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Dewetting of Liquid Filaments in Wedge-Shaped Grooves Krishnacharya Khare,† Martin Brinkmann,† Bruce M. Law,‡ Evgeny L. Gurevich,† Stephan Herminghaus,† and Ralf Seemann*,† Max Planck Institute for Dynamics and Self-Organization, D-37018 Go¨ttingen, Germany, and Department of Physics, Kansas State UniVersity, Manhattan, Kansas 66506 ReceiVed May 24, 2007. In Final Form: August 28, 2007 The dewetting of liquid filaments in linear grooves of a triangular cross section is studied experimentally and theoretically. Homogeneous filaments of glassy polystyrene (PS) are prepared in triangular grooves in a nonequilibrium state. At elevated temperatures, the molten PS restores its material contact angle with the substrate. Liquid filaments with a convex liquid-vapor interface decay into isolated droplets with a characteristic spacing depending on the wedge geometry, wettability, and filament width. This instability is driven by the interplay of local filament width and Laplace pressure and constitutes a wide class of 1D instabilities that also include the Rayleigh-Plateau instability as a special case. Our results show an accurately exponential buildup of the instability, suggesting that fluctuations have a minor influence in our system.
Introduction Liquid coatings on solid substrates may be subject to a wide variety of dynamical instabilities if they are prepared in nonequilibrium states. On homogeneous substrates, the liquid beads off either via the nucleation of circular dry patches or by spinodal dewetting (i.e., by the amplification of unstable surface waves1-4). On substrates with patterned wettability, completely different modes of instability can be observed that sensitively depend upon the imposed wettability pattern.5,6 In this article, we investigate such instabilities at topographic structures on solid substrates. Liquids are generally attracted to linear topographic defects such as steps and grooves,7-10 as one finds on almost all real surfaces (e.g., scratches or multiple vicinal steps). At such defects, stable liquid filamentous structures can form, in sharp contrast to perfectly planar substrates on which liquid filaments are generally unstable with respect to a peristaltic instability mode.5,11,12 The equilibrium shape of the resulting liquid filament is governed by a delicate interplay between the wettability and the specific substrate topography. If this balance is disturbed by a change in parameters, then the shape of the interface may become unstable and decay into a different liquid morphology. In a wedge formed by two planar surfaces, two principal equilibrium morphologies of the liquid can be found:10,13 (i) For * Corresponding author. E-mail:
[email protected]. † Max Planck Institute for Dynamics and Self-Organization. ‡ Kansas State University. (1) Oron, A.; Davis, S. H.; Bankoff, S. G. ReV. Mod. Phys. 1997, 69 931-980. (2) Seemann, R.; Herminghaus, S.; Jacobs, K. J. Phys.: Condens. Matter 2001, 13, 4925-4938. (3) deGennes, P.-G.; Brochard-Wyart, F.; Que´re´, D. Capillarity and Wetting Phenomena; Springer: Berlin, 2001. (4) Herminghaus, S.; Brochard, F. C. R. Phys. 2007, 7 1073-1081. Herminghaus, S.; Brochard, F. C. R. Phys. 2007, 8, (erratum). (5) Gau, H.; Herminghaus, S.; Lenz, S.; Lipowsky, R. Science 1999, 283, 46-49. (6) Scha¨fle, C.; Bechinger, C.; Rinn, B.; David, C.; Leiderer, P. Phys. ReV. Lett. 1999, 83, 5302-5305. (7) Brinkmann, M.; Blossey, R. Eur. Phys. J. E 2004, 14, 79-89. (8) Seemann, R.; Brinkmann, M.; Kramer, E. J.; Lange, F. F.; Lipowsky, R. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 1848-1852. (9) Roy, R. W.; Schwartz, L. W. J. Fluid. Mech. 1999, 391, 293-318. (10) Concus P. and Finn, R. Proc. Natl. Acad. Sci. U.S.A. 1969, 63, 292-299. (11) Lenz, P.; Lipowsky, R. Europ. Phys. J. E. 2001, 1, 249-262. (12) Koplik, J.; Lo, T. S.; Rauscher, M.; Dietrich, S. Phys. Fluids 2006, 18, 032104.
