Article pubs.acs.org/Langmuir
Thinning and Rupture of Liquid Films by Moving Slot Jets Christian W. J. Berendsen, Jos C. H. Zeegers, and Anton A. Darhuber* Mesoscopic Transport Phenomena Group, Department of Applied Physics, Eindhoven University of Technology, Den Dolech 2, 5612AZ Eindhoven, The Netherlands S Supporting Information *
ABSTRACT: We present systematic experiments of the rupture and dewetting of thin films of a nonvolatile polar liquid on partially wetting substrates due to a moving slot jet, which impinges at normal incidence. The relative motion was provided by a custom-built spin coater with a bidirectionally accessible axis of rotation that enabled us to measure film thickness profiles in situ as a function of substrate velocity using dual-wavelength interference microscopy. On partially wetting polymeric substrates, dry spots form in liquid films with a residual thickness well below 1 μm. We measured the density of dry spots as well as the density and size distribution of the residual droplets as a function of film thickness. In a certain parameter range, the droplet distributions exhibit pronounced anisotropy due to the effect of long-range shear stresses on the dewetting rim instability. We find robust power-law scaling relations over a large range of film thicknesses and a striking similarity to literature data obtained with ultrathin polymer melt layers on silicon substrates.
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function of film thickness and found that for a low film thickness hmin the dry spot density follows the power law NH ∼ −2 h−4 min, decreasing to NH ∼ hmin as a result of confinement effects. In the experiments described in this article, we rotate a transparent polymer substrate, coated with a thin liquid film, underneath a slot jet impinging at normal incidence. We measure the residual film thickness by dual-wavelength interference microscopy. We develop a corresponding numerical model and achieve quantitative agreement. Moreover, we determine the dry-spot density as well as the density and size distribution of the resulting residual droplets as a function of substrate speed. In a certain parameter range, the droplet distributions exhibit pronounced anisotropy resulting from the effect of long-range shear stresses on the dewetting rim instability.
INTRODUCTION The rupture of thin liquid films on partially wetting substrates has been a field of study for decades. Padday1 showed that a liquid film on a Teflon or wax surface ruptures below a critical film thickness. Taylor2 performed a systematic study of the effort needed to make a hole in a liquid film and used a stationary impinging air jet to achieve this. A similar technique was employed later to initiate the dewetting of thin liquid films on partially wetting substrates.3,4 Kadoura et al. studied the rupture of water films after spraying on a solid surface.5 The use of a slot jet or air knife to thin a liquid film is common in applications such as gas jet wiping in galvanization lines.6−16 Slot jets are also used for the drying of wafers in the semiconductor industry.17 In immersion lithography,18−20 a photoresist-covered wafer is typically exposed through a layer of ultrapure water in order to increase the effective numerical aperture and thus the achievable resolution of the lithography system. Air knives are utilized to confine the water meniscus laterally between the objective lens and a partially wetting wafer, which is moving at a relative speed of approximately 1 m/s. Above a critical speed of translation, however, a thin liquid film is left on the wafer, which spontaneously breaks up, dewets, and leaves undesirable residual droplets on the substrate. The corresponding droplet size distributions are of crucial importance because of the potential formation of air bubbles21,22 that can distort the illumination pattern or the appearance of so-called watermark defects.23 In earlier work,24 we studied the deformation and rupture of a thin liquid film thinned by an impinging jet emanating from a round nozzle. In these experiments, we created a narrow track of low film thickness by controlling the jet Reynolds number and substrate velocity. We analyzed the density of dry spots as a © 2013 American Chemical Society
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MATERIALS AND METHODS
Figure 1c shows a schematic drawing of the experimental setup, consisting of a custom-designed spin coater that provides optical access to the spinning substrate from both above and below. Above the substrate, a slot jet is attached to a pneumatic vertical translation stage that can be lowered into position during or shortly after the spin coating of the thin liquid film. Beneath the transparent substrate, we implemented a high-speed dual-wavelength interference imaging system to measure film-height profiles and visualize the film rupture. The top view of the substrate shows the circular trajectory of the slot jet with respect to the substrate (Figure 1a). Figure 1b shows a schematic side view of the slot jet deforming a liquid film. We discuss Received: October 14, 2013 Revised: December 3, 2013 Published: December 3, 2013 15851
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Figure 2. Influence of the stand-off distance H, as defined in Figure 1b, normalized by the slit width w = 100 μm, on the spanwise thickness uniformity for Rew = 242. The micrographs correspond to H/w = (a) 5, (b) 10, and (c) 20. The green dashed lines indicate the approximate location of the slot jet above the substrate. The stand-off distance H between the nozzle exit and the substrate is H ≈ 200 μm. A small distance H and high Rew are necessary to obtain a uniform residual film thickness. At larger H, spanwise film thickness corrugations appeared,26 as visualized for Rew = 242 in Figure 1. These striations are possibly related to Görtler vortices.8 At greater stand-off distances, we also observed a weak but audible whistling sound,27 decreasing in frequency with increasing H. At low Rew or very high rotational speeds Ωsub, the reproducibility of the residual film thickness was reduced. Dual-Wavelength Interference Microscopy. Setup. We measure the liquid film thickness distribution using dual-color interference microscopy. The setup consists of two high-power light-emitting diodes (LED, Luxeon I) with emission wavelengths of λ1 = (655 ± 11) nm (red) and λ2 = (466 ± 14) nm (blue), respectively. The light sources are pulsed alternatingly in synchronization with the frame rate (250 Hz) of the high-speed camera (Photron SA-4) such that two consecutive frames were recorded with different wavelengths. The camera is fitted with illumination and imaging optics, including a microscope objective (Olympus, Uplan 4× , NA 0.13) yielding a field of view of approximately 4 × 4 mm2. The liquid film deformation caused by the slot jet (length L = 10 mm) is characterized by separately measuring close to the inner and outer edges of the track. Data Analysis. After background subtraction, we analyze the dark and light fringes for each wavelength. From the position and intensity of these interference fringes, two relative height profiles are constructed using a film thickness difference of Δhdark−light = λ/(4n) between neighboring fringes, where n = 1.46 is the average refractive index of 3EG in the visible spectrum. By matching the fringe positions of both colors, we are able to establish the fringe order and thus the absolute film thickness profile. The film thickness of the residual film hres in the track is obtained by greyscale analysis. For the material system 3EG/PC, provided that the fringes are well resolved in the regions where the film thickness gradient (|∇h|) is highest, film thicknesses can be measured with an accuracy of 5−20 nm in the range of 30 nm < h < 1500 nm. The error bars correspond to the difference in greyscale interpolation results of the two colors as well as the variation in film thickness for the duration of an experiment. Figure 3a−c shows a typical dual-wavelength interference measurement. An overlay of the two subsequent red and blue images is presented in Figure 3a. The film thickness profiles h(x) at the positions indicated by the yellow lines in Figure 3a are plotted in Figure 3b,c. The corresponding local substrate speed is Usub ≡ Ωsubr, where Ωsub is the angular velocity and r is the distance from the axis of rotation. The residual film thickness values hres(r = 17.1 mm) = (56.6 ± 3) nm and hres(r = 20.1 mm) = (64.8 ± 6) nm reflect the fact that hres increases with radial coordinate r as a result of the higher speed at greater distances from the center of rotation.
