9189
2007, 111, 9189-9192 Published on Web 07/17/2007
Switching Direction of Laterally Ordered Monolayers Induced by Transfer Instability Antonio Raudino† and Bruno Pignataro*,‡ Dipartimento di Scienze Chimiche, UniVersita` di Catania, V.le A. Doria 6-95125, Catania, Italy, and Dipartimento di Chimica Fisica, UniVersita` di Palermo, V.le delle Scienze 90128, Palermo, Italy ReceiVed: June 1, 2007; In Final Form: June 29, 2007
Langmuir-Blodgett monolayers may show nanoscopic periodic patterns parallel and/or perpendicular to the transfer direction. The experimental findings are interpreted by a nonequilibrium model based on the stability of surfactant concentration and film thickness coupled fluctuations near the meniscus of a surfactant-covered receding thin film. In the high and low transfer speed limits, periodic fluctuations of the fluid subphase thickness, respectively perpendicular and parallel to the transfer, are selected. A qualitative phase diagram shows how transfer speed and film density manage the pattern shape.
Theoretical and experimental studies showed that instability at a moving front can lead to regular patterns. These can arise in directional solidification from coupled heat flow and mass diffusion gradients,1,2 uniaxial stress,3 and lattice mismatch, as in heteroepitaxy.4 Instability at a front of a fluid has been observed in gravity-driven falling films (viscous fingering)5 or in rising films under a temperature gradient.6-8 Mostly, the regular features extend in the same direction of the moving front, with periodicity ranging from the millimeter (gravity forces)5 to micron (temperature gradients)6-8 scales. Instability phenomena arising from intermolecular forces are expected to give periodic features too, but at a nanometric scale. Elegant experiments already reported in the literature demonstrated the formation of laterally ordered nanometric features by transferring monolayers from the water/air interface to a withdrawing solid by Langmuir-Blodgett (LB).9-16 Gleiche et al.9 showed the formation of dipalmitoylphosphatidylcholine (DPPC) with hundreds of nanometers of periodic stripes perpendicular to the transfer direction. In a recent paper, the same authors showed that periodic features in the direction parallel to the transfer can be obtained by lowering the transfer speed.10 We discovered that periodic stripes of a slightly shorter phospholipid, L-Rdimyristoylphosphatidylcholine (DMPC), can be obtained in the parallel rather than perpendicular direction at relatively low transfer speed and high film pressure. Our experiments were carried out by employing high purity materials from Fluka, a Multimode/Nanoscope IIIA (Digital Instruments) scanning force microscope (SFM) working in the dynamic mode,17 and a KSV minitrough apparatus18 for LB film preparation. According to results from other groups,9,10 the formation of anisotropic systems strongly depends on the transfer speed. Figure 1 reports 2D SFM images of DMPC monolayers transferred on mica at different speeds (10 °C, 30 mN/m), showing coexisting condensed (brighter/about 0.5 nm higher) and expanded (darker/ * To whom correspondence
[email protected]. † Universita ` di Catania. ‡ Universita ` di Palermo.
should
be
addressed.
