Dewetting of the Thin Liquid Bilayers on Topographically Patterned

(22) have also confirmed a resonance effect in dewetting of thin films on ... Polymer bilayers also serve as simple hydrodynamic models for the .... T...
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Langmuir 2008, 24, 14048-14058

Dewetting of the Thin Liquid Bilayers on Topographically Patterned Substrates: Formation of Microchannel and Microdot Arrays Dipankar Bandyopadhyay, Ashutosh Sharma,* and Chaitanya Rastogi† Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India ReceiVed July 26, 2008. ReVised Manuscript ReceiVed September 1, 2008 A long-wave nonlinear analysis of the defect induced instabilities engendered by van der Waals forces in thin ( 0. Further, an unstable upper layer on a stable lower layer is expected when S1S > 0, S1S2 > 0, and S21 < 0. Thus, the dewetting pathways of a bilayer can be profoundly influenced by the (i) choice of materials, which governs the nature of the intermolecular forces and (ii) thickness of the films, which governs the relative strength of the van der Waals forces at the two interfaces. Thus, the relative strength of the intermolecular interaction can also be spatially varied by introducing physicochemical heterogeneities on the substrate. In what follows, we classify the bilayers into three different cases based on the substrate surface energies and show the influence of a substrate defect or a patch on the dewetting pathway. Thereafter, we show different strategies to obtain ordered selforganized structures employing the pattern directed dewetting on topographically patterned substrates. A. Case 1: Bilayers on High Energy Substrates (γS > γ2 > γ1). An example bilayer of this case is silicon wafer (Si)/ polydimethylsiloxane (PDMS)/polystyrene (PS). The combination of surface energies (γS ) 0.27 N/m, γ2) 0.031 N/m, and γ1 ) 0.013 N/m) leads to S21 < 0 and S1S2 < 0 (A2 and A1 > 0), and therefore, both layers are unstable in this case. The destabilizing forces compete with each other, and the stronger force governs the instability. Further, the spreading coefficients SS1 > 0 and SS2 > 0 for this kind of bilayer indicate that in the absence of the upper layer (lower layer) the lower layer (upper layer) is completely stable on the rigid substrate.

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Figure 2. 2-D spatio-temporal evolution of a case 1 bilayer over an L ) 8Λ domain in the presence of a small defect of amplitude a ) 0.3 and width Lp ) Λ on the rigid substrate. In this simulation, h10 ) h30 ) 10 nm, µ1 ) µ2 ) 1 Pa · S, and ε1 ) ε2 ) 0.1. Images correspond to (a) T ) 0, (b) T ) 0.34, (c) T ) 0.51, and (d) T ) 1.21.

Figure 2 shows the evolution of free interfaces induced by a single defect in a large domain (L ) 8Λ). Here, Λ is the dimensionless dominant wavelength corresponding to the maximum growth rate of the instability obtained from the linear stability analysis65 of a bilayer resting on a flat surface. The base state film thicknesses (h10 ) h30 ) 10 nm) and viscosities (µ2 ) µ1 ) 1 kg/m s) ensure that the destabilizing intermolecular force in the lower layer is much stronger than that in the upper layer in the beginning of the evolution. Starting from a small amplitude random noise [image (a)], the lower layer ruptures first [image (b)] at the crest of the defect where the destabilizing intermolecular force is maximum in the lower layer. The hole formed in the lower layer grows because of the uncompensated Young’s force and increasingly displaces the lower layer from the solid surface. Consequently, the rim height adjacent to the hole increases, resulting in a local thinning of the upper layer. After the hole-rim crosses a certain threshold height, increased destabilizing intermolecular force in the upper layer now leads to its rupture. Following this, the growth of the dewetted region between the films leads to a zone of depression adjacent to it. As the depression grows in amplitude and the upper layer intrudes into the lower layer, the increased strength of the destabilizing intermolecular force in the lower layer creates further holes in the lower layer [image (c)]. The hole formed in the upper layer grows wider with time, and the rims adjacent to the hole finally coalesce to form droplets. The final morphology shows an array of holes in the lower layer filled by the upper layer liquid together with large droplets of the upper fluid sitting on the dewetted lower layer [image (d)]. Thus, a single defect can engender a rather longrange lateral order in the dewetting films. A comparison between Figure 2 and Figure 4 in ref 68 reveals that, other than the spatial location of the initial rupture, which is directed by the presence of the defect in Figure 2, the spatiotemporal evolutions are quite similar in both cases. However, this directed initial rupture can be exploited for generating selforganized patterns. In Figure 3, we show the evolution of a bilayer on a rigid substrate with periodic ridges and valleys. The length scale of the substrate pattern is chosen to be the same as the spinodal length scale of the instability, Lp ) Λ. The other simulation parameters are kept similar to those for the bilayer shown in Figure 2. Figure 3 shows that the stronger destabilizing intermolecular force at the lower layer causes rupture in bending mode on all the ridges of the patterned substrate [image (b)]. Thereafter, the holes expand and the adjacent rims coalesce to form droplets in the valleys of the substrate [image (c)]. Since the upper layer is stable on the patterned substrate, it merely fills the interstitial spaces between the droplets formed by dewetting

