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Domain relaxation in Langmuir films

Published online by Cambridge University Press:  04 January 2007

JAMES C. ALEXANDER ,
ANDREW J. BERNOFF ,
ELIZABETH K. MANN ,
J. ADIN MANN ,
JACOB R. WINTERSMITH  and
LU ZOU
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
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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.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 571 , 25 January 2007 , pp. 191 - 219
Copyright
Copyright © Cambridge University Press 2007

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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 4.7 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 1.4 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 3.5 MB