E-mail:
10.1021/jp074263s CCC: $37.00
lower) phases. At 2 mm/min (Figure 1a), these phases are randomly distributed. A speed of 7 mm/min (Figure 1b) leads to the formation of periodic bands (about 800 nm wide) separated by channels (about 200 nm) extending parallel to the transfer direction. By increasing the speed, the periodicity decreases, and the lateral order is progressively lost. At 20 mm/ min, parallel (upper part of the figure) and chaotic (lower part) structures separated by a perpendicular strip coexist, suggesting the occurrence of a transition (Figure 1c). Speeds as high as 60 mm/min give whole disordered structures. At variance with previous works showing equilibrium patterns in LB monolayers,18,19 the speed dependence indicates that, in this case, the periodic structures are induced by nonequilibrium phenomena. The transient nature of these features is further supported by the loss of periodicity after a few weeks of aging (data not shown). Up to date, no explanation about the switching of the structures’ orientation has been reported. Tentatively, the formation of parallel structures has been suggested to be related to fingering instabilities,10 even though these typically generate patterns on larger scales.5 On the other side, perpendicular patterns were treated by considering substrate-mediated condensation and meniscus oscillations.9 Hereafter, we report a model that allows one to predict the above features with a common physical base. We consider a surfactant-covered thin film (typically, a water subphase in LB) moving along the transfer direction onto a solid substrate where surfactants are irreversibly bound. The LB film is not a continuous homogeneous layer; rather, it behaves as a freely diffusing but concentrated 2D gas. Let Φ be the concentration (surface fraction) of an insoluble surfactant spread onto the subphase surface; the continuity equation imposes ∂Φ/∂t + Us‚∇sΦ ) -∇ s‚J, where Us is the velocity field of the fluid at the air-film interface, ∇s is the divergence operator, and J is the surfactants’ diffusive flux, J ) -Λ(Φ)Φ∇sµTOT, where µTOT is the surfactant chemical potential and Λ(Φ) ≈ Λo(1 - Φ) is its mobility coefficient, which decreases upon raising the number of occupied sites. Near the meniscus, the coordinates’ frame parallel to the film surface © 2007 American Chemical Society
9190 J. Phys. Chem. B, Vol. 111, No. 31, 2007
Letters
Figure 1. Pattern selection by speed transfer modulation (temperature 10 °C, surface pressures 30 mN/m) at (a) 2 mm/min (random features), (b) 7 mm/min (periodic structures), (c) 10 mm/min, and (d) 20 mm/min; Z-scale ) 3 nm.
can be rectified by introducing a new frame with the x-axis parallel to the substrate (Figure 2), ∇s ) (cosΘ∂/∂x, ∂/∂y), where Θ is the dynamic contact angle. We write the potential as
µS ) kT log
Φ - Ae-κH 1-Φ
(1)
The former term, with kT as the thermal energy, arises from the surfactant mixing entropy. The latter term accounts for the surfactant-solid interaction. The continuity equation becomes
A ∂Φ + Us‚∇sΦ ) Do∇s‚ ∇sΦ + κ(1 - Φ)Φe-κH∇sH ∂t kT (2)
(
)
where Do ) ΛokT is a diffusion coefficient. The nonlinear term (1 - Φ)Φ in eq 2 enables us to investigate concentrated monolayers. We partition Φ into a steady and a fluctuating part, Φ)Φ h (x) + φ(x,y,t), and look for a steady solution to eq 2, satisfying the proper boundary conditions: (a) far from the meniscus (x f xL), the surfactant concentration is constant
lim Φ h ) Φo
xfxL
where Φo depends on the LB surface lateral pressure; (b) at the meniscus (x ) 0), the flux of surfactants to the solid substrate is proportional to the solid/film concentration difference, JS ) K(Φ h |x)0 - ΦS), where ΦS is the surfactant concentration on the substrate and K is a transfer rate. Furthermore, ΦS ) Q/Σ, where Q is the amount of surfactants jumping from the meniscus
Figure 2. Schematic picture of a surfactant-covered thin fluid film on a solid substrate withdrawing by a rate U. The scheme includes the coordinate system. The curly arrow describes the motion field of the fluid.
and Σ is the newly formed substrate area in the time interval t what is related to the substrate transfer speed U and meniscus lateral length L by Σ ) LUt. Since Q ) LJSt, we combine the previous relationships to find another boundary condition
JS )
KU h |x)0 Φ h | ≈ UΦ K + U x)0
(3)
with K . U. A first integration of the stationary eq 2 is trivial;
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J. Phys. Chem. B, Vol. 111, No. 31, 2007 9191
Figure 3. Calculated surfactant concentration (dimensionless surface fraction) versus the distance from the meniscus (x) at (a) low speed (U f 0) and (b) high speed (10-3 m sec-1). Other parameters are A/kT ) 2.0, Φo ) 0.2, κ ) 109 m, Θ ) 25°, and D0 )10-12 m2 sec-1.
