Ind. Eng. Chem. Res. 1996, 35, 2955-2963
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Drainage of a Partially Wetting Film: Dodecane on Silicon An-Hong Liu, Peter C. Wayner, Jr., and Joel L. Plawsky* The Isermann Department of Chemical Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180-3590
Drainage of the partially wetting film dodecane on a vertical silicon substrate was investigated using image scanning ellipsometry (ISE). The ability of ISE to map every point on the surface simultaneously was demonstrated for the much more complicated case of the partially wetting fluid. Depending on the relative magnitude of the intermolecular, gravitational, and capillary forces, four flow regions were identified: the interfacial, transition, hydrodynamic, and meniscus regions. The partially wetting fluid began draining according to hydrodynamic theory, but after a short period the monotonically increasing film thickness profile became unstable and a mound or incipient drop began to form ∼1 mm from the leading edge of the fluid. Although the mound grew over time, differential drainage was never strong enough to nucleate a dry patch, and so the film drained to an adsorbed layer. Two-dimensional profiles show the thickness profile first becoming unstable in the flow direction and then evolving a secondary instability perpendicular to the flow direction. The junction line between the interfacial and transition regions moved down the plate with an apparent diffusivity: Dcl ) 4.9 × 10-13 m2/s. Introduction The dynamics of fluid flow in thin liquid films has been extensively studied both theoretically and experimentally, yet our understanding of this field is far from complete. The major impediment to enhancing our understanding is the lack of a suitable experimental tool for obtaining the film thickness profile in the appropriate range between 10 Å and 10 µm. The authors have developed an instrument capable of resolving the film thickness profile in this region. It has been called image scanning ellipsometry (ISE) and can operate in the steady state or transient state. This instrument was used to analyze solid films in the steady-state22 and draining films of a perfectly wetting, fluorocarbon liquid (FC-70) in the transient state.23 The first two studies demonstrated the usefulness of the instrument and confirmed the validity of theoretical models developed to describe the film thickness profile for perfectly wetting fluids. Alhough the behavior of perfectly wetting fluids is fairly well understood, such is not the case for partially wetting or nonwetting liquids, and more experimental data are needed. We discuss the application of image scanning ellipsometry to measure the film thickness profiles of a partially wetting fluid, dodecane, as it drains down a vertical silicon substrate. We will discuss the experimental technique, present the experimental results and discuss those results, within the framework of recent drainage theories. Experimental Technique Image Scanning Ellipsometry Technique. The details of the design and operation of ISE and its calibration are discussed in Liu et al.22,23 We will not attempt to reproduce that discussion here but will give an overview of the instrument’s design and operation so that the subsequent experiments can be fully evaluated. The image scanning ellipsometer, as shown in Figure 1, is based upon conventional null ellipsometry39 * Author to whom correspondence should be addressed. Telephone: (518) 276-6049. Fax: (518) 276-4030. E-mail:
[email protected].
S0888-5885(95)00720-2 CCC: $12.00
Figure 1. Schematic diagram of the image scanning ellipsometer (ISE).
and is related to the techniques of microscopic ellipsometry28 and dynamic imaging microellipsometry.5 It uses polarized light as a probe to determine the thickness and optical properties of the film. Although the ISE is based on a conventional null ellipsometer, its mode of operation is unique. On the surface of a substrate supporting a uniform thin film, the ellipticity or phase difference introduced in the components of the incident polarized light illuminating the surface can be just canceled by a phase difference that occurs upon reflection from the film. The entire surface appears dark, and we refer to this as the null point. Since we know the state of the incident polarized light and the state of the reflected light, we can determine the film thickness and refractive index at the null point. On the other hand, a nonuniform film produces an image consisting of a pattern of light and dark bands or null fringes. This pattern of fringes represents a contour map of the surface, charting its optical thickness (refractive index × thickness) as a function of position. Knowing the refractive index of the film, we can determine its thickness profile. Images of draining films illustrating this are shown in Figure 2. Here we see two separate images, one for a perfectly wetting fluid and the other for the partially wetting dodecane. The perfectly wetting fluid appears as a series of parallel bands, indicating a monotonically increasing film thickness profile. The partially wetting fluid has a more complicated profile. The bullseyes are indicative of © 1996 American Chemical Society
2956 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996
Figure 3. (a) Images of a perfectly wetting film at different polarizer settings. (b) Vertical slice through panel a showing the location of the intensity minima and movement of the minima across the film. Table 1. Physical Properties of FC-70 and Dodecane property (kg/m3)
density viscosity (Ns/m2) surface tension (N/m) refractive index boiling point (°C) modified Hamaker constant, A h a (Nm) adsorbed film thickness (m) oxide layer thickness (m) a
Figure 2. (a) Image of the draining fluorocarbon film, FC-70, approximately 1 min following the start of drainage. (b) Image of draining film of dodecane 1 h and 45 min following the start of drainage.
