Spontaneous Dewetting and Ordered Patterns in Evaporating Thin

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Spontaneous Dewetting and Ordered Patterns in Evaporating Thin Liquid Films on Homogeneous and Heterogeneous Substrates Kajari Kargupta, Rahul Konnur, and Ashutosh Sharma* Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India Received September 1, 2000. In Final Form: November 14, 2000 The growth of instabilities and the initial stages of dewetting of volatile thin aqueous films on partially wettable solid substrates are investigated based on 2D nonlinear simulations. Dewetting by the formation of holes occurs by a spinodal mechanism due to the hydrophobic attraction on chemically homogeneous surfaces. The number density of holes and, consequently, the rate of dewetting can be enhanced by as much as an order of magnitude by evaporation on a homogeneous surface. At moderate to high rates of evaporation, all the holes do not form at the same time uniformly over the surface but form gradually in a rather ordered way around the earliest holes which act as “seeds”. On a chemically heterogeneous substrate, spatial gradients of the interaction potential and the rate of evaporation engender the surface instability. A chemical heterogeneity can induce faster rupture at a higher mean thickness and, thus, control the hole size distribution and the pattern of drying very significantly. A locally ordered, complex pattern often forms that consists of a central giant “nucleated” hole surrounded by a few concentric rings of smaller spinodally created satellite holes. An increase in the rate of evaporation encourages the formation of a larger number of ringlike structures containing the satellite holes but reduces the size difference between the spinodal satellite holes and the heterogeneously nucleated holes. The results obtained are in accord with recent experimental observations.

Introduction The stability and dynamics of thin evaporating/ condensing fluid films on solid substrates have been of interest1-12 because of their importance in diverse scientific and technological applications such as solvent coating, drying, thin film heat transfer, and pattern formation in thin films. Most real surfaces encountered in practice are heterogeneous on nanometer to micrometer scales, for example, because of contamination, cavities, variable chain adsorption, uneven oxide layer, and so forth. The processes that are influenced by the substrate properties such as condensation13,14 and dewetting15-39 are profoundly * To whom correspondence should be addressed. E-mail: [email protected]. Fax: (91) 512 590104. (1) Burelbach, J. P.; Bankoff, S. G.; Davis, S. H. J. Fluid Mech. 1988, 195, 463. (2) Sharma, A.; Ruckenstein, E. Phys. Chem. Hydrodynam. 1988, 10, 675. (3) Oron, A.; Davis, S. H.; Bankoff, S. G. Rev. Mod. Phys. 1997, 69, 931. (4) Samid-Merzel, N.; Lipson, S. G.; Tannhauser, D. S. Phys. Rev. E 1998, 57, 2906. (5) Elbum M.; Lipson, S. G. Phys. Rev. Lett. 1994, 72, 3562. (6) Sharma, A. Langmuir 1998, 14, 4915. (7) Padmakar, A. S.; Kargupta, K.; Sharma, A. J. Chem. Phys. 1999, 110, 1735. (8) Lipson, S. G. Phys. Scr. 1996, T67, 63. (9) Thiele, U.; Mertig, M.; Pompe, W. Phys. Rev. Lett. 1998, 80, 2869. (10) Wayner, P. C. Langmuir 1993, 9, 294; AIChE J. 1999, 45, 2055. (11) Oron, A.; Bankoff, S. G. J. Colloid Interface Sci. 1999, 218, 152. (12) Erres, M. H.; Weidner, D. E.; Schwartz, L. W. Langmuir 1999, 15, 1859. (13) Kumar, A.; Whitesides, G. M. Science 1994, 263, 60. (14) Zhao, H.; Beysens, D. Langmuir 1995, 11, 627. (15) Pompe, T.; Frey, A.; Herminghaus, S. Langmuir 1998, 14, 2585. (16) Rockford, L.; Liu, Y.; Mansky, P.; Russell, T. P. Phys. Rev. Lett. 1999, 82, 2602. (17) Gleiche, M.; Chi, L. F.; Fuchs, H. Nature 2000, 403, 173. (18) Kataoka, D. E.; Troian, S. M. Nature 1999, 402, 794. (19) Nisato, G.; Ermi, B. D.; Douglas, J. F.; Karim, A. Macromolecules 1999, 32, 2356. (20) Boltau, M.; Walhelm, S.; Mlynek, J.; Krausch, G.; Steiner, U. Nature 1998, 391, 877.

affected by the presence of chemical heterogeneities. Deliberately tailored chemically heterogeneous substrates are also potential candidates for engineering of desired nano- and microscale patterns.14-23 For volatile films, the current theoretical understanding of the thin film stability and dynamics is limited to films on chemically homogeneous substrates. Although various theoretical frameworks1-7,10-12 are now available for the analysis of volatile films on homogeneous surfaces, the problem of pattern selection in drying films has not been investigated in the detail known for the nonevaporating films.36-39 A notable exception is the seminal 1D numerical study of Oron and Bankoff11 that indicates the possibility of interesting patterns in volatile films. Here, we will make use of a (21) Gau, H.; Herminghaus, S.; Lenz, P.; Lipowsky, R. Science 1999, 283, 46. (22) Lenz, P.; Lipowsky, R. Phys. Rev. Lett. 1998, 80, 1920. (23) Lenz, P.; Lipowsky, R. Eur. Phys. J. E 2000, 1, 249. (24) Herminghaus, S.; Jacobs, K.; Mecke, K.; Bischof, J.; Frey, A.; Ibn-Elhaj, M.; Schlagowski, S. Science 1998, 282, 916. (25) Herminghaus, S.; Jacobs, K.; Mecke, K.; Bischof, J. Phys. Rev. Lett. 1999, 83, 5302. (26) Stange, T. G.; Evans, D. F.; Hendrickson, W. A. Langmuir 1997, 13, 4459. (27) Guerra, J. M.; Srinivasrao, M.; Stein, R. S. Science 1992, 68, 75. (28) Konnur, R.; Kargupta, K.; Sharma, A. Phys. Rev. Lett. 2000, 84, 831. Kargupta, K.; Konnur, R.; Sharma, A. Langmuir 2000, 16, 10243. (29) Williams, M. B.; Davis, S. H. J. Colloid Interface Sci. 1982, 90, 220. (30) Sharma, A.; Ruckenstein, E. J. Colloid Interface Sci. 1986, 113, 456. (31) Teletzke, G. F.; Davis, H. T.; Scriven, L. E. Chem. Eng. Commun. 1987, 55, 41. (32) Sharma, A.; Jameel, A. T. J. Colloid Interface Sci. 1993, 161, 190. (33) Sharma, A.; Jameel, A. T. J. Chem. Soc., Faraday Trans. 1994, 90, 625. (34) Khanna, R.; Jameel, A. T., Sharma, A. Ind. Eng. Chem. Res. 1996, 35, 3081. (35) Sharma, A. Langmuir 1993, 9, 861. (36) Sharma, A.; Khanna, R. Phys. Rev. Lett. 1998, 81, 3463. (37) Sharma, A.; Khanna, R. J. Chem. Phys. 1999, 110, 4929. (38) Khanna R.; Sharma A. J. Colloid Interface Sci. 1997, 195, 42. (39) Singh J.; Sharma A. J. Adhes. Sci. Technol. 2000, 14, 145.