the integration constant is all but the flux JS. A second integration20 enables us to calculate the local surfactant concentration, which strongly depends on the transfer speed U, as shown in Figure 3. Curves evidence a velocity-related surfactant enrichment (small U) or depletion (high U) near the meniscus x ≈ 0; details will be published elsewhere. The surfactant concentration, however, contains two still unknown parameters, the film thickness profile H ≈ x‚tgΘ and the surface velocity Us. They are obtained by the continuity equation ∂H/∂t ) -∇‚ JTOT, where JTOT is the horizontal flux across a film section. JTOT contains two terms, (a) an influx of fluid -UH due to the motion of the film-supporting substrate and (b) an efflux J due to capillary and gravity forces. Neglecting gravity effects, playing a role only far from the meniscus, and employing the lubrication approximation, the efflux J reads J ) (H3/3η)∇‚ (γ∇2H - Π(H,Φ)) + (H2/2η)∇γ. The coefficients H3/3η and H2/2η, with η as the fluid viscosity, behave as diffusion coefficients. The first term describes the capillary force; here, γ ) γ(Φ) is the air-film interfacial energy, and ∇2H ≡ (∂2H/ ∂x2) + (∂2H/∂y2) is the curvature. The disjoining pressure Π(H,Φ) arises from the solid-film-surfactant interactions, Π(H,Φ) ) Πo(H) + Φ∂µS/∂H, where Πo(H) is the pressure in the absence of the surfactant and µS the surfactant chemical potential given by eq 1. Last, the third term is the tangential force related to surface tension gradients. The γ generally decreases with concentration, γ(Φ) ≈ γo(1 - σΦ) (where σ ≡ ∂γ/∂Φ|Φ)0), yielding, in the presence of a surface potential, to a Marangoni effect.21 Combining the above equations22
{
∂H 1 2 ) -∇‚ -UH H σγo∇Φ + ∂t 2η 1 3 H [∇‚(γ∇2H + Π(H,Φ))] (4) 3η
}
Equations 2 and 4 are the basic tools to investigate pattern formation. As for Φ, we partition H into a steady and a fluctuating part, H ) H h (x) + h(x,y,t). Steady solution to eq 4 yields the thickness H h and surface velocity Us. Near x ) 0, Us e -U, and H h ≈ x‚tgΘ. Let us expand the surfactant concentration and thickness fluctuations as φ ) σ(x)sin(qyy)exp(ωt) and h ) ξ(x)sin(qyy)exp(ωt), where σ(x) and ξ(x) are localized functions near the meniscus. Linearizing eqs 2-4 and performing a stability analysis,23 the solvability condition imposes
[
]
ω - C11 C12 C21 ω - C22 ) 0
(5)
with Cij being the complex functions of q. Solution to eq 5 gives a dispersion relationship, ω ) ω(q). It displays a fastest rate of
Figure 4. Theoretical surfactant-substrate normal pressure (P) versus the distance from the meniscus (x) calculated at (a) low speed (U f 0; parallel waves) and (b) high speed (10-3 m sec-1; perpendicular waves). Curve b has been enlarged (×3). The parameters used were as those in Figure 3.