Figure 1. Schematic drawings of the experimental setup. (a) Top view of the substrate showing a track of lower film thickness induced by the slot jet. (b) Side-view sketch of the nozzle and the liquid-film deformation. (c) Schematic representation of the experimental setup, including imaging optics and the optically accessible spin-coater. the details of the setup and the experimental procedures in the following sections. Sample Preparation. For the preparation of the liquid film and subsequent thinning by a moving slot jet, we have designed a spin coater that provides accessibility to the axis of rotation from both sides. The spin coater consists of a sample holder supported by a hollow cylindrical roller bearing. The rotation is transmitted by a tooth belt connected to a precision motor, which enables rotational speeds of between 0 and 6000 rpm with an accuracy of ±0.2 rpm. The partially wetting sample, a polymer plate of optical-quality polycarbonate (PC, Makrofol DE, thickness 750 μm) cut to the dimensions of 60 × 60 mm2, is attached to a ring-shaped supporting plate using double-sided tape (3M, Scotch tape 410B). A liquid film of triethylene glycol (3EG, Sigma, product number 95126, purity ≥99.5%, viscosity μliq = 49 mPa· s, and surface tension γ = 45.5 mN/m) is spin-coated at 3000 rpm for 20 s, resulting in a uniform film of thickness h0 = 3.3 μm. In the final seconds of the spin-coating process, the nozzle is brought into position using a pneumatic vertical translation stage while the jet is still off. Slot Jet. The slot jet nozzle, schematically depicted in Figure 1b, is made of stainless steel and has an exit slit of dimensions w × L = 110 μm × 10 mm. The entrance length from a small pressure chamber to the exit is Hin = 12.5 mm, and the interior walls are parallel and smooth with a roughness of less than 100 nm. The horizontal outflow plane surrounding the slit has a total width of 3w. To reduce reflections in the microscope image, the underside of the nozzle is painted black. Precautions were taken to avoid the penetration of paint into the nozzle opening, and the paint was dried thoroughly before the experiment. Pressurized air is filtered through a HEPA filter (GE Whatman, HEPA-CAP) and a particle filter (Headline filters, 25-64-50C, 99.99+ % removal of 0.1 μm particles and aerosols). A needle valve (Brooks NRS8514) controls the gas flow rate Q, which is measured using a mass flow meter (Bronkhorst EL-FLOW F-112AC). Using tilt stages, we positioned the jet nozzle parallel to the substrate at a radial position between r = 10.5 and 20.5 mm from the center of rotation. The flow rate was kept constant at Q = 8.6 L/min, corresponding to a Reynolds number of Rew = ρgas vw/μ gas ≈ 1020. Here, we have used the density ̅ ρgas = 1.17 kg/m3 and viscosity μgas = 1.83 × 10−5 Pa·s of air and the average air flow speed v ̅ = Q/(wL) ≈ 145 m/s. Considering that the hydraulic diameter for a narrow slit DH ≈ 2w, the hydraulic Reynolds number ReDH ≈ 2040, which is still in the laminar regime.25 15852
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Figure 3. Measurement of film thickness by dual-wavelength interferometry. (a) Example of a micrograph, taken at the outer radius of the track for Ωsub = 1.5 rpm. Two subsequent blue and red images have been overlaid. (b, c) The height profile measured at the positions indicated by the solid yellow lines in plot a. The solid green lines represent numerical simulations.
Droplet Imaging. After each measurement, the resulting droplet patterns were recorded using two techniques. First a photograph was taken of the entire substrate area using a digital photocamera (Sony NEX-5) with a zoom lens (Sony SE-18200, 18−200 mm). Subsequently, the droplet patterns were imaged in an upright microscope (Olympus BX51) using Olympus objectives of various magnifications (2.5×, 5×, 10×, and 20×) and a CCD camera (Pike, Allied Vision Technologies). The radial position of the image was kept constant by rotating the sample under the microscope on a rotational stage with a belt drive that was actuated by a dc motor (Maxon 118689) with a planetary gearhead (Maxon 134782).
Figure 4. Numerical simulations of the distributions of (a) stagnation pressure Pslot, (b) its first derivative ∂Pslot/∂x, and (c) shear stress τslot of an impinging slot jet at Rew = 1000 and H = 200 μm. The dashed lines indicate the width w of the slot jet. (d) Experimental and numerical results for the residual film thickness hres due to slot jet impingement as a function of substrate speed Usub = Ωsubr or capillary number Ca ≡ μliqUsub/γ for a 3EG film on PC.
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The evolution of the thin liquid film is evaluated on a domain with a length Ld = L1 + L2 = 12 mm. The initial condition is a flat film of uniform thickness h0. The boundary conditions are
NUMERICAL MODEL Our model of the deformation of a moving liquid film due to an impinging slot jet consists of two parts. First we obtain the shear stress and pressure profile on a stationary solid plate by solving the incompressible Navier−Stokes equation28 using the finite element software Comsol 3.5a. We use a 2D model for the cross section of a slot jet of width w = 110 μm. We prescribe the entrance velocity profile as a fully developed parabolic Poiseuille profile with a maximum velocity of 3v/2. ̅ The computational domain is 1 mm high and 2 cm wide. Figure 4a,c shows the resulting pressure profile Pslot(x) and shear stress profile τslot(x) for Rew = 1000. The gradient of the pressure ∂Pslot/∂x is plotted in Figure 4b. The dashed vertical lines in Figure 4a−c indicate the location and width w of the slot jet. The resulting pressure and shear stress profiles Pslot(x) and τslot(x) are used as input for the thin-film model based on the lubrication approximation29,30 −
⎤ ⎡ 2 ∂Q x h3 ∂P ∂ ⎢ h τslot ∂h = = − + Usubh⎥ ⎥⎦ 3μ liq ∂x ∂t ∂x ∂x ⎢⎣ 2μ liq
∂ 2h + ρliq gh + Pslot ∂x 2
(3)
h(x = − L 2) = h0 and P(x = −L 2) = ρgh0
(4)
representing an undisturbed film of thickness h0 that enters the domain at x = −L2. In the time-dependent simulations, the film thickness downstream from the jet quickly approaches a quasi-steadystate value, which we term hres. Two examples of simulated height profiles h(x) for Rew = 1000 are plotted together with the experimental height profiles in Figure 3b,c. The match between the experimental and numerical data is excellent.
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RESULTS AND DISCUSSION Residual Film Thickness. We measured the residual film thickness hres for different rotational speeds Ωsub near the inner and outer edges of the track (Figure 1a). For comparison, we also evaluated hres from the numerical simulations of the film height profiles for Rew = 1000 and different values of Usub. The initial film thickness in the numerical simulations was set to either h0 = 3 or 20 μm. Figure 4d shows the results, plotted as a function of the local substrate speed Usub ≡ Ωsubr. The top axis represents the capillary number Ca ≡ μliqUsub/γ.
(1)
where μliq is the liquid viscosity, Qx is the volume flux, and P is the augmented pressure P ≡ −γ
∂P (x = L1) = 0 and Q x(x = L1) = Usubh ∂x
(2) 15853
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Figure 5. Interference micrographs showing film rupture in liquid films of 3EG on PC due to slot jet impingement in the radial interval 11.9 < r < 14 mm for different substrate rotational rates: (a) Ωsub = 0.10 rad/s (1 rpm), (b) 0.16 rad/s (1.5 rpm), (c) 0.21 rad/s (2 rpm), (d) 0.31 rad/s (3 rpm), (e) 0.42 rad/s (4 rpm), (f) 0.63 rad/s (6 rpm), (g) 0.84 rad/s (8 rpm), and (h) 1.05 rad/s (10 rpm). The dashed lines indicate the position of the slot jet. The circled areas in panels a−d are magnified in the insets.
Ωsub and in which dry spots prominently appear in the track. With decreasing Ωsub and thus a lower residual film thickness, the density of holes or dry spots NH increases. The distance between neighboring interference fringes on the slopes decreases with decreasing angular velocity, indicating a smaller slope. At high rotational speeds (Figure 5f−h), a temporal variation of the film thickness is observed, represented by the azimuthal variation in greyscale value along the track, which may be due to air-flow-induced vibrations of the plate and corresponding variations of the stand-off distance H. The size of the dry spots in these images depends on the substrate speed. After nucleation, the dry spots grow at a more or less constant dewetting speed Vd ≈ 0.85 mm/s. The substrate speed Usub exceeds Vd in all cases. Because of the higher residence time in the field of view, the size of the dry spots in the pictures initially increases with decreasing Ωsub, as seen in Figure 3h−e. For even lower Ωsub, however, the dryspot size is limited by the high density of dry spots. The rims of the dry spots merge and break up into droplets before leaving the field of view (Figure 5a−d). We quantified the density of dry spots per unit area NH as a function of substrate speed Usub and thus residual film thickness hres. We analyzed the interference micrographs of all experiments and marked the location of each freshly nucleated dry spots. After marking the positions of all dry spots, we divided the field of view into 10 bands along the radial direction, each corresponding to a distinct range in local substrate speed Usub.