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Spontaneous Dewetting and Ordered Patterns

formalism6,7 that incorporates the effects of disjoining pressure and curvature on the rate of evaporation and will use a realistic disjoining pressure isotherm for aqueous films that consists of the van der Waals and the hydrophobic forces. In general, the surface instability leading to hole formation on a homogeneous substrate is engendered when the gradient of the conjoining pressure is negative. In particular, for an aqueous film on a partially or completely wettable substrate, the surface instability is derived from the polar hydrophobic attraction, as predicted theoretically6,7 and confirmed experimentally.5,8,9 The net van der Waals force is usually stabilizing for the aqueous films.40,41 The linear stability analysis and numerical simulations based on the 1D nonlinear thin film equation predict a significant influence of the rate of evaporation on the length scale and number density of holes.6,7,11 However, neither the linear theory nor 1D simulations can uncover the complete 3D morphology of an evaporating/condensing thin film during various stages of evolution of the instability. A systematic theoretical understanding of pattern formation in drying films on chemically heterogeneous surfaces is even sparser. We have recently proposed a new mechanism of instability of nonvolatile thin films on heterogeneous surfaces, where the surface instability leading to dewetting is derived from the gradient of wettability rather than from the nonwettability of the substrate itself, as in the spinodal dewetting.28 On the experimental front, a recent study of volatile unstable films on a graphite substrate shows systematic changes in dewetting patterns and hole size distribution with the rate of evaporation.9 At a higher rate of evaporation, holes of two widely different diameters are evolved. A decrease in the rate of evaporation reduces the number density of smaller holes and increases the diameter of the larger holes. A near complete suppression of smaller holes leads to monomodal hole size distribution at very low rates of evaporation. It was hypothesized that two different mechanisms of rupture, spinodal dewetting and heterogeneous nucleation at defects, were both operative.9 Here, we address the problems of stability, dynamics, morphology, and pattern selection for evaporating thin films based on the nonlinear 2D thin film equation. The effects of evaporation on the initial stages of dewetting and on the size distribution of holes are investigated both for the spinodal (homogeneous surface) and heterogeneous mechanisms. Heterogeneity is introduced as a patch having a different chemical potential or wettability compared to its surroundings.28 The dynamical and morphological features of the surface instability engendered by the two different mechanisms are clearly contrasted. Theoretical results are used to explain the recent experimental observation.9 Theory 1. Equation of Motion in Thin Film Approximation. We consider a thin ( 0, is satisfied everywhere.28 The local mass flux, m, from a thin liquid film is given by3,6,7,10,42

m ) Kg(pve - pv)

(2)

where Kg is the mass transfer coefficient for evaporation. An upper bound for Kg can be obtained from the kinetic theory, but it is best evaluated from evaporation experiments on thick flat films when the vapor phase resistance cannot be neglected. The equilibrium vapor pressure, pve, depends on the local curvature as well as on the disjoining pressure (-φ). The equilibrium vapor pressure, pve, for a thin fluid domain of finite curvature and thickness is obtained from the extended Kelvin equation6,7,10,43-45:

pve ) p0 exp{-(VL/RT)(γf∇2h - φ)}

(3)

where VL is the molar volume of the liquid, p0 is the equilibrium vapor pressure for a flat (hxx ) 0 and hyy ) (42) Scharge, R. W. A Theoretical Study of Interface Mass Transfer; Columbia University Press: New York, 1953. (43) Derjaguin, B. V. Theory of Stability of Thin Films and Colloids; Consultant Bureau/Plenum: New York, 1989. (44) Philip, J. R. J. Chem. Phys. 1977, 66, 5069. (45) Derjaguin, B. V.; Churaev, N. V. J. Colloid Interface Sci. 1978, 66, 389.

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0) bulk (φ ) 0) liquid surface, R is the universal gas constant, and T is the temperature. The classical Kelvin equation for bulk (φ ) 0) liquids already implies that the equilibrium vapor pressure for a convex (concave) liquid surface is higher (lower) than the equilibrium vapor pressure for a flat surface. In addition, the extended Kelvin equation also implies that the equilibrium vapor pressure of a thin fluid domain on a wettable surface (φ < 0) is lower than the corresponding vapor pressure p0 on a partially wettable surface (with φ > 0).6,7 Combining eqs 2 and 3 yields the following expression for the mass flux.6,7

m ) Kg[p0 exp{(VL/RT)(-γf∇2h + φ)} - pv]

(4)

Thus, the local rate of evaporation from a thin ( 0. 2. Excess Intermolecular Interactions. The total intermolecular body force potential per unit volume, φ, is related to the total free energy per unit area, ∆G, by the relation φ ) ∂∆G/∂h. ∆G can be represented as the sum of its apolar van der Waals and the polar hydrophobic components for the aqueous films:32,33,35,40,41

∆G ) SLW(d02/h2) + SP exp[(d0 - h)/lp]

(5)

where SLW and SP are the apolar and polar components of the total spreading coefficient; S ) γs - γf - γsf, where γs and γsf are the solid surface tension and the solid-fluid surface tension, respectively. d0 is an equilibrium cutoff distance (0.158 nm40,41,46), where Born repulsion takes over, and lp is a correlation length for the polar hydrophobic attraction, which is usually in the range of 0.4 nm to a few nanometers.40,41,46 For aqueous films, SLW is almost always positive because the apolar component of solid surface tension is usually in excess of the apolar component of surface tension of water (about 21.8 mN/m).35,40,41,46 SP for aqueous films, however, is always negative because of a large polar component of the free energy of adhesion of water, which signifies the polar hydrophobic attraction.32,34,35,44-46 Figure 1 shows the characteristic variation of the second derivative of excess free energy (φh) with the film thickness h for an aqueous film on a (46) Israelachvilli, J. H. Intermolecular and Surface Forces; Academic: New York, 1985.