growth ωmax which corresponds to a maximal wavelength qmax. The most likely patterns are those with q ≈ qmax, where qmax is obtained from ∂ω(q)/∂q ) 0. Limiting Case: Constant Height. When the film height is constant, the mean surfactant concentration does not change along the film surface; hence, H h (x) ) Ho, Φ h (x) ) Φo, and Θ ) 0. Assuming fast surfactant diffusion, we get, from eq 2, φ ≈ -const‚h, namely, surfactants concentrate in the valleys of the thickness fluctuations. From eq 5, the most likely pattern wavelength is
q2max ≈
1 [∂Πeff/∂Ho + BΦo(1 - Φo)] 2γ
(6)
where q2 ) q2x + q2y . The ∂Πeff/∂Ho ≡ ∂Πo/∂Ho + ΦoκAe-κHo is the average substrate-induced pressure acting on the surfactant + fluid system, and B ≡ (A/kT)κe-κHo((3/2Ho)σγo - Aκe-κHo). The last term in eq 6, vanishing at Φo f 0 and Φo f 1, measures the coupling between film thickness and surfactant concentration fluctuations; the film is unstable (positive righthand side of eq 6) for strong surface-active substances (σγo . 0) and small thicknesses. Switching the Orientation of the Meniscus Instability. When the surfactant-solid substrate energy is attractive, the surfactant concentration Φ(x) near the meniscus x ) 0 is large at small transfer speed U. This is no longer true at high U; the concentration Φ(x) near x ) 0 is small because of diffusive/ hydrodynamics effects. At small U, the surfactant-dependent pressure normal to the film is located at x ) 0 (Figure 4a). On the contrary, at high U, the pressure is small near x ) 0, reaches a maximum, and then decreases (Figure 4b). This different behavior is due to the surfactant concentration profile of Figure 3. The dramatic difference in the force profile may explain the speed-dependent patterning. At small U, the pressure is localized in a long and narrow strip along the meniscus. Here, because of the small x-averaged film height, the coupled thicknessconcentration fluctuations become unstable (eq 6, with q ≈ qy), forming small dots of greater surfactant concentration. Upon slowly raising the substrate, new instabilities arise behind the previous ones at the meniscus and, favored by surfactantsurfactant interactions, they pile up, yielding stripes of dense substrate-bound surfactants parallel to the transfer direction. The situation is different at high U. Here, the normal pressure is localized at some distance X from the meniscus because of competing attractive forces and diffusive/hydrodynamic effects (Figure 4b). The large thickness fluctuations at x ≈ X give rise to a valley behind the meniscus, which is filled by surfactants jumping from the nearby film. Upon further pulling out of the
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Letters structures are favored. Between these two regions, there is a transition zone where both parallel and perpendicular features coexist. Other factors affect the patterns. By changing the substrate chemistry (oxygen-plasma-treated silicon instead of mica) or varying the phospholipid length (DPPC instead of DMPC), one can destroy the lateral order. In conclusion, intermolecular and surface forces in LB monolayers may lead to nonequilibrium nanoscopic periodic structures. Our model explains, for the first time, how the directional features can be switched from parallel (high speed/ low pressure) to perpendicular (low speed/high pressure) with respect to the substrate motion. The above patterning strategy can be used for different molecular systems,9-16 thus allowing for several applications (surface wettability, chromatography, site-selective recognition processes, optical grids, etc.). Acknowledgment. The authors thank Professor P.-G. de Gennes for his critical reading. The Universities of Catania and Palermo (Cofinanziamento d’Ateneo) and MUR (PRIN 2006 Prot. 2006037708) are acknowledged for financial support.
Figure 5. Qualitative reconstructed pattern diagram of phospholipid LB monolayers. Representative AFM images are reported for each region (x,y scale, 4 µm). *Adapted by permission from Macmillan Publishers: Nature (ref 9), copyright 2000.9 Even if parallel and perpendicular patterns appear quite similar, at low pressure, the channels are empty, whereas at high pressure, they enclose surfactants in the expanded phase.
