Instability of Thin Polymer Films on Coated Substrates: Rupture, Dewetting, and Drop Formation

JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.

178, 383–399 (1996)

0133

Instability of Thin Polymer Films on Coated Substrates: Rupture, Dewetting, and Drop Formation ASHUTOSH SHARMA * ,1

AND

GU¨NTER REITER†

*Department of Chemical Engineering, Indian Institute of Technology at Kanpur, Kanpur-208016, India; and †Institut de Chimie des Surfaces et Interfaces, CNRS, 15 rue Jean Starcky, F-68057 Mulhouse, France Received November 7, 1994; accepted August 16, 1995

INTRODUCTION We present a direct comparison of theoretical predictions and experimental results on instabilities and various stages of dewetting of thin (õ60 nm) films on coated substrates. Thin (ú10 nm) polystyrene films, prepared on silicon wafers with three different nanosized (Ç1 nm) coatings, dewet spontaneously above the glass transition temperature by the growth of cylindrical holes with wavy rims. We could clearly distinguish, and theoretically explain, four different stages as dewetting proceeded: (a) rupture of the film; (b) expansion and coalescence of holes to form a polygonal ‘‘cellular’’ pattern; (c) disintegration of polymer ridges forming the polygon into spherical drops due to Rayleigh instability; and (d) fingering instability of hole rims during hole expansion witnessed only on low wettability coatings. The theory gives a clear understanding of roles of substrate (silicon) wettability, coating wettability, and the film thickness in various stages (phenomena). For films much thicker than the coating, the wettability (contact angle) of the coated substrate by polystyrene has no influence on the initial length scale of the instability (number density of holes). This is explained by the dominating influence of long-range Lifshitz–van der Waals (LW) interactions originating from the bulk substrate; the LW and short-ranged polar interactions with the coating determine only the contact angle after appearance of an ultrathin three-phase contact zone (rupture). In stage b also, the final polygon diameters are rather independent of the coating, which is explained by insensitivity of stage a to the coating properties and a competition between the Rayleigh instability (leading to droplets) and drainage from ridges (leading, to coalescence). Hole growth, fingering instability, and the drop diameters, on the contrary, are highly affected by the surface properties (wettability) of coatings. An increase in the film thickness (h) decreases the number density of initial holes (ah 04 ), increases the polygon diameter ( ah 2 ), increases the drop diameter ( ah q , q varies from 1 to 1.5 depending on the contact angle), and makes the fingering instability stronger. Theory and experiment are in excellent qualitative and quantitative agreement. The theory also suggests several interesting possibilities for the design of future experiments on the dewetting of thin films. q 1996 Academic Press, Inc. Key Words: coated substrates; thin film stability; dewetting; drop formation; disjoining pressure; contact angle

1

Author to whom correspondence should be addressed.

Stability and homogeneity of thin films (1–9) on solid supports are of technological and scientific importance in applications ranging from coatings (10–12), paints, adhesives, and photographic films to fundamental studies of multilayer adsorption (13), diffusion (14), wetting (15– 18), and film boiling/condensation (19–21). While relatively thick micrometer-sized films can rupture and dewet due to large-amplitude external disturbances exceeding a critical size (9, 22, 23), the free interfaces of ultrathin ( õ100 nm) films become unstable and deform spontaneously whenever the intermolecular disjoining pressure (24) increases with increased film thickness (1–9). The initial growth of instability may lead either to a true rupture (hole formation) with the appearance of a receding three-phase contact line (5, 6, 12, 25–28), or to a timestationary nonuniform film consistings of drops and films when repulsive forces become dominant (8, 9, 26–30). For relatively thick ( ú10 nm) charge-neutral films, the initial instability on uncoated substrates is governed largely by the longer ranged apolar Lifshitz–van der Waals (LW) forces (8, 26, 27) and the rupture occurs whenever the effective Hamaker constant is positive, i.e., the LW component of the spreading coefficient is negative (8, 26, 27), and the shortrange repulsive interactions (when present) are dominated by the LW interactions. Numerical solutions (5, 8, 12, 20, 25, 27, 31) of the nonlinear equations of motion for thin films on uncoated substrates have provided useful information on the lengthscale (wavelength) of the fastest growing (preferred) mode of the instability, and on the initial shape of 2-D, axisymmetric holes formed in the film. However, after appearance of a three-phase contact line, dynamic simulations are inadequate in tracking the evolution of complex 3-D patterns which result from the growth and interaction of neighboring holes. For stratified foam films (31, 32), it is known that even during the early stages of the hole formation, the hole profile may lose its radial symmetry, and its

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0021-9797/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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rim may evolve into a 3-D wavy pattern due to the curvature induced instability (31). While all of the previous theoretical analyses of thin film stability have been confined to films on uncoated single substrates, the surfaces of most substrates encountered in applications are variously modified by thin coatings. The role of surface coatings in the dewetting of thin films and in altering the wettability of composite substrates has not been systematically investigated. Recent experimental studies (28, 33) of thin polymer films heated above their glass transition temperature on coated silicon wafers present a fascinating catalogue of various types of instabilities and patterns during different stages of dewetting. Initially smooth polystyrene films ( õ100 nm) break up by the creation of holes, which repeatedly grow and coalesce until the drainage from their giant rims slows down greatly and a polygonal, 2-D foamlike structure is created. The polymer ribbons forming the polygon then decay into spherical drops, possibly due to the Rayleigh instability of cylindrical masses. This hypothesis is tested in the present work. During the hole growth, the generation of droplets due to fingering instabilities is also observed on the low wettability surfaces. Among other things, we identify the key mechanisms of the instability during different stages of dewetting and provide, for the first time, evidence for the nonaxisymmetric, wavy nature of the hole-rim during dewetting. For thin films on uncoated substrates, the initial formation of holes, as well as the equilibrium contact angles of resultant drops, is governed by the macroscopic parameters of wetting—the apolar and polar components of the spreading coefficient (8, 18, 26, 27, 29). Thus, for uncoated substrates, the film stability may be correlated with the equilibrium wettability of the surface (8, 18, 26, 27, 29). However, a puzzling aspect of the polymer film experiments (28, 33) on coated substrates is the way in which wettability of the surface affects different stages of dewetting. On silicon wafers with nano-sized coatings of different wettabilities, the early stages of dewetting and the diameters of holes and polygons are largely unaffected by the wettability (contact angle) of the polymer on the composite surface (substrate plus coating). However, diameters of final droplets depend significantly on the equilibrium contact angle of the polymer on coated substrates. An understanding of these phenomena requires quantification of the effects of surface coatings on the film stability and on the contact angle. While it is experimentally known that nano-sized coatings are sufficient to alter the equilibrium contact angles drastically (33–37), the influence of coating properties on the thin film stability has never been quantified. This aspect is important because in many applications (e.g., biomaterials, flotation), thin surface coatings are applied for tailoring of surface properties and wettability (10, 11). Even when coatings are not applied, few substrates may be considered chemically homogeneous

