Λm ) 2π/Km) of the fastest growing (dω/dK ) 0) linear mode is λm ) (-8π2γ/(∂2∆G/∂h2))1/2. On a chemically heterogeneous surface, Φ ) Φ(H,X,Y). Hydrodynamic interactions arise due to the coupling of the flow fields on the neighboring parts where potential, Φ, is different. At a constant film thickness, the variation of Φ in X and Y directions is modeled by a periodic step function of periodicity Lpx and Lpy, respectively. Gradient of force, ∇Φ, at the boundary of the blocks causes flow from the less wettable (higher pressure) to the more wettable (lower pressure) parts, even when the spinodal stability condition ∂Φ/∂H > 0 is satisfied everywhere.39,40 A chemically heterogeneous patterned surface introduces one additional spinodal length scale. The dominant spinodal length scales on the less wettable and more wettable parts are denoted as λml and λmm, respectively. It is known that a single less wettable block in the absence of its neighbors can cause rupture only if its width exceeds a critical length scale, WC , λCl. 39,40 2B. Excess Intermolecular Interactions. We consider a fairly general representation of antagonistic (attractive/repulsive) short-range and long-range intermolecular interactions applicable to the polymer films on high-energy surfaces such as silicon (eq 3)4,17,19,39-41 and to the aqueous films (eq 4).38,43,44 In both cases, the longrange 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 the critical are metastable. 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 is17,19,40
-12π∆G ) [(As - Ac1)/(h + dc1 + dc2)2 + (Ac1 - Ac2)/(h + dc2)2 + Ac2/h2] (3) where h is the local film thickness. A negative value of the effective Hamaker constant (Asm ) Asl ) -1.88 × 10-20 J) signifies a long-range repulsion, whereas a positive value on coating (Ac1m ) Ac1l ) 1.13 × 10-20 J) represents an intermediate-range attraction.17,19,39,40 The nonwettable coating (e.g., oxide) thickness (dc1m ) 0.5 nm, dc1l ) 4 nm) is increased which causes the macroscopic contact angle (41) Kim, H. I.; Mate, C. M.; Hannibal, K. A.; Perry, S. S. Phys. Rev. Lett. 1999, 82, 3496. (42) Thiele, U.; Mertig, M.; Pompe, W. Phys. Rev. Lett. 1998, 80, 2869. (43) Padmakar, A. S.; Kargupta, K.; Sharma, A. J. Chem. Phys. 1999, 110, 1735. (44) Van Oss, C. J.; Chudhury, M. K.; Good, R. J. Chem. Rev. 1988, 88, 927.
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was approximated as a continuous function by the differencing scheme. The convergence was checked by reducing the grid size and also by introducing nonuniformly spaced grids at the boundary of the blocks in the initial trials. To a very good approximation, the checkerboard pattern of Figure 1c is constructed by using the following Fourier series for the square patterns48 (eq 6). The functional dependence of the potential, Φ, on the local film thickness, H(X,Y) and on the spatial position (X,Y) due to substrate heterogeneity are introduced as follows Figure 2. Variation of excess Gibbs free energy, ∆G, and spinodal parameter φh ()∂∆G2/h2) for an island forming system (eq 4): Asl ) -1.41 × 10-20 J; Spl ) -0.2317 mJ/m2; lp ) 1.5 nm.