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Figure 3. 2-D spatio-temporal evolution of a case 1 bilayer over an L ) 4Λ domain on a substrate with periodic ridges and valleys of amplitude a ) 0.3 and Lp ) Λ. In this simulation, h10 ) h30 ) 10 nm, µ1 ) µ2 ) 1 Pa · S, and ε1 ) ε2 ) 0.1. Images correspond to (a) T ) 0, (b) T ) 0.23, (c) T ) 0.29, and (d) T ) 6.4 × 106.

Figure 4. 2-D spatio-temporal evolution of a case 1 bilayer over an L ) 4Λ domain on a substrate with periodic ridges and valleys of amplitude a ) 0.3 and Lp ) Λ. In this simulation, h10 ) 10 nm and h30 ) 5 nm, µ1 ) µ2 ) 1.0 Pa · S, and ε1 ) ε2 ) 0.1. Images correspond to (a) T ) 0.072, (b) T ) 0.076, (c) T ) 0.085, and (d) T ) 0.1.

of the lower layer. At the late stages of dewetting, the unstable upper layer now retracts from the top of the droplets formed in the lower layer [image (d)]. Thus, at near equilibrium, an array of lower layer droplets is formed in the valleys of the patterned substrate and the upper layer liquid fills out the dewetted interstitial spaces formed because of the dewetting of the lower layer [image (d)]. Interestingly, freezing the evolution at stages (b) to (c) and removal of the lower layer material by a selective solvent can produce an array of micro/nanochannels, but the same procedure at stage (d) yields an array of micropillars. It is interesting to note here that the dewetting pathway can change significantly if the upper layer is unstable from the beginning. Figure 4 shows the spatio-temporal evolution of such a bilayer. Images (a)-(c) show that in short time the bilayer interfaces evolve in the squeezing mode and the upper layer first dewets the lower layer because of the larger destabilizing force. However, in the later stages of dewetting, the lower layer also ruptures on the ridges of the patterned substrate and the final morphology again consists of an array of upper and lower layer droplets on the ridges and valleys, respectively [image (d)]. A comparison of Figures 3 and 4 indicates that although the dewetting pathway and the intermediate morphologies are clearly distinct in these two bilayers, the final morphologies are similar. Figures 3 and 4 based on 2-D simulations indicate that topographically patterned substrates can be very useful in aligning the dewetted structures when the substrate periodicity is nearly commensurate with the spinodal length scale (Lp ) Λ) of the interfacial instability. In Figure 5, we show an example in 3-D

Dewetting of Thin Liquid Bilayers

Figure 5. 3-D spatio-temporal evolution of a case 1 bilayer over an L ) 3Λ domain on a substrate with periodic ridges and valleys of amplitude a ) 0.3 and Lp ) Λ. The first, second, and third rows in this figure depict the liquid-air interface, the liquid-liquid interface, and the composite images, respectively. Image (I) in the third row depicts the substrate surface. In this simulation, h10 ) h30 ) 10 nm, µ1 ) µ2 ) 1.0 Pa · S, and ε1 ) ε2 ) 0.1. Images correspond to (I) T ) 0.03, (II) T ) 0.098, and (III) T ) 0.55. Darker shades of gray represent the lower thickness regions. A linear gray scale is used between the minimum and maximum thicknesses in each panel.

where the dewetted morphologies align on a surface consisting of an array of parallel ridges and valleys [image (I), third row]. It may be noted that all 3-D figures in this study show the liquid-air interface, the liquid-liquid interface, and the composite image in the first, second, and third rows, respectively, with the exception of the first image, (I) in the third row, which shows the surface topography. The 3-D morphologies in Figure 5 show that, in a short time, the lower layer dewets the solid at the top of the ridges [image (I)], but the three-phase contact line thus formed remains straight in the direction parallel to the ridges. Thereafter, as the dewetted region grows, the rims formed adjacent to the dewetted zones coalesce and the lower layer forms a periodic array of ridges in the valleys of the patterned substrate [image (III)]. Subsequently, the upper layer dewets at the top of the ridges formed in the lower layer [image (III)]. The final morphology shows a periodic array of upper and lower layer ridges alternating on the ridges and the valleys of the patterned substrate, respectively. In essence, the cross-sectional morphologies in this case mirror the structures shown in Figure 3 at all stages. An interesting aspect is that suppression of instability is in the direction parallel to the substrate-ridges, which allows generation of structures that are completely straight and unruptured in the transverse direction, but periodic in the span-wise direction. The growth of instability is suppressed in the transverse direction, y, because there is no gradient of potential in that direction, namely, f ) f(x) in eqs 1 and 2. The spinodal time scale on a flat uniform substrate governs the growth of instability in the transverse direction, which is much larger than the time scale of rupture induced by the heterogeneous mechanism.13-19 The interfaces thus deform quickly and rupture along the spanwise direction. Once the final structures shown in Figures 3 and 4 form, it becomes a stable configuration even at long times. This is because now both the lower and the upper layers rest on the substrate on which both are thermodynamically stable. The final morphology thus produced is rather interesting in that if a selective solvent now removes the upper layer, the lower layer material forms an array of straight and open microchannels on top of the ridges of the substrate. In contrast, upon removal of the lower layer material, the upper layer material forms open microchannels located in the valleys of the patterned substrate. An array of closed microchannels can also be obtained by freezing