Figure 1. Variation of spinodal parameter (φh ) ∂2∆G/∂h2) with thickness: curve 1, homogeneous substrate; curve 2, heterogeneous patch. The inset resolves the φh curve close to the critical thickness, hc1. For the homogeneous surroundings, SLW ) 106 mJ/m2, SP ) -159 mJ/m2, lp ) 0.55 nm, and hc1 ) 6.038 nm, and for the patch, ShLW ) 106 mJ/m2, ShP ) - 200 mJ/m2, lp ) 0.55 nm, and hc1 ) 6.235 nm.

Figure 2. Schematic diagram of a thin film on a heterogeneous substrate.

partially wettable substrate. The longer-range van der Waals repulsion wins beyond a certain critical thickness, hc1, 7,35 and thus a thicker film is mechanically stable on a homogeneous substrate because φh > 0. On a homogeneous substrate, evaporation thins the film until its thickness reduces below hc1, at which time the surface becomes unstable and the growth of the instability leads to dewetting by the formation of holes.7 To introduce heterogeneity, the polar component of the spreading coefficient (ShP) on a small part of the substrate (patch) is varied from that of its surroundings (curve 2 of Figure 1). A more negative value of ShP indicates a less wettable patch of higher contact angle. The contact angle is obtained from the extended Young-Dupre equation cos θ ) 1 + ∆G(he)/γf.46-49 ∆G(he) denotes the free energy per unit area evaluated at the equilibrium adsorbed thickness.7 A more negative value of the polar spreading coefficient shifts the critical thickness to a higher value (hc1) (curve 2 of Figure 1). Figure 2 shows the schematic of the patch and its surroundings. On a heterogeneous substrate, instability may also arise at a thickness above hc1.28 This will be confirmed here for volatile films. Simulations are reported for a few sets of parameters chosen to illustrate the important phenomena while keeping some contact with a recent set of experiments.9 3. Nondimensional Thin Film Equation. The following nondimensional thin film equation obtained from its dimensional counterpart, eq 1, governs the stability and spatiotemporal evolution of an evaporating thin film subjected to excess intermolecular interactions.1-3,6,7 (47) Brochard-Wyart, F.; di Megilo, J. M.; Quere, D.; de Gennes, P. G. Langmuir 1991, 7, 335. (48) Sharma, A. Langmuir 1993, 9, 3580. (49) Hirasaki, G. J. Contact Angle, Wettability and Adhesion; VSP: Amsterdam, 1993.


∂H/∂T + ∇‚[H3∇(∇2H)] - ∇‚[H3∇Φ] + C1M ) 0

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(6)

The following scalings have been used to obtain the above equation:

H ) h/hI

d ) d0/hI

3µ ∂h1/∂t + (3µKgVLp0/FRT) exp(VLφ(h0)/RT) [-γf∇2h1 + φ(h0)h1] + h03{γf ∇‚∇(∇2h1) φh(h0)(∇2h1)} ) 0 (15)

l ) lp/hI T ) (A2Ca)tν/hI2 (7a)

X ) (|A|Ca)1/2x/hI

Y ) (|A|Ca)1/2y/hI

(7b)

In the above, H(X,Y,T) is the nondimensional film thickness scaled with the initial mean thickness hI. d, l, X, Y, and T are the nondimensional cutoff distance, correlation length, lateral spatial coordinates, and time, respectively. The nondimensional parameters are defined as follows:

Ca ) (3Fν2)/(γfhI)

and

A ) (-6SLWd2)/(γfCa) P ) -SP/(6d2l2SLW) (8)

where ν is the kinematic viscosity (µf/F). In eq 6, the nondimensionalised mass flux M and the excess intermolecular interaction Φ are defined as7

M ) (1/p0)(m/Kg) ) exp[C2{-sgn(A) ∇ H + Φ}] - R1 (9) 2

Φ ) (-φhI)/(6SLWd2) ) (H-3/3) - Pl exp{(d - H)/l} (10) where

C1 ) Kg(p0hI/Fν)(A2Ca)-1

C2 ) (3AFν2/hI2)(VL/RT) R1 ) pv/p0 (11)

Here, the term sgn(A) refers to the sign of nondimensional parameter A; this is negative for the case of apolar repulsion (SLW > 0) considered here. 4. Equilibrium Thickness. At steady state, the mass flux is zero and eq 4 for a flat equilibrium adsorbed film gives

φ(he) ) (RT/VL) ln(pv/p0)

(12)

The dependence of the equilibrium thickness he on the degree of subsaturation pv/p0 can be evaluated from the above equation and results in the familiar adsorption isotherm.6,7,44,45 Further details may be found elsewhere.7 5. Linear Stability Analysis. Different methods for the linear stability analysis of the 1D counterpart of eq 1 have been developed and compared earlier for evolution on a homogeneous substrate.7 The complete linear analysis of the 2D thin film eq 1 predicts a dominant length scale of instability for the system, which is the diagonal length of a square cell. Following the procedure given in Padmakar et al.,7 we linearize eq 1 together with eqs 3 and 4 around the mean thickness and seek a solution of the form

h ) h0(t) + h1(x,y,t)

where h1, h0

(13)

Here, h0(t) is the instantaneous mean film thickness. Linearization of the evolution equation, eq 1, decomposes it into two equations describing the evolution of the mean thickness and the growth of instability, which are, respectively,

dh0(t)/dt + (Kg/F)[p0 exp{VLφ(h0)/RT} - pv] ) 0 (14)

φ(h0) and φh(h0) denote the intermolecular potential function and its derivative (spinodal parameter) evaluated at the mean thickness, respectively. The solution of the above eq 15 with periodic boundary conditions is of the form

h1 ) f(t) cos(kxx + kyy) or h1 ) f(t) sin(kxx + kyy) (16) Substituting this in eq 15 results in an ordinary differential equation describing the growth of the amplitude of initial perturbations with time, which is given by

3µ df/dt + [k2h03 + (3µKgVLp0/FRT) exp(VLφ(h0)/RT)] [γfk2f + φh(h0)f] ) 0 (17) where

k ) (kx2 + ky2)1/2

(18)