to increase from 0° on the completely wettable part to 1.7° on the less wettable part. A still shorter range repulsion may rise due to chemically adsorbed or grafted layer of the polymer (Ac2 ) -0.188 × 10-20 J, dc2 ) 1 nm are the film Hamaker constant and the thickness of the adsorbed layer, respectively).40 In this example, the films are spinodally unstable below the critical thickness (hc) of 12.4 nm on the less wettable part of the substrate. For the simulations shown below the values of viscosity (µ) and interfacial tension (γ) for polymer film are taken: µ ) 1 kg/m‚s and γ ) 38 mJ/m2. The representation, eq 3, is chosen for illustration without affecting the underlying physics, which we have verified for many different potentials composed of antagonistic short- and long-range interactions. The following expression (eq 4) has also been frequently used to describe the long-range van der Waals and the shorter range non van der Waals excess interaction energy per unit area.42-44
∆G ) [-(As/12π h2) + Sp exp(-h/lp)]
(4)
The effective Hamaker constant, As, and the polar spreading coefficient, Sp, are both negative (long-range van der Waals repulsion combined with shorter range attraction, e.g., hydrophobic attraction44). A completely wettable island forming system43,45-47 is considered (Asl ) -1.41 × 10-20 J, Spl ) -0.2317 mJ/m2, and lp ) 1.5 nm) where ∆G is always positive but varies nonmonotonically with the film thickness (Figure 2). Thus the macroscopic contact angle is everywhere zero. However, the film can still be unstable when the spinodal parameter (φh) ∂2∆G/ ∂h2 < 0 (Figure 2). It is known that on a homogeneous substrate, instability in such a system leads to coexisting thick and thin macroscopic island-like regions.43,45 A purely repulsive potential (Asm ) -1.41 × 10-20 J, Spm ) 0.0)46 is considered on the other part of the substrate. For the simulations shown below, the values of viscosity (µ) and interfacial tension (γ) for water film are taken: µ ) 0.001 kg/m‚s and γ ) 72.8 mJ/m2. 2C. Numerical Methods. Equation 1 was directly solved for the 2D and 3D morphologies starting with an initial volume preserving small amplitude random perturbation and periodic boundary conditions. A central differencing scheme with half node interpolation is used to discretize the above thin film equation,17-19,38-40 and the time integration is done with the Gear algorithm to adequately address the numerical stiffness of the problem. The excess nondimensional potential, Φ, for the substrate pattern shown in parts a and b of Figure 1 was taken as a step function across the block boundary, which (45) Samid-Merzel, N.; Lipson, S. G.; Tannhauser, D. S. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1998, 57, 2906. (46) Sharma, A. Langmuir 1993, 9, 861. (47) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827.
Φ(H,X,Y) ) Φ(H) F(X,Y)
(5)
F(X,Y) ) [1 + 4H{sin(KHX) + sin(3KHX)/3 + sin(5KHX)/5} × {sin(KHY) + sin(3KHY)/3 + sin(5KHY)/5}/π] (6) where Φ(H) is the conjoining pressure evaluated for the mean value of parameters (As, Ac1, Ac2, Sp), H is the dimensionless amplitude of heterogeneity (conjoining pressure) and the wavenumber for the pattern, KH ) 2π(2πγ/|As|)1/2ho2/Lp; Lp is the pattern periodicity. The convergence of eq 6 was found to be adequate, since the overshoot at the point of discontinuity due to Gibb’s phenomena48 remains negligible with respect to the amplitude of heterogeneity (H) for as few as three terms. 3. Results and Discussion For illustrating the important results, three kinds of patterns were considered as described in Figure 1. The pattern periodicity or the (center to center) separation distance between two blocks of the same wettability in two orthogonal directions x and y are termed as Lpx and Lpy, respectively. The block width in x and y directions are termed as Wx and Wy, respectively. For an isotropic pattern, as shown in parts b and c of Figure 1, Lpx ) Lpy ) Lp and Wx ) Wy ) W. We investigate the morphological patterns that develop spontaneously in a thin film subjected to a long-range repulsion and a short/intermediate-range attraction on these periodic heterogeneous substrates based on 3D simulations and thereby explore the conditions for the ideal or nearly ideal templating. The term near-ideal templating can be defined as follows:17-19 (a) the less wettable parts are dewetted almost fully without any liquid remnants of the film remaining there; (b) liquid columns or posts of constant cross-sectional curvature form on the more wettable blocks; (c) the base of the liquid columns or ridges formed on the more wettable parts cover them completely, i.e., the contact lines are straight and reside close to the patch-boundaries; (d) the liquid ridges on the more wettable parts are almost flat. A few examples of ideal templating are given below before exploring in detail the conditions for ideal templating. In each picture of Figure 3, as well as in the subsequent figures, the first image represents the substrate pattern and the remaining four images depict the evolution of the thin film pattern at different times. In the first image, more (/completely) wettable and the less wettable parts are represented by black and white, respectively. For the other images, a continuous gray scale between the maximum and minimum thickness in each picture is used. Figure 3 depicts (48) Riley, K. F.; Hobson, M. P.; Bence, S. J. Mathematical methods for physics and engineering; Cambridge University Press: Cambridge, 1998.