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Figure 6. 3-D spatio-temporal evolution of a case 1 bilayer over an L ) 3Λ domain on a substrate with periodic, cross-patterned ridges and valleys of amplitude a ) 0.3 and Lp ) Λ. The first, second, and third rows in this figure depict the liquid-air interface, the liquid-liquid interface, and the composite images, respectively. Image (I) in the third row depicts the substrate surface. In this simulation, h10) h30 ) 10 nm, µ1 ) µ2 ) 1.0 Pa · S, and ε1 ) ε2 ) 0.1. Images correspond to (I) T ) 0.23, (II) T ) 0.5, (III) T ) 0.76, and (IV) T ) 1.18. Darker shades of gray represent the lower thickness regions. A linear gray scale is used between the minimum and maximum thicknesses in each panel.

the evolution midway before the upper film ruptures and by the removal of the lower layer material, as discussed previously for Figure 3. Clearly, the formation of embedded microchannels is encouraged by starting with thicker upper layers that are more robust against the growth of instability at its upper free surface. In such cases, the liquid-liquid interface forms a perfectly outof-phase replica of the lower layer morphology on the substrate topography. Figure 6 shows bilayer micropatterning on a 3-D substrate configuration consisting of a cross-pattern of periodic ridges and valleys having Lp ) Λ [image (I), third row]. Images (I)-(IV) in this figure show that the lower layer dewets the solid on the crests of the rigid substrate [image (II)]. Subsequent growth of the holes leads to the formation of an array of droplets [image (IV)] on the valleys of the patterned substrate. The upper layer simultaneously dewets on top of these droplets, resulting in a final morphology consisting of a periodic array of lower layer droplets in the valleys of the patterned substrate with the upper layer filling the interstitial spaces [image (IV)]. These 3-D intermediate and equilibrium morphologies can be further manipulated by selective etching of one of the materials. For example, removal of the lower layer material at equilibrium leaves a thin upper layer as a well-ordered membrane [image (IV)]. If the upper layer is kept sufficiently thick to prevent its dewetting, the same procedure would result in an array of microwells. B. Case 2: Bilayers on Low Energy Substrates (γS < γ2 < γ1). An example of this class of bilayers is SiO2/PMMA/PS (γS ) 0.026 N/m, γ2 ) 0.031 N/m, and γ1) 0.036 N/m), where SiO2 is quartz and PMMA is polymethylmethacrylate. In this case, the spreading coefficients are S21 > 0 (A1 > 0) and S1S, S2S, and S1S2 < 0 (A1S, A2S, and A1 > 0). The lower layer is unstable under the influence of S1S2 < 0 (A2 > 0) and S1S (A1S > 0). The upper layer is stable when floating on the lower layer because S21 > 0; however, it becomes unstable when it meets the solid substrate because of S2S < 0. This opens up an interesting possibility of the formation of secondary structures by dewetting of the upper layer on the substrate after the lower layer ruptures and recedes from some spots. Figure 7 shows the 3-D morphological evolution of this type of bilayer in the presence of a single isolated defect on a rigid substrate (L ) 8Λ). The square shaped defect has the dimension of Λ × Λ in the span-wise and transverse spatial directions. It is already known for a single film that when the heterogeneous patch is very small (,Λ), the film rupture is initiated by the

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Figure 7. 3-D spatio-temporal evolution of a case 2 bilayer over an L ) 8Λ domain in the presence of a small defect of amplitude a ) 0.3 and dimension Λ × Λ on the rigid substrate. The first, second, and third rows in this figure depict the liquid-air interface, the liquid-liquid interface, and the composite images, respectively. Image (I) in the third row depicts the substrate surface. In this simulation, h10 ) h30 ) 10 nm, µ1 ) µ2 ) 1.0 Pa · S, and ε1 ) ε2 ) 0.1. Images correspond to (I) T ) 33, (II) T ) 81, and (III) T ) 255. Darker shades of gray represent the lower thickness regions. A linear gray scale is used between the minimum and maximum thicknesses in each figure.