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right up to their surfaces. Surface contamination, hydration, and reactions (e.g., Si r SiO2 ) can frequently change the surface properties rather profoundly. In such cases, top layers of the substrate may be thought of as pseudo-coatings. The main goals of the paper are the following: (a) identification of possible mechanisms for the evolution of various types of instabilities during different stages of dewetting of films on coated substrates; (b) theoretical descriptions of the film instability, dewetting, and equilibrium contact angle on coated substrates; and (c) a direct comparison of the present theory with previously reported experimental results (28, 33), which should also help our understanding of the unanswered questions/apparent anamolous behavior observed experimentally. In view of these goals, we believe it is necessary to briefly summarize and partially repeat the key experimental observations reported previously (28, 33). In what follows, we first briefly identify various mechanisms for the unfolding of the experimentally observed (28, 33) instability of thin polymer films on variously coated silicon substrates. The influences of the film thickness and the surface properties of the substrate, coatings, and the film on the film stability and wetting are then theoretically explored and compared with experiments (28, 33). EVOLUTION OF THE INSTABILITY: QUALITATIVE CONSIDERATIONS

While the details concerning the experiments are given elsewhere (33), we summarize here the key experimental observations necessary for formulating the theory of thin film stability on coated substrates. In addition, we also identify here the dominant physical mechanisms that appear to control the evolution of instability at various stages of dewetting. Experiments on the dewetting of thin films are best done with amorphous polymer films because (a) the high viscosity of the polymer allows time-resolved experiments, (b) the vapor pressure is practically zero, thus ensuring minimal mass loss, and (c) thin and smooth films can be initially prepared and stored below the glass transition temperature. Briefly, relatively smooth thin films of varying thickness ( Ç10–60 nm) were deposited by spin coating solutions of polystyrene (Mw Å 28 and 660 Kg/mol) onto three different types of surfaces—two types of silicon wafers (wafers A and B) with unknown surface coatings (as received from Wacker Chemitronics, Germany) and silicon wafers with self-assembled octadecyltrichlorosilane (OTS) coatings. Xray reflectrometry was used to measure the film thickness and the roughness of various interfaces with an accuracy of 0.2 nm. The thickness of various coatings (A, B, and OTS) on silicon wafers, as well as the roughness of the polymer– air and the polymer–coating interfaces were all less than 1 nm. The film surface also did not show any structure or

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FIG. 1. Schematic of the growth of a hole in a thin film on a coated substrate.

defects within the resolution of interference light microscopy (laterally 1 mm and vertically 1 nm). Figure 1 is the schematic presentation of the initially smooth film (broken line) on a coated substrate. Solid lines represent the initial stages in the formation of a hole. The film samples were annealed in a vacuum oven above the glass transition temperature (135 to 1707C) for periods ranging from 15 min to 24 h. The structural details concerning the film breakup and dewetting did not depend on the annealing temperature, but the kinetics of dewetting slowed down at lower temperatures which is mostly due to higher viscosity. The morphological details at various stages of dewetting were examined by an optical-phase interference microscope (OPIM) after samples were quenched to the room temperature. The real-time measurements were also performed where the sample was heated directly under a standard microscope (Zeiss, Germany). The initially smooth films broke up due to creation of holes, which repeatedly grew and coalesced until a polygonal ‘‘cellular’’ structure resembling a 2-D foam was created. The polymer ribbons forming the polygonal edges then decayed into spherical drops, which may be due to the Rayleigh instability (38) of cylindrical threads. The contact angles for PS on different coatings were deduced from micrographs of droplets and are summarized in Table 1. The equilibrium contact angle of the polymer is a measure of its wettability on the surface. Coatings of wafer B and OTS wafer are much less wettable by PS compared to the coating on wafer A. X-ray reflectrometry and OPIM confirmed the absence of any residual film on the substrate where dewetting occurred. Micrographs reveal the following detailed sequence of events in the film breakup and dewetting of substrates. First, a collection of randomly distributed, fairly uniform in size circular holes appear in the film and expose the substrate at their bases (Fig. 2). As is shown later, the initial formation of holes is engendered by the growth of polymer surface deformations (Fig. 1), which are fuelled by the LW component of the conjoining pressure in thin films. Once a

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hole is formed, it grows on the substrate due to the conjoining pressure near the contact line and also because of the surface tension force engendered by the circumferential curvature (perpendicular to the plane of the drawing in Fig. 1) of the hole. The surface tension force due to the circumferential curvature is proportional to ( g f /r) ( Ì H/ Ìr), where g f is the polymer surface tension, H(r, t) is the local thickness of the deformed film, and r is the radial coordinate describing the hole/rim surface, measured from the center of the hole. This induces liquid flow away from the crater and drives the rim surrounding the hole outward. Figure 2 is a typical micrograph of uniform, growing holes with their associated rims that appear dark. The excess polymer displaced by the growing hole forms a thick rim around the hole, because the viscous resistance for the flow through the film builds up as mH 03 (6, 7), thus preventing an easy escape of the polymer. However, the rim does not grow indefinately because the surface tension force engendered by the transverse curvature (proportional to Ì 2H/ Ìr 2 ) hinders the growth of rim to large heights, as is also shown for hole formation in stratified foam films (31). For wafer A, the number of holes ( NH ) per reference area were counted when the holes first became detectable by light microscopy. Another important observation that has not been previously reported is that the rim of an expanding hole develops circumferential corrugations as seen in Fig. 3. Apparently, once a ring-like rim grows and acquires a sufficiently large cross-sectional curvature, it can disintegrate into uneven pockets of the fluid due to the capillary instability in the azimuthal direction. The process appears to be analogous to the Rayleigh instability of long cylindrical threads (38), except that in our case, the thread (rim) is circular and moving. Bergeron et al. (31) have also proposed the possible existence of this secondary instability during the process of hole-sheeting in stratified foam films. Closely related to the generation of satellite polymer pockets of uneven heights, Fig. 3 for wafer A also depicts another type of instability in its rudimentary form. The top view of the lone expanding hole shows development of fingers/small polymer pockets on the substrate from the retracting polymer rim, which is reminiscent of fingering instabilities in driven wetting films (39, 40) and in displacement of fluids (41). The Rayleigh instability of the rim should be related to its cross-sectional curvature (steepness). The curvature in turn is determined by the difference between the amount of disTABLE 1 Average Contact Angle (u) of PS on Various Coatings of Silicon Substrate

u

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Wafer B

OTS wafer

22 { 4

42 { 5

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FIG. 2. Micrograph of formation of holes in a 40-nm-thick polystyrene film on wafer A. The rims are dark. The length of the bar is 100 mm. Polystyrene samples shown in all micrographs have a molecular weight of 28 K, except the micrographs in Figs. 7 and 8.