mounds on the surface representing incipient droplet formation from the once continuous film. The film thickness at the center of the null regions can be determined knowing the refractive index of the film and the angle settings on the polarizer, compensator, and analyzer. Since the film thickness is a periodic function of the null points, we need to know where the leading edge of the film is so we can determine where the first fringe appears. Once we located the first fringe, we can reference all other film thicknesses. A single image of the film gives a useful but relatively crude contour map of the surface. By taking a series of contour maps at different polarizer settings, we can sweep the null points across the surface and build up a more detailed picture of the film thickness profile. Figure 3 shows three images of a perfectly wetting film indicating how the null points change with polarizer setting.
FC-70
dodecane
1940 0.02716 0.018 1.303 215 -4.4 × 10-23
748.7 1.35 × 10-3 0.027 1.422 216
5.8 × 10-9 2 × 10-9
∼5.0 × 10-10 2 × 10-9
A h ) Aslv/6π.
Experimental Draining System. The drainage system used in this study was described previously in Liu et al.23 We chose fluids having a very low vapor pressure at room temperature (20-30 °C) to avoid problems with evaporation and condensation. Some evaporation and condensation undoubtedy occurred since we did not isolate the system to achieve thermal equilibrium. Moreover, evaporation and condensation would occur along with drainage to try and restore mechanical equilibrium to the fluid. The low vapor pressures of our fluids ensured that drainage was the dominant phenomenon observed. The perfectly wetting fluid we used in our previous study was a fluorocarbon, FC-70. In this study we used dodecane. Dodecane had the advantage of being partially nonwetting, relatively nonvolatile, and relatively free of dissolved gases. The physical properties of both fluids are listed in Table 1. The experimental apparatus was the same as that used in the perfectly wetting fluid experiments.23 It consisted of a closed system formed from a fused silica cell (Figure 4). The cell was formed in the shape of a trapezoid to ensure that the incident and reflected light would leave the cell perpendicular to the cell walls. Since the clarity of the reflectivity fringes depends upon
Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 2957
Figure 4. Diagram of the sample holder showing the fused silica cell containing dodecane, the Si wafer, and the hinged support.
maximizing the refractive index difference between the fluid and substrate, a high-index, single-crystal silicon wafer with an ultra-thin (6.6 mm), which is not easily seen in Figure 5, is the interfacial region consisting of the adsorbed film. The interfacial region and hydrodynamic draining region are connected via a transition region. We define the “junction line” as the intersection of the interfacial and transition regions. Beyond the junction line, a disturbance is set up by the disjoining pressure, and in the transition region, this disturbance becomes a “hump” that appears to grow over time and travel toward the liquid pool. Figure 5b shows an expanded view of the development of this transition region over time. We believe this is the first time investigators have been able to fully resolve the shape of the film thickness profile for a draining film through the transition region. The slope of the film thickness profile in Figure 5b appears to be steep, but it is actually very shallow, on the order of 4 × 10-3 m/m. The hump appears to start out from a series of disturbances that are present in the transition region at 30 min. The small disturbances in the thicker region of the film (>400 nm) appear to die out while those in the thinner region (∼200-300 nm) grow to form hump sometime between 30 and 45 min after drainage began. The third draining region, where a parabolic thickness profile exists, extends for another 2-3 mm down from the transition region. This is the hydrodynamic draining region where gravitational forces determine the rate of drainage of the film. In this region, the film
2958 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996
Figure 6. Verification of hydrodynamic drainage during the early stages of drainage of dodecane.