formation of a hole at the center of the heterogeneity.17 However, as the size of the heterogeneous patch increases, the initial rupture of the film gradually shifts toward the periphery of the patch and the dewetting starts by the formation of several holes or a ring surrounding a small droplet at the center of the patch.17 Image (I) in the Figure 7 shows that a relatively large heterogeneity (Λ × Λ) causes the first rupture of the unstable lower layer near the periphery of the patch and creates a small droplet remaining at the center of the elevated patch. The rupture of the lower layer then brings the upper layer in contact with the solid. The upper layer now also dewets the solid [image (I)] because it is unstable on the substrate, S2S < 0. Interestingly, the equilibrium contact angle (θ1) of the dewetting front resulting from the rupture of the lower layer on the solid in the presence of the upper layer is smaller than the equilibrium contact angle (θ2) of the upper layer liquid on the solid substrate in air because |S2S| > | S1S2|. Thus, as the hole formed in the lower layer grows to achieve the equilibrium contact angle [images (II) and (III)], the front created after the dewetting of the upper layer on the solid forces the contact line of the ruptured lower layer to recede beyond its equilibrium [image (III)]. The eventual size of the holes formed in this case becomes much larger than the spinodal length scale, which opens up the possibility of generating ordered structures having periodicities larger than the spinodal length scale. Images (a)-(d) in Figure 8 show 2-D spatio-temporal evolution of this type of bilayer on a patterned substrate with periodic ridges and valleys having Lp ) Λ. The figure clearly shows that the bilayer initially evolves in the bending mode and the lower layer eventually ruptures on all the crests of the substrate pattern [image (b)]. Consequently, the upper layer meets the substrate and dewets it. Thereafter, the holes grow to achieve equilibrium contact angle, and the rims adjacent to the holes coalesce to form periodic droplets in the valleys of the substrate [image (d)]. The upper layer finally encapsulates the drops in the lower layer [image (d)] as the evolution approaches the equilibrium. It is interesting to note here that images (e) and (f) show equilibrium morphologies similar to that in image (d) even when Lp is changed to Λ/2 and 2Λ, respectively. Images (d)-(f) suggest the important role of substrate pattern induced lateral confinement which governs the length scale of the instability and can produce periodic structures that are much larger or much smaller than the spinodal length scale. This can be useful from the technological point of

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Figure 8. 2-D spatio-temporal evolution of a case 2 bilayer on a substrate with periodic ridges and valleys of amplitude a ) 0.3. Images (a)-(d) correspond to Lp ) Λ, image (e) corresponds to Lp ) Λ/2, and image (f) corresponds to Lp ) 2Λ. In this simulation, h10 ) h30 ) 10 nm, µ1 ) µ2) 1.0 Pa · S, and ε1 ) ε2 ) 0.1. Images correspond to (a) T ) 0, (b) T ) 0.7, (c) T ) 0.78, (d) T ) 1.02, (e) T ) 5.2 × 103, and (f) T ) 2.75.

Figure 9. 3-D spatio-temporal evolution of a case 2 bilayer over an L ) 4Λ domain on a substrate with periodic ridges and valleys of amplitude a ) 0.3 and Lp ) Λ. The first, second, and third rows in this figure depict the liquid-air interface, the liquid-liquid interface, and the composite images, respectively. Image (I) in the third row depicts the substrate surface. In this simulation, h10 ) h30 ) 10 nm, µ1 ) µ2) 1.0 Pa · S, and ε1 ) ε2 ) 0.1. Images correspond to (I) T ) 0.6, (II) T ) 1.18, and (III) T ) 3.76. Darker shades of gray represent the lower thickness regions. A linear gray scale is used between the minimum and maximum thicknesses in each figure.

view because it not only opens up the new avenues for pattern miniaturization but also provides a larger window of substrate pattern periodicity for the generation of ordered microstructures as compared to a single film. The dewetted structures shown in the images (d)-(f) can also be useful in micro- or nanofluidic applications because they can produce embedded or encapsulated nano/microchannels upon selective removal of the lower layer by a solvent. In Figure 9, we show 3-D spatio-temporal evolution where the dewetted morphologies align on a surface having an array of parallel ridges and valleys with Lp ) Λ [image (I), third row]. Images (I)-(III) in this figure show that the lower layer dewets the solid at the ridges of the patterned substrate and forms a periodic array of ridges in the valleys of the topographically patterned substrate. After dewetting of the lower layer, the upper layer comes in contact and dewets the solid on the ridges of the

Dewetting of Thin Liquid Bilayers

Figure 10. 3-D spatio-temporal evolution of a case 2 bilayer over an L ) 4Λ domain on a substrate with periodic, cross-patterned ridges and valleys of amplitude a ) 0.3 and Lp ) Λ. The first, second, and third rows in this figure depict the liquid-air interface, the liquid-liquid interface, and the composite images, respectively. Image (I) in the third row depicts the substrate surface. In this simulation, h10 ) h30 ) 10 nm, µ1 ) µ2) 1.0 Pa · S, and ε1 ) ε2 ) 0.1. Images correspond to (I) T ) 0.55, (II) T ) 1.03, and (III) T ) 2.62. Darker shades of gray represent the lower thickness regions. A linear gray scale is used between the minimum and maximum thicknesses in each figure.

