Energy-driven pattern formation in planar dipole-dipole systems in the presence of weak noise

We study pattern formation in planar fluid systems driven by intermolecular cohesion (which manifests as a line tension) and dipole-dipole repulsion, which are observed in physical systems including ferrofluids in Hele-Shaw cells and Langmuir layers. When the dipolar repulsion is sufficiently strong, domains undergo forked branching reminiscent of viscous fingering. A known difficulty with these models is that the energy associated with dipole-dipole interactions is singular at small distances. Following previous work, we demonstrate how to ameliorate this singularity and show that in the macroscopic limit only the scale of the microscopic details relative to the macroscopic extent of a system is relevant and develop an expression for the system energy that depends only on a generalized line tension $\Lambda$ that in turn depends logarithmically on that scale. We conduct numerical studies that use energy minimization to find equilibrium states. Following the subcritical bifurcations from the circle, we find a few highly symmetric stable shapes, but nothing that resembles the observed diversity of experimental and dynamically simulated domains. The application of a weak random background to the energy landscape stabilizes a wide range of domain morphologies recovering the diversity observed experimentally. With this technique, we generate a large sample of qualitatively realistic shapes and use them to create an empirical model for extracting $\Lambda$ with high accuracy using only a shape's perimeter and morphology.

Figure 1

Examples of two-dimensional dipole-mediated systems in experiments. (a) and (b) Ferrofluid enclosed in a Hele-Shaw cell. (Images provided by D. P. Jackson [10, 11, 12].) (c) and (d) Octylcyanobiphenyl Langmuir films, or monolayers of polymer molecules, condensed into their fluid phase. (Images provided by E. K. Mann [21].) (e) and (f) Results of our numeric simulations.

Figure 2

Shown on the bottom is a plot of the perimeters of the first five harmonic bifurcations from a circular domain. The black dots represent the theoretical bifurcation points ${\Lambda }_n$, the solid lines denote stable numeric solutions, and the dashed lines denote unstable numeric solutions. The top shows a circular domain alongside those bifurcations. These shapes were taken with $\Lambda$ values of (a) $-1.2$, (b) $-1.38$, (c) $-1.52$, (d) $-1.65$, (e) $-1.69$, and (f) $-1.77$.

Figure 3

Subcritical bifurcation of the dogbone (stripe) from the circle. The solid lines denote stable numeric solutions and the dashed lines denote unstable ones. The subcritical branch of solutions regains stability in a fold bifurcation at $\Lambda \simeq -1.227$.

Figure 4

Representatives of (a) stripe, (b) forked, and (c) doubly forked domain morphologies at $\Lambda =-2$. These appear to be the only stable morphologies in the range of $\Lambda$ we have investigated in the absence of a random energy background.

Figure 5

Asymptotic behavior of the perimeter of the three stable domain morphologies for $N=8196$. (a) Perimeter of each morphology as a function of $\Lambda$. (b) Relative error between the perimeter of each morphology and $L_{\mathrm{rec}}$, the asymptotic rectangle perimeter.

Figure 6

Ratios of the rectangle width $w_{\mathrm{rec}}$ to linear combinations of the stable perimeters that correspond, in the limit of negative $\Lambda$, to the constants (a) $c$ and (b) $m$.

Figure 7

Example of two branches in a more complex branching domain.

Figure 8

Sampling of stable solutions to our numeric model over random external potentials. Traveling down the vertical axis corresponds to decreasing $\Lambda$ and traveling to the right on the horizontal axis corresponds to increasing background intensity.

Figure 9

(a) Perimeter $L$ of stable domains as a function of $\Lambda$. The color of each point denotes the value of ${log}_{10}\left(a_0\right)$, the order of magnitude of the random background, as detailed by the legend. The solid black line is a plot of $L_{\mathrm{rec}}\left(\Lambda \right)$. (b) Relative error of each perimeter from $L_{\mathrm{rec}}$.

Figure 10

Average number of junctions in numeric domains as a function $\Lambda$. The error bars denote standard error.

Figure 11

Difference between the generating value $\Lambda$ and the mean predicted value ${\Lambda }^{\prime }$ for sets of 50 domains. The error bars denote standard error.