Domain Relaxation in Langmuir Films

James C. Alexander
Case Western Reserve University
Andrew J. Bernoff
Harvey Mudd College
Elizabeth K. Mann
Kent State University
Jacob Wintersmith and the BAM
Jacob Wintersmith in Elizabeth Mann's Lab in the
Physics Department at Kent State. The Brewster
Angle Microscope (BAM) used for our experiments
is in the background.
J. Adin Mann, Jr
Chemical Engineering
Case Western Reserve University
Jacob R. Wintersmith
Physics '06
Harvey Mudd College
Lu Zou
Kent State University


We report on both experimental and theoretical studies of  molecularly-thin Langmuir films on the surface of a quiescent subfluid. The film covers the entire fluid surface, but several 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, primarily that of Stone & McConnell (1995) and Lubensky & Goldstein (1996), identifying the dominant effects via dimensional analysis, and reducing the system to a more tractable form. The result is a free boundary problem where 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.

Link to Article:  Domain Relaxation in Langmuir Films in the Journal of Fluid Mechanics, Volume 571, January 2007, pp 191- 219.

Experimental Results for Bola Formation and Relaxation

Below we show a series of Brewster Angle Microscopy photos (by Lu Zou) from  Prof. Elizabeth Mann's laboratory at Kent State University.

Figure 1: Relaxation of a bola. The brighter domains consist of about 5 layers of 8CB (Octylcyanobiphenyl), while the dark background consists of 3 layers of 8CB. A shear field was established which distorted the domain to a bola with a thin tether. The shear field was then shut off and the domain allowed to relax as shown in the series of images. The time interval is 0.5 seconds. The scale bar in the right frame spans 0.2 mm.

Picture of a Relaxing Bola (Link to MPEG Video)
Figure 2: This is an 8 second movie of the relaxation of a bola (click on the picture). The image is approximately 4mm by 5mm. Note that the image is distorted as it is filmed at the Brewster Angle (approximately 53 degrees).

Numerical Simulation of Bola Formation and Relaxation with a Boundary Integral Method

These numerical simulations were developed by Jacob Wintersmith under the supervision of Prof. Andrew Bernoff at Harvey Mudd College. They use a boundary integral method for a hydrostatic Langmuir layer driven by a line tension at the boundary and damped by a Stokesian subfluid. Details can be found in the preprint above.

Boundary Integral Simulation of Bola Relaxation Relaxation of a bola

Figure 3: A series of snapshots of 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.  It gets stretched out to a length of 60. The stretching snapshots are taken every 0.16 time units, and the relaxation snapshots are showed every 13.1 time units.. 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.
Boundary Integral Simulation (click for MPEG Video)
Figure 4: This is a movie of the numerical simulation of the straining and subsequent relaxation of the bola depicted in Figure 3 (click on the picture above to see the film). The domain is subject to a straining flow for 5 units of time and is allowed to relax for approximately 40 units of time. There are 32 frames per unit of time in the motion picture.

Image Processing to Determine the Line Tension

We can determine the line tension from experimental observations by digitizing the experimental images to determine the initial location of the boundary, evolving the boundary forward with the boundary integral code and then comparing the numerical results to the experiemntal observations. The line tension determines how fast the Langmuir domain relaxes;  by minimizing the error between the experimental results and the numerical evolution we can determine the line tension within a few percent. The experiments were conducted by Lu Zou and the image processing and numerics were produced by Jacob Wintersmith

Frame 1 Frame 2
Frame 3 frame 4

Below is a motion picture showing a comparison between the numerics and experiments.

Experiments vs. Numerics

These results have now appeared in an article:

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