substrate. Near equilibrium, the upper layer encapsulates the dewetted structures of the lower layer. The lower layer structures in the valleys run parallel to the substrate patterns. This is because of the absence of the gradient of the intermolecular interactions in the transverse direction, which suppresses the growth of instability. Image (III) clearly indicates that this kind of bilayer is capable of generating embedded micro/nanochannels once the lower layer is selectively removed. In Figure 10, another example of bilayer micropatterning is shown on a substrate consisting of cross-patterned periodic ridges and valleys with Lp )Λ [image (I), third row]. Images (I)-(III) in this figure show that the lower layer dewets the solid at the crest of the ridges on the substrate. Subsequent growth and coalescence of holes lead to the formation of an array of droplets in the valleys. Thereafter, the upper layer meets the solid at the ridges of the patterned substrate and dewets the substrate. At the late stages of dewetting, an ordered array of the lower layer droplets encapsulated by the upper layer in the valleys is formed [image (IV)]. C. Case 3: Bilayers on Intermediate Energy Substrates (γ1 > γS > γ2). An example of this case is SiO2/PS/PDMS (γS ) 0.026 N/m, γ2 ) 0.013 N/m, and γ1 ) 0.031 N/m). The spreading coefficients are S21 and S2S > 0 (A1 and A2S < 0); S1S and S1S2 < 0 (A1S and A2 < 0). Thus, the instability is caused by the attractive components, S1S and S1 S2. From the free energy point of view, the only difference with case 2 is that S2S > 0, which ensures the stability of the upper layer on the solid substrate after the rupture of the lower layer. Figure 11 shows 3-D morphological evolution in the presence of a defect of dimension Λ × Λ at the center of a rigid flat substrate (image (I), third row; domain size, L ) 8Λ). The lower layer ruptures at the periphery of the defect and creates a tiny droplet on the defect [image (I)]. Thereafter, the growth of the dewetted region causes the rim to retract from the defect. Several secondary holes decorated around a ring now appear from the zone of depression adjacent to the retracting rim [image (II) and (III)]. The rupture of lower layer around the defect and the subsequent formation of satellite holes around the primary hole are also observed experimentally,79 as well as predicted theoreti(79) Becker, J.; Gru¨n, G.; Seemann, R.; Mantz, H.; Jacobs, K.; Mecke, K. R.; Blossey, R. Nat. Mater. 2003, 2, 59.

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Figure 11. 3-D spatio-temporal evolution of a case 3 bilayer over an L ) 8Λ domain in the presence of a small defect of amplitude a ) 0.3 and dimension Λ × Λ on the rigid substrate. The first, second, and third rows in this figure depict the liquid-air interface, the liquid-liquid interface, and the composite images, respectively. Image (I) in the third row depicts the substrate surface. In this simulation, h10 ) h30 ) 10 nm, µ1 ) µ2 ) 1.0 Pa · S, and ε1 ) ε2 ) 0.1. Images correspond to (I) T ) 104, (II) T ) 609, (III) T ) 863, and (IV) T ) 1201. Darker shades of gray represent the lower thickness regions. A linear gray scale is used between the minimum and maximum thicknesses in each figure.

Figure 12. 2-D spatio-temporal evolution of a case 3 bilayer on a substrate with periodic ridges and valleys of amplitude a ) 0.3. Images (a)-(d) correspond to Lp ) Λ, image (e) corresponds to Lp ) Λ/2, and image (f) corresponds to Lp ) 2Λ. In this simulation, h10 ) h30 ) 10 nm, µ1 ) µ2) 1.0 Pa · S, and ε1 ) ε2 ) 0.1. Images correspond to (a) T ) 0, (b) T ) 0.75, (c) T ) 1.8, (d) T ) 2.8, (e) T ) 3.2 × 108, and (f) T ) 9.39.

cally17 for a single film. Following this, the holes grow in size to achieve the equilibrium contact angle and the later time morphology appears as a random ensemble of droplets and ribbons [image (IV)]. The combination of surface energies for this case ensures that the upper layer is stable on the rigid substrate as well as on the dewetted structures of the lower layer. Thus, in contrast to case 2 discussed earlier, the dewetted substrate and the equilibrium structures formed in the lower layer are thus always covered with a thin cushion of the upper layer liquid. As in the case of the case 2 systems, the case 3 bilayers can also generate patterns when directed by the prepatterned substrates. Images (a)-(d) in Figure 12 show the 2-D morphologies of a case 3 bilayer on a patterned substrate with periodic ridges and valleys with Lp ) Λ. The underlying pattern on the rigid substrate directs the initial rupture of the lower layer in the bending mode on the crests of the ridges [image (c)]. Following this, the holes expand and the rims of the adjacent holes coalesce to form periodic droplets in the valleys [image (d)]. The upper layer always forms a thin cushion on the dewetted morphologies