placed polymer added to the rim from the expanding hole and the amount which flows radially outward from the rim region in the same time. For high wettability systems (e.g., wafer A), the velocity, £ of the hole growth is low since £ a u 3 (15, 33, 42) and the small fingers seen in Fig. 3 remain underdeveloped and attached to the main body of the moving rim-mass. The faster growth of holes for large contact angles should, however, produce less diffused, steeper rims, resulting in the stronger instability of the rim. This is indeed the case for the low wettability systems as shown in Fig. 4 for wafer B (see also Fig. 14 of Ref. (33)). Thus, for low wettability systems (high u systems, e.g., wafer B and OTS wafer) fingers are more developed and they become long more quickly. The long fingers which are unable to keep pace with the more rapidly moving rims eventually pinch off in the form of drops due to the Rayleigh instability of long cylinderical masses. The formation of droplets left behind during hole growth is also shown in Fig. 4 for wafer B of low wettability (see also Fig. 14 of Ref. (33) for evolution of fingering instability). An interesting but previously unreported observation is

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that the fingering cannot be induced on wafer A even when the velocity of hole growth is increased by increasing the annealing temperature, thereby decreasing the polymer viscosity. However, a decrease in viscosity not only increases the velocity of hole growth, but also facilitates a faster flow away from the rim. This may leave the curvature of the rim rather unaffected, at least in the temperature range investigated. The rims of growing holes eventually contact each other, the opposing faces flatten out due to increased viscous resistance encountered by the escaping polymer and a thin liquid ribbon is formed between neighboring holes by a merger of their rims (Figs. 5a–5c). The coalescence occurs (Fig. 5c) as the polymer ribbon thins due to the capillary pressure (as in 3-D foams) engendered by the lateral (in-plane) curvature. Figures 3b, 5c, and 6 further emphasize the wavy and uneven shape of rims at the time of coalescence. In contrast to the micrographs of relatively thin films in Figs. 3 and 5, the micrograph of a thicker (200 nm) film in Fig. 6 shows greatly increased complexity of the rim-shape. For thick films, the rim appears to be more wavy in the azimuthal as

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FIG. 3. Micrographs and 3-D constructed profiles of wavy, uneven rims of expanding holes in a 38-nm-thick polystyrene film on wafer A. Vertical ˚. scale is in A

well as the radial directions. The formation of secondary fluid pockets in the rim becomes more pronounced. This observation is also in accord with the reasoning that since the viscous resistance to the flow varies as mH 03 (6, 7), the growth of hole (as well as its rim) should be slower for thinner films. Also, less material is available for the growth of rims in thinner films. These qualitative observations, as well as the quantitative interpretation of the data presented later in this paper, support the view that the fingering instability, whenever it can occur, becomes more important for thicker films. According to this interpretation, there is also a possibility that the fingering may largely cease for extremely thin films on less wettable substrates (e.g., wafer B) also. However, the behavior of films thinner than 10 nm has not yet been investigated. The dispersion of holes in polymer (2-D foam) becomes dense due to the repeated growth and coalescence of holes (Fig. 7). Tertiary coalescences also become more frequent during the later stages of ripening. Eventually, a network-like polygonal structure with thin long polymer ribbons is formed (Fig. 8). At this stage, further coalescence of giant holes slows down considerably because the time (tD ) for drainage (to allow for coalescence) in foams varies as D m (43), where D is the diameter of hole (bubble) and the exponent m for 2-D foams is reported to be as high as 7 (43). As the drainage from polymer threads and coalescence slows down, the Rayleigh instability engendered by the cross-sectional curvature of long ribbons becomes dominant and the polymer threads (polygonal

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edges) disintegrate into droplets, which appear neatly arranged in polygonal patterns (Fig. 9). All of the above-described stages in the evolution of the instability were also found for less wettable systems—wafer B and OTS wafer (28, 33). The only additional feature was the generation of droplets due to fingering instabilities during the growth of holes (as discussed previously for Fig. 4; see also Figs. 4 and 14 of Ref. (33)). In such cases, the empty spaces visible in Fig. 9 are also filled with a uniform distribution of droplets that are left behind (28, 33). The diameter of polygons ( Dp ) and the diameter of droplets (Dd ) were experimentally quantified for all wafers from micrographs by averaging at least 10 different spots from each sample. The distribution of these diameters were found to be rather narrow, and the statistical error bars were about 15% of the averages. Interestingly, while diameters of polygons were found to be rather independent of wettability ( u ), diameters of drops decrease substantially for low wettability systems (results shown later). Both Dp and Dd increased with increased initial film thickness. In what follows, we first formulate theories of hole formation and decay of ribbons into drops. The comparisons of theory with experiments is provided in the Results and Discussion section. THEORY

The excess pressure (compared to the bulk) in a thin ( õ100 nm) film is termed as the ‘‘disjoining/conjoining

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FIG. 4. Micrographs of the evolution of fingering instabilities and generation of drops from an expanding hole in a 50-nm-thick polystyrene film on wafer B. The length of the bar is 50 mm.

pressure’’ (24), which is engendered by the long-range intermolecular interactions, and therefore depends on the film thickness. The disjoining pressure is defined as the negative derivative of the free energy (per unit area) with respect to the film thickness. Repulsive interactions in a thin film produce positive disjoining pressures, whereas attractive interactions leading to the film instability produce negative disjoining pressures. The negative of the disjoining pressure is also termed as the conjoining pressure. As an example, the excess Lifshitz–van der Waals energy per unit area of a uniform film of thickness h on uncoated substrate can be written in terms of an effective Hamaker constant (A) as ( 0 A/12ph 2 ). Clearly, the LW component of the disjoining pressure, P equals ( 0 A/16ph 3 ), which is positive for A õ 0 (repulsion) and negative for A ú 0 (attraction and possible film instability). Similarly, other components of the disjoining pressure for interactions other than LW (e.g., polar interactions) may be defined. It is important to note, however, that the functional form of P depends greatly on the geometry of the film, e.g., for films between curved surfaces like spheres or coated planar substrates to be considered here, the equations for P are different. If the total disjoining pressure decreases with a decrease in the film thickness, i.e., the total conjoining pressure increases, a spontaneous deformation of the film surface results due to the flow of material from the thinner to thicker regions of the film (1– 4). In the absence of significant repulsive forces, the growth of surface instability eventually results in the formation of

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holes with the appearance of receding contact lines on the substrate (1–9, 12, 25–27). The relevant aspects of the thin film stability are now summarized. The surface instability (as shown in Fig. 1) evolves on a length scale (wavelength) that is large compared to the mean film thickness. The linearized equations of motion incorporating the intermolecular forces, when simplified in the lubrication approximation, admit space periodic solutions for the film thickness, H(x, t), H Å h / e exp( ikx / vt),

i 2 Å 01,

[1]

where h is the mean film thickness, e ! h is the initial amplitude of surface nonhomogeneties (defects), k is the wavenumber of instability, x is the coordinate parallel to the film surface, t is time, and the growth rate, v, of the instability for axisymmetric (or 2-D) perturbations is given by (1– 8, 25–27, 44) v Å Ck 2[ 0 gf k 2 0 ( Ìf / Ìh)],