The initial conditions for eqs 14 and 17 are h0 ) hc1 and f ) at t ) 0, repectively, where hc1 is the critical thickness.7 The wavenumber, for which the total time of rupture, tr, is the minimum, is the dominant wavenumber. The dominant length scale of instability, λm ) 2π/km, obtained from this analysis gives the length of the diagonal of a unit square cell of length λm/x2. 6. Numerical Method. We directly solved the 1D and 2D forms of nonlinear thin film eq 6 over an area of nΛ2 with periodic boundary conditions, starting with an initial small amplitude volume conserving random perturbation. Λ is the dimensionless dominant wavelength, evaluated based on linear analysis. Equation 6, discretized using central differencing in space with half node interpolation, was solved using Gear’s Algorithm (especially suitable for stiff equations) for time marching.36-39 To address the problem of 3D pattern selection, we used at least 30 × 30 grids per Λ × Λ domain simulation applying periodic boundary conditions. To study the influence of heterogeneity on the evolution, the linear stability analysis of eq 6 based on the properties of a homogeneous substrate (surroundings) was carried out and the dominant spinodal lengthscale was computed. The total length of the domain was taken to be 3-5 times the value predicted by this analysis. Results and Discussion 1. Morphological Pathway of the Evolution of Instability on a Homogeneous Substrate. In this section, we discuss the morphological and dynamical features of the evolution of instability in an evaporating thin film on a homogeneous substrate. For comparison, the evolution of an unstable nonvolatile film of thickness (6 nm) slightly smaller than the critical thickness, hc1, is first simulated. Figure 3 depicts the morphology at different times during the evolution of a nonevaporating film on a homogeneous substrate in a domain size of area 9Λ2. The initial random perturbation is rapidly reorganized into a small-amplitude undulating pattern on a length scale close to the linear theory prediction, Λ. Long “hills” of structure undergo some fragmentation, while the “valleys” thin locally to produce largely circular full

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Figure 3. Evolution of surface instability in a 6.023 nm thick nonvolatile film on a homogeneous substrate. Pictures from left to right correspond to increasing times (t) of 0, 2.85, 5.27, 5.74, 9.08, 14.2, 24.6, 28.36, 33, and 103.54 s. The five increasing shades of gray used in the pictures represent nondimensional thickness ranges: 1.1. The conjoining pressure parameters and the spinodal parameter for this figure, as well as for Figures 5-7, correspond to curve 1 of Figure 1. For this figure as well as for Figures 4-7, 9, 10, 12, 13, 18, and 19, the dimensional times are evaluated on the basis of the following values: µ ) 5 kg/(ms) and γf ) 72.8 mJ/m2.

thickness holes surrounded by circular rims. As the holes expand and new holes form, their cross sections increasingly tend to deviate from that of a circle because of increased viscous resistance to their growth. A repeated coalescence of holes suggests rudiments of a larger structure, that should eventually fragment into droplets (long time results not shown).36,37 The most interesting feature is a slight tendency for the formation of secondary holes at the periphery of the holes that appear first. The genesis of doublets may be understood as follows. Because of an “explosive” nonlinear phase of very short duration, not all spinodally created holes appear at precisely the same time.36-37 The holes that appear first begin to grow and develop an elevated rim. The far side of the rim where it meets the undisturbed film develops a slight depression50 which acts as a “seed” for the formation of secondary holes prior to the appearance of other spinodal holes far away. A cut-sectional 2D view (Figure 4) taken from the 3D simulations of Figure 3 shows this process most clearly. The destabilizing term -∇‚{h3∇φ} in eq 1 is responsible for engendering a destabilizing flow from thinner to thicker regions leading to local thinning whenever the spinodal parameter, ∂φ/∂h, is negative. An early growth of the depression bordering the far side of the rim results because of the maximum value of the term ∇‚{h3∇φ} ∼ h03(∂φ/ ∂h)∇2h there, because curvature (or the gradient of slope) is large at the corner near the rim-border where it joins the largely flat film of very small curvature. This causes liquid flow from the far border of the rim toward the interior of the rim whenever the net force is attractive and grows stronger with local thinning; that is, the spinodal parameter, ∂φ/∂h, is negative. Figure 4 clearly shows the fast growth of the depression associated with the rim of an existing hole. The growth at later times is further accelerated because of the nonlinear effects. The time scale for the growth of this depression depends on its amplitude and on the magnitude of the spinodal parameter. In contrast to the satellite holes, the true spinodal holes are separated by a mean distance of the order of Λ, which is rather accurately predicted by the linear analysis.36,37 Even though holes do not form uniformly over the surface at the same instant of time but rather by a gradual ordering or layering around the existing holes, the hole density (per unit area) in the areas filled by the holes is still rather accurately predicted by (50) Ghatak, A.; Khanna, R.; Sharma A. J. Colloid Interface Sci. 1999, 212, 483.

Figure 4. Cut-sectional views of the 3D images shown in Figure 3. Curves 1-3 show nonlinear growth of the depression behind the rim of an existing hole at times 5.27, 5.74, and 5.9 s, respectively.

the linear theory (as in the last image of Figure 3). However, as discussed later the linear theory is no longer adequate to describe the hole density at high rates of evaporation and on heterogeneous surfaces. The induction of a hole by the presence of a rim has also been reported recently in a 1D simulation for an evaporating film.11 The discussion below for evaporating films shows that the phenomenon of the successive layering of holes initiated by the earliest holes occurs even more readily as the film becomes more unstable, for example, when the rupture is initiated at a lower mean thickness by the higher rate of evaporation. Figures 5-7 depict the evolution of evaporating thin films for decreasing rates of evaporation. The bulk evaporation rate has been varied from 20 to 180 nm/s by varying the ratio of partial pressure pv (humidity) to equilibrium vapor pressure p0. Figures 5 and 6 summarize


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Figure 5. Evolution of surface instability in a 6.023 nm thick volatile film on a homogeneous substrate with a high bulk evaporation rate of Ev ) 180 nm/s and pv/p0 ) 0.1. Pictures from left to right correspond to increasing times (t) of 0, 1.55, 1.615, 1.6255, 1.6328, 1.641, 1.645, 1.66, 1.6745, and 1.703 ms. In this as well as in the subsequent images, a continuous linear gray scale between the minimum and maximum thickness in each picture has been used.

Figure 6. Evolution of surface instability in a 6.023 nm thick volatile film on a homogeneous substrate with an intermediate bulk evaporation rate of Ev ) 100 nm/s and pv/p0 ) 0.5. Pictures from left to right correspond to times (t) of 0, 0.0483, 1.0657, 2.675, 2.6913, 2.705, 2.715, 2.7246, 2.7368, and 2.75246 ms.