liquid contact angles θ smaller than the wedge angle ψ, stable liquid filaments exist with a cylindrical liquid-vapor interface curved toward the wedge bottom (Figure 1 a). In mechanical equilibrium, these homogeneous liquid filaments extend along the entire length of the wedge. (ii) For large contact angles (θ > ψ), the stable liquid morphology consists of droplets whose shape is given by a segment of a sphere.13,14 If the contact angle is rapidly changed from a value corresponding to (i) into a value corresponding to (ii), then the filament becomes unstable and decays in a characteristic way into a chain of isolated droplets. It is this instability that we investigate here. Experimental Section Substrates with linear triangular grooves, 2 µm wide by 7 mm long, were purchased from Mikromasch, Spain. The grooves were fabricated in silicon using standard photolithography and anisotropic wet etching of the Si surface resulting in a wedge angle of ψ ) 54.7°. To modify the wettability, the silicon surface is hydrophobized by various self-assembled monolayers (ABCR, Germany). Octadecyltrichlorosilane (OTS) was deposited from solution,15 leading to a contact angle of θ ) (58 ( 1)° for polystyrene and excellent surface quality. The contact angles were inferred directly from atomic force micrographs (AFM) on plane parts of the substrate for advancing and receding liquid fronts. To obtain larger contact angles, semifluorinated chlorosilanes (3-heptafluoroisopropoxy)-propyltrichlorosilane (HTS) and heptadecafluoro-1,1,2,2-tetrahydrodecyl)dimethylchlorosilane (HMS) were vapor deposited onto the silicon substrate, resulting in contact angles of (64 ( 2)° and (74 ( 2)°, respectively. Intermediate contact angles were achieved at the cost of increased contact angle hysteresis by the coevaporation of either HTS or HMS with (3-methacryloyloxypropyl)-trichlorosilane (MTS), resulting in contact angles of (57 ( 3)° and (60 ( 3)°, respectively. As the wetting liquid, we used short-chain polystyrene, fabricated by anionic polymerization with a molecular weight of Mw ) 1.89 kg/mol and a polydispersity of Mw/Mn ) 1.06 (PSS, Mainz, Germany). The polymer has one butyl and one styrene end group and no remaining monomers or other additives. The polymer is precipitated from methanol and dried under high vacuum. The material is glassy (13) Concus, P.; Finn, R.; McCuan, J. Indiana UniV. Math. J. 2001, 50, 411442. (14) The dimensions of the wedge have to be sufficiently large compared to the size of the droplet to avoid pinning of the contact line at the upper edges of the groove walls. (15) Sagiv, J. J. Am. Chem. Soc. 1980, 102, 92-98.
10.1021/la701515u CCC: $37.00 © 2007 American Chemical Society Published on Web 10/26/2007
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Figure 1. (a) Sketched cross section of a liquid filament in contact with a linear wedge. (b) Different types of behavior presented in the (θ, ψ) plane. The bold part of the θ axis represents the peristaltic instabilities of cylindrical liquid filaments on planar substrate. at room temperature and can be considered to be a Newtonian liquid well above the glass-transition temperature of about 60 °C.16-18 The polystyrene was deposited onto the grooved substrates via spin casting a toluene solution. During this spin-coating process, the polystyrene solution is removed from the ridges because of its rather large contact angle. Polystyrene deposits, however, remain within the triangular grooves. The polystyrene filling height h and filling width w within the grooves were adjusted via the spin-coating parameters and the concentration of the polystyrene solution, respectively.