The experimental data suggest a power-law relation hres ∼ Usub0.93. This agrees with our results for a round nozzle in ref 24, where we observed that the exponent decreases slightly with the viscosity of the liquid, ranging from 1 for water to 0.85 for glycerol. The numerical simulations and experimental data match very well in the experimentally accessible range. Moreover, the numerical simulations indicate a transition in the scaling behavior at high substrate speeds. This transition is related to the relative importance of the contribution of the jet pressure Pslot and the shear stress τslot to the thinning dynamics. Power laws can be derived from the lubrication equation (eq 1) in steady state (∂h/∂t = 0). In the shear-dominated regime (i.e., for low values of Usub and hres), a balance between the first and third terms on the right-hand side of eq 1 yields a power law hres ∼ μUsub/τslot. For higher Usub and thus higher hres, the pressure term prevails, yielding hres ∼
μUsub ∂Pslot ∂x
∼
Usub (5)
Because the residual film thickness hres cannot exceed the initial film thickness h0, the green curve in Figure 4d levels off at hres = h0 = 3 μm. Until that point, hres is essentially independent of the initial film thickness h0 as demonstrated by the overlap of the two curves for h0 = 3 and 20 μm. Dry-Spot Nucleation. Figure 5a−h shows a series of interference images, obtained with different rotational speeds 15854
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In Figure 6a,b, we plot NH as a function of the local film thickness hres, determined from Ωsub according to the power law fit h ∼ Usub0.93 = (Ωsubr)0.93 (Figure 4d).
on silicon (Si) wafers. Despite the difference in the material system, the data are surprisingly close to our measurements. The data of Reiter31,36,37 overlap perfectly with our data, but the data of Meredith et al.32 at low film thicknesses and Verma et al.33 at higher film thickness also match our experiments well. The reported contact angles of PS on Si were θ ≈ 10° in ref 33 and θ ≈ 22 ± 4° in ref 31, a value comparable to the value24 of the receding contact angle of triethylene glycol (3EG) on polycarbonate (PC), θrec ≈ 29 ± 2°. Variations in the oxide layer thickness on the silicon wafers are a likely reason for the deviations between the reported results. In the experimental data of Jacobs et al.,34 the density of dry spots NH is higher than the others. This is very likely due to the higher contact angle36 θ ≈ 50° of PS on the octadecyltrichlorosilane (OTS) selfassembled monolayer applied to the silicon wafer. If the density of dry spots is a result of a disjoining pressure caused by long-range interactions37−39 similar to nonretarded van der Waals forces, then the overlap in measurement data despite the difference in materials systems could be interpreted to be due to the fact that Hamaker constants of most organic materials systems do not vary significantly.35,40 However, the small time constants of rupture indicate that defects govern the rupture process.34,41 However, the similarity of these film rupture results raises the question of what the nature of these defects might be. Residual stresses were found to be important in polymer melt layers,41 but they are an unlikely cause of rupture in a low-viscosity Newtonian fluid such as 3EG. Droplet Patterns. After film rupture, the dry spots expand until their dewetting rims coalesce and leave a pattern of droplets behind on the substrate. After each experiment, we first recorded images of these droplet patterns. Figure 7 shows sections of these photographs with droplet patterns resulting from experiments at different substrate rotational rates. The droplets are primarily located in the track region, indicated by the light-blue color in Figure 1a. With increasing rotational speed Ωsub and thus a higher residual film thickness hres, the droplets become larger and their density decreases, consistent with the decreasing density of dry spots discussed above. The initial phase of the experiment (i.e., when the jet was turned on) is contained in the field of view of Figure 7a,g. In Figure 7f, the jet was turned off. After photographing the substrates, we imaged the droplet patterns in an upright microscope using a rotational stage. Figure 8a−e shows a series of such microscope images that provide a higher contrast and magnification than the photographs in Figure 7. The lines in Figure 8a−e represent the slot jet trajectory. We analyzed all microscope images of the droplet patterns and extracted the droplet density and droplet radii by image analysis using the image-processing toolbox of Matlab. On the basis of these data, we plot histograms of the droplet radius distributions in Figure 8f−j, corresponding to the measurement series in Figure 8a−e. The histograms are based on 30 logarithmically spaced bins in the range 0.1 μm ≤ Rd ≤ 10 mm. The dashed curves represent the droplet radius distribution for the corresponding images, whereas the solid lines are based on all images in each measurement. The dashed vertical line corresponds to the average droplet radius for the images in Figure 8a−e, whereas the average radius ⟨Rd⟩ for the entire measurement series is indicated by the colored bars and the symbols above the histograms. In the images in Figure 8d,e, corresponding to a residual film thickness of 100 ≲ hres ≲ 500 nm (8 ≲ Ωsub ≲ 16 rpm), welldefined polygonal droplet patterns are present and are very
Figure 6. (a) Density of dry spots NH in a thin film of 3EG on PC as a function of the local film thickness hres. The black squares indicate the film rupture of freshly spin-coated films without slot jet impingement. (b) Comparison of the data in plot (a) to literature results obtained for polystyrene melts on oxidized silicon31−33 or silicon grafted with a selfassembled monolayer.34
The density of dry spots in Figure 6a appears to follow a power law NH ∼ hresα, where the exponent α lies between −3 and −4. The spread in the measurement data increases with higher film thickness. The value of α = −4 agrees with the exponent associated with a nonretarded van der Waals type disjoining pressure isotherm. A power law exponent of α = −3 was also observed by Verma et al. for polymer films immersed in water33 and associated with double-layer interactions.35 Figure 6a also contains data (squares with vertical error bars) for 3EG films spin coated on 60 × 60 mm2 PC substrates (i.e., without the presence of an impinging slot jet). Depending on the film thickness, single or several dry spots appear within a few seconds after the spin coater stops rotating. A film of 3EG on PC with an initial film thickness h0 = 5 μm is stable most of the time. However, a film of h0 = 3 μm always exhibits one to three rupture events shortly after spin coating. Interestingly, the density of dry spots per unit area in this case is close to the extrapolation of the power law for exponent α = −3. In Figure 6b, we compare the experimental data of Figure 6a for films of the polar liquid triethylene glycol on polycarbonate to those described in the literature for polystyrene (PS) melts 15855
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Figure 7. Segments of photographs of the center area of the substrates after experiments performed at different sample rotational rates: (a) Ωsub = 0.10 rad/s (1 rpm), (b) 1.5 rpm, (c) 2 rpm, (d) 3 rpm, (e) 4 rpm, (f) 6 rpm, (g) 8 rpm, (h) 10 rpm, (i) 12 rpm, (j) 16 rpm, (k) 20 rpm, and (l) 30 rpm.
