Reorientation of a hexagonal pattern under broken symmetry: The hexagon flip

An unexpected pattern transition has been found experimentally in the transformation from hexagons to stripes caused by an applied anisotropy effect. The particular system studied is the surface instability of a horizontal layer of magnetic liquid in a tilted magnetic field. Two orthogonal Helmholtz pairs of coils provide a vertical and a tangential magnetic field. Whereas the vertical component destabilizes the flat layer, the tangential one preserves its stability. The ensuing surface patterns comprise regular hexagons, anisotropic hexagons, and stripelike ridges. The phase diagram for the tilted field instability is measured using a radioscopic technique. The investigation reveals an interesting effect: the flip from one hexagonal pattern to another under an increasing tangential field component, which is explained in terms of amplitude equations as a saddle-node bifurcation.

Figure 1

Surface protuberances on magnetic fluid, where the view is directed along the $y$ axis. Hexagonal pattern for the vertical induction $B_z=19.4\mathrm{mT}$ (a). Increasing the horizontal component from zero to $B_x=7.7\mathrm{mT}$ results in stretched hexagons (b) and finally at $B_z=22.4\mathrm{mT}$ in ridges (c).

Figure 2

Scheme of the experimental setup with two Helmholtz pairs of coils for the vertical and horizontal components of the induction, $B_z$ and $B_x$, respectively.

Figure 3

(Color) Series of surface profiles recorded by the radioscopic technique. The surface elevation is color coded by hue, saturation, value (HSV). H is increased proportional to the height from zero to 360 degree, whereas S and V were fixed to 100%. The horizontal component $B_x$ of the induction is oriented towards the right. Underneath, the surface patterns are shown in Fourier space. The vertical induction is fixed to $B_z=18.6\mathrm{mT}$. The horizontal induction $B_x$ increases from (a) $3.0\mathrm{mT}$, via (b) $3.8\mathrm{mT}$, (c) $5.3\mathrm{mT}$, (d) $5.8\mathrm{mT}$, (e) $6.4\mathrm{mT}$, to (f) $7.7\mathrm{mT}$.

Figure 4

The transversal $\left(T\right)$ and longitudinal $\left(L\right)$ modes and the other contributing modes before $\left(C\right)$ and after $\left(D\right)$ the hexagon flip.

Figure 5

The measured amplitudes $C$ (black solid circles), $T$ (black open squares), $L$ (gray open squares), and $D$ (gray solid circles) for increasing $B_x$.

Figure 6

Evolution of the amplitudes $L$ and $D$ under an increase of $B_x$ for hexagons already properly aligned at $B_x=0\mathrm{mT}$.

Figure 7

Phase diagram for the transition to liquid ridges and to hexagons upon increase (solid line) and decrease (dotted line) of $B_z$ in steps of $75\mu T$ for 60 different values of $B_x$. The open circles are indicating the hexagon flip for increasing $B_x$ and fixed $B_z$. To guide the eyes the latter are connected by lines (dashed line).

Figure 8

The calculated amplitudes of the selected modes vs the increasing horizontal induction (cf. Fig. 5). The dashed lines indicate the unstable branches.