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Figure 3. Ideal templating of complex surface patterns in thin films. The substrate energy pattern is represented by eq 4, by a combination of island-forming parameters (on less wettable parts) and pure repulsion (complete wettability of more wettable parts). In all the figures, the mean thickness is h0 ) 6.65 nm. Lp ) λml. In parts a-c, the less wettable area fractions are Af ) 0.4375, 0.25, and 0.5, respectively. The first image of each figure represents the substrate pattern. The rest of the images depict the thin film evolution with time. The second to fourth images of Figure 3a correspond to the nondimensional time, T ) 4.0, 21.45, 57.7, and 217.44, respectively. The images from left to right in Figure 3b are at time T ) 14.17, 54.28, 301.38, and 3396.72, respectively; in Figure 3c T ) 9.5, 53.9, 279.9, and 3750.53, respectively. For Figure 3c, the amplitude of heterogeneity, H ) 0.5.
Figure 4. Surface profiles of ideally templated thin film patterns. Parts a-c correspond to the last images of parts a-c of Figure 3, respectively.
the time evolutions of thin film patterns when the conditions are chosen such that they lead to a near ideal replication of the isotropic surface energy pattern in all the three different cases. Figure 4 shows the ideally templated surface profiles at lager times for the cases depicted in Figure 3. The ideal templating of the substrate pattern shown in the first image of Figure 3a leads to the formation of arranged liquid posts or columns on the more wettable blocks connected by a very thin remnant liquid film (Figure 3a, Figure 4a). For the pattern of Figure 3b, an XY array of holes, surrounded by almost flat liquid ridges (Figure 3b and Figure 4b), is formed on the less wettable blocks (Figure 1 b). Recently, dewetting experiments have been performed on this kind of pattern (Figure 1b).6 Similarly, alternating liquid columns and holes (Figure 3c and 4c) ideally template the checkerboard kind of substrate pattern. However, ideal templating occurs only when some specific conditions are met. Interestingly, as it will be shown later, these conditions are largely invariant for different types of potentials. An interesting scenario is presented in Figures 3 and 4 that show the evolution of a thin film on a substrate where the macroscopic contact angle of the liquid is zero everywhere (complete macroscale wettability), but subtle variations of the potential introduce instability (without
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Figure 5. Morphological evolution of a 5 nm thick film pattern on a coated substrate (eq 3) consisting of completely wettable blocks (Lp ) 0. 0.82, λm ) 1.75 µm). In parts a to d, the less wettable area fractions, Af, are 0.36, 0.51, 0.84, and 0.96, respectively. The second to fifth images of Figure 5a correspond to the nondimensional time, T ) 4.0, 24.4, 64.13, and 165.46, respectively. The images from left to right in Figure 5b are at time T ) 22.31, 54.48, 185.11, and 282, respectively; in Figure 5c are at T ) 122.42, 297.85, 442.45, and 706.89, respectively; in Figure 5d are at T ) 334.91, 394.0, 531.81, and 712.46, respectively.