Figure 8. Microscope images of droplet patterns and corresponding droplet radius histograms for (a, f) Ωsub = 1 rpm and 24 ≤ hres ≤ 25 nm (in the field of view), (b, g) 2 rpm and 46 ≤ hres ≤ 49 nm, (c, h) 4 rpm and 85 ≤ hres ≤ 97 nm, (d, i) 8 rpm and 129 ≤ hres ≤ 220 nm, and (e, j) 16 rpm and 287 ≤ hres ≤ 463 nm. (a−e) The solid green lines represent the jet trajectory. (f−j) The solid red lines represent the histogram of droplet radii Rd for the full measurement, whereas the dashed lines represent the histogram for the images in a−e. The red symbols and error bars in f−j correspond to the mean value and standard deviation throughout the measurement. The vertical dotted line indicates the mean value for the measurements in a−e.
similar to the patterns reported for polymer films.31,36 The corresponding droplet radius histograms (Figure 8i,j) are broad because the droplet patterns consist of large droplets and smaller satellite droplets. In the intermediate substrate speed range corresponding to film thicknesses of 40 ≲ hres ≲ 100 nm in Figure 8b,c, this polygonal pattern exhibits a pronounced statistical anisotropy. It originates from the interaction between the long-ranged shear stress of the boundary layer of the slot jet and the instability of the dewetting rims.24,42,43 Because of the low substrate speed and relatively low dry-spot density, these instabilities had time to develop while still being affected by the shear stress of the air flow. The phenomenon of droplet shedding in the intermediate substrate speed range is illustrated in Figure 9. The first image shows four freshly nucleated dry spots. In the second picture, recorded 80 ms later, a rim instability can be observed.43 This instability gives rise to the droplet shedding on the right edge of the dry- spot, closest to the slot jet, while the dry spot grows. The final frame shows the resulting droplet pattern on the substrate. The larger droplets become more monodisperse, but an extra peak appears in the histograms at very low droplet radius (Figure 8g,h). For even lower substrate speed and thus thinner films (Figure 8a,f), again an isotropic dense pattern of small droplets is left on the substrate.
Figure 10a,b presents the average droplet radius ⟨Rd⟩ and density of droplets Nd as a function of residual film thickness hres. Both graphs show a clear power law dependence with residual film thickness. The power law exponent for the average droplet radius ⟨Rd⟩ as a function of hres is 1.1. The density of droplets (Figure 10b) follows a power law Nd ∼ hres−2.8. Data obtained using a round air jet24 are included as gray symbols in the graphs in Figure 10. Within error bars, the corresponding power law exponents are comparable. The data for the round air jet appear to have a lower density of droplets and a higher average droplet radius than the current data obtained with the slot jet. This is due to the fact that in the analysis of the round jet measurement data the minimum film thickness in the narrow track was used for the horizontal axis, whereas rupture also occurred in the neighboring region of slightly higher film thickness. 15856
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that the influence of the wall shear stress dominates over the stagnation pressure of the slot jet. The density of dry spots NH follows the power law NH ∼ hresα with exponent −4 < α < −3 and overlaps with literature data obtained for other material systems. Finally, we determined the density and size distribution of resulting droplet patterns as a function of hres and compared them to experimental results obtained for a narrow track in a thin liquid film.
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ASSOCIATED CONTENT
* Supporting Information S
Video of film thinning and rupture for different substrate speeds as well as atomic force microscopy surface characterization data for polycarbonate substrates. This material is available free of charge via the Internet at http://pubs.acs.org/
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected].
Figure 9. Dry-spot growth and coalescence showing the directional release of droplets by the combined action of rim instability and airflow-induced shear stress. The last frame is a microscope image of the final droplet distribution.
Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work is part of the research programme “Contact Line Control during Wetting and Dewetting” (CLC) of the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). The CLC programme is cofinanced by ASML and Océ. We thank Laurens van Bokhoven (ASML) and Avraham Hirschberg (Eindhoven University of Technology) for fruitful conversations on nozzle design and instabilities related to impinging slot jets and Jim Overkamp, Rogier Cortie, Michel Riepen, Ramin Badie, and Nico Kemper (ASML) for discussions of the interaction between gas jets and liquid films and residual droplet patterns in immersion lithography. Finally, we thank Henny Manders and Jørgen van der Veen for their help in the design and realization of the experimental setup.
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REFERENCES
(1) Padday, J. Cohesive properties of thin films of liquids adhering to a solid surface. Spec. Discuss. Faraday Soc. 1970, 1, 64−74. (2) Taylor, G. I.; Michael, D. H. On making holes in a sheet of fluid. J. Fluid Mech. 1973, 58, 625−639. (3) Redon, C.; Brochard-Wyart, F.; Rondelez, F. Dynamics of dewetting. Phys. Rev. Lett. 1991, 66, 715−718. (4) Bower, C. L.; Simister, E. A.; Bonnist, E.; Paul, K.; Pightling, N.; Blake, T. D. Continuous coating of discrete areas of a flexible web. AIChE J. 2007, 53, 1644−1657. (5) Kadoura, M.; Chandra, S. Rupture of thin liquid films sprayed on solid surfaces. Exp. Fluids 2013, 54, 1−11. (6) Ellen, C. H.; Tu, C. V. Jet stripping of molten metallic coatings. Phys. Fluids 1985, 28, 1202−1203. (7) Lacanette, D.; Gosset, A.; Vincent, S.; Buchlin, J.-M.; Arquis, E. Macroscopic analysis of gas-jet wiping: numerical simulation and experimental approach. Phys. Fluids 2006, 18, 042103. (8) Tu, C.; Wood, D. Wall pressure and shear stress measurements beneath an impinging jet. Exp. Therm. Fluid Sci. 1996, 13, 364−373. (9) Tuck, E.; Vanden-Broeck, J.-M. Influence of surface tension on jet-stripped continuous coating of sheet materials. AIChE J. 1984, 30, 808−811. (10) Tuck, E. O. Continuous coating with gravity and jet stripping. Phys. Fluids 1983, 26, 2352−2358.
Figure 10. (a) Density Nd and (b) average radius ⟨Rd⟩ of residual droplets as a function of residual film thickness hres. The gray data points are based on earlier work.24
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SUMMARY We performed a systematic experimental study of film rupture in thin liquid films of triethylene glycol on partially wetting polycarbonate substrates. The liquid film was deposited by spin coating and subsequently thinned using a slot jet impinging at normal incidence while moving the substrate. We measured the film thickness profile by dual-wavelength interference microscopy and recorded microscope images of the film rupture and compared these to numerical simulations. The local residual film thickness hres depends almost linearly on the substrate speed Usub in the experimentally accessible range, indicating 15857
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(37) Sharma, A.; Reiter, G. Instability of thin polymer films on coated substrates: rupture, dewetting, and drop formation. J. Colloid Interface Sci. 1996, 178, 383−399. (38) Vrij, A. Possible mechanism for the spontaneous rupture of thin, free liquid films. Discuss. Faraday Soc. 1966, 42, 23−33. (39) Reiter, G.; Sharma, A.; Casoli, A.; David, M.-O.; Khanna, R.; Auroy, P. Thin film instability induced by long-range forces. Langmuir 1999, 15, 2551−2558. (40) Parsegian, V. A. Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists; Cambridge University Press: New York, 2006. (41) Reiter, G.; Hamieh, M.; Damman, P.; Sclavons, S.; Gabriele, S.; Vilmin, T.; Raphaël, E. Residual stresses in thin polymer films cause rupture and dominate early stages of dewetting. Nat. Mater. 2005, 4, 754−758. (42) Brochard-Wyart, F.; Redon, C. Dynamics of liquid rim instabilities. Langmuir 1992, 8, 2324−2329. (43) Münch, A.; Wagner, B. Contact-line instability of dewetting thin films. Physica D 2005, 209, 178−190.