Figure 8. (a) Latter stages of draining for dodecane. (b) Enlarged view of the interfacial and transition regions.
Figure 7. Interfacial and transition regions during drainage of dodecane and FC-70. Lines through the experimental points are smoothed spline fits for the eye to follow.
thickness should increase with the square root of position along the plate. This function is plotted in Figure 6, and the experimental data do follow the straight line dependence that theory predicts. The final region of drainage lies near the liquid pool. Here, the film thickness increases rapidly as the liquid pool is approached and exhibits a high curvature. This is the start of the intrinsic meniscus region where surface tension dominates the drainage. During the first few minutes of drainage, the film behaves as if there were no intermolecular or surface forces and drains in a purely hydrodynamic fashion. Figure 7 shows a comparison between dodecane drainage at 30 min and FC-70 drainage that illustrates this point. As both films become thinner, the intermolecular forces become pronounced and start a wave that appears to begin in the interfacial region. In the partially
wetting film, this wave grows in amplitude and wavelength and starts to travel from the interfacial region to the liquid pool. In the perfectly wetting fluid, this wave is suppressed as the intermolecular forces cause the fluid to adhere to the plate. The nature of how the intermolecular forces promote adhesion or cohesion and how those forces affect the film thickness profile has been treated in may references; for example Chapter 11 in the textbook by J. Israelachvili.16 As drainage progresses, the “wave” in the partially wetting fluid continues to travel toward the liquid pool. This can be seen by comparing Figures 5 and 8. The wavelength increases between 1 h and 1 h and 45 min but thereafter decreases as liquid drains out of or evaporates from the hump. Drainage or evaporation from the hump also causes a decrease in amplitude, and eventually, the film drains down to an adsorbed film after 6 h. The apparent velocity of the wave, calculated from the centerline movement, between 30 min and 1 h is ∼1.3 × 10-7 m/s. This decreases to ∼9.8 × 10-8 m/s in the period from 1 h to 1:45 h and 45 min and thereafter decreases sharply and even appears to change direction as the wave dies out. The hydrodynamic region continually shrinks as drainage progresses. The meniscus region steadily grows at the expense of the hydrodynamic region. More information about the drainage process can be obtained by referring to Figure 9a-f. These are twodimensional profiles derived from the experimental images. Here we can see that the wave starts out as a one-dimensional disturbance. The amplitude and wavelength grow until 1 h but starting at 1 h a disturbance
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Figure 9. Dodecane drainage: (a) t ) 45 min; (b) t ) 1 h; (c) t ) 1 h and 45 min; (d) t ) 3 h; (e) t ) 4 h and 30 min; (f) t ) 6 h and 35 min.
Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 2961
Dcl )
6πA h µδo
(1)
where A h is the modified Hamaker constant (Aslv/6π), µ is the fluid viscosity, δo is adsorbed film thickness, and Dcl is the diffusion coefficient for the junction line. Assuming the junction line “penetrates” down the plate as the fluid drains, we can use a classic penetration depth expression to extract the junction line “diffusivity” from the experimental data involving the location of the junction line at constant film thickness. Here that film thickness was the adsorbed film thickness, δo:
(ho - h) ) 3.643xDclt
Figure 10. Full drainage profiles of dodecane (30 min) and FC70 (4 h and 40 min) showing the similarities in the thickness profiles for the hydrodynamic and meniscus regions.