Figure 7. Evolution of surface instability in a 6.023 nm thick volatile film on a homogeneous substrate with a low bulk evaporation rate, Ev ) 20 nm/s and pv/p0 ) 0.9. Pictures from left to right correspond to times (t) of 0, 10.2181, 10.3166, 10.3868, 10.4814, 10.5255, 10.7639, and 10.9212 ms. The simulation domain area is 111 556 nm2 (4.7Λ2 on the basis of linear theory).

the major events in the time evolution of a film at high (low humidity, pv/p0 ) 0.1) and moderate rates of evaporation (humidity, pv/p0 ) 0.5), respectively. Both the simulations are in a domain of 116 964 nm2 area (this size roughly equals 19Λ2 and 11Λ2 for Figures 5 and 6, respectively, on the basis of linear theory). The initial undulating patterns quickly undergo further reorganization resulting in an almost flat film surface, which evolves mainly because of evaporative thinning. The mean thickness reduces from the initial critical thickness (hc1 ) 6.038 nm) to a certain mean thickness (hr ) 3.10 and 3.5 nm for Figures 5 and 6, respectively), at which the instabilities grow explosively and rupture occurs via the formation of holes surrounded by largely circular and uneven liquid rims. A nearly uniform film close to the thickness of the equilibrium adsorbed layer at the cor-

responding degree of subsaturation is left behind at the base of an expanding hole. As in the nonvolatile case, depressions form ahead of the far sides of the rim and grow into satellite holes. Compared to the nonvolatile case, the growth rate of depressions is enhanced as a result of the increased spinodal parameter because the mean film thickness is decreased as a result of evaporation. In the simulations shown here, the rate of evaporation at the location of a depression was found to be about the same as elsewhere, notwithstanding a negative curvature of the depression which should decrease evaporation locally as discussed in the Theory section (eq 4). This is because of a compensatory effect of φ on the rate of evaporation because a lower thickness at the depression increases φ, thereby increasing evaporation, as also discussed in the Theory section (eq 4).

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Figure 8. Dependence of number density of holes/depressions per 100 µm2 on the bulk rate of evaporation. Curves 1 and 2 correspond to 2D simulations and the linear analysis, respectively.

Both Figures 5 and 6 at high to moderate rates of evaporation show the formation of successive generations of satellite holes near the periphery of the rims of existing holes (e.g., starting with image 4 in Figure 5). This leads to the growth of a locally ordered morphology that continuously enlarges by the addition of holes at its periphery. However, at low rates of evaporation (Figure 7, pv/p0 ) 0.9), the formation of satellite holes is minimized because the rupture occurs at a higher mean thickness (hr ) 4 nm) closer to the critical thickness, where the spinodal parameter is weak. In such cases, the depression ahead of the rim is not sufficiently prominent to engender a satellite hole rapidly prior to the appearance of spinodal holes correlated on the scale Λ. Figures 5 and 6 also indicate that at a decreased rate of evaporation, the average hole size becomes larger, and the number density of holes is reduced. At a lower rate of evaporation, the less negative value of the spinodal parameter, φh, at a higher height of rupture (hr) leads to weaker instability caused by intermolecular interactions (eq 4) and a larger length scale for the instability (decreased number density).7 2. Effect of Evaporation on Number Density of Holes. The linear analysis correctly predicts the time of rupture (tr) and the mean film thickness (hr) at the instant of rupture.7 For example, mean deviations between the simulations and the linear theory for hr and tr are about 1% and 5%, respectively, for the systems shown in Figures 5-7. However, the number of holes/depressions formed during the evolution is always higher than the number density predicted by the linear theory because of the formation of ordered doublets and satellite holes that are not properly accounted for in the linear analysis. Figure 8 compares the number density of holes per 100 µm2 substrate (Nd) area obtained from the 2D nonlinear simulations and the linear stability analysis.7 The discrepancy between the predictions of a purely spinodal mechanism (linear analysis) and simulations increases with increased rate of evaporation because of the greater population of satellite holes. Thus, simulations show a somewhat greater dependence of number density (dominant wavelength) on the bulk rate of evaporation (Ev ∝ (1 - pv/p0)) compared to the linear theory. The values of the exponent n in the relation Nd ) Evn are found to be 0.56 and 0.64 from the linear analysis and 2D simulations, respectively, for the range of parameters investigated here. 3. Effect of Evaporation on Kinetics of Dewetting. The dewetted area was identified in the simulations where the film thickness reaches the flat equilibrium film thickness. Figure 9 compares the variation of the fractional dewetted area (dewetted area/total area ) Ad) with time

Figure 9. Variation of fractional dewetted area, Ad, with time: (a) is for a nonvolatile film, and (b) and (c) correspond to volatile films with lower (pv/p0 ) 0.9) and higher (pv/p0 ) 0.1) rates of evaporation.

for a nonvolatile film (Figure 9a) with that of a volatile film with a lower (Figure 9b, pv/p0 ) 0.9) and a higher rate (Figure 9c, pv/p0 ) 0.1) of evaporation. All of these curves show a sharp increase in the area immediately after the time of initial rupture. The fractional dewetted area continues to increase very sharply because of the growth of existing holes and also because of the formation of new holes. Following the initial sharp incline, the slope of the curve decreases. At the later stage, when a sufficient number of holes are formed, the adjacent rims of holes interact with each other, followed by the coalescence of these holes. Consequently, the fractional dewetted area during this stage of evolution increases less rapidly. Increased evaporation leads to a more rapid onset of dewetting (as in Figure 9). The instantaneous rate of dewetting, defined as the time rate of change of the fractional dewetted area (Rd ) dAd/dt), varies significantly with the rate of evaporation, as shown in Figure 10. For a nonvolatile film, the rate of increase in the dewetted area is very slow and it reaches a maximum of 0.0067 s-1. Evaporation enhances the rate of dewetting by many orders of magnitude, for example, rates of 1500 and 8004 s-1 in parts b and c of Figure 10, respectively. The growth rate of a hole depends on its contact angle and is largely unaffected by the rate of evaporation. As discussed earlier, increased evaporation forces rupture at a lower mean thickness where the spinodal parameter is higher, that is, the instability is stronger. Thus, a larger number density of holes (per unit area) at a higher rate of evaporation (Figure 8) enhances the rate of dewetting. 4. Evolution of Instability on a Chemically Heterogeneous Substrate. In this section, we discuss the different initial stages of evolution of a thin film on a chemically heterogeneous substrate. For illustration, a substrate containing a single less wettable patch has been considered. Figure 11a shows the initial stages of instability starting from a nearly uniform thickness above hc1 (hI ) 42 nm) until the appearance of a primary hole on the heterogeneity. For comparison, the evolution on a homogeneous substrate is depicted in Figure 11b. Figure


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Figure 10. Variation of instantaneous rate of dewetting, Rd, with time: (a) is for a nonvolatile film, and (b) and (c) correspond to volatile films with lower (pv/p0 ) 0.9) and higher (pv/p0 ) 0.1) rates of evaporation.