(11) Yoneda, H. Analysis of Air-Knife Coating. M.Sc. Thesis, University of Minnesota, 1993. (12) Hocking, G.; Sweatman, W.; Fitt, A.; Breward, C. Deformations during jet-stripping in the galvanizing process. J. Eng. Math. 2011, 70, 297−306. (13) Elsaadawy, E.; Hanumanth, G.; Balthazaar, A.; McDermid, J.; Hrymak, A.; Forbes, J. Coating weight model for the continuous hotdip galvanizing process. Metall. Mater. Trans. B 2007, 38, 413−424. (14) Buchlin, J.-M. Modelling of Gas Jet Wiping. Thin Liquid Film and Coating Processes; Lecture Series; Von Karman Institute for Fluid Dynamics: Rhode St. Genèse, Belgium, 1997. (15) Myrillas, K.; Rambaud, P.; Mataigne, J.-M.; Gardin, P.; Vincent, S.; Buchlin, J.-M. Numerical modeling of gas-jet wiping process. Chem. Eng. Process. 2013, 68, 26−31. (16) Zhang, Y.; Cui, Q.-P.; Shao, F.-Q.; Wang, J.-S.; Zhao, H.-Y. Influence of air-knife wiping on coating thickness in hot-dip galvanizing. J. Iron Steel Res. Int. 2012, 19, 70−78. (17) Tamaddon, A. H.; Belmiloud, N.; Doumen, G.; Struyf, H.; Mertens, P. W.; Heyns, M. M. Evaluation of high-speed linear air-knife based wafer dryer. Solid State Phenomena 2013, 195, 239−242. (18) Switkes, M.; Rothschild, M. Immersion lithography at 157nm. J. Vac. Sci. Technol., B 2001, 19, 2353−2356. (19) French, R. H.; Tran, H. V. Immersion lithography: photomask and wafer-level materials. Annu. Rev. Mater. Sci. 2009, 39, 93−126. (20) Mulkens, J.; Streefkerk, B.; Jasper, H.; de Klerk, J.; de Jong, F.; Levasier, L.; Leenders, M. >Defects, overlay, and focus performance improvements with five generations of immersion exposure systems. Proc. SPIE 2007, 6520, 652005. (21) Keij, D. L.; Winkels, K. G.; Castelijns, H.; Riepen, M.; Snoeijer, J. H. Bubble formation during the collision of a sessile drop with a meniscus. Phys. Fluids 2013, 25, 082005. (22) Winkels, K. G. Fast Contact Line Motion: Fundamentals and Applications. Ph.D. Thesis, University of Twente, Enschede, The Netherlands, 2013. (23) Wei, Y.; Brainard, R. L. Advanced Processes for 193 nm Immersion Lithography; SPIE: Bellingham, WA, 2009. (24) Berendsen, C. W. J.; Zeegers, J. C. H.; Darhuber, A. A. Deformation and dewetting of thin liquid films induced by moving gas jets. J. Colloid Interface Sci. 2013, 407, 505−515. (25) Blevins, R. D. Applied Fluid Dynamics Handbook; Van Nostrand Reinhold Company: New York, 1984. (26) Yoon, H. G.; Ahn, G. J.; Kim, S. J.; Chung, M. K. Aerodynamic investigation about the cause of check-mark stain on the galvanized steel surface. ISIJ Int. 2009, 49, 1755−1761. (27) Arthurs, D.; Ziada, S. Self-excited oscillations of a high-speed impinging planar jet. J. Fluids Struct. 2012, 34, 236−258. (28) Berendsen, C. W. J.; Zeegers, J. C. H.; Kruis, G. C. F. L.; Riepen, M.; Darhuber, A. A. Rupture of thin liquid films induced by impinging air-jets. Langmuir 2012, 28, 9977−9985. (29) Williams, M. B.; Davis, S. H. Nonlinear theory of film rupture. J. Colloid Interface Sci. 1982, 90, 220−228. (30) Oron, A.; Davis, S. H.; Bankoff, S. G. Long-scale evolution of thin liquid films. Rev. Mod. Phys. 1997, 69, 931−980. (31) Reiter, G. Dewetting of thin polymer films. Phys. Rev. Lett. 1992, 68, 75−78. (32) Meredith, J. C.; Smith, A. P.; Karim, A.; Amis, E. J. Combinatorial materials science for polymer thin-film dewetting. Macromolecules 2000, 33, 9747−9756. (33) Verma, A.; Sharma, A. Submicrometer pattern fabrication by intensification of instability in ultrathin polymer films under a water− solvent mix. Macromolecules 2011, 44, 4928−4935. (34) Jacobs, K.; Herminghaus, S.; Mecke, K. R. Thin liquid polymer films rupture via defects. Langmuir 1998, 14, 965−969. (35) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: Burlington, MA, 2011. (36) Reiter, G. Unstable thin polymer films: rupture and dewetting processes. Langmuir 1993, 9, 1344−1351. 15858
dx.doi.org/10.1021/la403988n | Langmuir 2013, 29, 15851−15858
Thinning and Rupture of Liquid Films by Moving Slot Jets Christian W. J. Berendsen, Jos C. H. Zeegers, and Anton A. Darhuber* Mesoscopic Transport Phenomena Group, Department of Applied Physics, Eindhoven University of Technology, Den Dolech 2, 5612AZ Eindhoven, The Netherlands S Supporting Information *
ABSTRACT: We present systematic experiments of the rupture and dewetting of thin films of a nonvolatile polar liquid on partially wetting substrates due to a moving slot jet, which impinges at normal incidence. The relative motion was provided by a custom-built spin coater with a bidirectionally accessible axis of rotation that enabled us to measure film thickness profiles in situ as a function of substrate velocity using dual-wavelength interference microscopy. On partially wetting polymeric substrates, dry spots form in liquid films with a residual thickness well below 1 μm. We measured the density of dry spots as well as the density and size distribution of the residual droplets as a function of film thickness. In a certain parameter range, the droplet distributions exhibit pronounced anisotropy due to the effect of long-range shear stresses on the dewetting rim instability. We find robust power-law scaling relations over a large range of film thicknesses and a striking similarity to literature data obtained with ultrathin polymer melt layers on silicon substrates.
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function of film thickness and found that for a low film thickness hmin the dry spot density follows the power law NH ∼ −2 h−4 min, decreasing to NH ∼ hmin as a result of confinement effects. In the experiments described in this article, we rotate a transparent polymer substrate, coated with a thin liquid film, underneath a slot jet impinging at normal incidence. We measure the residual film thickness by dual-wavelength interference microscopy. We develop a corresponding numerical model and achieve quantitative agreement. Moreover, we determine the dry-spot density as well as the density and size distribution of the resulting residual droplets as a function of substrate speed. In a certain parameter range, the droplet distributions exhibit pronounced anisotropy resulting from the effect of long-range shear stresses on the dewetting rim instability.