complete rupture) and synchronized deformations of the film on some parts of the substrate (eq 4, Figure 2). In such cases, the resulting structure consists of largely flattopped islands rather than the curved liquid cylinders that form in partially wetting systems. 3A. Thin Film Patterns on Arrays of More Wettable Blocks. A. Liquid Bridging. Depending on its periodicity and the less wettable area fraction, more wettable blocks on a less wettable substrate can induce a large variety of ordered patterns different from the wettability pattern. For example, liquid bridges, arrays of liquid columns, arrays of columns and cylinders on a flat liquid sheet, etc. can all result. Figure 5 summarizes the effects of increasing less wettable area fraction on the resulting thin film morphology, on a less (partially) wettable substrate consisting of isotropic completely wettable blocks of periodicity, Lp close to λml. For a fixed periodicity interval, i.e., the distance between the two blocks (Lpx ) Lpy ) Lp ∼ 0.82λml), the area fraction is varied by varying the width (Wx ) Wy ) W) of the more wettable blocks. For this substrate, the area fraction of less wettable part, Af ) [1 - (WxWy/LpxLpy)]. A large ratio of W/Lp ()0.8), i.e., a small area fraction, Af ()0.36) leads to the formation of arranged arrays of liquid columns (image 5 of Figure 5a). Instability due to the chemical potential gradient (∇φ) at the block boundaries leads to depressions and dewetting on the less wettable parts of the substrate. Depression and dewetting start at the boundary of the four edges of the block. The underlying mechanism of heterogeneous dewetting has been discussed in greater details elsewhere.17-19,39,40 Accumulation of the drained liquid results in the formation of liquid columns on the wettable blocks. The liquid contact line remains at the boundary of the blocks. Thus, in this case, the substrate energy pattern is faithfully replicated in the film morphology. An increase in the less wettable area fraction (Af ) 0.51) by reducing the width of the completely wettable blocks (W/Lp ) 0.7) causes the spilling of liquid on the less wettable part of the substrates (image
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Figure 6. Morphological evolution of 5 nm thick film pattern on a coated substrate (eq 3) consisting of completely wettable blocks (Lp ∼ 2λm ) 4.25 µm). In parts a and b, the less wettable area fractions, Af, are 0.4375 and 0.64, respectively. The second to fourth images of Figure 6a correspond to the nondimensional time, T ) 119.26, 186.70, 394.05, and 1401.36, respectively. The second to fourth images of Figure 6b are at time T ) 101.14, 231.34, 292.04, and 779.6, respectively. The 3D surface profiles corresponding to the third image of Figure 6a and the fourth image of Figure 6b are also shown.
5 of Figure 5b). Some of the liquid columns on the more wettable part now remain connected by continuous liquid ridges on the less wettable part and form liquid bridges. Further increase in Af (Figure 5c,d) leads to increasing disorder that completely destroys the column-like structure. These liquid ridges and their fragments show preferred orientations (image 5 of Figure 5c). At larger times, these cylindrical liquid bridges further disintegrate into droplets due to the Rayleigh instability and the order of structure may be completely destroyed. With the increase in the less wettable area fraction, this transition of discrete liquid columns to continuous liquid bridges is found for all the different kind of potentials we investigated. However, the transition value of the area fraction differs depending on the system details. Simply put, spillage occurs due to high-energy penalty on the accumulation of a large amount of liquid on the completely wettable parts, when the area fraction of partially wettable regions is relatively large. In such a case, very high liquid surface areas and curvatures are avoided by the presence of some liquid on the less wettable parts. Experimentally, Sehgal et al.5 have observed droplet bridging across the less wettable stripes at the late stages of dewetting. Interestingly, their data5 also support the prediction that this happens when the less wettable area fraction becomes more than 0.5 (for example, Figure 7 of ref 5).5 The phenomenon of liquid spilling in the transverse direction over the less wettable stripes was also reported7,14-16 on the substrates consisting of alternating hydrophilic and hydrophobic stripes when the effective liquid volume per hydrophilic stripe becomes large. In contrast to the liquid spilling on a striped patterned substrate, two transitions occur on the isotropic more wettable blocks considered here, to reduce the energy penalty when the volume of liquid on the more wettable areas is increased: (1) the first transition produces ordered spilling of liquid in either of the two directions resulting in the liquid cylinders that have about the same width as the width of the more wettable blocks (Figure 5b), and (2) the second transition leads to spilling in both the directions and the formation of a bulge (Figure 5d). It will be shown below that one can obtain perfectly ordered stripe-like liquid bridges even on an anisotropic rectangular block pattern. B. Secondary Structures: A Novel “Block Mountain-Rift Valley” Pattern. Figure 6 summarizes the effect of separation distance between blocks (Lpx ) Lpy ) Lp), keeping the less wettable area fraction constant (constant W/Lp). Increase in the separation distance or
