pubs.acs.org/Langmuir © 2009 American Chemical Society
Gasous Hole Closing in a Polymer Langmuir Monolayer Lu Zou† Department of Physics, Kent State University, Kent, Ohio 44242
Andrew J. Bernoff Department of Mathematics, Harvey Mudd College, Claremont, California 91711
J. Adin Mann, Jr. Department of Chemical Engeering, Case Western Reserve University, Cleveland, Ohio 44106
James C. Alexander Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106
Elizabeth K. Mann* Department of Physics, Kent State University, Kent, Ohio 44242. †Currently at the Liquid Crystal Institute, Kent State University, Kent, OH. Received August 7, 2009. Revised Manuscript Received October 2, 2009 The hole-closing phenomenon is studied in a polymer Langmuir film with coexisting gaseous and liquid phases both as a test of hydrodynamic theories of a two-dimensional fluid embedded in a three-dimensional one and as a means to accurately determine line tension, an important parameter determining size, shape, and dynamics within these and other membrane model systems. The hole-closing curve consists of both a universal linear regime and a history-dependent nonlinear one. Improved experimental technique allows us to explore the origin of the nonlinear regime. The linear regime confirms previous theoretical work and yields a value λ=(0.69 ( 0.02) pN for the line tension of the boundary between the gaseous and liquid phases. The observed hole closing also demonstrates that the two-dimensional polymer gas must be taken as having a small, probably negligible elasticity, so that line-tension measurements assuming that both phases are incompressible should be re-evaluated.
1. Introduction The hydrodynamics of a quasi-two-dimensional interfacial fluid coupled to surrounding bulk fluids are essential to describe motion in biological membranes and membrane model systems. In treating such motion, it is generally assumed that the problem can be reduced to two-dimensional flow, parallel to the interface1-3 We recently argued4 that vertical flow, perpendicular to the interface, may be necessarily present. For example, if the surface layer consists of two fluid phases, any contrast between these two phases, in surface viscosity or compressibility, leads to flow in the surrounding fluids perpendicular to the interface.4 Perhaps the clearest example where such flow must exist is that of a quasi two-dimensional gaseous hole in a monomolecular liquid layer at the air/water interface (Langmuir layer). The minimization of boundary energy, or line tension, will favor layer motion to close the hole and minimize its boundary. However, such motion in the Langmuir layer will entrain bulk liquid toward the hole center. Since the bulk fluid is incompressible, it must
plunge away from the surface. We presented an analytic solution for this motion4 and predicted a constant hole closing rate, in the limit of zero elasticity for the two-dimensional gas at the interior of the hole and neglecting evaporation and condensation at the liquid hole edge. Spontaneous hole closing in a polymer Langmuir layer was observed qualitatively 15 years ago.5,6 Preliminary measurements of the closing rate for noncircular holes within a polymer monolayer at gas/liquid coexistence show an initial history-dependent nonlinear decrease in hole size, followed by a universal linear regime consistent with the hydrodynamic model. Comparison of the linear regime with the model gave an estimate for the line tension of the gas/liquid boundary within a factor of 2 of the line tension deduced from hydrodynamic relaxation of deformed liquid or gaseous domains toward a circular equilibrium shape. Here we present systematic studies of the hole closing rate for nearly circular holes in monolayers of poly(dimethylsiloxane) at gas/liquid coexistence. These experiments should be distinguished from the very interesting ones7,8 concerning cavitation within
*To whom correspondence should be addressed. (1) Hughes, B. D.; Pailthorpe, B. A.; White, L. R. J. Fluid Mech. 1981, 110, 349– 372. (2) Stone, H. A.; McConnell, H. M. Proc. R. Soc. London Ser. A-Math. Phys. Sci. 1995, 448, 97–111. (3) Lubensky, D. K.; Goldstein, R. E. Phys. Fluids 1996, 8, 843–854. (4) Alexander, J. C.; Bernoff, A. J.; Mann, E. K.; Mann, J. A., Jr.; Zou, L. Phys. Fluid 2006, 18, 062103.
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(5) Mann, E. K.; Henon, S.; Langevin, D. J. Phys. II 1992, 2, 1683–1704. (6) Mann, E. K. PDMS films at water surfaces - texture and dynamics, Thesis, University de Paris VI, 1992. (7) Khattari, Z.; Hatta, E.; Heinig, P.; Steffen, P.; Fischer, T. M.; Bruinsma, R. Phys. Rev. E 2002, 65, 041603. (8) Muruganathan, R. M.; Khattari, Z.; Fischer, T. M. J. Phys. Chem. B 2005, 109, 21772–21778.
