Langmuir 2009, 25, 1919-1922
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Controlled Pattern Formation in Thin Liquid Layers Rodica Borcia* and Michael Bestehorn Lehrstuhl Statistische Physik/Nichtlineare Dynamik, Brandenburgische Technische UniVersita¨t, Erich-Weinert-Strasse 1, 03046 Cottbus, Germany ReceiVed NoVember 13, 2008. ReVised Manuscript ReceiVed January 9, 2009 We examine the fully nonlinear behavior of a thin liquid film on a hydrophobic/hydrophilic solid support in three dimensions using a phase field model. For flat homogeneous substrates, the stability of thin liquid layers is investigated under the action of gravity. The coarsening process at the solid boundary can be controlled on inhomogeneous substrates. On substrates chemically patterned in an adequate way with hydrophobic and hydrophilic spots, one can obtain stable, regular liquid droplets and even design liquid structures (PACS numbers: 47.54.-r, 68.18.Jk, and 05.70.Np).
The spatiotemporal behavior of thin liquid layers on a solid support has widespread technological applications (e.g., in coating/wetting phenomenal; (see refs 1 and 2 and references therein) and opens new strategies for designing smart microfluidic devices.3 If the free surface of the flat film is unstable to spatial disturbances, then pattern formation sets in and drops, holes, and eventually film rupture may occur. In a thin liquid layer, one finds after a while the formation of larger and larger structures, a phenomenon known as coarsening. The dynamics converges to a stationary state that consists of a single elevation (drop) or suppression (hole) of the surface. This development can be interrupted by rupture of the film. Rupture is obtained if the surface touches the substrate and the thickness reaches zero in a certain domain. One can avoid rupture by introducing a repelling disjoning pressure (caused by van der Waals forces) acting in very thin liquid layers with heights in the range of 1-100 nm. In this situation, a completely dry region cannot exist. The substrate is covered by a so-called precursor film. The precursor film is stabilized by the disjoining pressure in the evolution equation for the film thickness and is controlled by the Hamaker constant. Including the repelling disjoining pressure at the solid substrate, pattern formation in two and three dimensions was intensively studied in the long-time limit using the long-wave approach.4-8 Using a lubrication approximation, this approach is valid only for small velocities and small contact angles (hydrophilic surfaces). Our aim is to study the behavior of thin liquid films on both hydrophilic and hydrophobic surfaces using a phase field model. The phase field models introduce an order parameter that thermodynamically describes the phases. For a liquid-vapor system, we chose the density F as an order parameter. F ) 1 denotes the liquid phase, and F ) 0 denotes the vapor phase. The wettability properties at the substrate as well as the static contact
angle θ and the thickness of the precursor film are controlled by the density at the solid boundary FS, a free parameter between 0 and 19,10
cos θ ) -1 + 6FS2 - 4FS3 FS ) 0 means that there is no precursor film at the substrate (no-wetting case), and FS ) 1 corresponds to the largest precursor film (complete wetting). Large values of FS (FS > 0.5) correspond to small contact angles (θ < 90°) and vice versa. No restrictions regarding small or large contact angles appear within the phase field formalism. Phase Field Equations. The interface is introduced implicitly by gradients of the phase field in the free-energy functional of the system9,11,12
F [F] )
∫V [f(F) + K2 (∇F)2] dV
(1)
where the first term represents the free-energy density for the homogeneous phases. For a system in equilibrium and without interfacial mass exchange, the free-energy density has the form of a double-well potential with two minima corresponding to the two alternative phases: F ) 1 (liquid) and F ) 0 (vapor bulk). We take
C f(F) ) F2(F - 1)2 2
(2)
with C being a characteristic value of the free-energy density f(F). The second term in eq 1 is a gradient energy, which is a function of the local state. The specific interfacial free energy γ is, by definition, the difference per unit area of the interface between the actual free energy of the system and that which it would have if the properties of the phases were continuous throughout. Hence, the free-energy excess of the interface takes the form of13
* Corresponding author. E-mail:
[email protected]. (1) de Gennes, P. G. ReV. Mod. Phys. 1985, 57, 827. (2) Starov, V. M.; Velarde, M. G.; Radke C. J. Wetting and Spreading Dynamics; CRC Press: Boca Raton, FL, 2007. (3) Alexeev, A.; Balazs, A. C. Soft Matter 2007, 3, 1500. (4) Bestehorn, M.; Neuffer, K. Phys. ReV. Lett. 2001, 87, 046101. (5) Scheid, B.; Oron, A.; Colinet, P.; Thiele, U.; Legros, J. C. Phys. Fluids 2002, 14, 4130. (6) Bestehorn, M.; Pototsky, A.; Thiele, U. Eur. Phys. J. 2003, B33, 457. (7) Pototsky, A.; Bestehorn, M.; Thiele, U. Physica D 2004, 199, 138. (8) Pismen, L. M.; Thiele, U. Phys. Fluids 2006, 18, 042104. (9) Pismen, L. M.; Pomeau, Y. Phys. ReV. 2000, E62, 2480. (10) Borcia, R.; Borcia, I. D.; Bestehorn, M. Phys. ReV. 2008, E78, 066307.