periodicity introduces additional interesting features in dewetting patterns. In contrast to Figure 5, an increased width of the less wettable channels in Figure 6 forces the onset of dewetting at the four edges of the more wettable blocks. Cylindrical liquid ridges are formed at the center of the less wettable channels. Columnar structures with largely flat tops form on the more wettable blocks. However, there is also some accumulation of liquid on the less wettable channel itself, leading to the formation of secondary liquid structures on it. Thus, on each less wettable channel, two narrower dewetted channels separated by a central liquid ridge are formed (image 3 of Figure 6a and images 3 and 4 of Figure 6b). In particular, the channel intersections trap substantial amounts of liquid, forming a network structure of nearly spherical droplets (at intersections) connected by the straight liquid cylinders at the center of the channels. The formation of secondary droplets at the intersection of liquid channels on a block-patterned substrate has recently been experimentally observed.6 At larger times, the long connecting liquid cylinders disintegrate into droplets by the Rayleigh instability. For the case, where the thickness of the columns is higher than the entrapped cylinders (for larger W/Lp), Oswald ripening causes these tiny droplets of high curvature (higher Laplace pressure) to gradually disappear and merge with the liquid columns (image 5 of Figure 6a). For larger less wettable area fractions, the thickness of these cylinders can be higher than the height of the columns (image 5 Figure 6b). The final “block mountain-rift valley” structure consists of an ordered array of liquid droplets on less wettable channels coexisting with the flat-topped columns on completely wettable blocks. Earlier studies on a single less wettable heterogeneous patch (or stripe) have already established17,19,39,40 that the rupture is initiated at the patch center for small patch sizes, whereas for larger sizes, dewetting starts at the patch boundaries. The former results in complete dewetting of the patch, but the latter leads to the formation of partially dewetted regions containing accumulated liquid drops (or cylinders).17-19,39,40 Recently, Sehgal et al.5 experimentally found the transition of confined state to doublet droplet state on each less wettable stripe with the increase of stripe width. The secondary structures appear due to the shifting of the onset of dewetting from the center to the boundary of the stripes width. Investigation here for the block patterns also follows this general observation. However, the droplet at the intersection of
Mesopatterning of Thin Liquid Films
Figure 7. Morphological phase diagram for the isotropic substrate pattern, consisting of more wettable blocks on less wettable substrate, for the potential of eq 3. The lines 1, 2, 3, and 4 denote the boundaries between different regimes at the initial stage of dewetting. The dark-shaded region corresponds to the conditions for ideal templating. The 3D microstructures corresponding to the five symbols are also shown.
the channels is a unique feature of patterns on a blockpatterned substrate. C. Morphological Phase Transitions. Figure 7 summarizes the important morphological phase transitions on an isotropically patterned substrate consisting of completely wettable blocks, with respect to the two normalized parameters: dimensionless periodicity and dimensionless block width. A large number of simulations show that decrease in Lp below a characteristic length scale, λh (∼0.7λm ∼ λc), cannot produce heterogeneous rupture and true dewetting but can only cause partial thickness deformations that however synchronize with the substrate pattern (left of boundary 1 in Figure 7). Recently, Sehgal et al. found experimentally such synchronous deformations in wetting films.5 For a substrate pattern consisting of parallel stripes, it was earlier observed that the necessary condition for the ideal templating requires that the pattern periodicity must be greater than λh.17-19 This study confirms it as a general condition for the ideal templating of complex 2D substrates as well. For the less wettable area fraction greater than 0.5 or the block width below a transition length (boundary 2, Figure 7 where W/λml ∼ 0.7Lp/λml; Af ∼ 0.5) liquid bridging across the less wettable regions occurs for Lp less than 1.5 λm (boundary 3, Figure 7). An increase in Lp beyond the boundary 3 eliminates the bridge formation at smaller W (larger Af) and produces trapped liquid domains on the less wettable part, resulting in the “block mountain-rift valley” pattern. For Lp/λml > 2 (boundary 4), the liquid ridges on the more wettable block do not remain flat but form a curved surface. Also, the secondary structures appear even at a very large value of block width (small value of Af). In view of all of the above, the ideal templating is possible (the darker shaded region confined by the boundaries 1 and 4) for the periodicity between λh ( λh; Lpy > λh. In this case too, a unique morphological feature on a larger periodicity substrate is the presence of secondary droplets at the intersection of the less wettable channels (image 5 of Figure 8c). Anisotropy initially induces an abacus-like microstructure (images 3 and 4 of Figure 8c) consisting of a series of droplets on a single string. Finally, it disintegrates to form a series of secondary droplets.