INTRODUCTION The rupture of thin liquid films on partially wetting substrates has been a field of study for decades. Padday1 showed that a liquid film on a Teflon or wax surface ruptures below a critical film thickness. Taylor2 performed a systematic study of the effort needed to make a hole in a liquid film and used a stationary impinging air jet to achieve this. A similar technique was employed later to initiate the dewetting of thin liquid films on partially wetting substrates.3,4 Kadoura et al. studied the rupture of water films after spraying on a solid surface.5 The use of a slot jet or air knife to thin a liquid film is common in applications such as gas jet wiping in galvanization lines.6−16 Slot jets are also used for the drying of wafers in the semiconductor industry.17 In immersion lithography,18−20 a photoresist-covered wafer is typically exposed through a layer of ultrapure water in order to increase the effective numerical aperture and thus the achievable resolution of the lithography system. Air knives are utilized to confine the water meniscus laterally between the objective lens and a partially wetting wafer, which is moving at a relative speed of approximately 1 m/s. Above a critical speed of translation, however, a thin liquid film is left on the wafer, which spontaneously breaks up, dewets, and leaves undesirable residual droplets on the substrate. The corresponding droplet size distributions are of crucial importance because of the potential formation of air bubbles21,22 that can distort the illumination pattern or the appearance of so-called watermark defects.23 In earlier work,24 we studied the deformation and rupture of a thin liquid film thinned by an impinging jet emanating from a round nozzle. In these experiments, we created a narrow track of low film thickness by controlling the jet Reynolds number and substrate velocity. We analyzed the density of dry spots as a © 2013 American Chemical Society
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MATERIALS AND METHODS
Figure 1c shows a schematic drawing of the experimental setup, consisting of a custom-designed spin coater that provides optical access to the spinning substrate from both above and below. Above the substrate, a slot jet is attached to a pneumatic vertical translation stage that can be lowered into position during or shortly after the spin coating of the thin liquid film. Beneath the transparent substrate, we implemented a high-speed dual-wavelength interference imaging system to measure film-height profiles and visualize the film rupture. The top view of the substrate shows the circular trajectory of the slot jet with respect to the substrate (Figure 1a). Figure 1b shows a schematic side view of the slot jet deforming a liquid film. We discuss Received: October 14, 2013 Revised: December 3, 2013 Published: December 3, 2013 15851
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Figure 2. Influence of the stand-off distance H, as defined in Figure 1b, normalized by the slit width w = 100 μm, on the spanwise thickness uniformity for Rew = 242. The micrographs correspond to H/w = (a) 5, (b) 10, and (c) 20. The green dashed lines indicate the approximate location of the slot jet above the substrate. The stand-off distance H between the nozzle exit and the substrate is H ≈ 200 μm. A small distance H and high Rew are necessary to obtain a uniform residual film thickness. At larger H, spanwise film thickness corrugations appeared,26 as visualized for Rew = 242 in Figure 1. These striations are possibly related to Görtler vortices.8 At greater stand-off distances, we also observed a weak but audible whistling sound,27 decreasing in frequency with increasing H. At low Rew or very high rotational speeds Ωsub, the reproducibility of the residual film thickness was reduced. Dual-Wavelength Interference Microscopy. Setup. We measure the liquid film thickness distribution using dual-color interference microscopy. The setup consists of two high-power light-emitting diodes (LED, Luxeon I) with emission wavelengths of λ1 = (655 ± 11) nm (red) and λ2 = (466 ± 14) nm (blue), respectively. The light sources are pulsed alternatingly in synchronization with the frame rate (250 Hz) of the high-speed camera (Photron SA-4) such that two consecutive frames were recorded with different wavelengths. The camera is fitted with illumination and imaging optics, including a microscope objective (Olympus, Uplan 4× , NA 0.13) yielding a field of view of approximately 4 × 4 mm2. The liquid film deformation caused by the slot jet (length L = 10 mm) is characterized by separately measuring close to the inner and outer edges of the track. Data Analysis. After background subtraction, we analyze the dark and light fringes for each wavelength. From the position and intensity of these interference fringes, two relative height profiles are constructed using a film thickness difference of Δhdark−light = λ/(4n) between neighboring fringes, where n = 1.46 is the average refractive index of 3EG in the visible spectrum. By matching the fringe positions of both colors, we are able to establish the fringe order and thus the absolute film thickness profile. The film thickness of the residual film hres in the track is obtained by greyscale analysis. For the material system 3EG/PC, provided that the fringes are well resolved in the regions where the film thickness gradient (|∇h|) is highest, film thicknesses can be measured with an accuracy of 5−20 nm in the range of 30 nm < h < 1500 nm. The error bars correspond to the difference in greyscale interpolation results of the two colors as well as the variation in film thickness for the duration of an experiment. Figure 3a−c shows a typical dual-wavelength interference measurement. An overlay of the two subsequent red and blue images is presented in Figure 3a. The film thickness profiles h(x) at the positions indicated by the yellow lines in Figure 3a are plotted in Figure 3b,c. The corresponding local substrate speed is Usub ≡ Ωsubr, where Ωsub is the angular velocity and r is the distance from the axis of rotation. The residual film thickness values hres(r = 17.1 mm) = (56.6 ± 3) nm and hres(r = 20.1 mm) = (64.8 ± 6) nm reflect the fact that hres increases with radial coordinate r as a result of the higher speed at greater distances from the center of rotation.
Figure 1. Schematic drawings of the experimental setup. (a) Top view of the substrate showing a track of lower film thickness induced by the slot jet. (b) Side-view sketch of the nozzle and the liquid-film deformation. (c) Schematic representation of the experimental setup, including imaging optics and the optically accessible spin-coater. the details of the setup and the experimental procedures in the following sections. Sample Preparation. For the preparation of the liquid film and subsequent thinning by a moving slot jet, we have designed a spin coater that provides accessibility to the axis of rotation from both sides. The spin coater consists of a sample holder supported by a hollow cylindrical roller bearing. The rotation is transmitted by a tooth belt connected to a precision motor, which enables rotational speeds of between 0 and 6000 rpm with an accuracy of ±0.2 rpm. The partially wetting sample, a polymer plate of optical-quality polycarbonate (PC, Makrofol DE, thickness 750 μm) cut to the dimensions of 60 × 60 mm2, is attached to a ring-shaped supporting plate using double-sided tape (3M, Scotch tape 410B). A liquid film of triethylene glycol (3EG, Sigma, product number 95126, purity ≥99.5%, viscosity μliq = 49 mPa· s, and surface tension γ = 45.5 mN/m) is spin-coated at 3000 rpm for 20 s, resulting in a uniform film of thickness h0 = 3.3 μm. In the final seconds of the spin-coating process, the nozzle is brought into position using a pneumatic vertical translation stage while the jet is still off. Slot Jet. The slot jet nozzle, schematically depicted in Figure 1b, is made of stainless steel and has an exit slit of dimensions w × L = 110 μm × 10 mm. The entrance length from a small pressure chamber to the exit is Hin = 12.5 mm, and the interior walls are parallel and smooth with a roughness of less than 100 nm. The horizontal outflow plane surrounding the slit has a total width of 3w. To reduce reflections in the microscope image, the underside of the nozzle is painted black. Precautions were taken to avoid the penetration of paint into the nozzle opening, and the paint was dried thoroughly before the experiment. Pressurized air is filtered through a HEPA filter (GE Whatman, HEPA-CAP) and a particle filter (Headline filters, 25-64-50C, 99.99+ % removal of 0.1 μm particles and aerosols). A needle valve (Brooks NRS8514) controls the gas flow rate Q, which is measured using a mass flow meter (Bronkhorst EL-FLOW F-112AC). Using tilt stages, we positioned the jet nozzle parallel to the substrate at a radial position between r = 10.5 and 20.5 mm from the center of rotation. The flow rate was kept constant at Q = 8.6 L/min, corresponding to a Reynolds number of Rew = ρgas vw/μ gas ≈ 1020. Here, we have used the density ̅ ρgas = 1.17 kg/m3 and viscosity μgas = 1.83 × 10−5 Pa·s of air and the average air flow speed v ̅ = Q/(wL) ≈ 145 m/s. Considering that the hydraulic diameter for a narrow slit DH ≈ 2w, the hydraulic Reynolds number ReDH ≈ 2040, which is still in the laminar regime.25 15852
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Figure 3. Measurement of film thickness by dual-wavelength interferometry. (a) Example of a micrograph, taken at the outer radius of the track for Ωsub = 1.5 rpm. Two subsequent blue and red images have been overlaid. (b, c) The height profile measured at the positions indicated by the solid yellow lines in plot a. The solid green lines represent numerical simulations.
Droplet Imaging. After each measurement, the resulting droplet patterns were recorded using two techniques. First a photograph was taken of the entire substrate area using a digital photocamera (Sony NEX-5) with a zoom lens (Sony SE-18200, 18−200 mm). Subsequently, the droplet patterns were imaged in an upright microscope (Olympus BX51) using Olympus objectives of various magnifications (2.5×, 5×, 10×, and 20×) and a CCD camera (Pike, Allied Vision Technologies). The radial position of the image was kept constant by rotating the sample under the microscope on a rotational stage with a belt drive that was actuated by a dc motor (Maxon 118689) with a planetary gearhead (Maxon 134782).
Figure 4. Numerical simulations of the distributions of (a) stagnation pressure Pslot, (b) its first derivative ∂Pslot/∂x, and (c) shear stress τslot of an impinging slot jet at Rew = 1000 and H = 200 μm. The dashed lines indicate the width w of the slot jet. (d) Experimental and numerical results for the residual film thickness hres due to slot jet impingement as a function of substrate speed Usub = Ωsubr or capillary number Ca ≡ μliqUsub/γ for a 3EG film on PC.