γ)
∫-∞+∞ K ( ∂F ∂z )
2
dz
(3)
which gives a direct connection between the surface tension coefficient γ and the gradient energy term K (∂F/∂z)2/2 (where K is the square gradient parameter). (11) Bray, A. J. AdV. Phys. 1994, 43, 357. (12) Jasnow, D.; Vinals, J. Phys. Fluids 1996, 8, 660. (13) Cahn, J. W.; Hilliard, J. E. J. Chem. Phys. 1958, 28, 258.
10.1021/la803687v CCC: $40.75 2009 American Chemical Society Published on Web 01/23/2009
1920 Langmuir, Vol. 25, No. 4, 2009
Letters
∂F + ∇ · (FV b) ) 0 ∂t
(5)
bV)/(dt) represents the total derivative of the velocity In eq 4, (d field b V, p ) F∂f(F)/∂F - f(F) is the thermodynamic pressure, η is the dynamic viscosity, and g the gravitational acceleration. The second term on the right-hand side denotes the Korteweg stress introduced first in ref 14 and studied latter in refs 15-18 and references therein. This term describes the contribution of capillary forces and substitutes into the phase field model the classical boundary condition for tangential stresses at the liquid-vapor interface.15 We scale the variables by using d, d2/νl, and νl/d as units for the length, time, and velocity where d ) √K ⁄C represents the characteristic thickness of the diffuse interface and νl is the liquid kinematic viscosity. The following nondimensional parameters appear: Ca ) K /Flν2l (the capillary number) and G ) gd3/ν2l (the gravitational number; Fl is the liquid density. The surface tension coefficient can be related to C and K as γ ≈ √KC.9 For the numerical results presented in this letter, we have d ) 1 nm, γ ) 0.05 N/m, Fl ) 1000 kg/m3, and νl ) 0.001 cm2/s. One obtains C ) 5 × 107 N/m2, K ) 5 × 10-11 N, and Ca ) 5. Chemical Potential: Maxwell Construction. The stability of a flat liquid film on a solid support can be understood in terms
Figure 1. Chemical potential µ vs film thickness h for substrates with different wettabilities: (a) Fs ) 0.9 and (b) Fs ) 0.5 and G ) 10- 4. The unit d represents a characteristic length of the diffuse interface. The Maxwell construction gives the binodal points that separate the partial wetting area from the complete wetting area. (c) Binodals hmin and hmax as a function of Fs.
The basic equations consist of the Navier-Stokes equations, including phase field gradient terms for assuring the shear stress balance at the interface
F
dV b ) - ∇ p + F ∇ (∇ · (K ∇ F)) + ∇ · (η ∇ b V) dt η b (4) V + Fg + ∇ ∇ ·b 3
(
and the continuity equation
)
Figure 2. Transitions in a thin, unstable liquid film on a flat, homogeneous solid substrate under gravity effects (h0 ) 16d, G ) 10-4, and Fs ) 0.3): (a) t ) 50, (b) t ) 250, (c) t ) 550, (d) t ) 1300, (e) t ) 1900, (f) t ) 5200, (g) t ) 9250, and (h) t ) 21 250.
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Langmuir, Vol. 25, No. 4, 2009 1921
Figure 3. Transitions in a thin, unstable liquid layer lying on a hydrophobic substrate (Fs1 ) 0.2) chemically patterned with hydrophilic square spots (Fs2 ) 0.9, h0 ) 16d, and G ) 10-4): (a) t ) 50, (b) t ) 150, (c) t ) 450, (d) t ) 1050, (e) t ) 2200, and (f) t ) 5400.
of the dependence of the chemical potential on the film thickness. Pismen and Pomeau formulated a continuum model describing the liquid layer behavior in the vicinity of the three-phase contact line (solid-liquid-vapor).9 Starting from the Euler-Lagrange formalism, they considered an expression for the chemical potential
µ)
∂f ∂2F + Gz ∂F ∂z2
(6)
We solved eq 6 numerically under the boundary conditions F(z ) 0) ) Fs and F(zf∞) ) 0. Figure 1a,b illustrates the Maxwell construction for two different wettabilities of the solid substrate. Details regarding the numerical (kink) solutions F(z) of eq 6 are given in ref 10. From the equal-area tie-line representation, one computes the liquid thicknesses hmin and hmax (binodal points) that separate the absolutely stable films (for h < hmin and h > hmax) from the unstable/metastable films (hmin < h < hmax). The liquid-solid interaction force depends on the thickness h. The stability in very thin liquid films (h < 0.1 µm) is explained by the presence of the attractive molecular component or/and the electrical component of the disjoning pressure, acting at the solid boundary. In thick liquid layers (J0.1 µm), the influence of van der Waals forces is negligible. In this case, for large enough liquid depths, the gravitational force stabilizes the layer. (14) Korteweg, D. J. Arch. Sci. Phys. Nat. 1901, 6, 1. (15) Borcia, R.; Bestehorn, M. Phys. ReV. 2003, E67, 066307. (16) Bessonov, N.; Pojman, J. A.; Volpert, V. J. Eng. Math. 2004, 49, 321. (17) Borcia, R.; Bestehorn, M. Eur. Phys. J. 2005, B44, 101. (18) Pojman, J. A.; Whitmore, C.; Turco Liveri, M. L.; Lombardo, R.; Marszalek, J.; Parker, R.; Zoltowski, B. Langmuir 2006, 22, 2569. (19) Bestehorn, M. Hydrodynamik und Strukturbildung; Springer-Verlag: Berlin, 2006; p 347.