begins to form perpendicular to the drainage direction. By the time we reach 1 h and 45 min, the original wave appears to be a two-dimensional disturbance, leading to incipient droplet formation. At this time, the wavelength of the disturbance in the flow direction and perpendicular to the flow direction are approximately equal at ∼1.1 mm. The tendency to droplet formation is even more pronounced at 3 h, but by this time, the disturbance has nearly passed and the “droplets” have drained or evaporated away. We pick up a new disturbance at 4 h and 30 min next to the liquid pool. This too eventually drains away and leads to a flat, adsorbed film (at distances larger than 3 mm) beyond 6 h. The contact angle was measured for the profile at 6 h and found to be ∼0.2°( 0.1°. This value is representative of the receding contact angle for all the measured profiles shown in Figure 9a-f. The profiles of a draining film have been measured using interference microscopy and interferometry by Muller26 and by Bascom et al.1 The hydrodynamic draining region was observed in both works, but the authors could obtain only limited information at the transition between the hydrodynamic and interfacial thin film regions. The shape of our profiles in the hydrodynamic region agree with the parabolic profiles observed by their work and in our previous work with a wetting fluid (Figure 10).23 The evaporating and equilibrium meniscus region on an inclined flat substrate have been characterized by interference microscopy by Cook et al.6 The shape of our profile in this region also agrees with their previous work. The junction line between the interfacial and transition regions appears to move down the plate toward the liquid pool. This behavior was also observed in our work with perfectly wetting fluids.23 Theory predicts that the junction line should “diffuse” down the plate, and expressions for this diffusion coefficient have been developed in the context of spreading drops by borrowing from Frenkel’s hopping model for surface diffusion.8 Lopez et al.24 and Teletzke et al.38 have shown that the diffusion coefficient should be related to the Hamaker constant, the viscosity of the fluid, and the adsorbed film thickness according to:
(2)
In eq 2, ho represents the original location of the junction line, relative to the liquid pool, at the start of the experiment, and h is the position of the junction line at any time, t. Based on our limited amount of data, the diffusivity is ∼4.9 × 10-13 m2/s. Using the physical property data from Table 1 this leads to a modified Hamaker constant of ∼2.0 × 10-26 Nm. We note that the measured diffusion coefficient is not directly analogous to the surface diffusion coefficient of eq 1 since the draining liquid already pre-wets the surface. However, successful measurements of surface diffusion coefficients for spreading drops are extremely difficult, and it is only the spreading action of the draining film coupled with the ability of the ISE to follow the junction line through time and space, that allows us to measure a diffusion coefficient at all. The value for the modified Hamaker constant is fairly small and may not be large enough to account for the shape of the film thickness profile and development of the transition region. Most partially wetting fluids and nonwetting fluids have Hamaker constants with absolute magnitudes much larger than this. The theoretical diffusion coefficient expression we used is based on a perfectly wetting fluid, and the development of the transition region may retard the value of the diffusion coefficient in a way not accounted for in the surface diffusion model. There have been numerous attempts to model the thickness profile of a draining film since Jeffreys18 first considered the problem. Two drainage scenarios are usually considered. Continuous withdrawal theory3,10,15,29,31,32 assumes the substrate to be infinitely long. Either the substrate is removed from the fluid at a constant velocity or the fluid is withdrawn at constant velocity. The substrate and liquid bath are always in contact. Drainage theory4,7,12,18,33,37 considers a short substrate completely removed from the fluid bath. In this case, substrate loses contact with the liquid pool. Our experimental