[2]

where g f is the film surface tension and f is the intermolecular free energy per unit volume (derivative of the free energy per unit area) of molecules located at the free surface of the film. The parameter C depends inversely on the film viscosity, on rheology of the film fluid (44), and on the film thickness, but not on k. Clearly, for thin films, f is identical

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389

FIG. 4—Continued

to the negative of the disjoining pressure, 0 P (conjoining pressure). Clearly, from Eq. [2], the growth of instability ( v ú 0) is possible only for ( Ìf / Ìh) õ 0, and for wavenumbers in the range of (0, kc ), where kc is the critical wavenumber satisfying v Å 0. This conclusion can also be obtained from purely thermodynamic considerations (1, 2), regardless of the film rheology (44). The preferred (dominant) mode of the instability is the fastest growing mode. The lengthscale

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( l ) of the most probable mode is therefore obtained from [2] by Ìv / Ìk Å 0. l Å 2p /k Å 2p(2g f ) 1 / 2[ 0 ( Ìf / Ìh)] 01 / 2

[3]

Extensive numerical simulations (5, 8, 25, 27) of nonlinear equations of motion also confirm that the lengthscale of the fastest growing mode is indeed given by [3] whenever the initial amplitude of the disturbance (roughness of the

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FIG. 5. Micrographs of different stages of interaction and coalescence of neighboring holes in a 35-nm-thick polystyrene film on wafer A. The length of the bar is 50 mm.

film) is small ( e õ 0.1 h). This requirement is met in experiments with PS films reported here. In the absence of significant electrical double layer effects, the excess free energy ( f ) arises due to the long-range apolar Lifshitz–van der Waals interactions and also due to the much shorter ranged polar ‘‘acid–base’’ interactions (8, 17, 18, 24, 26, 27, 45), which are often described as hydrophobic interaction, hydration pressure, etc. The acid–base (AB) interactions can only occur between molecules that display conjugate polarities as measured by their electron (proton) donor and acceptor capabilities (45). The AB interactions among the molecules of PS are therefore ruled out because PS is a weakly electron-donor monopolar material (45) which, unlike dipolar substances like water, cannot form hydrogen bonds. It is because of this that the polar acid– base component of the surface tension of PS is indeed zero and its total surface tension equals its LW component ( g f Å g LW Å 38 mJ/m 2 ) (45). While the conjugate AB interacf tions among molecules of PS and coatings cannot be ruled out completely, they should be weak and short-ranged contact forces, owing to the lack of a mechanism for their propagation away from the interface. Further, there is no known organic/polymeric material with significant electron acceptor properties which can have significant conjugate AB interactions with the extremely weak electron donor PS (45). Nonlinear numerical simulations (8, 27) and the linear result [3] (26) show that even for dipolar substances like water, where AB interactions become relatively strong and long ranged (45), the free energy ( f ) and the scale of

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instability are governed solely by the LW interactions for relatively thick films ( ú10 nm). All PS films considered in our study satisfy this requirement, and therefore we consider only the LW interactions in formulation of f (or conjoining pressure). The LW interaction potential in [3] is evaluated at the mean film thickness (h), and it is made up of three different LW interactions, namely: (a) interactions among the molecules of the thin film of thickness h, (b) interactions among the molecules of the film and the underlying coating, and finally (c) interactions among the molecules of the film and the substrate on which the coating is deposited (Fig. 1). The total LW potential energy per unit volume at the free surface of the film is readily obtained by the pairwise summation of intermolecular potentials in the microscopic approach of London and Hamaker, f LW Å

1 1 (Aff 0 Acf ) 0 (Asf 0 Acf ), [4] 6ph 3 6p(h / d ) 3

where d is the thickness of the coating (Fig. 1), Aij denotes the Hamaker constant for LW interactions between molecules of type i and j, and subscripts f, c, and s denote the film, coating, and the substrate (silicon or fused quartz in our experiments), respectively. As expected, [4] reduces to the form A/6ph 3 in the limiting cases of d r 0 or ` . For d r 0 (absence of coating), the effective Hamaker constant, A is (Aff 0 As f ), whereas for d r ` , the influence of the substrate is completely screened and A equals (Aff 0 Acf ).

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FIG. 5—Continued

The lengthscale of the instability is obtained from Eqs. [3] and [4] as

F

l Å 4p( pg f ) 1 / 2h 2 (Aff 0 Acf )

/

S D 1/

d h

04

(Acf 0 As f )

G

01 / 2

.

[5]

Interestingly, the effect of coating on the film stability de-

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pends on the ratio of the coating thickness and the film thickness. l varies as h 2 both for ( d /h) r 0 and ( d /h) r ` , but the effective Hamaker constants for these two situations are different, as described earlier. Also in the case when the difference (Acf 0 As f ) is small, the film is governed by a single effective Hamaker constant, A Å Aff 0 Acf á Aff 0 As f . A similarity of LW interactions for the substrate and coating, however, does not necessarily imply similar wetting characteristics. As is discussed later, the wettability (u ) is profoundly altered by the short-range polar ‘‘acid–base’’ interactions.

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FIG. 6. Micrograph of the formation of complex, wavy patterns and secondary fluid pockets in the rim of a coalescing hole. The sample is a 200-nm-thick polystyrene film on wafer A. The length of the bar is 70 mm.

Due to the unknown nature of coatings in our experiments, it is not possible to estimate Acf . However, ( d /h) in Eq. [5] may be neglected as a first approximation because d õ 1 nm and h is in the range of 15–60 nm. This approximation is also supported by experimental data reported in the Results and Discussion section. The full expression [5] should be useful for the design and interpretation of future experiments involving thicker coatings and thinner films, where the LW interactions with the coating also influence the film stability. For ( d /h) ! 1, the effective Hamaker constant, Aff 0 As f in [5] may be related to the individual Hamaker constants (45, 46), q

q

q

A Å Aff 0 Asf Å Af f ( Aff 0 Ass ),

[6]

where (45, 46) Aii Å 24pd 20 g LW . i

[7a]

Relations [6] and [7a] give the relationship between the effective Hamaker constant and the LW component of the spreading coefficient, S LW of the film material on the substrate (8, 18, 26, 45), A Å 012pd 20 S LW ,

[7b]

where S LW for the film material on the substrate (silicon) is defined as (18, 26, 45) 0 g LW 0 g LW S LW Å g LW s f sf

[8a]

1/2 1/2 1/2 Å 2( g LW [( g LW 0 ( g LW ]. f ) s ) f )

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[8b]