Published on Web 11/03/2009
Langmuir 2010, 26(5), 3232–3236
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Figure 1. Schema of the monolayer at the air/water interface. A small gaseous hole in a much larger domain of polymer liquid, surrounded by additional gaseous monolayer.
monolayers outside of the coexistence regime, where formation of the cavity, by either laser heating or microbubble coalescence, must lead to out-of-equilibrium pressures in the surrounding liquid or solid. Improved experimental technique allows circular holes in accord with the model, more precise results, and an exploration of the origin of the nonlinear, history-dependent regime.9 The linear regime gives us an accurate measure of the line tension in this system. The line tension is a critical parameter in determining domain size, shape, and dynamics in, for example, biological membranes10 and their model systems such as vesicles,11-13 supported mono- and bilayers,14 and Langmuir layers.7,15
2. Theory Recently, based on previous hydrodynamic models, Alexander et al. developed a model, the Inviscid Langmuir Layer Stokesian Subfluid (ILLSS) model,4,16 for two coexisting two-dimensional fluids within a layer coupled to a Stokesian fluid bulk substrate, in the limit where viscous energy loss is dominated by the substrate. This is the case for many fluid monolayers.4,16 Domination of surface viscosities by substrate viscosity was explicitly demonstrated for the PDMS system used here.17 This ILLSS model was applied to a circular gaseous hole in the middle of a large but finite circular liquid domain embedded in a gaseous outer region. An infinite and flat surface, containing gaseous and liquid phases, sits on top of an incompressible and infinitely deep subfluid (Figure 1). The liquid surface phase is considered incompressible with a Gibbs elasticity Kg f ¥, with a vanishing elasticity Kg f 0 for the gaseous phase. The Gibbs elasticity is defined analogously to the usual three-dimensional one by Kg Γ
dΠ dΓ
ð1Þ
with Γ the density of the surface layer and Π its surface pressure. (9) Lee, L. T.; Mann, E. K.; Langevin, D. L.; Meunier, J. Langmuir 1991, 7, 3076. (10) Simons, K.; Ikonen, E. Nature 1997, 387, 569–572. (11) Baumgart, T.; Hess, S. T.; Webb, W. W. Nature 2003, 425, 821–824. (12) Amar, M. B.; Allain, J.-M.; Puff, N.; Angelova, M. I. Phys. Rev. Lett. 2007, 99, 044503. (13) Honerkamp-Smith, A. R.; Cicuta, P.; Collins, M. D.; Veatch, S. L.; Den Nijs, M.; Schick, M.; Keller, S. L. Biophys. J. 2008, 95, 236–246. (14) Goksu, E. I.; Vanegas, J. M.; Blanchette, C. D.; Lin, W.-C.; Longo, M. L. Biochim. Biophys. Acta-Biomembr. 2009, 1788, 254–266. (15) Benvegnu, D. J.; McConnell, H. M. J. Phys. Chem. 1992, 96, 6820–6824. (16) Alexander, J. C.; Bernoff, A. J.; Mann, E. K.; Mann, J. A., Jr.; Pugh, J. M.; Zou, L. J. Fluid Mech. 2007, 571, 191–219. (17) Mann, E. K.; Henon, S.; Langevin, D.; Meunier, J.; Leger, L. Phys. Rev. E 1995, 51, 5708–5720.
Langmuir 2010, 26(5), 3232–3236
Figure 2. The surface pressure isotherm graph of PDMS monolayer at the air-water interface. All the measurements were performed when the area per monomer was between 0.38 and 0.46 nm2 corresponding to the two arrows, well within the gas-liquid coexistence regime.
Most liquid Langmuir monolayers can be considered incompressible. However, the situation is less clear for gas monolayers than for three-dimensional gases. In flow driven by line tension, the surface phase may be considered incompressible if the Gibbs elasticity Kg . λ/L, where λ is line tension and L is the characteristic size of the domain. The compressibility of the polymer gas monolayer is likely to be particularly low, because of its low molecular number density. The polymer considered here is poly(dimethylsiloxane), also known as PDMS, which can form a gaseous and liquid coexisting layer at the air-water interface. The viscosity of both surface phases is indeed negligible compared to the bulk viscosity.17 For the PDMS liquid phase, the Gibbs elasticity has been directly measured as Kg =40 mN/m.18 Generally, the accessible range of domain size is 2 μm < L < 5 mm, depending on the experimental setup. The line tension λ of the boundary between the twodimensional liquid and the gaseous phases was deduced17 as λ= (1.1 ( 0.3) 10-12 N with a model assuming both phases are incompressible, or about twice that from preliminary data for hole-closing rates in the linear regime, assuming that the polymer gas has negligible elasticity. Therefore, independent of assumptions about the elasticity of the gaseous phase, λ/L < 10 -3 mN/ m , Kg =40 mN/m. The PDMS liquid phase can be treated as incompressible. The Gibbs elasticity for the PDMS gaseous phase on the water surface can not be directly determined from the surface pressure vs molecular area (see Figure 2), since here both the directly measured elasticity and the surface pressure are too small to distinguish from zero (i.e., Π < 2 10-2 mN/m18). Data from lower molecular weights,19 where the surface pressure is measurable, extrapolates to a much smaller surface pressure, Π , 5 10-4 mN/m. We can attempt to estimate the elasticity assuming an ideal gas, for which the Gibbs elasticity is just the surface pressure. Since the formation of a liquid phase indicates attractive interactions, the Gibbs elasticity of the real gaseous phase will be even less. Thus Kg , 510 -4 mN/m. From the discussion above, λ/L is between 10-7 and 10-3 mN/m. Therefore, the gaseous phase can not be considered incompressible, and is much more likely to behave in the opposite limit of zero elasticity. (18) Mann, E. K.; Langevin, D. Langmuir 1995, 7, 1112. (19) Granick, S. Macromolecules 1985, 18, 1597–1602.