Results and Discussion As the solvent evaporates during the spin coating process, the polymer freezes in a nonequilibrium morphology, forming filaments with a concave polymer-vapor interface (i.e., curved toward the wedge bottom). When heated above the glass-transition temperature, the polystyrene (PS) very rapidly re-establishes its material contact angle θ with the substrate. The cross section of the liquid-vapor interface is found to relax to a circular arc, while the filament width, w, remains longitudinally homogeneous (z-direction, cf. Figure 1). If the contact angle is smaller than the wedge angle θ < ψ, then the polymer forms concave filaments that are homogeneous in the longitudinal direction. Such filaments were always found to be stable in our experiments. If, however, the contact angle is larger than the wedge angle θ > ψ, then the polymer forms convex homogeneous filaments (Figure 2a). These are found to be unstable such that initial fluctuations are amplified and a chain of regularly spaced droplets is formed at long times (Figure 2b). The time scale of the longitudinal reorganization of liquid filaments into liquid droplets exceeds the time scale of the transverse equilibration (which proceeds over much shorter distance) by orders of magnitude. The optical micrograph in Figure 2c shows the final pattern of polystyrene droplets after filament decay. As shown in the inset of Figure 2c, the droplet spacing is well described by a Gaussian distribution. The variance of the distribution is in the range of 20%. The preferred separation distance 〈d〉 is obtained by averaging over all droplet centerto-center distances, d. As depicted in Figure 3a, we observe a linear dependence of the preferred distance 〈d〉 on the filling width w. In other words, the normalized droplet distance dh ) 〈d〉/w, represented by the slope of the straight lines, is a function of the wettability only. This relation is plotted in Figure 3b. The data suggest that the preferred droplet distance diverges as the contact angle θ approaches the wedge angle ψ from above. The basic mechanism for the filament instability, occurring at contact angles θ > ψ, can be explained by arguments similar (16) Seemann, R.; Herminghaus, S.; Jacobs, K. Phys. ReV. Lett. 2001, 87, 196101. (17) Becker, J.; Gru¨n, G.; Seemann, R.; Mantz, H.; Jacobs, K.; Mecke, K.; Blossey, R. Nat. Mat. 2003, 2, 59-63. (18) Fetzer, R.; Jacobs, K.; Mu¨nch, A.; Wagner, B.; Witelski, T. P. Phys. ReV. Lett. 2005, 95, 127801.
Figure 2. AFM micrographs of (a) an unstable polystyrene filament and (b) an isolated polystyrene droplet after filament decay. (c) Optical micrograph of polystyrene droplets resulting from the complete decay of polystyrene filaments. (Inset) Distribution of counts for different center-to-center distances between neighboring droplets, fitted by a Gaussian distribution.
to the peristaltic instability of a liquid filament on a planar surface5 and is also related to the Rayleigh-Plateau instability of a freestanding liquid cylinder.19 The mean curvature H and the excess free energy γ of the liquid surface determine the Laplace pressure P of the liquid filament via P ) 2Hγ. By virtue of the wedge geometry, an increase in the filling width w reduces the mean curvature. Hence, the Laplace pressure will drive the liquid from regions with smaller filling width toward regions with larger filling width, resulting in a morphological instability. Very short corrugations are effectively suppressed by the surface tension of the liquid, but all fluctuations above a critical longitudinal wavelength λ* are amplified. Because viscous resistance suppresses liquid flux over large distances, we expect to find a wavelength λmax > λ* where the growth rate is maximal. We expect this preferred wavelength to prevail in the spatial pattern until a late stage of the dewetting process such that 〈d〉 ) λmax. We find experimentally that the lateral equilibration toward a filament with a cylindrical surface is much faster than the development of the instability by liquid transport along the groove. This implies that the liquid will be locally equilibrated, although it may be far from the global mechanical equilibrium. Because the characteristic droplet spacing 〈d〉 is much larger than w, the long-wavelength approximation can be used to solve the Stokes equation.20 We assume that the Laplace pressure P depends only upon the longitudinal (z) coordinate (Figure 1). If this variation is weak, as our experiments suggest, then we may use the linear relation -µ∂zP ) Q between the pressure gradient ∂zP and the volumetric flow rate Q along the z direction, where µ is the liquid mobility in the groove. Because the liquid is incompressible, we furthermore have ∂zQ ) -∂tA, where A is the cross-sectional (19) Lord Rayleigh. Proc. London Math. Soc. 1878, 10, 4-13. (20) Romero, L. A. J. Fluid. Mech. 1996, 322, 109-129.