11a shows that the initial evolution (until rupture) on a heterogeneous substrate consists of three successive stages: (a) the film thins by evaporation from the initial thickness to some intermediate thickness, hin, without significant concurrent growth of instability; (b) instability initiates with a local depression at the location of the patch, while the rest of the film remains undisturbed and flat; (c) rupture on the patch occurs much earlier than on the spinodal time scale. The time of rupture in the presence of a heterogeneity depends on the potential difference introduced by the heterogeneity. A detailed account of the underlying physics for the heterogeneous mechanism for nonevaporating films can be found elsewhere.28 The rate of evaporation is also enhanced on a less wettable patch (with higher potential, φ) compared to the surroundings. The outward flow across the patch boundary, from the less wettable to more wettable regions, acts in a cooperative manner and causes a depression to occur even at a thickness higher than hc1. However, the position of hin depends on the relative time scale of evaporation and of instability engendered by the heterogeneity. The amplitude of the depression grows, while the mean thickness continues to decrease from hin to hr. Rupture at the location of the patch occurs at a shorter time (th ) 0.09017 s) and higher thickness (hr ) 28 nm) compared to that on a homogeneous substrate (tr ) 0.125117 s and hr ) 24.5 nm). No surface undulation (related to the spinodal length scales) is seen on the remaining part of the film. The heterogeneously formed hole grows across the patch. At this stage, the mean thickness remains almost constant and dewetting occurs mainly because of the growth of holes. At later stages of evolution, spinodal holes form around the primary holes. This issue will be discussed in detail later. As mentioned earlier and seen in Figure 11b on a homogeneous substrate, the initial evolution consists of two successive stages: (a) evaporative thinning from the initial thickness to a thickness hr, without any significant onset of instability, and (b) the appearance of periodically placed initial depressions, on a length scale close to the dominant wavelength as

Figure 11. Initial evolution of instability in a volatile film on (a) a heterogeneous substrate and (b) a homogeneous substrate. Length of the simulation domain, L ) 98 µm ) 12λ where λ is the dominant wavelength as calculated from the linear analysis based on the following properties of the surroundings. SLW ) ShLW ) 20 mJ/m2, SP ) -20 mJ/m2, ShP ) -30 mJ/m2, lp (both for the patch and its surroundings) ) 2.5 nm, hc1 (surroundings) ) 36.218 nm, hc1 (patch) ) 37.65 nm, bulk evaporation rate Ev ) 100 nm/s, and λ (surroundings) ) 8 µm. The length of the heterogeneity is 7.84 µm (∼λ). Profiles 1-6 in (a) correspond to 14.59, 47.45, 62.849, 74.254, 88.98, and 90.11 ms, respectively. Profiles 1-5 in (b) correspond to 14.59, 30.8033, 105.481, 123.365, and 125.077 ms, respectively. In this figure as well as in Figures 14-17, the dimensional times are evaluated on the basis of the following values: µ ) 10-3 kg/(ms) and γf ) 72.8 mJ/m2.

calculated from linear analysis, followed by an explosive growth of instability, while the mean thickness remains almost constant. On a homogeneous substrate, the difference between hin and hr is negligible, and thus the intermediate step b as discussed earlier for a heterogeneous surface is absent here. Thus, heterogeneity effectively accelerates the onset of dewetting and introduces two additional time scales of instability: the time scale of heterogeneous instability/rupture and the time scale of evaporation on the patch. These time scales, together with the time scales for the spinodal dewetting and the formation of satellite holes, determine the morphological features of the pattern that evolves on a heterogeneous substrate. 5. Morphological Patterns on a Heterogeneous Substrate at Different Rates of Evaporation. On the basis of nonlinear 2D simulations, we now illustrate the effects of the rate of evaporation on the initial dynamical features and morphological patterns in a volatile film on a heterogeneous substrate. To obtain the effect of evaporation on the morphology, the rate of evaporation has been varied by changing the partial pressure (i.e., pv/p0) from 0.1 to 0.9. The substrate is considered to have a small (0.025 µm) less wettable patch. The total area of the

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Figure 12. Evolution of surface instability in a 10 nm thick volatile film on a heterogeneous substrate with a centrally located less wettable patch. Pictures from left to right correspond to times (t) of 0, 0.5385, 0.738, 0.74418, 0.7516, 0.7888, 0.7915, 0.79707., 0.82127, and 0.926 ms. The disjoining pressure isotherm for this figure as well as for Figure 13 are the same as used in Figure 1 (see curves 1 and 2 of Figure 1), and the bulk evaporation rate is Ev ) 900 nm/s. The simulation domain area is 81225 nm2 (11.7Λ2 on the basis of linear theory).

Figure 13. Evolution of surface instability in a 10 nm thick volatile film on a heterogeneous substrate with a centrally located less wettable patch. Pictures from left to right correspond to times (t) of 0, 2.31, 2.538, 2.73, 2.749, 2.762, 2.778, 2.787, 2.7987, and 2.813 ms. The bulk evaporation rate is Ev ) 100 nm/s. The simulation domain area is 3.2Λ2.