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Figure 13. 3-D spatio-temporal evolution of a case 3 bilayer over an L ) 4Λ domain on a substrate with periodic ridges and valleys of amplitude a ) 0.3 and Lp ) Λ. The first, second, and third rows in this figure depict the liquid-air interface, the liquid-liquid interface, and the composite images, respectively. Image (I) in the third row depicts the substrate surface. In this simulation, h10 ) h30 ) 10 nm, µ1 ) µ2) 1.0 Pa · S, and ε1 ) ε2 ) 0.1. Images correspond to (I) T ) 2.5, (II) T ) 3.7, and (III) T ) 11.4. Darker shades of gray represent the lower thickness regions. A linear gray scale is used between the minimum and maximum thicknesses in each figure.

Figure 14. 3-D spatio-temporal evolution of a case 3 bilayer over an L ) 4Λ domain on a substrate with periodic, cross-patterned ridges and valleys of amplitude a ) 0.3 and Lp ) Λ. The first, second, and third rows in this figure depict the liquid-air interface, the liquid-liquid interface, and the composite images, respectively. Image (I) in the third row depicts the substrate surface. In this simulation, h10 ) h30 ) 10 nm, µ1 ) µ2) 1.0 Pa · S, and ε1 ) ε2 ) 0.1. Images correspond to (I) T ) 1.0, (II) T ) 3.12, and (III) T ) 4.8. Darker shades of gray represent the lower thickness regions. A linear gray scale is used between the minimum and maximum thicknesses in each figure.

of the lower layer and on the dewetted substrate. Image (e) for Lp ) Λ/2 shows that decreased pattern periodicity increases the frequency of the structures formed. However, image (f) shows that the interfaces fail to trace the substrate patterns when Lp ) 2Λ. In contrast to the observations for case 2, ordering of structures takes place when the pattern periodicity is near or below the linear length scale (Λ) of the instability. Images (d) and (e) also show that this type of bilayer is also capable of producing ordered micro/nanochannels once the lower layer is selectively removed. In Figure 13, 3-D morphologies of dewetting on a surface having an array of parallel ridges and valleys [image (I), third row] with Lp ) Λ is shown. Images (I)-(III) in this figure show that the lower layer dewets the solid at the ridges of the rigid substrate and forms a periodic array of ridges in the valleys of the patterned substrate. The upper layer liquid completely covers the lower layer structures as well as the portions of the substrate from where the lower layer has retracted. The morphology shown in image (III) is interesting because a selective etching of the lower layer can produce an ordered array of embedded micro/ nanochannels. It is also interesting to note here that the liquid-air interface in this case forms an in-phase replica of the morphology of the dewetted lower layer on the solid substrate. In Figure 14, another example of bilayer micropatterning is shown on a cross-patterned rigid substrate consisting of periodic ridges and valleys with the spacing Lp ) Λ [image (I), third row]. Images (I)-(III) in this figure show that the lower layer dewets the solid at the top of the ridges on the patterned substrate and the subsequent growth and coalescence of the holes lead to the formation of an array of droplets on the valleys of the topographically patterned substrate. The upper layer covers the dewetted structures formed by the lower layer liquid. Thus, the dewetted morphology shows a collection of ordered lower layer microdroplets embedded inside a thin layer of the upper liquid. It is interesting to note here that experiments80 with a single film on topographically cross-patterned substrates also show an ordered collection of droplets at the valleys as seen for the dewetted lower layer shown in image (III). The final morphology indicates that, in parallel to the creation of dewetted structures in the lower layer, the liquid-air interface of the upper

layer also forms an in-phase replica of the lower layer patterns. Thus, the case 3 bilayers not only are useful in generating ordered micro/nanochannels or micro/nanodots but also can simultaneously create the replica mold of the substrate patterns in a polymer. D. Influence of Initial Condition on the Dewetting Pathway and Final Morphology. Experimental evidence shows that, while coating the polymer films on the topographically patterned substrates, the interfaces of the polymer films do not always remain flat but can acquire roughness correlated to the substrate patterns.81-84 Thus, the initial configuration of the film interfaces vis-a`-vis the substrate pattern is another parameter that could possibly govern the dewetting pathway. It is expected that the initial film roughness would depend on a host of factors such as spin speed, polymer concentration, solution viscosity and surface tension, rate of solvent evaporation, and, most importantly, film thickness vis-a`-vis the amplitude of the substrate corrugations. However, there is as yet no systematic study of the topography of the film interfaces produced by spin coating on periodically rough substrates. Limited experimental data81 on topographically patterned substrates indeed indicate that the initial amplitude of spin coated surfaces becomes rather small as the polymer completely fills the substrate depressions and the mean film thickness becomes more than about 4 times the substrate amplitude. Further, the amplitude of the first layer coated directly on the substrate is expected to be larger than that of the second layer coated on top.82-84 In order to assess the effects of initial amplitude on the dewetting pathways, we consider here some extreme cases of correlated interfaces where the initial amplitudes at film interfaces are comparable to the amplitude of the substrate heterogeneity. Images (a)-(c) in Figure 15 show the evolution of a case 1 bilayer similar to the bilayer shown in Figure 3, except that the initial configuration of the lower layer is assumed to fully conformal to the substrate with the same amplitude as the substrate pattern (a ) 0.3 and Lp ) Λ) and the top layer is assumed flat.