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LW LW g LW s , g f , and g s f are the LW components of surface and interfacial tensions, respectively, and d 0 in [7a] and [7b] is the minimum cut-off equilibrium distance between surfaces where the extremely short-range Born repulsion climbs to infinity (hard-sphere model). The best estimate for d 0 is 0.158 nm (45) for an array of condensed-phase materials, including PS. In fact, the relation between A and S LW implied by Eq. [7b] can be derived directly as follows, without taking recourse to relations [6] and [7a]. For d ! h, the LW component of the excess free energy per unit area is given by ( 0 A/12ph 2 ), as determined from Eq. [4] together with the definition * fdh Å excess energy per unit area. The change in the free energy accompanying a reduction in the film thickness from infinity to the molecular cut-off distance (absence of the film) is therefore given by A/12pd 20 . The same change can also be represented in terms of the LW components of the interfacial tensions as ( g LW f LW / g LW s f ) 0 g s . Equation [7b] follows immediately by equating the two definitions of the change in the free energy. The reason it is advantageous to write A in terms of S LW is because while A is known only for a few standard substrates and fluids, S LW can be determined for any substrate by facile measurements of the equilibrium contact angles of the apolar (LW) liquids of known surface tensions (8, 18, 26, 45). The relationship of the effective Hamaker constant to the macroscopic parameter of wetting 0 S LW and its implications for the contact angle and film stability have been discussed in greater detail elsewhere (8, 18, 26, 45). Here it suffices to note that the LW forces can engender the film instability only if A ú 0 (Aff ú Ass ), in which case the LW component of the spreading coefficient defined by [7b] is negative and the surface nonwettability is promoted by LW interactions õ g LW in [8b]. because g LW s f The most probable area of the unit cell formed by the fastest growing two-dimensional surface instability is l 2 , and the initial number of holes per reference area of the film, ar , are thus obtained by combining [5] with [7] to give ( d /h ! 1).

NH Å ar / l 2 Å 03ar d 20 S LW /4p 2gf h 4;

S LW õ 0. [9]

S LW is also related to the equilibrium contact angle ( us ) of PS on uncoated silicon substrate by the Young–Dupre equation, S LW Å (cos us 0 1) gf 0 S AB , where S AB is the polar ‘‘acid–base’’ component of the spreading coefficient (8, 18, 26, 45) of PS on the uncoated substrate. In the Young–Dupre equation, (cos us 0 1) g f Å S, S is the total spreading coefficient which equals S LW / S AB , where S AB is defined analogous to Equation [8a] for S LW , viz. S AB Å AB AB AB AB g AB 0 g AB s 0 gf s f . Here g s , g f , and g s f denote the polar ‘‘acid–base’’ components of the respective surface and interfacial tensions (45).

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FIG. 7. Micrograph of tertiary coalescences during the later stages of ripening of holes-in-polymer dispersion. The sample is a 15-nm-thick PS film of molecular weight 660 K on wafer A. The length of the bar is 7 mm.

Equation [9] is also in accord with the intuitive reasoning that the thinner films on less wettable (large É S LW É) substrates should be more unstable, in that they disintegrate into a larger number of holes. Nonlinear simulations (5, 8, 25, 27) also confirm the expectation of the linear theory that the growth of instability continues unhindered (because Ìf / Ìh Å 0A/2ph 4 remains negative for all h) until a single hole with its rim is formed in each unit cell. Derivations leading to Eq. [9] also clarify that the dominant wavelength (initial number density of holes) for relatively thick films on thin coatings is governed largely by the long-range LW interactions of the film with the underlying substrate, regardless of the nature of LW and AB interactions with the coating. This is because initially, (h/ d ) is large, LW interactions with the coating are not fully developed (Eq. [4]), and AB interactions with the coating are too short-ranged to matter in comparison to the long-ranged LW interactions with the substrate. Interestingly, while the initial hole formation in rela-

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tively thick films of apolar or monopolar materials on substrates with thin coatings is governed entirely by the net LW interactions with the substrate (Eq. [9]), the equilibrium contact angle (wettability) after formation of equilibrium three-phase contact lines is largely determined by the LW and polar AB interactions between the coating and the film. Indeed, Eqs. [4] and [5] already show that as the local film thickness declines near the point of rupture (h r d 0 ), the effective Hamaker constant in the contact zone (h r d 0 ) nearly equals (Aff 0 Acf ), which is affected only by the LW properties of the coating. Further, the short-range AB interactions of the film material with the coating also become important in the contact zone. The equilibrium contact angle on the coated substrate, u, can be obtained from the augmented Young–Dupre equation (18, 24, 29, 47, 48) cos u Å 1 0

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*

`

d0

fdh.

[10a]

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FIG. 8. Micrograph of final polygonal pattern with largely intact rims in a 10-nm-thick PS film on wafer A ( Mw Å 660 K). The length of the bar is 7 mm.

The total potential (per unit volume), f equals f LW / f AB , where f LW is given by Eq. [4] and f AB is the potential due to short-range surface AB interactions of the coating with the film. We note that f AB is related to the AB component of the excess free energy, DG AB , of the film on the coating ` by the definition f AB Å d DG AB /dh. Thus the integral *d 0 fdh equals [ DG AB (` ) 0 DG AB (d 0 )]. This free energy change in reducing the film thickness from infinity (bulk film) to the molecular cut-off (no film) can also be written in terms of AB components of surface and interfacial tenAB AB / g AB 0 g AB is the AB sions as ( g AB f cf c ) Å 0Sc . Sc component of the spreading coefficient for the film material on the coating. Further, we note (analogous to Eq. [7b]) that another effective Hamaker constant in Eq. [4] may be LW is the LW written as Aff 0 Acf Å 012pd 20 S LW c , where S c component of the spreading coefficient of the film material on coating. Thus, S LW is defined as ( g LW 0 g LW 0 g LW c c f cf ).

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`

(8, 18, 26) and Finally, by noting that *d0 f AB dh Å 0S AB c using Eqs. [4] and [7b], Eq. [10a] gives u on a coated substrate, cos u Å 1 /

1 gf

F

S LW / S AB c c

/

G

d 20 (S LW 0 S LW c ) , ( d / d0 ) 2

[10b]

where S LW / S AB Å Sc is the total spreading coefficient of c c the film materials on the coating, and its AB component, Å g AB 0 g AB 0 g AB S AB c c f cf , is engendered by the surface AB interactions with the coating. Clearly, even for very thin coatings thicker than a few d 0 , u is determined solely by interactions with the coating, and

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395

FIG. 9. Micrograph of final polygonal pattern after decay of rims into spherical droplets. The initial sample was a 45-nm-thick PS film on wafer A. The length of the bar is 70 mm.

the influence of the underlying substrate is screened out, i.e., for d @ d 0 , cos u Å 1 / (S LW / S AB c c )/ g f .

