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Domain relaxation in Langmuir films

Published online by Cambridge University Press:  04 January 2007

JAMES C. ALEXANDER
Affiliation:
Department of Mathematics, Case Western Reserve University, Cleveland, OH 44106, USA
ANDREW J. BERNOFF
Affiliation:
Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA
ELIZABETH K. MANN
Affiliation:
Department of Physics, Kent State University, Kent, OH 44242, USA
J. ADIN MANN
Affiliation:
Department of Chemical Engineering, Case Western Reserve University, Cleveland, OH 44106, USA
JACOB R. WINTERSMITH
Affiliation:
Department of Physics, Harvey Mudd College, Claremont, CA 91711, USA
LU ZOU
Affiliation:
Department of Physics, Kent State University, Kent, OH 44242, USA

Abstract

We report on theoretical studies of molecularly thin Langmuir films on the surface of a quiescent subfluid and qualitatively compare the results to both new and previous experiments. The film covers the entire fluid surface, but domains of different phases are observed. In the absence of external forcing, the compact domains tend to relax to circles, driven by a line tension at the phase boundaries. When stretched (by a transient applied stagnation-point flow or by stirring), a compact domain elongates, creating a bola consisting of two roughly circular reservoirs connected by a thin tether. This shape will then relax slowly to the minimum-energy configuration of a circular domain. The tether is never observed to rupture, even when it is more than a hundred times as long as it is wide. We model these experiments by taking previous descriptions of the full hydrodynamics, identifying the dominant effects via dimensional analysis, and reducing the system to a more tractable form. The result is a free boundary problem for an inviscid Langmuir film whose motion is driven by the line tension of the domain and damped by the viscosity of the subfluid. Using this model we derive relaxation rates for perturbations of a uniform strip and a circular patch. We also derive a boundary integral formulation which allows an efficient numerical solution of the problem. Numerically this model replicates the formation of a bola and the subsequent relaxation observed in the experiments. Finally, we suggest physical properties of the system (such as line tension) that can be deduced by comparison of the theory and numerical simulations to the experiment. Two movies are available with the online version of the paper.