Figure 4. Controlled pattern formation in an unstable liquid film on a functional surface (h0 ) 13d and G ) 10-4). A functional surface is numerically simulated by alternating hydrophobic (Fs1 ) 0.2) with hydrophilic spots (Fs2 ) 0.9) in an adequate way. The patterned substrate is covered with a thin liquid film. The thin film breaks up and decodes the surface information (HI): (a) t ) 50, (b) t ) 150, (c) t ) 250, (d) t ) 400, (e) t ) 650, and (f) t ) 1400.
For a liquid thickness between hmin and hmax, a small disturbance in the system is enough to trigger pattern formation and droplet nucleation. By increasing Fs, the two equal areassabove and below the Maxwell linesbecame smaller and finally reduced to a single point for hydrophilic surfaces Fs f 1. Repeating the Maxwell construction for different wettabilities at the solid substrate, one can display the complete diagram of liquid film stability (Figure 1c). The binodal points hmin and hmax depicted in Figure 1c (as functions of Fs) separate the partial wetting area characterized by dewetting phenomena/drop formation from the complete wetting area where the films are absolutely stable. Coarsening and Controlled Pattern Formation. Now we investigate in three dimensions the breakup of a liquid film into drops for homogeneous solid supports. Furthermore, we design liquid structures on heterogeneous solid supports that are adequately patterned chemically. We solve eqs 4 and 5 numerically using a code based on a finite-difference method19 with a mesh of 200 points × 200 points × 100 points. A similar code was developed earlier for 2D phase field models describing floating liquid droplets with an applied temperature gradient.20 For the density, we impose periodic lateral boundary conditions. At the top boundary, we assume that F ) 0. The solid boundary is placed on the bottom where F ) FS. As the initial condition, we take a flat liquid layer of thickness h0 with density F ) 1. The flat liquid film is destabilized, adding to the density field a small disturbance aξ with noise amplitude (20) Borcia, R.; Bestehorn, M. Phys. ReV. 2007, E75, 056309.
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a ) 0.001 and a uniformly random distribution ξ(x, y, z) between 0 and 1. For h0 ) 16d and a homogeneous solid boundary with Fs ) 0.3, the liquid layer is unstable (Figure 1c). The thin film breaks up into small droplets that nucleate until one drop remains. The transition from film to drop is illustrated in Figure 2 through isodensity surface snapshots. Very recent experiments show that coarsening can be avoided or controlled by pinning effects on heterogeneous supports that are chemically patterned.21 After the film breakup, the dewetted droplets tend to occupy the substrate regions with higher wettability. For thin liquid films (h0 < 20d), stable regular structures (e.g., perfect squares (Figure 3f) or designed liquid structures (Figure 4f)) can be obtained as the final dewetted morphology on an “intelligent” or functional surface. A functional surface is numerically realized by alternating hydrophobic (Fs < 0.5) and hydrophilic (Fs > 0.5) spots in an adequate way. Desirable prepatterns can be obtained by manipulating the flow on the microscopic scale. By the appropriate design of chemically patterned substrates, one can effectively program microfluidic devices so that the system can perform a number of functions in an autonomous manner. The patterned surface constitutes a code. The unstable fluid flowing over the surface can be used to decode this information. Once the surface has been patterned,
Letters
no external controls (other than an imposed flow) are needed to steer the system. This is an alternative to laboratory-on-a-chip methods where liquid droplets are actively manipulated by an array of electrodes in the substrate.22,23 In conclusion, we explained the formation of 3D drops near a three-phase contact line (solid-liquid-vapor) using a phase field model. We investigated the stability of a thin liquid film on a flat homogeneous solid support with variable wettability under gravity effects. We determined the critical height below which a flat film becomes unstable on a hydrophobic/hydrophilic surface. Finally, for adequately designed solid substrates, our phase field simulations are able to model the controlled pattern formation of liquid structures. Acknowledgment. This work was partially supported by the European Space Agency under the AO-99-110 CIMEX project (Convection and Interfacial Mass Exchange) and partially by the Deutsche Forschungsgemeinschaft. LA803687V (21) Mukherjee, R.; Bandyopadhyay, D.; Sharma, A. Soft Matter 2008, 4, 2086. (22) Pollack, M. G.; Shenderov, A. D.; Fair, R. B. Lab Chip 2002, 2, 96. (23) Schwartz, J. A.; Vykoukal, J. V.; Gascoyne, P. R. C. Lab Chip 2004, 4, 11.