[10c]

Indeed, it is experimentally well known (33–37) that even a monolayer of coating material can profoundly alter the contact angle, and for continuous coatings thicker than about 1 nm, u attains a value characteristic of the coating material. Thus, differences in contact angles of PS (Table 1) are mainly due to different values of S LW and S AB for coatings c c on wafer A, wafer B, and OTS wafer. To summarize, the initial density of holes (Eq. [9]) for relatively thick films on coated substrates is largely governed by the LW component of the wettability (S LW ) of the substrate regardless of the coating. However, after breakup of the film, the equilibrium contact angle is largely determined

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by the wettability (Sc ) of the coating, regardless of the substrate. We now show that the diameter of polygons also depends only on the substrate wettability. Thus, one may anticipate the absence of any correlation between wettability ( u ) of the coated substrate and the film instability. This is indeed the case in experiments reported here. Interestingly, as is shown later, the diameters of final drops depend on wettability ( u ) of coating and also on S LW . As discussed earlier, the final polygonal pattern (Fig. 8) results from the growth and coalescence of initial holes. The circle equivalent diameter of the polygon is obtained from pD 2p /4 Å Nc l 2 , where Nc is the average number of initial holes that combine to form a polygonal cell. With the help of Eq. [9], Dp is found to be Dp Å (4h 2 /d 0 )[ 0 Nc pg f /3S LW ] 1 / 2 .

[11]

As discussed earlier, the final degree of coalescence (Nc )

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before disintegration of ribbons (edges) into droplets is determined by a competition between the time scales of drainage ( tD ) and the Rayleigh instability ( tR ) of the slowly draining polymer ribbon formed between holes. An a priori prediction of Nc would therefore require an understanding of the complete 3-D dynamics of these processes occuring simultaneously, which has never been attempted. However, as a first approximation, Nc may be obtained by the condition that the time for drainage (leading to coalescence) becomes larger than the time for the Rayleigh instability (leading to drop formation), i.e., tD ú tR . Whereas tD in 2-D foams shows a very strong dependence on Dp ( tD a D pm , m ranges from 5 to 7) (43) and tR is a relatively weak function (38) of the cross-sectional perimeter of the ribbon ( tR a ( ur) 2 / 3 ), where r is the cross-sectional curvature of the polymer ribbon (thread). Thus, the condition tD Ç tR implies that Dp (and Nc from Eq. [11]) should vary little for systems that have the same value of S LW but somewhat different values of r and u. As is shown later, experiments indeed confirm the dependence implied by Eq. [11] and the invariance of Nc for different coatings on the same substrate. For low wettability (high u ) systems, the mass of polymer collected in polygonal edges (Fig. 8) is less than the initial mass contained in the film because of generation of drops by fingering instabilities during the growth of holes (Fig. 4). If n f is the fraction of the original mass collected in polygonal edges, the polymer volume in each polygonal edge (between two polygonal holes) equals ( pD 2p h/4np ) (2n f ), where Dp is the average number of edges. Assuming a circular cross section for the cylindrical edges (ribbons), the volume of a single edge can also be written as r 2 (2u 0 sin 2u ) Dp p /2np , where r is the radius of curvature of the cylindrical polygonal edge (rim) and u is the contact angle. Thus from the mass conservation, r 2 (2u 0 sin 2u ) Å nf Dph.

FIG. 10. Double logarithmic plot of the average number of initial holes per 10 4 mm2 for samples on wafer A. The line and open circles represent the theory and experiments, respectively.

from Eqs. [12] and [13], the final diameter of the spherical drop (Dd ) is obtained as q

D 3d Å 24 2(Dphn f ) 3 / 2 u sin 3u[(2 / cos u ) 1 (1 0 cos u ) 2 (2u 0 sin 2u ) 1 / 2 ] 01 ,

[15]

where Dp is obtained from Eq. [11], n f Å 1 for high wettability systems (wafer A) that lack the fingering instability, and n f õ 1 for low wettability systems (wafer B and OTS wafer). Equations [9] and [15] relate the observed variables of the film instability (NH , Dp , and Dd ) to the film thickness and the macroscopic parameters of wetting 0 S LW (or us ) and u.

[12] RESULTS AND DISCUSSION

Finally, assuming that the polygonal edges decay into drops due to the Rayleigh instability (Fig. 9), the dominant wavelength (length ofq fragments) of the instability should approximately equal 2 1 circular circumference of the thread (38). Thus, the volume of each cylindrical fragment is q

V f Å (2 2 ur)[r 2 (2u 0 sin 2u )/2],

[13]

which recedes to form a spherical drop of volume, Vd , Vd Å

pD 3d (1 0 cos u ) 2 (2 / cos u ). 24 sin 3u

[14]

From the mass conservation, Vf Å Vd , and by eliminating r

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Figure 10 shows the dependence of the average number of holes per 10 4 mm2 (NH ) on the film thickness for wafer A. The best linear fit to the data (not shown) on double logarithmic plot gives NH } h p , where p Å 04.1 { 0.15 (33) agrees with the predicted value of 04 from Eq. [9]. The best theoretical fit to the data (solid line) is obtained for S LW Å 08.1 mJ/m 2 in Eq. [9]. The theoretical fit is almost as good as the linear regression (33) of the data (on a log–log plot) in the entire range of the film thickness ( Ç10–60 nm). The value of S LW Å 08.1 mJ/m 2 appears to be realistic for silicon–PS system, and it corresponds to an effective Hamaker constant of 7.6 1 10 021 J (from Eq. [7b]). Since the Hamaker constant, Af f , for PS is 6.6 1 10 020 J (45), Eq. [6] gives a theoretical estimate of the Hamaker constant for the silicon substrate, Ass Å 5.2 1 10 020

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J. This is in agreement with the measured values of the Hamaker constant for fused quartz, which range from 5 1 10 020 to 6 1 10 020 J (46). The calculated value of S LW for c the coating of wafer A is much smaller ( á02.8 mJ/m 2 from Eq. [10c] with S AB á 0 and u Å 227, Table 1). c It is also interesting to note that dewetting of PS films did not occur on wafers when the oxide layer was stripped by hydrofluoric acid. This is because the oxide removal exposes the higher energy Si surface, thus increasing the value of Ass , and decreasing A (from Eq. 6). Indeed, based on the contact angles of apolar liquids, Zhao et al. (30) reported (or Ass from Eq. [7a]) of the about a 22% increase in g LW s oxide-stripped Si surface. An increase in Ass of similar magnitude for our substrates would make A very small (even negative), thus ensuring the stability despite the presence of thin nonwettable coatings with finite u. Interestingly, much thinner films (when d /h cannot be neglected in Eq. [9]) can still rupture due to nonwettability of coatings even if the substrate is wettable. This possibility, however, does not apply to experiments reported here. The variation of the polygon diameter (Dp ) with h for all three coatings is summarized in Fig. 11. The best linear fit of all data on a log–log plot gives the slope as 1.99 { 0.07, which is in agreement with Dp } h 2 predicted from Eq. [11]. Lines represent the best fit of data using Eq. [11] in the form log Dp (in mm) Å log C / 2 log h (in nm),

[16]

where C Å (4/d 0 )[ 0 Nc pg f /3S LW ] 1 / 2 1 10 03 nm01 and d 0 Å 0.158 nm. The best values of C for all coatings are

FIG. 11. Double logarithmic plot of the average diameter of polygons for samples on wafer A (filled squares), wafer B (open circles), and OTS wafer (filled triangles). The lines are the best theoretical fit to the data using values of C reported in Table 2. Lines for wafers A and B concide (solid line); OTS wafer is represented by the broken line.