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Figure 3. Normalized droplet distance d ) dh/w as a function of θ. The neutrally stable mode λ*/w (---) and the fastest growing mode λmax ) x2λ*/w (-) are shown according to eq 3. (Inset) Preferred droplet distance d vs filling width w. The lines show the fastest growing mode λmax according to eq 3.
area of the liquid filament. We thus obtain ∂z(µ∂zP) ) ∂tA, which may be linearized to
µo∂zz2δP ) ∂tδA
(1)
where δP(z, t) and δA(z, t) are small variations around the constant values Po and Ao, respectively, corresponding to the unperturbed filament. Furthermore, µo ) cA2o/η with liquid viscosity η and a dimensionless number c that depends upon the geometry of the groove and the flow boundary conditions.20 We assume everywhere a circular cross section with radius R(z, t), making Young’s angle θ with the substrate. This is justified experimentally because we could not detect any difference between the contact angle of the advancing and the receding liquid front within an accuracy of about 2°. There is thus a direct relation between the cross-sectional radius R and the height of the crest of the filament h. We solve eq 1 with the Fourier ansatz R ) Ro + δRk exp((ikz + t/τ), where Ro is the radius of the unperturbed (cylindrical) filament surface and τ is the characteristic time scale of the decay. To obtain the local Laplace pressure in the filament, we consider the mean curvature of the varying liquid-vapor interface along the center line. Constructing the deformation modes, we find a variation δc⊥ ) -δR/R2o of the normal curvature c⊥ parallel to the xy plane. We approximate the variation of the initially zero normal curvature c| on top of the filament in the z direction by the second derivative of the height variation δh of the center line, δc| ) -∂2z δh. As a result, we obtain a relation between the
Figure 4. (a) Time series of in-situ AFM line scans at 80 °C showing the profile of a decaying liquid filament along the center line of a triangular groove. (b) Amplitude of the preferred wavelength as revealed from Fourier analysis as a function of time. The marked points (corresponding to room temperature and the first hole, respectively) were not included in the fit.
amplitude δHk of the mean curvature variation and δRk. Together with the amplitude δAk corresponding to the variation of the cross-sectional area, it is then straightforward to derive the dispersion relation between the normalized time constant τj ) τγµo/w5 and the normalized wave number kh ) kw
hk2 sin [4 cos ψ sin2 - kh2(cos ψ - cos θ)] cos ψ( - sin cos + tan ψ sin2 )
) τj-1 (2)
where ) θ - ψ (cf. Figure 1). We see that for concave meniscus < 0, we have τj-1 < 0, and the filament is stable. For a convex meniscus (i.e., for > 0), however, the filament is unstable.21 We find not only exponentially damped modes with k > k*, where k* is the zero of the dispersion relation (eq 2) but also growing modes, with k < k*. For the wavelength of the neutrally stable mode, λ* ) 2π/k*, we find
λ* π cos ψ - cos θ 1/2 ) w sin cos ψ
(
)
(3)
The preferred wavelength, where the growth rate τj-1 is maximal, is determined by λmax ) x2λ*. From the parametrization of the deformation mode by scaled circular arcs, it is not clear whether the liquid-vapor interface exhibits a mean curvature variation that depends solely on the z coordinate, as assumed in our model. To quantify these deviations, we additionally considered deformation modes that are exact to linear order in the variation. Details of the calculation (21) Note that the denominator is quadratic in to leading order.
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are provided in Supporting Information. The main result of this calculation is the dispersion relation
-
4kh2q2 sin3 [cos(q) + q tan θ sin(q)] ) τj-1 sin(q) - q tan θ sin(q) - cos(q) q
[
]
(4)
with a transverse wave number of q2 ) 1 - ((kh/2)sin )2 that can assume either real or purely imaginary values depending on the normalized longitudinal wavenumber kh. The wavelength at neutral stability τ-1 ) 0 according to eq 4 is identical to the result given by Roy and Schwartz.9 A numerical comparison between the simple dispersion relation (eq 2) and the exact solution in linear order (eq 4) reveals only small relative differences for the particular values considered here. However, the relative difference may become large for ψ f 0 and θ f 0. Except for these rather extreme cases, the numerical values of the neutrally stable wave number k* compare very well. Figure 1b shows an overview of the morphological stability, presented in the (ψ, θ) plane. The bold part of the θ axis corresponds to the peristaltic mode on a planar substrate (ψ ) 0), which contains, for θ ) π/2 and full slip boundary conditions, the