Determination of interphase line tension in Langmuir films

A Langmuir film is a molecularly thin film on the surface of a fluid; we study the evolution of a Langmuir film with two coexisting fluid phases driven by an interphase line tension and damped by the viscous drag of the underlying subfluid. Experimentally, we study a $4^{\prime }$-8-alkyl[1,$1^{\prime }$-biphenyl]-4-carbonitrile (8CB) Langmuir film via digitally imaged Brewster angle microscopy in a four-roll mill setup which applies a transient strain and images the response. When a compact domain is stretched by the imposed strain, it first assumes a bola shape with two tear-drop shaped reservoirs connected by a thin tether which then slowly relaxes to a circular domain which minimizes the interfacial energy of the system. We process the digital images of the experiment to extract the domain shapes. We then use one of these shapes as an initial condition for the numerical solution of a boundary-integral model of the underlying hydrodynamics and compare the subsequent images of the experiment to the numerical simulation. The numerical evolutions first verify that our hydrodynamical model can reproduce the observed dynamics. They also allow us to deduce the magnitude of the line tension in the system, often to within 1%. We find line tensions in the range of $200–600\mathrm{pN}$; we hypothesize that this variation is due to differences in the layer depths of the 8CB fluid phases.

Figure 1

Schematic of the four-roll-mill flow profile. The geometry of the rollers, $a=6.6\mathrm{mm}$ and $b=10.5\mathrm{mm}$, produces maximum homogeneity in the extension rate [18, 19, 20]. The shaded ellipse shows the area illuminated by the laser beam.

Figure 2

(Color online) A series of snapshots (for data set B) showing the relaxation of a stretched domain to a circle. The snapshots are separated by $2.85\mathrm{s}\approx 0.558T_{*}$. The first photo is superimposed with a curve marking the boundary determined by our edge-finding algorithm; this is the initial condition for the simulation of the domain relaxation. In subsequent photos the superimposed curve shows the simulated shape of the bola, based on that initial condition. The simulation accurately reproduces the observed dynamics even for complex, asymmetrical domains. The line tension of this domain is found to be $390\pm 3\mathrm{pN}$, where the uncertainty corresponds to the standard deviation of $T_{*}$.

Figure 3

(Color online) Three images showing the same photo overlayed with simulated boundary curves from three different times in the simulation. The center image shows the simulated curve at the time, $T_{\mathit{\text{best}}}$, when it most closely matches the photo; the left and right images show the simulated curve at $T=T_{\mathit{\text{best}}}-0.1T_{*}$ and $T=T_{\mathit{\text{best}}}+0.1T_{*}$, respectively.