DOI: 10.1021/la902939e
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A gaseous hole created in the liquid layer is a nonequilibrium initial state. For mechanical equilibrium, it should reduce in size until the pressure inside the hole is larger than the pressure in the liquid phase outside of it, with the pressure difference given by the line tension λ and the radius of curvature R through the Laplace equation, ΔΠ = λ/R, just as it would be in a three-dimensional balloon or an air bubble in water. Here, ΔΠ = Πi - Πo, is the surface pressure jump over the hole boundary. As the hole starts closing, the surface phases drag the subfluid toward the hole center and cause a three-dimensional flow inside the subfluid, which must include both horizontal and vertical components. The hydrodynamics of the two fluid surface phases and the bulk subfluid must both be considered. Thermodynamic equilibrium, as the pressure in the hole rises, should also be considered, but in the limit of zero elasticity, the internal pressure will remain unchanged for a considerable portion of the collapse process. This point will be discussed further in the Conclusion and results section. From the assumptions discussed above and the coupling between the surface phases and the bulk subfluid, the ILLSS model provides an explicit expression of the hole-closing rate At λ 1 At ¼ - 2 0 1 η λ
rffiffiffiffi ! A ΔΠ π
ð2Þ
Here, η0 is the viscosity of the subfluid, A=A(t) is the area of the hole and ΔΠ=Πi - Π¥ is the difference of the surface pressures in the hole and at ¥. Note that this equation predicts that the hole closing rate vanishes if the pressure in the interior of the hole is large enough to act against the elastic energy of the boundary, as given by the two-dimensional Laplace equation ΔΠ=λ/R. If ΔΠ is small enough that ΔΠ , λ(π/A)1/2, then At ¼ - 2
λ η0
ð3Þ
which suggests that the area of the hole decreases linearly and provides a new way to measure the line tension.
3. Experimental Setup Poly(dimethylsiloxane), i.e., (OSi(CH3)2)N (Mw = 31 600 and polydispersity Mw/MN=1.06; N=427), was used as received from Polymer Source Inc. in a hexane (OPTIMA, Fisher) spreading solvent. The solution was deposited on a pure water surface in a Langmuir trough (mini-trough, KSV). An ultrathin PDMS liquid monolayer, ∼0.6 nm thick, partially covers the water surface, coexisting with a PDMS gaseous phase. The gas-liquid coexisting surface is observed with a Brewster angle microscope and videos are captured by a CCD camera. In this work, the image quality was improved by substituting an argon laser (Innova 70C, 488 nm, 400 mw, Coherence Inc.) for the previous semiconductor laser. Further details about the PDMS monolayer preparation and the microscope system can be found in ref 4. A platinum wire (diameter=0.13 mm) is cleaned with chloroform and with a KOH/ethanol solution, then rinsed with pure water. In the initial experiments, the wire was mounted on a wire holder and manually manipulated to touch the PDMS monolayer vertically and removed carefully. After some trials, nearly circular gaseous holes were created by moving the platinum wire vertically and smoothly. In preliminary work,4 holes were not circular as assumed by the model and did not relax to the circular shape before the holes closed. There were several uncontrolled factors in the manipulation, such as the speed of the wire when its tip touches and leaves the monolayer, how long the tip stays under the surface and how deep the tip goes. We will see that some of 3234 DOI: 10.1021/la902939e
Figure 3. Platinum wire mounted on a dipper controlled by a PC.
Figure 4. Area of a hole in a PDMS liquid monolayer as a function of time, for 12 circular or near circular domains. The solid symbols are data from manually made holes and the hollow ones are from dipper-made holes. The time for each curve is shifted by Δt so that the curves overlap over the widest range of areas for A e 0.02 mm2. The inset expands the region A e 0.02 mm2 on a linear scale. The slope of the solid fitting line is (-1.30 ( 0.03) 10-3 mm2/s. these parameters effect the initial size of holes and the hole-closing process. In order to obtain a quantitative control on these unknown factors, the wire was mounted on a dipper (M1006, KSV) well-controlled by a computer, as shown in Figure 3; it could move vertically at speeds between 1 and 40 mm/min, with a depth precision of 0.1 mm. The Reynolds number associated with the tip insertion is