substrate is kept constant to clearly compare the relative size of the structures as the rate of evaporation is varied. Figure 12 depicts different stages of evolution of a thin volatile film on a heterogeneous substrate for a high evaporation rate (pv/p0 ) 0.1). For comparison, Figure 13 summarizes the different stages of evolution on the same substrate (same total area and patch characteristics) for a lower rate of evaporation (pv/p0 ) 0.9). In both cases, the initial film thickness is 10 nm, which is higher than the critical thickness (6.038 nm). Figure 12 confirms the three stages of overall evolution discussed earlier. Heterogeneity causes a rapid depression at the location of the patch, while the mean thickness of the patch surroundings continues to reduce from hin (5.6 nm) to thickness at rupture hr (3.3 nm) (image 2 of Figure 12). This is followed by the rupture of the film by the formation of a full thickness primary hole (image 3 of Figure 12). The bulk part of the substrate (surroundings) remains almost flat at the time of rupture. The thickness where the instability/ depression starts is much higher (5.6 nm) than on a homogeneous substrate (3.6 nm, predicted by the linear theory). The growth of the primary depression is assisted by the flow caused by the heterogeneity, by the higher rate of evaporation on the patch, and also by the negative spinodal parameter (φh < 0). The heterogeneously formed hole continues to expand across the patch boundary. A ringlike depression surrounding the elevated rim now begins to form (image 5 in Figure 12) and grow in amplitude by the spinodal mechanism. The depression provides the preferred site for the formation of several satellite holes surrounding the primary hole. The growth

of the initially “nucleated” hole is stopped because of the local ordering of holes around the periphery of its rim. This issue is discussed later in greater detail. Further dewetting now continues by the formation and growth of a large number of small diameter satellite holes. Thus, the presence of heterogeneity leads to a more pronounced bimodal distribution of hole sizes, where the size of the larger (heterogeneously formed) holes depends on the difference between the time of rupture on the heterogeneity and the time of formation of satellite holes; i.e.that is, the primary hole continues to grow until the satellite holes appear. Finally, the holes coalesce and the circular liquid rim disintegrates into droplets because of the Rayleigh instability of the circular threads. An increase in the partial pressure (decrease in the rate of evaporation) affects both the dynamical and morphological features of evolution (Figure 13). The onset of instability is at a higher mean thickness, 7.9 nm, which is higher than hc1. At this mean thickness, the spinodal parameter, φh, is positive, and therefore the instability grows solely by the heterogeneous mechanism initially. As the mean thickness declines below the critical, the spinodal mechanism also accelerates the growth of instability. Rupture occurs at a mean thickness of 3.7 nm and at a time that is much shorter than the spinodal time scale. The hole expands across the patch. In this case, the difference in rupture time on the patch and on its surroundings is more than the previous case, because rupture occurs at a higher mean thickness. The difference in rupture times on the patch and on its surroundings suppresses the formation of satellite holes for a long time,


Figure 14. Evolution of instability in a volatile film on a heterogeneous substrate, which is identical to the substrate in Figure 11 (of the same dimensional length and characteristics (SLW, SP, ShP, lp)). The bulk evaporation rate Ev ) 180 nm/s, and λ ) 6.7 µm. Profiles 1-5 correspond to times of 56.3934, 60.8428, 64.951, 67.646, and 70.961 ms. Evolution on the half domain (L/2 ) 49 µm) is shown, where the heterogeneity is at the righthand corner and is of length 3.9 µm.

so that the primary hole becomes much bigger (last image of Figure 13) than in the case of high rates of evaporation. Clearly, at low evaporation rates the population of heterogeneously grown holes becomes more prominent and the relative population density (per unit area) of the satellite holes decreases. Thus, on a substrate containing many heterogeneous sites, an increase in the rate of evaporation tends to make the hole size distribution increasingly bimodal, consisting of two distinct peaks at large diameter primary, heterogeneously nucleated holes and at the smaller diameter of satellite holes. This prediction is in accordance with the experimental finding,9 where it was observed that a bimodal distribution of hole sizes at lower humidity shifts to an increasingly monomodal distribution at higher humidity. With the lowering of evaporation rate, the large diameter peak shifts to even larger diameters and the peak corresponding to the smaller diameter nearly vanishes.9 Simulations therefore support the hypothesis9 that there are indeed two distinct populations of holes that are engendered by two distinct mechanisms (heterogeneous and spinodal) during dewetting of aqueous films. In addition, the simulations and the theory presented here provide a clear rationale for the observed shift in the hole size distribution with the rate of evaporation. 6. Ordered Arrangement of Holes around a Heterogeneity. In the following section, we discuss the phenomenon that leads to an ordered arrangement of holes of different sizes around a larger hole that forms rapidly, for example, because of heterogeneity.28 In particular, we address the role of evaporation in producing an ordered arrangement of holes. Figure 14 depicts the evolution of different sized holes around a heterogeneously formed hole, as obtained from a 1D simulation. The figure shows the formation of four successively formed holes of gradually decreasing size around the initially nucleated hole. The height of the rim associated with the holes also decreases with the size of the holes. The succession of holes is formed from growth of the depression adjacent to the rim of the preceding

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Figure 15. Evolution of instability in a volatile film on a heterogeneous substrate for a bulk evaporation rate of Ev ) 100 nm/s and λ ) 8 µm. For this as well as for Figure 16, the substrate properties and dimensional length are the same as in Figure 14. Profiles 1-5 correspond to times of 90.155, 93.106, 104.145, 111.24, and 119.37 ms.

Figure 16. Evolution of instability in a volatile film on a heterogeneous substrate for a bulk evaporation rate of Ev ) 20 nm/s and λ ) 12.3 µm. Profiles 1-4 correspond to times of 332.26, 337.334, 356.17, and 378.83 ms.

hole. Purely spinodal holes on the surroundings are not yet formed. A decrease in the rate of evaporation (Figures 15 and 16) decreases the number of satellite holes (per unit length), increases the size of the primary hole and the height of its rim, and makes the population of satellite holes less prominent. An increased rate of evaporation increases the density (per unit length) of the satellite holes for the following reasons. The number density of satellite holes (per unit length) depends on how far an existing hole can grow on the substrate before the depression ahead of its rim produces another satellite hole. Increased evaporation forces rupture at a lower film thickness, where the instability is stronger, and thus the amplitude of the depression increases faster with increased rate of evaporation. However, the growth rate of an existing hole, which depends only on the equilibrium contact angle, remains almost unaffected by the rate of evaporation. Thus, increased evaporation causes a larger number of satellite holes per unit length of the substrate. As shown below for the 2D situations, this translates into a larger number of satellite holes per unit area and also a larger number of layers containing the satellite holes around the primary hole or the seed.

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Figure 17. Evolution of surface instability in a 39 nm thick volatile film on a heterogeneous substrate with a centrally located less wettable patch. The bulk evaporation rate is Ev ) 36 nm/s. Pictures from left to right correspond to times of 193.3, 198.26, 241.34, 253.55, 256.89, 261.24, 271.77, 275.67, 281.61, and 287.9 ms. SLW ) ShLW ) 10 mJ/m2, SP ) -20 mJ/m2, ShP ) -40 mJ/m2, lp ) 2.5 nm (both for the patch and its surroundings), hc1 (surroundings) ) 38.58 nm, and hc1 (patch) ) 40.9 nm. The total simulation domain length L ) 97.5 µm ≈ 7.8λ. The length of the heterogeneity at the center is 9.75 µm (0.78λ).