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Figure 9. Morphological evolution of a 6.65 nm thick film on completely wettable substrates (dark regions in the first images) with less wettable blocks (eq 4). Lp ) λml. Af ) 0.04, 0.25, and 0.64, in parts a, b, and c, respectively. The second to fourth images, in Figure 9a, correspond to the nondimensional time, T ) 25.5, 429.6, 619.12, and 1280.28, respectively; in Figure 9b times correspond to T ) 14.17, 54.28, 301.4, and 3396.7, respectively; in Figure 9c times correspond to T ) 6.38, 75.5, 134.4, and 785.4, respectively.
3B. Thin Film Patterns on Arrays of Less Wettable Blocks. Interestingly, our simulations showed that it is far easier to template the substrate pattern consisting of less wettable blocks on a completely wettable substrate (pattern of Figure 1b) than the earlier discussed case of more wettable blocks in a partially wettable matrix (pattern of Figure 1a). Figure 9 shows the morphological patterns formed on a completely wettable substrate (eq 4) with less wettable blocks of different widths, i.e., for different less wettable area fractions, Af ()WxWy/LpxLpy). The corresponding cross-sectional surface (H-Y) profiles at a fixed X are also shown. Ideal templating is readily obtained for small block widths (Figure 9a; W ) 0.2Lp), i.e., for the low area fractions (Af ) 0.04), where dewetting starts at the center of the patch17-19,39,40 and stops at the patch boundaries. For a larger patch width (Figure 9b, where W ) 0.5Lp, Af ) 0.25), dewetting starts at the boundaries of the patch across which the chemical potential gradient is induced. This leads to the formation of an entrapped droplet at the center of the patch (images 2 and 3 of Figure 9b). The mechanism of the formation of such a “castle-moat” structure was described in detail elsewhere39,40 for a single less wettable patch. Such entrapped drops at the center of square blocks have recently been observed by Han and co-workers.6 The droplets thus formed on less wettable blocks gradually disappear and the liquid ridges surrounding the dewetted blocks reach a constant flat height. The resulting membrane-like pattern, comprising of a 2D array of holes on a flat liquid sheet, almost ideally replicates the substrate pattern. Further increase in the patch width (W ) 0.8Lp, Af ) 0.64) results in the formation of increasingly circular (rather than more square like) holes on dewetted blocks. Less wettable regions in this case are partially covered by the liquid especially near the sharp patch corners. Thus, as was also seen as a general rule in the earlier discussed cases, the smaller area fraction of the more wettable region leads to the liquid spilling over the less wettable parts that destroys ideal templating. This transition of microstructures with the increase of the less wettable area fraction is independent of the exact form of the potential chosen.
Figure 10. Morphological phase diagram for the isotropic pattern consisting of less wettable blocks on completely wettable substrate for the potential of eq 3. The lines 1, 2, and 3 denote the boundaries between different regimes at the initial stage of dewetting. The shaded region corresponds to the conditions for ideal templating. The 3D microstructures corresponding to the five symbols are also shown.