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Papers
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Adamson, A. W. & Gast, A. P. 1998 Physical Chemistry of Surfaces, 6th Edn. John Wiley and Sons.Google Scholar
Almgren, R. 1996 Singularity formation in Hele-Shaw bubbles. Phys. Fluids 8, 344352.CrossRefGoogle Scholar
Almgren, R., Bertozzi, A. & Brenner, M. P. 1996 Stable and unstable singularities in the unforced Hele-Shaw cell. Phys. Fluids 8, 13561370.CrossRefGoogle Scholar
Aris, R. 1990 Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Dover.Google Scholar
Benvegnu, D. J. & McConnell, H. M. 1992 Line tension between liquid domains in lipid monolayers. J. Phys. Chem. 96, 68206824.CrossRefGoogle Scholar
Brochard-Wyart, F. 1990 Stability of a liquid ribbon spread on a liquid surface. C. R. Acad. Sci Ii 311, 295300.Google Scholar
Constantin, P., Dupont, T. F., Goldstein, R. E., Kadanoff, L. P., Shelley, M. J. & Zhou, S.-M. 1993 Droplet breakup in a model of the Hele-Shaw cell. Phys. Rev. E 47, 41694181.CrossRefGoogle Scholar
DeKoker, R. & McConnell, H. M. 1993 Circle to dogbone—shapes and shape transitions of lipid monolayer domains. J. Phys. Chem. 97, 1341913424.CrossRefGoogle Scholar
DeKoker, R. & McConnell, H. M. 1996 Stripe phase hydrodynamics in lipid monolayers. J. Phys. Chem. 100, 77227728.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability, 2nd Edn. Cambridge University Press.CrossRefGoogle Scholar
Dupont, T. F., Goldstein, R. E., Kadanoff, L. P. & Zhou, S.-M. 1993 Finite-time singularity formation in Hele-Shaw systems. Phys. Rev. E 47, 41824196.CrossRefGoogle ScholarPubMed
Edidin, M. 2003 The state of lipid rafts: From model membranes to cells. Annu. Rev. Biophys. Biomol. Structure 32, 257283.CrossRefGoogle ScholarPubMed
Gaines, G. L. Jr 1966 Insoluble Monolayers at Liquid-Gas Interfaces. Interscience.Google Scholar
Glasner, K. 2003 A diffuse interface approach to Hele-Shaw flow. Nonlinearity 16, 4966.CrossRefGoogle Scholar
Goldstein, R. E., Pesci, A. I. & Shelley, M. J. 1993 Topology transitions and singularities in viscous flows. Phys. Rev. Lett. 70, 30433046.CrossRefGoogle ScholarPubMed
Goodrich, F. C. 1981 The theory of capillary excess viscosities. Proc. R. Soc. Lond. A 374, 341370.CrossRefGoogle Scholar
Gottlieb, D. & Orszag, S. A. 1977 Numerical Analysis of Spectral Methods: Theory and Applications. Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Heinig, P., Helseth, L. E. & Fischer, T. M. 2004 Relaxation of patterns in 2D modulated phases. New J. Phys. 6, 189.CrossRefGoogle Scholar
Hou, T., Lowengrub, J. & Shelley, M. 1994 Removing the stiffness from interfacial flow with surface tension. J. Comput. Phys. 114, 312338.CrossRefGoogle Scholar
Hughes, B. D., Pailthorpe, B. A. & White, L. R. 1981 The translational and rotational drag on a cylinder moving in a membrane. J. Fluid Mech. 110, 349372.CrossRefGoogle Scholar
Joly, M. 1972 Rheological properties of monomolecular films: Part ii: Experimental results. theoretical interpretation. applications. Surface Colloid Sci. 5, 79193.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Lauger, J., Robertson, C. R., Frank, C. W. & Fuller, G. G. 1996 Deformation and relaxation processes of mono- and bilayer domains of liquid crystalline langmuir films on water. Langmuir 12 (23), 56305635.CrossRefGoogle Scholar
Lubensky, D. K. & Goldstein, R. E. 1996 Hydrodynamics of monolayer domains at the air-water interface. Phys. Fluids 8, 843854.CrossRefGoogle Scholar
Lucassen, J., Akamatsu, S. & Rondelez, F. 1991 Formation, evolution, and rheology of 2-dimensional foams in spread monolayers at the air-water interface. J. Colloid Interface Sc. 144, 434448.CrossRefGoogle Scholar
Mann, E. K. 1992 PDMS films at water surfaces: texture and dynamics. PhD thesis, Université de Paris VI.Google Scholar
Mann, E. K., Hénon, S. & Langevin, D. 1992 Meunier molecular layers of a polymer at the free water surface: Microscopy at the Brewster angle. J. Phys. Paris 2, 16831704.Google Scholar
Mann, E. K., Hénon, S., Langevin, D., Meunier, J. & Léger, L. 1995 The hydrodynamics of domain relaxation in a polymer monolayer. Phys. Rev. E 51, 57085720.CrossRefGoogle Scholar
Mann, E. K. & Primak, S. 1999 The stability of two-dimensional foams in Langmuir monolayers. Phys. Rev. Lett. 83, 53975400.CrossRefGoogle Scholar
Mann, J. A. Jr 1985 Dynamics, structure and function of interfacial regions. Langmuir 1, 1023.CrossRefGoogle Scholar
Mann, J. A. Jr, Crouser, P. D. & Meyer, W. V. 2001 Surface fluctuation spectroscopy by surface-light-scattering spectroscopy. Appl. Optics 40 (24), 40924112.CrossRefGoogle ScholarPubMed
Mayor, S. & Rao, M. 2004 Rafts: Scale-dependent, active lipid organization at the cell surface. Traffic 5, 231240.CrossRefGoogle ScholarPubMed
de Mul, M. N. G. & Mann, J. A. Jr 1998 Determination of the thickness and optical properties of a langmuir film from the domain morphology by Brewster angle microscopy. Langmuir 14, 24552466.CrossRefGoogle Scholar
Parton, R. G. & Hancock, J. F. 2004 Lipid rafts and plasma membrane microorganization: Insights from Ras. Trends Cell Biol. 14 (3), 141147.CrossRefGoogle ScholarPubMed
Powers, T. R., Huber, G. & Goldstein, R. E. 1990 Fluid-membrane tethers: Minimal surfaces and elastic boundary layers. Phys. Rev. E 65 (4), 041901.CrossRefGoogle ScholarPubMed
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.CrossRefGoogle Scholar
Pugh, J. M. 2006 Numerical simulation of domain relaxation in Langmuir films. Senior Thesis, Department of Physics, Harvey Mudd College.Google Scholar
Simons, K. & Ikonen, E. 1997 Functional rafts in cell membranes. Nature 387, 569572.CrossRefGoogle ScholarPubMed
Steffen, P., Wurlitzer, S. & Fischer, T. M. 2001 Hydrodynamics of shape relaxation in viscous langmuir monolayer domains. J. Phys. Chem. A 105 (36), 82818283.CrossRefGoogle Scholar
Stone, H. A. & McConnell, H. M. 1995 Hydrodynamics of quantized shape transitions of lipid domains. Proc. R. Soc. Lond. A 448, 97111.CrossRefGoogle Scholar
Trefethen, L. N. 2000 Spectral Methods in MATLAB. Society for Industrial and Applied Mathematics (SIAM).CrossRefGoogle Scholar
Tryggvason, G. & Aref, H. 1983 Numerical experiments on Hele–Shaw flow with a sharp interface. J. Fluid Mech. 136, 130.CrossRefGoogle Scholar
Wurlitzer, S., Steffen, P. & Fischer, T. M. 2000a Line tension in langmuir monolayers probed by point forces. J. Chem. Phys. 112, 59155918.CrossRefGoogle Scholar
Wurlitzer, S., Steffen, P., Wurlitzer, M., Khattari, Z. & Fischer, T. M. 2000b Line tension in langmuir monolayers probed by point forces. J. Chem. Phys. 113, 38223828.CrossRefGoogle Scholar
Zou, L., Basnet, P., Wang, J., Kooijman, E. E. & Mann, E. K. 2006 Line tension in smectic liquid crystal Langmuir multilayers. Preprint.Google Scholar