Rayleigh-Plateau instability of a liquid jet as a special case. Our experiments were performed at finite ψ (ψ ) 0.955, as indicated by the arrow), where the transition between the stable and the unstable regime is accessible by varying θ. This dependence upon the contact angle is given by eq 3 and displayed as solid curve in Figure 3b. As we can see, it is in excellent agreement with the preferred distance 〈d〉 of the droplets after filament decay, as represented by the data points. It is finally of interest to investigate the temporal behavior of the instability. First, one may quantitatively determine the mobility µ and thereby the geometry constant c that allows for the determination of the microscopic slip length at the liquid/solid interface. Second, as the theoretical model suggests, one would expect a purely exponential rise in the amplitude of initial perturbations as long as nonlinearities may be neglected. However, it has been predicted that in small-scale systems such as dewetting scenarios (the present one), thermal fluctuations may strongly affect this behavior and lead to deviations from simple exponential behavior.22-24 These effects were meanwhile experimentally confirmed for the spinodal dewetting of thin polystyrene films (i.e., a 2D system). It is well known that fluctuations are of even more noticeable impact in lower dimensions. In fact, it has been shown that in triangular grooves and conical troughs, fluctuations maybecome completely dominant in wetting-phase transitions.24-26 (22) In this case, we have ω ) D sin , which yields the Rayleigh result λ* ) πD, where D ) 2R is the diameter of a cylinder with the same curvature. (23) Mecke, K.; Rauscher, M. J. Phys.: Condens. Matter 2005, 17, S3515S3522. (24) Fetzer, R.; Rauscher, M.; Seemann, R.; Jacobs, K.; Mecke, K. Phys. ReV. Lett. 2007, 99, 11450.
We thus have directed our attention to the time dependence of the amplitude of the fastest growing mode, which is directly accessible via in-situ AFM experiments at elevated temperature. A typical result is shown as a line-scan time series in Figure 4a, taken along the central axis of a molten filament. The sinusoidal undulation of these scan lines confirms the mode selection process and the fact that the characteristic droplet distance 〈d〉 indeed coincides with the fastest growing wavelength λmax. This becomes very obvious when taking a Fourier analysis of the various line scans (not shown). We clearly find one sharp peak at 1/λ ) 1/λmax rising in time. Plotting the amplitude of the fastest growing mode (i.e., the peak of the Fourier transformation at 1/λmax), as done in Figure 4b, reveals that the amplitude of the fastestgrowing mode λmax increases exponentially with time over almost two orders of magnitude. This confirms the growth law predicted by our simple model with high confidence. We see that the growth law can be followed over a wide range, thus enabling measurements of potential deviations from the exponential growth law with high accuracy.
Conclusions In this article, we presented an experimental and theoretical study of how liquid filaments may dewet from triangular grooves. Liquid filaments with contact angle θ smaller than the wedge angle of the groove ψ are stable and form homogeneous liquid wedges with negative mean curvature. Liquid filaments with contact angle θ > ψ, however, are in a nonequilibrium state and are curved toward the vapor phase. They are morphologically unstable and decay into isolated droplets with a preferred separation. The latter depends on the degree of filling, the contact angle of the liquid, and the wedge angle of the triangular groove. The instability is driven by the interplay between the surface tension and the Laplace pressure and can be modeled precisely with linear stability analysis. This type of dynamic instability constitutes a wide class of 1D instabilities, which also includes the Rayleigh-Plateau instability as a special case. Acknowledgment. We are grateful to A. Parry for very useful comments. This work was supported by DFG priority program 1164 under grant number Se 1118/2. B.M.L. acknowledges support for this work through the U.S. National Science Foundation under grant number DMR-0603144. Supporting Information Available: Detailed derivation of dispersion relation 4 for unstable liquid filaments in wedge-shaped grooves. This material is available free of charge via the Internet at http://pubs.acs.org. LA701515U (25) Bruschi, L.; Carlin, A.; Parry, A. O.; Mistura, G. Phys. ReV. E 2003, 68, 021606. (26) Parry, A. O.; Rascon, A.; Wood, A. J. Phys. ReV. Lett. 2005, 85, 345348. (27) Parry, A. O.; Wood, A. J.; Rascon, C. J. Phys.: Condens. Matter 2001, 13, 4591-4613. (28) Brinkmann, M.; Kierfeld, J.; Lipowsky, R. J. Phys. A: Math. Gen. 2004, 37, 11547-11573.