Figure 17 shows one such ordered morphological pattern formed during the evolution of a 39 nm thick film on a heterogeneous substrate having a single heterogeneity. The substrate area for this simulation is taken to be very large (61Λ2, based on the characteristics of the surroundings) to show clearly the large-scale pattern that evolves at late times. The equilibrium contact angles on the patch and its surroundings are 59° and 37°, respectively. The figure shows a large primary hole surrounded by two layers of satellite holes even for a bulk evaporation rate as small as 36 nm/s. The initial rupture on the patch occurs rapidly at 0.193 s, which is short compared to the rupture time (0.248 s) on the patch surroundings. A ringlike depression forms around the uneven rim of the primary hole (image 3 in Figure 17). The spinodal instability preferentially engenders several satellite holes (as a layer) in the depression (images 4-6 of Figure 17). The mean distance between the satellite holes within an ordered layer is close to the spinodal length scale, Λ, as predicted by the linear theory. The satellite holes grow and coalesce to form a flowerlike morphology which slowly transforms into an annular dewetted region with its own associated rim. A second generation of satellite holes now forms again starting from the depression at the far side of the rim (images 8-10 in Figure 17). As mentioned earlier, the hole size and the rim height gradually decrease from the inner to outer layers. The width of the annular dewetted zone depends on the time for the formation of a satellite hole and on the growth velocity of the dewetted zone. The number of holes increases in the successive layers to maintain the same average distance, Λ, between two adjacent holes. The process of ordering continues until the true spinodal holes start appearing on the surroundings (not shown). In the simulation shown, the ordering is already very long ranged, exceeding about 3.5Λ. We have recently reported28 such an ordered morphology during the evolution of nonvolatile thin films on heterogeneous substrates. Although a heterogeneity initiates ordering by introducing a seed or the primary hole rapidly, evaporation further strengthens the process of long-range ordering by accelerating the growth of the depression associated with the rim of a seed. 7. Dynamics of Dewetting on a Heterogeneous Substrate. To study the effect of evaporation on the dynamics of dewetting on a heterogeneous substrate, we have computed the fractional dewetted area. Figure 18 compares the variation of the fractional dewetted area (Ad) for a higher (Figure 18a) and a lower rate (Figure 18b) of evaporation, for the system discussed in Figures

Figure 18. Variation of fractional dewetted area, Ad, with time during the evolution of a volatile film on heterogeneous substrates described in Figures 12 and 13: (a) is for a higher rate of evaporation (pv/p0 ) 0.1, Ev ) 900 nm/s), and (b) corresponds to a lower (pv/p0 ) 0.9, Ev ) 100 nm/s) rate of evaporation.

12 and 13. The dewetted area increases sharply immediately after the onset of rupture due to the growth of the first nucleated hole on the heterogeneity. Rupture time decreases with the increase in the rate of evaporation. Contrary to the case of dewetting on a homogeneous surface (Figure 9), the instantaneous rate of dewetting (slopes of the curves at any time) remains unchanged at the initial stage of dewetting (before the satellite hole formation) at different rates of evaporation (Figure 19). This is because the initial hole density is governed solely by the number of heterogeneous sites, and the velocity of hole growth is largely independent of the rate of evaporation. Figure 19a shows that the maximum rate of dewetting occurs immediately after the time of satellite hole formation. The maximum rate of dewetting is slightly lower at low rates of evaporation (Figure 19b), where the formation of satellite holes is suppressed. The reason the maximum rate of dewetting does not increase significantly despite the formation of satellite holes is because the growth of the primary hole is arrested by the formation of secondary


Figure 19. Variation of instantaneous rate of dewetting, Rd,, during the evolution of a volatile film on a heterogeneous substrate with time (t): (a) is for a higher rate of evaporation (pv/p0 ) 0.1, Ev ) 900 nm/s), and (b) corresponds to a lower rate of evaporation (pv/p0 ) 0.9, Ev ) 100 nm/s).

holes. On a homogeneous surface, increased evaporation significantly increases the rate of dewetting by producing a higher density of spinodally created holes (Figure 9). Thus, a weak dependence of the rate of dewetting on the rate of evaporation indicates dewetting dominated by the heterogeneous mechanism. Conclusion The problems of pattern selection, morphology, and dynamics during dewetting of unstable volatile thin films on partially wettable homogeneous and heterogeneous substrates have been studied based on 1D and 2D nonlinear simulations. On a chemically homogeneous surface, holes form by a spinodal mechanism due to increased attractive forces with a local decline in the film thickness. Thus, on a homogeneous surface an increased rate of evaporation engenders stronger instability at a

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lower thickness, increases the number density of holes, and thereby increases substantially the rate at which the substrate is dewetted. On a chemically heterogeneous surface, holes can form more rapidly by another heterogeneous nucleation mechanism, which is derived from a microscale wettability contrast, that is, because of the flow of liquid from the less wettable to more wettable regions. The rate of dewetting of a heterogeneous surface therefore is much less sensitive to changes in the rate of evaporation or humidity. Contrary to the predictions of the purely spinodal model, even on a homogeneous surface, holes are not always formed uniformly over the entire surface at some mean correlated distance predicted by the linear analysis, Λ. Satellite holes in the immediate proximity of the far side of the rim of an existing hole can also form readily because of the formation of a depression that serves as a seed. When the film is spinodally unstable, the depression grows into a secondary hole before the formation of the Λ-correlated holes of the purely spinodal mechanism. The formation of the satellite holes leading to a locally ordered structure becomes increasingly more prominent by the following: (a) increased time difference between the appearance of the first set of seed holes and the spinodal time scale for the appearance of Λ-correlated holes and (b) stronger instability leading to faster growth of a depression ahead of the rim of a seed hole. In view of these two conditions, the local ordering brought about by the satellite holes becomes more important for more unstable thinner films, for higher rates of evaporation, and for chemically heterogeneous surfaces. Thus, at low rates of evaporation on a heterogeneous surface, the population of satellite holes is minimized or can even be suppressed depending on the distance between the heterogeneities. At high rates of evaporation, a more bimodal distribution of hole sizes results because of the presence of larger heterogeneously grown holes and their smaller satellite holes. This theoretical result is in accord with the experimental observation.9 Acknowledgment. Discussions with S. G. Lipson, G. Reiter, S. Herminghaus, and R. Khanna are gratefully acknowledged. LA0012586

Spontaneous Dewetting and Ordered Patterns in Evaporating Thin

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