system had elements of both scenarios. The substrate was always in contact with the liquid pool, but liquid was essentially instantaneously removed from the substrate during the withdrawal phase. Much of the work on drainage of partially wetting films concerns spreading and rupture of a drop on a horizontal surface. Williams and Davis,40 Hwang et al.,13 Ruckenstein and Jain,30 Sharma and Ruckenstein,36 Burrelbach and Bankoff,2 Reisfeld and Bankoff,27 Kheshgi and Scriven,20 Gumerman and Homsy,11 Jameel and Sharma,17 Sharma and Jameel,34 Sharma et al.,35 and Mitlin25 have considered this problem, concentrating on describing the critical film thickness at rupture rather than describing the thick-
2962 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996
ness profile for an entire draining film. Their results for the primary wavelength leading to rupture did not agree with our experimental results. Gravitational forces on our fluid probably contributed to this. In addition, our fluid never ruptures. Goodwin and Homsy9 and Khesgi et al.19 have considered the inverse of the draining problem when the fluid is flowing along a surface it had never been in contact with before. This “wetting” problem is quite different than the problem we studied, and the authors did not consider intermolecular forces in the thin film region. Therefore, they were left with a condition of infinite stress at the contact line between the fluid and the dry surface. They did consider the inverse problem, their film thickness profiles had much in common with our observed profiles, especially the “hump” that formed at the transition region. Li and Slattery21 considered drainage of a partially wetting fluid and included intermolecular forces. They concluded that the fluid drains and leaves behind a series of microdrops. The formation of these microdrops relieves the stress at the “apparent” contact line. The microdrops were less than molecular dimensions high and perhaps several microns or so wide. These “drops” cannot be seen with our ISE and represent what happens up to the junction line of our profiles not within the transition region. Clearly, new models must be developed that can describe what is occurring within the transition region. Conclusions An optical instrument, the image scanning ellipsometer (ISE), whose operation is based on null ellipsometry and microellipsometry, was successfully applied to a system consisting of a vertically draining film of a partially wetting fluid: dodecane. The experimental profiles identified four clear draining regions: the interfacial region, the transition region, the hydrodynamic region, and the meniscus or capillary region. The ISE resolved the complete thickness profile from 1 nm to >20 µm with particular sensitivity in the interfacial and transition regions. Intermolecular forces are very strong in these regions, and this work represents one of the first collections of data available for the film thickness profile of a partially wetting film in these regions. The partially wetting film behaved initially as a perfectly wetting fluid and drained in a hydrodynamic fashion. As the film thinned, intermolecular forces became dominant, and an instability, leading to a pronounced hump, formed within the transition region. This hump grew in amplitude and wavelength as the disturbance wave propagated toward the liquid pool. Finally, drainage or evaporation from the hump caused it to disappear and leave behind an adsorbed film. The two-dimensional film profiles obtained from the experimental data showed the hump resulting from a onedimensional disturbance in the direction of drainage. As the film continued to drain, the initial one-dimensional disturbance became two-dimensional, and structures looking like incipient droplets began to form. The junction line separating the interfacial and transition regions “diffused” toward the liquid pool with an apparent diffusivity of 4.9 × 10-13 m2/s. This diffusivity was smaller than that observed with the perfectly wetting fluid FC-70, but still within the range of surface-type diffusivities.