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The evolution of the thin liquid film is evaluated on a domain with a length Ld = L1 + L2 = 12 mm. The initial condition is a flat film of uniform thickness h0. The boundary conditions are
NUMERICAL MODEL Our model of the deformation of a moving liquid film due to an impinging slot jet consists of two parts. First we obtain the shear stress and pressure profile on a stationary solid plate by solving the incompressible Navier−Stokes equation28 using the finite element software Comsol 3.5a. We use a 2D model for the cross section of a slot jet of width w = 110 μm. We prescribe the entrance velocity profile as a fully developed parabolic Poiseuille profile with a maximum velocity of 3v/2. ̅ The computational domain is 1 mm high and 2 cm wide. Figure 4a,c shows the resulting pressure profile Pslot(x) and shear stress profile τslot(x) for Rew = 1000. The gradient of the pressure ∂Pslot/∂x is plotted in Figure 4b. The dashed vertical lines in Figure 4a−c indicate the location and width w of the slot jet. The resulting pressure and shear stress profiles Pslot(x) and τslot(x) are used as input for the thin-film model based on the lubrication approximation29,30 −
⎤ ⎡ 2 ∂Q x h3 ∂P ∂ ⎢ h τslot ∂h = = − + Usubh⎥ ⎥⎦ 3μ liq ∂x ∂t ∂x ∂x ⎢⎣ 2μ liq
∂ 2h + ρliq gh + Pslot ∂x 2
(3)
h(x = − L 2) = h0 and P(x = −L 2) = ρgh0
(4)
representing an undisturbed film of thickness h0 that enters the domain at x = −L2. In the time-dependent simulations, the film thickness downstream from the jet quickly approaches a quasi-steadystate value, which we term hres. Two examples of simulated height profiles h(x) for Rew = 1000 are plotted together with the experimental height profiles in Figure 3b,c. The match between the experimental and numerical data is excellent.
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RESULTS AND DISCUSSION Residual Film Thickness. We measured the residual film thickness hres for different rotational speeds Ωsub near the inner and outer edges of the track (Figure 1a). For comparison, we also evaluated hres from the numerical simulations of the film height profiles for Rew = 1000 and different values of Usub. The initial film thickness in the numerical simulations was set to either h0 = 3 or 20 μm. Figure 4d shows the results, plotted as a function of the local substrate speed Usub ≡ Ωsubr. The top axis represents the capillary number Ca ≡ μliqUsub/γ.
(1)
where μliq is the liquid viscosity, Qx is the volume flux, and P is the augmented pressure P ≡ −γ
∂P (x = L1) = 0 and Q x(x = L1) = Usubh ∂x
(2) 15853
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Figure 5. Interference micrographs showing film rupture in liquid films of 3EG on PC due to slot jet impingement in the radial interval 11.9 < r < 14 mm for different substrate rotational rates: (a) Ωsub = 0.10 rad/s (1 rpm), (b) 0.16 rad/s (1.5 rpm), (c) 0.21 rad/s (2 rpm), (d) 0.31 rad/s (3 rpm), (e) 0.42 rad/s (4 rpm), (f) 0.63 rad/s (6 rpm), (g) 0.84 rad/s (8 rpm), and (h) 1.05 rad/s (10 rpm). The dashed lines indicate the position of the slot jet. The circled areas in panels a−d are magnified in the insets.
Ωsub and in which dry spots prominently appear in the track. With decreasing Ωsub and thus a lower residual film thickness, the density of holes or dry spots NH increases. The distance between neighboring interference fringes on the slopes decreases with decreasing angular velocity, indicating a smaller slope. At high rotational speeds (Figure 5f−h), a temporal variation of the film thickness is observed, represented by the azimuthal variation in greyscale value along the track, which may be due to air-flow-induced vibrations of the plate and corresponding variations of the stand-off distance H. The size of the dry spots in these images depends on the substrate speed. After nucleation, the dry spots grow at a more or less constant dewetting speed Vd ≈ 0.85 mm/s. The substrate speed Usub exceeds Vd in all cases. Because of the higher residence time in the field of view, the size of the dry spots in the pictures initially increases with decreasing Ωsub, as seen in Figure 3h−e. For even lower Ωsub, however, the dryspot size is limited by the high density of dry spots. The rims of the dry spots merge and break up into droplets before leaving the field of view (Figure 5a−d). We quantified the density of dry spots per unit area NH as a function of substrate speed Usub and thus residual film thickness hres. We analyzed the interference micrographs of all experiments and marked the location of each freshly nucleated dry spots. After marking the positions of all dry spots, we divided the field of view into 10 bands along the radial direction, each corresponding to a distinct range in local substrate speed Usub.
The experimental data suggest a power-law relation hres ∼ Usub0.93. This agrees with our results for a round nozzle in ref 24, where we observed that the exponent decreases slightly with the viscosity of the liquid, ranging from 1 for water to 0.85 for glycerol. The numerical simulations and experimental data match very well in the experimentally accessible range. Moreover, the numerical simulations indicate a transition in the scaling behavior at high substrate speeds. This transition is related to the relative importance of the contribution of the jet pressure Pslot and the shear stress τslot to the thinning dynamics. Power laws can be derived from the lubrication equation (eq 1) in steady state (∂h/∂t = 0). In the shear-dominated regime (i.e., for low values of Usub and hres), a balance between the first and third terms on the right-hand side of eq 1 yields a power law hres ∼ μUsub/τslot. For higher Usub and thus higher hres, the pressure term prevails, yielding hres ∼
μUsub ∂Pslot ∂x
∼
Usub (5)
Because the residual film thickness hres cannot exceed the initial film thickness h0, the green curve in Figure 4d levels off at hres = h0 = 3 μm. Until that point, hres is essentially independent of the initial film thickness h0 as demonstrated by the overlap of the two curves for h0 = 3 and 20 μm. Dry-Spot Nucleation. Figure 5a−h shows a series of interference images, obtained with different rotational speeds 15854
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In Figure 6a,b, we plot NH as a function of the local film thickness hres, determined from Ωsub according to the power law fit h ∼ Usub0.93 = (Ωsubr)0.93 (Figure 4d).