Alexander et al. supplementary movie

Movie 1. Brewster Angle Microscopy images of a relaxing Langmuir layer. This movie shows a bola relaxing to a circular Langmuir domain. The brighter domains consist of about 5 layers of 8CB (Octylcyanobiphenyl), while the dark background consists of 3 layers of 8CB. First the fluid is sheared distorting the domain to a bola with a thin tether; this bola then slowly relaxes back to a circular shape. This is an 8 second movie of the relaxation of a bola. The image is approximately 4mm by 5mm. Note that the image is distorted as it is filmed at the Brewster Angle (approximately 53 degrees)

Download Alexander et al. supplementary movie(Video)
Video 5 MB

Alexander et al. supplementary movie

Movie 1. Brewster Angle Microscopy images of a relaxing Langmuir layer. This movie shows a bola relaxing to a circular Langmuir domain. The brighter domains consist of about 5 layers of 8CB (Octylcyanobiphenyl), while the dark background consists of 3 layers of 8CB. First the fluid is sheared distorting the domain to a bola with a thin tether; this bola then slowly relaxes back to a circular shape. This is an 8 second movie of the relaxation of a bola. The image is approximately 4mm by 5mm. Note that the image is distorted as it is filmed at the Brewster Angle (approximately 53 degrees)

Download Alexander et al. supplementary movie(Video)
Video 1 MB

Alexander et al. supplementary movie

Movie 2. A numerical evolution of the Inviscid Langmuir Layer Stokesian Subfluid Model computed via a boundary integral method. The domain is originally a circle of radius 3. The domain is subject to a straining flow for 5 units of time and is allowed to relax for approximately 40 units of time. It gets stretched out to a length of 60. After the straining field is released, the domain assumes the classic bola shape, and eventually relaxes back to an ellipse approaching the energy-minimizing circular configuration. There are 32 frames per unit of time in the motion picture.

Download Alexander et al. supplementary movie(Video)
Video 26 MB

Alexander et al. supplementary movie

Movie 2. A numerical evolution of the Inviscid Langmuir Layer Stokesian Subfluid Model computed via a boundary integral method. The domain is originally a circle of radius 3. The domain is subject to a straining flow for 5 units of time and is allowed to relax for approximately 40 units of time. It gets stretched out to a length of 60. After the straining field is released, the domain assumes the classic bola shape, and eventually relaxes back to an ellipse approaching the energy-minimizing circular configuration. There are 32 frames per unit of time in the motion picture.

Download Alexander et al. supplementary movie(Video)
Video 3 MB
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