substrate, the valley defines the new meniscus that, in the steady condition, must stay at x ) 0. The process self-replicates during the transfer, yielding stripes perpendicular to the transfer direction. The decrease of the contact angle Θ with U in LB deposition24 further favors the formation of perpendicular stripes. In agreement with the experiments, the model leads to a picture where periodicity (q-1 max) decreases with the speed at high surfactant concentration. The concentration is a key factor managing the patterning mechanism. Both lateral pressure and temperature are related to Φ by an equation of state. Thus, by maintaining constant transfer speed and temperature (5 mm/ min, 10 °C), we prepared LB monolayers at pressures ranging from few mN/m (expanded phases) to 40 mN/m (condensed film).25 Low-pressure results in disordered layers, while at 30 mN/m expanded/condensed periodic structures, are observed. Above 40 mN/m, only homogeneous compact monolayers appear, in agreement with our model (use of eq 6 for Φo f 1 ad wetting film (∂Πo/∂Ho < 0)). The above trend has been reproduced by maintaining constant pressure (30 mN/m) and lowering the temperature from 37 (disordered layer) to 6 °C (compact layer). Figure 5 shows a qualitative phase diagram of the patterns obtained by managing LB density and transfer speed. The diagram has been reconstructed by considering both theoretical and experimental findings. SFM patterns of phospholipid monolayers are also shown. As a general feature, under certain limits, high speed and low lateral pressure drive the pattern toward periodic structures perpendicular to the transfer. In contrast, at lower speed and/or higher pressure, parallel
References and Notes (1) Politi, P.; Grenet, G.; Marty, A.; Ponchet, A.; Villain, J. Phys. Rep. 2000, 324, 271. (2) (a) Mullins, W. W.; Sekerka, R. F. J. Appl. Phys. 1963, 34, 323. (b) Mullins, W. W.; Sekerka, R. F. J. Appl. Phys. 1963, 35, 444. (3) Cantat, I.; Kassner, K.; Misbah, C.; Mu¨ller-Krumbhaar, H. Phys. ReV. E 1998, 58, 6027. (4) Rost, M.; Krug, J. Phys. ReV. Lett. 1995, 75, 3894. (5) Thiele, U.; Knobloch, E. Phys. Fluids 2003, 15, 892. (6) Kataoka, D. E.; Troian, S. M. Nature 1999, 402, 794. (7) Golovin, A. A.; Rubinstein, B. Y.; Pismen, L. M. Langmuir 2001, 17, 3930. (8) Cerro, R. L. J. Colloid Interface Sci. 2003, 257, 276. (9) Gleiche, M.; Chi, L. F.; Fuchs, H. Nature 2000, 403, 173. (10) Chen, X.; Lu, N.; Zhang, H.; Hirtz, M.; Wu, L.; Fuchs, H.; Chi, L. J. Phys. Chem. B 2006, 110, 8039. (11) Huang, J.; Tao, A. R.; Connor, S.; He, R.; Yang, P. Nano Lett. 2006, 6, 524. (12) Purrucker, O.; Fo¨rtig, A.; Lu¨dtke, K.; Jordan, R.; Tanaka, M. J. Am. Chem. Soc. 2005, 127, 1258. (13) Kovalchuk, V. I.; Bondarenko, M. P.; Zholkovskiy, E. K.; Vollhardt, D. J. Phys. Chem. B 2003, 107, 3486. (14) Eriksson, L. G. T.; Claesson, P. M.; Ohnishi, S.; Hato, M. Thin Solid Films 1997, 300, 240. (15) Mahnke, J.; Vollhardt, D.; Sto¨ckelhuber, K. W.; Meine, K.; Schulze, H. J. Langmuir 1999, 15, 8220. (16) Moraille, P.; Badia, A. Langmuir 2002, 18, 4414. (17) Pignataro, B.; Chi, L.; Gao, S.; Anczykowski, B.; Niemeyer, C.; Adler, M.; Fuchs, H. Appl. Phys. A 2002, 74, 447. (18) Pignataro, B.; Sardone, L.; Marletta, G. Langmuir 2003, 19, 5912. (19) Keller, S. L.; McConnell, H. M. Phys. ReV. Lett. 1999, 82, 1602. (20) A first integration of eq 2 leads to a Riccati equation with hypergeometric solutions. The approximation Φ(1 - Φ) ≈ Φ gives nonphysical concentrations (Φ > 1) even at moderate substrate-surfactant interactions (1-2 kT), while our solutions always satisfy Φ e 1. (21) Chauduri, M. K.; Whitesides, G. M. Science 1992, 256, 1539. (22) Equation 4 is a generalization of similar formulas already reported; for example, see: Oron, A.; Davis, S. H.; Bankoff, S. G. ReV. Mod. Phys. 1997, 69, 931. (23) Sharma, A.; Jameel, A. T. J. Colloid Interface Sci. 2003, 161, 190. (24) Petrov, J. G.; Petrov, P. G. Langmuir 1998, 14, 2490. (25) Albrecht, O.; Gruler, H.; Sackmann, E. J. Phys. (France) 1978, 39, 301.