Acknowledgment This material is based on work supported by the Department of Energy under Grant DE-FG02-89ER14045. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of DOE. Literature Cited (1) Bascom, W. D.; Cottington, R. L.; Singleterry, C. R. In Contact Angle, Wettability, and Adhesion; Gould, R. F., Ed.; American Chemical Society: Washington, DC 1964. (2) Burelbach, J. P.; Bankoff, S. G.; Davis, S. H. Nonlinear Stability of Evaporating/Condensing Liquid Films J. Fluid Mech. 1988, 195, 463. (3) Cerro, R. L.; Scriven, L. E. Rapid Free Surface Film Flows. An Integral Approach Ind. Eng. Chem. Fundam. 1980, 19, 40. (4) Chase, C. A.; Gutfinger, C. The Free Coating Process and Its Relationship to Drainage AIChE J. 1967, 13, 393. (5) Cohn, R. F.; Wagner, J. W.; Kruger, J. Dynamic Imaging Microellipsometry: Proof of Concept Test Results. J. Electrochem. Soc. 1988, 135, 1033. (6) Cook, R.; Tung, C. Y.; Wayner, P. C., Jr. Use of Scanning Microphotometer to Determine the Evaporative Heat Transfer Characteristics of the Contact Line Region. J. Heat Transfer 1981, 103, 325. (7) DeKee, D.; Schlesinger, M.; Godo, M. N. Postwithdrawal Drainage of Different Types of Fluids. Chem. Eng. Sci. 1988, 43, 1603. (8) Frenkel, J. Kinetic Theory of Liquids; Oxford University Press: Oxford, 1946. (9) Goodwin, R.; Homsy, G. M. Viscous Flow Down a Slope in the Vicinity of a Contact Line. Phys. Fluids A 1991, 3, 515. (10) Groenveld, P. Drainage and Withdrawal of Liquid Films. AIChE J. 1971, 17, 489. (11) Gumerman, R. J.; Homsy, G. M. The Stability of Radially Bounded Thin Films. Chem. Eng. Commun. 1975, 2, 27. (12) Gutfinger, C.; Tallmadge, J. A. Some Remarks on the Problem of Drainage of Fluids on Vertical Surfaces. AIChE J. 1964, 10, 774. (13) Hwang, C,; Chang, S.; Chen, J. On the Rupture Process of Thin Liquid Films. J. Colloid Interface Sci. 1993, 159, 184. (14) Higgins, B. G.; Scriven, L. E. Interfacial Shape and Evolution Equations for Liquid Films and Other Viscocapillary Flows. Ind. Eng. Chem. Fundam. 1979, 18, 208. (15) Higgins, B. G.; Silliman, W. J.; Brown, R. A.; Scriven, L. E. Theory of Meniscus Shape in Film Flows. A Synthesis. Ind. Eng. Chem. Fundam. 1977, 16, 393. (16) Israelachvili, J; Intermolecular and Surface Forces: With Applications to Colloidal and Biological Systems; Academic Press: London, 1985. (17) Jameel, A. T.; Sharma, A. Morphological Phase in Thin Liquid Films. J. Colloid Interface Sci. 1994, 164, 416. (18) Jeffreys, H. The Draining of a Vertical Plate. Proc. Cambridge Phil. Soc. 1930, 26, 204. (19) Khesgi, H. S.; Kistler, S. F.; Scriven, L. E. Rising and Falling film Flows: Viewed From a First-Order Approximation. Chem. Eng. Sci. 1992, 47, 683. (20) Kheshgi, H. S.; Scriven, L. E. Dewetting: Nucleation and Growth of Dry Regions. Chem. Eng. Sci. 1991, 46, 519. (21) Li, D.; Slattery, J. C. Analysis of the Moving Apparent Contact Line and Dynamic Contact Angle Formed by a Draining Film. J. Colloid Interface Sci. 1991, 143, 382. (22) Liu, A. H.; Wayner, P. C., Jr.; Plawsky, J. L. Image Scanning Ellipsometry for Measuring Non-uniform Film Thickness Profiles. Appl. Opt. 1994, 34, 3. (23) Liu, A. H.; Wayner, P. C., Jr.; Plawsky, J. L. Draining Wetting Films: An Application of Image Scanning Ellipsometry Phys. Fluids A 1994, 6, 1963. (24) Lopez, J.; Miller, C. A.; Ruckenstein, E. Spreading Kinetics of Liquid Drops on Solids. J. Colloid Interface Sci. 1976, 56, 460. (25) Mitlin,V. S. Dewetting of Solid Surface: Analogy with Spinodal Decomposition J. Colloid Interface Sci. 1993, 156, 491. (26) Muller, R. H. Optical Studies of Electrolyte Films on Gas Electrodes. J. Electrochem. Soc. 1966, 113, 943.
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(36) Sharma, A.; Ruckenstein, E. An Analytical Nonlinear Theory of Thin Film Rupture and Its Application to Wetting Films. J. Colloid Interface Sci. 1986, 113, 456. (37) Tallmadge, J. A. Draining Films in Unsteady Withdrawal. AIChE J. 1971, 17, 760. (38) Teletzke, G. F.; Davis, H. T.; Scriven, L. E. How Liquids Spread on Solids. Chem. Eng. Comm. 1987, 55, 41. (39) Tompkins, H. G. A User’s Guide to Ellipsometry; Academic Press: New York, 1993. (40) Williams, M. B.; Davis, S. H. Nonlinear Theory of Film Rupture. J. Colloid Interface Sci. 1982, 90, 220.
Received for review December 6, 1995 Revised manuscript received June 17, 1996 Accepted June 18, 1996X IE950720S
X Abstract published in Advance ACS Abstracts, August 15, 1996.