(80) Mukherjee, R.; Bandyopadhyay, D.; Sharma, A. Soft Matter 2008, 4, 2086.

(81) Mukherjee, R.; Pangule, R. C.; Sharma, A.; Banerjee, I. J. Chem. Phys. 2008, 127, 064703. (82) Muller-Buschbaum, P.; Gutmann, J. S.; Lorenz, C.; Schmitt, T.; Stamm, M. Macromolecules 1998, 31, 9265. (83) Muller-Buschbaum, P.; Stamm, M. Macromolecules 1998, 31, 3686. (84) Kraus, J.; Muller-Buschbaum, P.; Bucknall, D. G.; Stamm, M. J. Polym. Sci., Part B: Polym. Phys. 1999, 37, 2862.

Dewetting of Thin Liquid Bilayers

Figure 15. 2-D spatio-temporal evolution of case 1 bilayers with different initial configurations. Images (a)-(c) show the evolution of bilayer instability on a substrate having periodic ridges and valleys of amplitude a ) 0.3 and Lp ) Λ. Image (a) shows that the initial configuration at the liquid-liquid interface is same as the substrate pattern whereas the upper interface is assumed flat. Images (d)-(f) show that the initial configuration at the liquid-liquid interface is in-phase with the substrate pattern (a ) 0.3 and Lp ) Λ); however, the substrate pattern has a higher amplitude (a ) 0.5). Images correspond to (a) T ) 0, (b) T ) 1.76, (c) T ) 2.15, (d) T ) 0, (e) T ) 0.72, and (f) T ) 0.97. In all the images, h10 ) h30 ) 10 nm and µ1 ) µ2 ) 1.0 Pa · S.

Further, images (d)-(f) show the evolution of another bilayer identical to that in images (a)-(c), except that the substrate topography now has an amplitude (a ) 0.5) higher than the amplitude of the lower interface (a ) 0.3). The third case shown in Figure 16a–c is yet another scenario where both of the film interfaces have initial patterns (a ) 0.3) correlated to the substrate surface (a ) 0.5). In all of these rather extreme cases, the change in the initial configuration of the film interfaces does not alter the dewetting pathway and the resulting morphology and the instability follow the same path as shown in Figure 3 with initially flat interfaces. This is because, in all of these cases, rupture of the lower layer is followed by the breakup of the upper layer. We now consider one final extreme example where the dewetting pathway does change with the initial configuration. The bilayer shown in Figure 16d-f has an initial amplitude (a ) 0.5), which is equally high both at the liquid-liquid interface and the substrate. In contrast to the bilayer of Figure 16a-c, the surface of the upper layer is now assumed flat. In contrast to all the other cases shown thus far, the upper layer in this case ruptures first and the dewetting pathway and the dewetted structures change significantly as shown in Figure 16d-f. The increased local thinning of the upper layer now can cause the upper layer to rupture faster than the lower layer. At the late stages of dewetting, the lower layer also dewets the solid to form periodic upper layer structures on the valleys of the substrate and lower layer structures on the ridges of the substrate. Even though the initial configuration in Figure 16d is not very realistic, it does point to the importance of initial configuration in some extreme cases. Although it is beyond the scope of present work to systematically study the effects of initial configuration because of a very large number of possibilities, some qualitative ideas already emerge. When the initial amplitude of the film interfaces is much smaller than that of the substrate pattern, the dewetting pathway and morphologies are not likely to be greatly influenced. An essential qualitative change in dewetting behavior occurs only when the sequence of rupture of the two layers is altered. The

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Figure 16. 2-D spatio-temporal evolution of case 1 bilayers with different initial configurations. Images (a) and (d) show two different initial configurations of the films with respect to the substrate pattern in the beginning of the simulations. Images (a)-(c) show that the initial configurations at the interfaces are in-phase with the substrate pattern (a ) 0.3 and Lp ) Λ); however, the substrate pattern has a higher amplitude of patterns (a ) 0.5 and Lp ) Λ). Images (d)-(f) show the evolution of bilayer instability on a substrate having patterns of amplitude a ) 0.5 and Lp ) Λ. The liquid-liquid interface also has the initial configuration a ) 0.5 and Lp ) Λ, whereas the upper film is considered flat. Images correspond to (a) T ) 0, (b) T ) 0.42, (c) T ) 0.65, (d) T ) 0, (e) T ) 0.7, and (f) T ) 1.2. In all the images, h10 ) h30 ) 10 nm and µ1 ) µ2 ) 1.0 Pa · S.

analysis presented here with case 1 bilayers also indicates that the initial configuration will have even less influence for the case 2 and case 3 bilayers because the instability in those two cases is dominated by the unstable lower layer alone and the upper layer is initially thermodynamically stable. Thus, the upper layer ruptures, whenever it happens, after the breakup of the lower layer.