on silicon (Si) wafers. Despite the difference in the material system, the data are surprisingly close to our measurements. The data of Reiter31,36,37 overlap perfectly with our data, but the data of Meredith et al.32 at low film thicknesses and Verma et al.33 at higher film thickness also match our experiments well. The reported contact angles of PS on Si were θ ≈ 10° in ref 33 and θ ≈ 22 ± 4° in ref 31, a value comparable to the value24 of the receding contact angle of triethylene glycol (3EG) on polycarbonate (PC), θrec ≈ 29 ± 2°. Variations in the oxide layer thickness on the silicon wafers are a likely reason for the deviations between the reported results. In the experimental data of Jacobs et al.,34 the density of dry spots NH is higher than the others. This is very likely due to the higher contact angle36 θ ≈ 50° of PS on the octadecyltrichlorosilane (OTS) selfassembled monolayer applied to the silicon wafer. If the density of dry spots is a result of a disjoining pressure caused by long-range interactions37−39 similar to nonretarded van der Waals forces, then the overlap in measurement data despite the difference in materials systems could be interpreted to be due to the fact that Hamaker constants of most organic materials systems do not vary significantly.35,40 However, the small time constants of rupture indicate that defects govern the rupture process.34,41 However, the similarity of these film rupture results raises the question of what the nature of these defects might be. Residual stresses were found to be important in polymer melt layers,41 but they are an unlikely cause of rupture in a low-viscosity Newtonian fluid such as 3EG. Droplet Patterns. After film rupture, the dry spots expand until their dewetting rims coalesce and leave a pattern of droplets behind on the substrate. After each experiment, we first recorded images of these droplet patterns. Figure 7 shows sections of these photographs with droplet patterns resulting from experiments at different substrate rotational rates. The droplets are primarily located in the track region, indicated by the light-blue color in Figure 1a. With increasing rotational speed Ωsub and thus a higher residual film thickness hres, the droplets become larger and their density decreases, consistent with the decreasing density of dry spots discussed above. The initial phase of the experiment (i.e., when the jet was turned on) is contained in the field of view of Figure 7a,g. In Figure 7f, the jet was turned off. After photographing the substrates, we imaged the droplet patterns in an upright microscope using a rotational stage. Figure 8a−e shows a series of such microscope images that provide a higher contrast and magnification than the photographs in Figure 7. The lines in Figure 8a−e represent the slot jet trajectory. We analyzed all microscope images of the droplet patterns and extracted the droplet density and droplet radii by image analysis using the image-processing toolbox of Matlab. On the basis of these data, we plot histograms of the droplet radius distributions in Figure 8f−j, corresponding to the measurement series in Figure 8a−e. The histograms are based on 30 logarithmically spaced bins in the range 0.1 μm ≤ Rd ≤ 10 mm. The dashed curves represent the droplet radius distribution for the corresponding images, whereas the solid lines are based on all images in each measurement. The dashed vertical line corresponds to the average droplet radius for the images in Figure 8a−e, whereas the average radius ⟨Rd⟩ for the entire measurement series is indicated by the colored bars and the symbols above the histograms. In the images in Figure 8d,e, corresponding to a residual film thickness of 100 ≲ hres ≲ 500 nm (8 ≲ Ωsub ≲ 16 rpm), welldefined polygonal droplet patterns are present and are very
Figure 6. (a) Density of dry spots NH in a thin film of 3EG on PC as a function of the local film thickness hres. The black squares indicate the film rupture of freshly spin-coated films without slot jet impingement. (b) Comparison of the data in plot (a) to literature results obtained for polystyrene melts on oxidized silicon31−33 or silicon grafted with a selfassembled monolayer.34
The density of dry spots in Figure 6a appears to follow a power law NH ∼ hresα, where the exponent α lies between −3 and −4. The spread in the measurement data increases with higher film thickness. The value of α = −4 agrees with the exponent associated with a nonretarded van der Waals type disjoining pressure isotherm. A power law exponent of α = −3 was also observed by Verma et al. for polymer films immersed in water33 and associated with double-layer interactions.35 Figure 6a also contains data (squares with vertical error bars) for 3EG films spin coated on 60 × 60 mm2 PC substrates (i.e., without the presence of an impinging slot jet). Depending on the film thickness, single or several dry spots appear within a few seconds after the spin coater stops rotating. A film of 3EG on PC with an initial film thickness h0 = 5 μm is stable most of the time. However, a film of h0 = 3 μm always exhibits one to three rupture events shortly after spin coating. Interestingly, the density of dry spots per unit area in this case is close to the extrapolation of the power law for exponent α = −3. In Figure 6b, we compare the experimental data of Figure 6a for films of the polar liquid triethylene glycol on polycarbonate to those described in the literature for polystyrene (PS) melts 15855
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Figure 7. Segments of photographs of the center area of the substrates after experiments performed at different sample rotational rates: (a) Ωsub = 0.10 rad/s (1 rpm), (b) 1.5 rpm, (c) 2 rpm, (d) 3 rpm, (e) 4 rpm, (f) 6 rpm, (g) 8 rpm, (h) 10 rpm, (i) 12 rpm, (j) 16 rpm, (k) 20 rpm, and (l) 30 rpm.
Figure 8. Microscope images of droplet patterns and corresponding droplet radius histograms for (a, f) Ωsub = 1 rpm and 24 ≤ hres ≤ 25 nm (in the field of view), (b, g) 2 rpm and 46 ≤ hres ≤ 49 nm, (c, h) 4 rpm and 85 ≤ hres ≤ 97 nm, (d, i) 8 rpm and 129 ≤ hres ≤ 220 nm, and (e, j) 16 rpm and 287 ≤ hres ≤ 463 nm. (a−e) The solid green lines represent the jet trajectory. (f−j) The solid red lines represent the histogram of droplet radii Rd for the full measurement, whereas the dashed lines represent the histogram for the images in a−e. The red symbols and error bars in f−j correspond to the mean value and standard deviation throughout the measurement. The vertical dotted line indicates the mean value for the measurements in a−e.
similar to the patterns reported for polymer films.31,36 The corresponding droplet radius histograms (Figure 8i,j) are broad because the droplet patterns consist of large droplets and smaller satellite droplets. In the intermediate substrate speed range corresponding to film thicknesses of 40 ≲ hres ≲ 100 nm in Figure 8b,c, this polygonal pattern exhibits a pronounced statistical anisotropy. It originates from the interaction between the long-ranged shear stress of the boundary layer of the slot jet and the instability of the dewetting rims.24,42,43 Because of the low substrate speed and relatively low dry-spot density, these instabilities had time to develop while still being affected by the shear stress of the air flow. The phenomenon of droplet shedding in the intermediate substrate speed range is illustrated in Figure 9. The first image shows four freshly nucleated dry spots. In the second picture, recorded 80 ms later, a rim instability can be observed.43 This instability gives rise to the droplet shedding on the right edge of the dry- spot, closest to the slot jet, while the dry spot grows. The final frame shows the resulting droplet pattern on the substrate. The larger droplets become more monodisperse, but an extra peak appears in the histograms at very low droplet radius (Figure 8g,h). For even lower substrate speed and thus thinner films (Figure 8a,f), again an isotropic dense pattern of small droplets is left on the substrate.
Figure 10a,b presents the average droplet radius ⟨Rd⟩ and density of droplets Nd as a function of residual film thickness hres. Both graphs show a clear power law dependence with residual film thickness. The power law exponent for the average droplet radius ⟨Rd⟩ as a function of hres is 1.1. The density of droplets (Figure 10b) follows a power law Nd ∼ hres−2.8. Data obtained using a round air jet24 are included as gray symbols in the graphs in Figure 10. Within error bars, the corresponding power law exponents are comparable. The data for the round air jet appear to have a lower density of droplets and a higher average droplet radius than the current data obtained with the slot jet. This is due to the fact that in the analysis of the round jet measurement data the minimum film thickness in the narrow track was used for the horizontal axis, whereas rupture also occurred in the neighboring region of slightly higher film thickness. 15856
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that the influence of the wall shear stress dominates over the stagnation pressure of the slot jet. The density of dry spots NH follows the power law NH ∼ hresα with exponent −4 < α < −3 and overlaps with literature data obtained for other material systems. Finally, we determined the density and size distribution of resulting droplet patterns as a function of hres and compared them to experimental results obtained for a narrow track in a thin liquid film.
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ASSOCIATED CONTENT
* Supporting Information S
Video of film thinning and rupture for different substrate speeds as well as atomic force microscopy surface characterization data for polycarbonate substrates. This material is available free of charge via the Internet at http://pubs.acs.org/
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected].
Figure 9. Dry-spot growth and coalescence showing the directional release of droplets by the combined action of rim instability and airflow-induced shear stress. The last frame is a microscope image of the final droplet distribution.
Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work is part of the research programme “Contact Line Control during Wetting and Dewetting” (CLC) of the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). The CLC programme is cofinanced by ASML and Océ. We thank Laurens van Bokhoven (ASML) and Avraham Hirschberg (Eindhoven University of Technology) for fruitful conversations on nozzle design and instabilities related to impinging slot jets and Jim Overkamp, Rogier Cortie, Michel Riepen, Ramin Badie, and Nico Kemper (ASML) for discussions of the interaction between gas jets and liquid films and residual droplet patterns in immersion lithography. Finally, we thank Henny Manders and Jørgen van der Veen for their help in the design and realization of the experimental setup.
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REFERENCES
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Figure 10. (a) Density Nd and (b) average radius ⟨Rd⟩ of residual droplets as a function of residual film thickness hres. The gray data points are based on earlier work.24
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SUMMARY We performed a systematic experimental study of film rupture in thin liquid films of triethylene glycol on partially wetting polycarbonate substrates. The liquid film was deposited by spin coating and subsequently thinned using a slot jet impinging at normal incidence while moving the substrate. We measured the film thickness profile by dual-wavelength interference microscopy and recorded microscope images of the film rupture and compared these to numerical simulations. The local residual film thickness hres depends almost linearly on the substrate speed Usub in the experimentally accessible range, indicating 15857
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