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TABLE 2 The Best Fit Values of Parameter C in Equation [16] for Various Coatings of Silicon Substrate

Ca

Wafer A

Wafer B

OTS Wafer

0.0896 { 0.002

0.090 { 0.002

0.074 { 0.003

a

The mean value of C for all data is 0.088, and the mean percent error in prediction of the data from Eq. [16] is about 14%.

summarized in Table T2. The mean errors in prediction of the data using Eq. [16] are 8%, 13%, and 15% for wafer B, wafer A, and OTS wafer, respectively. For wafers A and B, S LW Å 08.1 mJ/m 2 , because both wafers had identical silicon substrates. An identical value of C for both wafers in Table 2 implies the same degree of coalescence (Nc ) of initial holes, despite very different wettabilities for coatings A and B. The best fit value of Nc for wafer A and B equals 2.7, which signifies a larger number of tertiary coalescences, as compared to binary coalescences, in generation of final polygonal network structure. Micrographs such as 8 and 9 do show a bimodel character of polygons with two different sizes. It therefore appears that the degree of coalescence of holes, before the Rayleigh instability of edges becomes operative, is rather independent of coating wettability ( u ), but depends only on the substrate wettability (S LW or us ). The latter also determines the initial density of holes. As discussed earlier, the Rayleigh instability can manifest when tD ( a D pm ) becomes comparable to tR ( a( ur) 2 / 3 ). Based on Eq. [12] it can be shown that ( ur) and therefore tR are indeed weak functions of u. The condition tD Ç tR for contact angles u1 and u2 implies (Dp1 /Dp2 ) Ç ( tR1 / tR2 ) 1 / m where m is large (5 to 7). Thus Dp1 á Dp2 is indeed expected in the range of contact angles investigated here. For OTS wafer, the value of C is about 17% smaller. While the silicon substrates for wafer A and B were identical, and they were used as received with coatings; coatings for OTS wafers were assembled in the laboratory on bare silicon wafers. A lower value of C for OTS wafers may therefore be either because of a somewhat higher É S LW É, or alternatively it may be because of statistical errors introduced by the smaller number ( Å4) of data available for OTS wafers. In fact, choosing an identical mean value of C ( Å0.088) for all three coatings does not increase the mean error in prediction ( Ç14%) of all data substantially. Thus, the value of C (or Nc for a given substrate) is largely invariant for different coatings that differ widely in their wettabilities. Finally, the diameter of drops produced by the decay of polygonal edges due to the Rayleigh instability is given by Eq. [15]. The data for the drop diameters are summarized in Fig. 12 for different coatings. For high wettability systems (e.g., wafer A), the fingering instabilities during the drop

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FIG. 12. Double logarithmic plot of the average diameter of droplets for samples on wafer A (open squares), wafer B (open circles), and OTS wafer (filled squares). The solid lines represent theoretical predictions from Eq. [15], with n f Å 1 for wafer A (no adjustable parameter), nfh Å 5.2 nm for wafer B, and nfh Å 4.2 nm for OTS wafer.

retraction are absent, and the volume of the polygonal ribbons is the same as the volume of the initial film (n f Å 1). As discussed earlier, the absence of fingering on wafer A is due to low velocities of hole growth (low cross-sectional curvature of the rim) for high wettability systems. For wafer A, the best linear fit on double logarithmic plot gives Dd a h q , where q Å 1.54 { 0.08 agrees well with the predicted value of q Å 1.5 from Eqns. [11] and [15]. Predictions of Dd from Eq. [15], without any further unknown parameters, are in remarkable agreement with the data for wafer A (Fig. 12). Predictions virtually concide with the best linear fit (33) of the data on a log–log scale in this range of film thicknesses. The agreement between the theory and experiments supports the hypothesis that the final decay of polygonal rims into droplets indeed occurs by the Rayleigh instability engendered by the cross-sectional curvature of polymer threads. Interestingly, for low wettability systems (data for wafer B and OTS wafer in Fig. 12), the best fit value of the exponent q is close to 1 ( Å1.05 { 0.05). Since Dpah 2 (from Eq. [11]), Eq. [15] in such cases implies that the product, hn f , should be independent of h, at least in the range of film thicknesses investigated experimentally. Physically, it means that the generation of drops due to fingering becomes more important (n f declines) as the film thickness increases. As discussed previously, an inverse dependence of n f on h may indeed be anticipated, because holes in thicker films should grow faster as the viscous resistance for the flow through the film builds up as H 03 . This should lead to rims of higher curvature and therefore a greater propensity for formation of thin fingers which eventually detach from the faster moving retracting mass. Thus, while dynamic details of the process are currently not well understood, the analysis of data together with Eq. [15] gives the following important clues. The quantity 0 (nfh) should in general depend on u, and for high wettability (low u ) systems, n f r 1 so that Ddah 1.5 ,

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but for low wettability systems, nf ah 01 so that Ddah from Eq. [15]. Indeed, for wafer B and OTS wafer, the predictions of Eq. [15] (solid lines in Fig. 12) are in complete agreement with the best fit for the data if nf h is assumed to be independent of h. The best fit values of nf h for wafer B and OTS wafer are 5.2 and 4.2 nm, respectively. A slightly lower value of nf h for OTS wafer ( u Å 48 { 8) also appears consistent with the slightly lower wettability of OTS wafer compared to wafer B ( u Å 42 { 5). For wafer B and OTS wafer, constancy of nf h also implies that n f should become close to one for films thinner than about 5 nm; fingering should largely cease, and Dd should become proportional to h 1.5 , as in the case of high wettability systems (wafer A). While this prediction cannot be tested with the aid of available data, it should help in the design of future experiments. In summary, thin coatings affect different stages of dewetting differently. For films that are much thicker than the coating, the initial stages of dewetting (e.g., number density of holes) are governed almost entirely by the wetting properties of the underlying substrate, regardless of the wettability of the coating. The equilibrium contact angles on the coated substrates, however, depend almost entirely on the coating properties, regardless of the substrate. After the formation of holes, at least four different time scales are involved in the dewetting process, the average time for contact ( tc Ç l / £ Ç h 2u 03 ) between rims of growing holes culminating in drainage, the time for drainage of the contact lines (threads) of length L between holes ( tD Ç L m , 5 õ m õ 7), the time for finger formation ( tF ), and the time for the decay of the polygonal threads into droplets due to the Rayleigh instability ( tR ). The competition between tc and tF determines if long fingers can develop before drainage begins. Fingering ( tF õ tc ) leading to drop formation from the rims of expanding holes is more pronounced for thicker films on less wettable substrates (large u ). Finally, as long as tD õ tR , coalescence of holes will occur, but once the contact line becomes sufficiently long, tD exceeds tR , and the polygonal edge decays into drops before coalescence can occur. The criterion tD Ç tR for the final polygonal pattern predicts Dp to be rather insensitive to the coating wettability ( u ), a fact also supported by experiments. Thus, the initial instability of the film, time of rupture, density of holes, degree of coalescence of holes, and diameter (density) of polygons, all depend largely only on the substrate used, independent of coating properties. For relatively thick films, application of thin coatings is rather ineffective in changing these parameters of the film instability. Wettability (contact angle) of the coating, however, exerts the key influences on two important events in the process of dewetting. The first is the generation of droplets via fingering instability during the growth of holes, which is encouraged by the higher velocity of growth (higher rim curvature) in low wettability