Figure 10 summarizes the various morphology transitions on a completely wettable substrate consisting of nonwettable blocks as the pattern periodicity and less wettable block width are varied. The dark-shaded region represents the parameters for ideal templating. As in the earlier case, decrease in periodicity below a characteristic length scale, λh (∼0.7λm) (boundary 1), produces templated deformations without true dewetting. In this case, increase in periodicity ensures ideal templating as long as the block width remains below a transition width (Wt, boundary 2). For a pattern periodicity larger than λm, increasing block width produces arrays of a “castle moat” pattern (in parameter regime shown by lighter shade) in which a liquid drop is trapped within a square-shaped dewetted region on each nonwettable block, while the liquid surface remains flat on the remaining completely wettable surroundings (pattern shown by the symbol unfilled circle).
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Figure 11. Morphological evolution of a 5 nm thick film on a partially wettable coated substrate with less wettable blocks (eq 3). The substrate (black regions) is characterized by: Asm ) -1.88 × 10-20 J, Ac1m ) 1.13 × 10-20 J, and dc1m ) 2.5 nm. Lp ∼ 1.5λml. The second to fourth images in Figure 11a correspond to the nondimensional time, T ) 245.34, 46057, 766.9, and 1560.85, respectively; in Figure 11b they correspond to T ) 233.26, 311.5, 669.9, and 1222, respectively; and in Figure 11c they correspond to T ) 190, 323.25, 978, and 1501.15, respectively. Af ) 0.0625, 0.25, and 0.5625, in parts a, b, and c, respectively.
Figure 12. Morphological evolution of a 6.65 nm thin film on a checkerboard patterned substrate having a large periodicity (Lp ) 2.5λm). The mean substrate energy is characterized by eq 4 with As ) -1.41 × 10-20 J, Sp ) -22.22 mJ/m2, and lp ) 1.5 nm. The amplitude of energy heterogeneity, H ) 0.5 (eq 6). The second to fourth images correspond to T ) 0.000257, 0.000314, 0.000476, and 0.0006095, respectively.
At larger less wettable area fractions (Af > 0.5), i.e., at larger block widths (W/λml ∼ 0.7Lp/λml; Af ∼ 0.5; boundary 3 in Figure 10), liquid spilling occurs and secondary structures (hills/droplets) appear at the intersection of the more wettable channels separating the nonwettable blocks. Thus at large periodicity and at very large lesswettable block width, combination of secondary droplets on the nonwettable blocks and liquid hill on the more wettable part produces more complex microstructure (shown by unfilled square symbol). Comparison of Figure 10 with Figure 7 clearly shows that the parameter region for ideal templating is larger for a completely wettable substrate with nonwettable blocks than for a nonwettable substrate with completely wettable blocks. On a less wettable substrate, ideal templating is confined only to a very large width of blocks. Thus the inversion of a substrate pattern (replacing the completely wettable part by nonwettable and vice versa) does not always lead to an efficient inversion of the pattern, i.e., replacement of droplets by holes. Figure 11 depicts the pattern induced dewetting and transitions in morphologies on a partially wettable, coated substrate (eq 3), instead of completely wettable substrates discussed above. As the less wettable area fraction is raised, the following sequence of morphological transitions is observed: single rupture on each block f multiple ruptures on each block with an entrapped drop and a liquid “cross-wire” pattern f liquid spilling and deformed dewetting on less wettable blocks. Intriguingly, for a partially wettable-partially wettable system, regardless of the fractional width of the block, the contact line of the dewetted region continues to grow beyond the boundaries
of the blocks and take the almost circular shapes (Figure 11a,b). On the basis of these observations, as well as simulations for many other substrate patterns (not shown), it is inferred that complete wettability of the more wettable part of the substrate instead of partial wettability is a far better choice for the transfer of substrate patterns in a thin film. Moreover, the liquid that collects over the more wettable regions after dewetting is of uneven thickness unless these regions are completely wettable at all film thicknesses. In summary, on a substrate comprising of less wettable blocks, the thin film pattern ideally replicates the substrate pattern when (i) the less wettable block width is below a transition width (