4. Conclusions 2-D and 3-D long-wave nonlinear analyses of the pathways of dewetting and resulting morphologies of thin liquid bilayers on topographically patterned substrates were investigated. Different strategies were devised to generate self-organized patterns such as arrays of open and closed micro/nanochannels, encapsulated and isolated micro/nanodroplets, and membranes. The major conclusions of this study are as follows: (i) Case 1 Bilayers. (A) Simulations show that physical heterogeneity on the rigid substrate influences the first rupture when the lower layer is more unstable. In contrast, if the film thicknesses ensure that the upper layer is more unstable in the beginning of evolution, the dewetting pathway after the rupture of the upper layer is profoundly influenced by the substrate patterns. (B) The dewetting of bilayers on substrates with an array of parallel ridges and valleys produces structures of the lower and upper layer materials along the valleys and ridges of the patterned substrate, respectively. Selective removal of any of these layers can thus be a technique for the fabrication of an array of open micro/nanochannels. The simulations also indicate that the dewetting of the upper layer can be prevented by an appropriate choice of film thickness and other conditions. In such cases, dewetting of the lower layer produces embedded micro/nanochannels upon a selective removal of the lower layer material. (C) The dewetting of bilayers on a cross-patterned substrate show that the lower layer can dewet to produce an

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ordered 2-D array of micropillars. A thin upper layer on top of these microdroplets is always unstable and fills out the spaces from where the lower layer recedes from the patterned substrate. Thus, if the lower layer patterns are etched out, the upper layer transforms to a membranelike structure. The upper layer also shows an out-of-phase replica of the dewetted structures formed at the lower layer. (ii) Case 2 Bilayers. The most interesting aspect is the dewetting of the upper layer secondary to the rupture of the lower layer, which exposes the substrate to the upper layer liquid. Deep into the nonlinear regime, the movement of two dewetting fronts displaying different contact angles leads to holes that are much larger than the spinodal length scale of instability. These bilayers can still generate periodic self-organized structures on the length scale of the substrate periodicity even when the pattern periodicity, Lp, is in the range of Λ/2 to 2Λ, where Λ is the spinodal length scale of the instability. Simulations uncovered many interesting structures such as an ordered array of straight micro/nanochannels and micro/nanodroplets of the lower layer encapsulated by the upper layer. (iii) Case 3 Bilayers. These systems are ideal candidates for the formation of embedded microchannels by the growth of instability. The stable upper layer in this case fully covers the dewetted structures formed in the lower layer, and the substrate regions uncovered by the dewetting of the lower layer are filled by the upper layer liquid. In addition, the liquid-air interface of these bilayers always traces out the morphologies that are formed at the liquid-liquid interface and, in the process, generates an in-phase replica of the lower layer morphologies on the patterned substrate. Therefore, these bilayers can also be useful in the pattern transfer applications. (iv) It is also interesting to note that bilayers follow the substrate patterns because the wavelength of the fastest growing mode is greatly influenced by the substrate patterning, especially when the pattern periodicity is less than the spinodal length scale. In

Bandyopadhyay et al.

addition, as compared to single film systems, the lowered interfacial tension at the liquid-liquid interface in the bilayers reduces the wavelength and hence reduces the spacing between the dewetted structures formed at the lower layer. Therefore, in conjunction with a closely spaced pattern periodicity, the reduced interfacial tension at the liquid-liquid interface opens up the possibility of further miniaturization of patterns using bilayers as compared to the single film systems. (v) The effects of the initial configuration of the bilayer interfaces, which are correlated to the substrate pattern, on the dewetting pathway and morphology are briefly considered. Dewetting is not qualitatively influenced when the initial amplitudes at the film interfaces are much smaller than the substrate amplitude, which should be the case for relatively thicker films. However, in some extreme cases, the dewetting pathway and the resulting morphology may change whenever the sequence of rupture of the upper and lower layers is altered by a change in the initial configuration. In summary, a comprehensive understanding of the bilayer instability on topographically patterned substrates based on nonlinear simulations is presented, which points to many interesting scenarios of dissipative pattern formation that are of interest in applications ranging from multilayer coatings, microfluidics, and self-organized patterning to a basic understanding of bilayer dewetting. It is hoped that the results obtained here may stimulate controlled experiments on dewetting of bilayers on topographically patterned substrates. Acknowledgment. A.S. acknowledges the support of the DST through its grants to the Unit on Nanosciences at IIT Kanpur, J. C. Bose Fellowship, and an IRHPA grant. C.R. was supported by a joint SURF/SURGE program between IIT Kanpur and the California Institute of Technology. LA802404S