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systems. Second, the final diameter of drops resulting from the decay of polygonal rims is also influenced by the coating wettability (Eq. 15). The influence of substrate wettability on Dd , however, also remains important because Dp in Eq. [15] is determined by the substrate properties (Eq. 11). The present theory and the interpretation of experiments should also be an aid to the better design of thin film experiments in the future. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

Sheludko, A., Adv. Colloid Interface Sci. 1, 391 (1967). Vrij, A., and Overbeek, J. Th. G., J. Am. Chem. Soc. 90, 3074 (1968). Ruckenstein, E., and Jain, R. K., Faraday Trans. 70, 132 (1974). Maldarelli, C., Jain, R. K., Ivanov, I., and Ruckenstein, E., J. Colloid Interface Sci. 78, 118 (1980). Williams, M. B., and Davis, S. H., J. Colloid Interface Sci. 90, 220 (1982). Sharma, A., and Ruckenstein, E., J. Colloid Interface Sci. 113, 456 (1986). Teletzke, G. F., Davis, H. T., and Scriven, L. E., Chem. Eng. Commun. 55, 41 (1987). Sharma, A., and Jameel, A. T., J. Colloid Interface Sci. 161, 190 (1993). Mitlin, V. S., J. Colloid Interface Sci. 156, 491 (1993). Wicks, Z. W., Jones, F. N., and Pappas, S. P., ‘‘Organic Coatings: Science and Technology.’’ Wiley-Interscience, New York, 1992. Garbassi, F., Morra, M., and Occhiello, E., ‘‘Polymer Surfaces.’’ Wiley, Chichester, 1994. Kheshgi, H., and Scriven, L. E., Chem. Eng. Sci. 46, 519 (1991). Philip, J. R., J. Chem. Phys. 66, 5069 (1977). Karim, A., Mansour, A., Felcher, G. P., and Russell, T. P., Phys. Rev. B. 42, 6846 (1990). de Gennes, P. G., Rev. Mod. Phys. 57, 827 (1985). Cazabat, A.-M., Contemp. Phys. 28, 347 (1987). Teletzke, G. F., Davis, H. T., and Scriven, L. E., Revue Phys. Appl. 23, 989 (1988). Sharma, A., Langmuir 9, 3580 (1993). Neogi, P., and Berryman, J. P., J. Colloid Interface Sci. 88, 100 (1982). Burelbach, J. P., Bankoff, S. G., and Davis, S. H., J. Fluid Mech. 195, 463 (1988). Wayner, P. C., Jr., Colloids Surf. 52, 71 (1991).

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22. Taylor, G. I., and Michael, D. H., J. Fluid Mech. 58, 625 (1973). 23. Sharma, A., J. Colloid Interface Sci. 156, 96 (1993). 24. Derjaguin, B. V., Kollid. Zh. 17, 191 (1955); Derjaguin, B. V., Churaev, N. V., and Muller, V. M., ‘‘Surface Forces.’’ Consultants Bureau, New York, 1987. 25. Yiantsios, S. G., and Higgins, B. G., J. Colloid Interface Sci. 147, 341 (1991). 26. Sharma, A., Langmuir 9, 861 (1993). 27. Sharma, A., and Jameel, A. T., J. Chem. Soc. Faraday Trans. 90, 625 (1994). 28. Reiter, G., Phys. Rev. Lett. 68, 75 (1992). 29. Jameel, A. T., and Sharma, A., J. Colloid Interface Sci. 164, 416 (1994). 30. Zhao, W., Rafailovich, M. H., Sokolov, J., Fetters, L. J., et al., Phys. Rev. Lett. 70, 1453 (1993). 31. Bergeron, V., Jimenez-Laguna, A. I., and Radke, C. J., Langmuir 8, 3027 (1992). 32. Kralchevsky, P. A., Nikolov, A. D., Wasan, D. T., and Ivanov, I. B., Langmuir 6, 1180 (1990). 33. Reiter, G., Langmuir 9, 1344 (1993). 34. Langmuir, I., Trans. Faraday Soc. 15, 62 (1920). 35. Zisman, W. A., Adv. Chem. Soc. 43, 1 (1964). 36. Whitesides, G. M., and Laibinis, P. E., Langmuir 6, 87 (1990). 37. Extrand, C. W., Langmuir 9, 475 (1993). 38. Rayleigh, Lord, Proc. London Math. Soc. 10, 4 (1978); Sekimoto, K., Oguma, R., and Kawaski, K., Ann. Phys. 176, 359 (1987). 39. Cazabat, A. M., Heslot, F., Troian, S. M., and Carles, P., Nature 346, 824 (1990). 40. Brochard-Wyart, F., and Redon, C., Langmuir 8, 2324 (1992). 41. Saffman, P. G., and Taylor, G. I., Proc. Roy. Soc. London Ser. A. 245, 312 (1958). 42. Tanner, L. H., J. Phys. D: Appl. Phys. 12, 1473 (1979). 43. Hartland, S., in ‘‘Thin Liquid Films’’ (I. B. Ivanov, Ed.), Surfactant Science Series, pp. 663–766. Dekker, New York, 1988. 44. Sathyagal, A. N., and Narasimhan, G., Chem. Eng. Commun. 111, 161 (1992). 45. van Oss, C. J., Chaudhury, M. K., and Good, R. J., Chem. Rev. 88, 927 (1988); Sep. Sci. Technol. 24, 15 (1989); van Oss, C. J., Colloids Surf. A. 78, 1 (1993); and references therein. 46. Israelachvili, J. H., ‘‘Intermolecular and Surface Forces.’’ Academic Press, New York, 1985. 47. Frumkin, A. N., Zh. Fiz. Khim. 12, 337 (1938). 48. Wong, H., Morris, S., and Radke, C. J., J. Colloid Interface Sci. 148, 317 (1992).

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