Hexagons: Wave Numbers & Defects

Patterns occur in a wide variety of physical systems, from giant volcanic basalt formations to nanometer-scale anodization pores. Diverse unrelated systems often share similar patterns. The challenge is to understand what common rules govern their behavior.  Experiments in fluids provide a useful model with which to study these phenomena. The underlying equations are well understood and advances in measurement techniques allow accurate measure of the dynamics of the fluid at multiple spatial points in time. However, one drawback is that it is usually difficult to control initial conditions. In our experiments in Benard-Marangoni convection we use a thermo-optical technique to impose a wide variety of initial conditions.

The experiments focus primarily on hexagonal patterns, which are the stable planform in Benard-Marangoni convection where the dominant driving mechanism is surface tension. We investigate wavenumber selection  and construct the Benard-Marangoni analogy of the Busse balloon. We also impose a single penta-hepta defect in an otherwise perfect hexagonal pattern to study defect dynamics.

As in the case of the Busse balloon for rolls, a range of stable wave numbers of hexagons can exist for given parameter values. The figures illustrate two different allowed wave numbers.

The figure below shows  the measured range of  stable wave numbers plotted with theoretical predictions by Michael Bestehorn [M. Bestehorn, Phys. Rev. E 48, 3622 (1933)]. For  epsilon < 0.5 experiment and theory show reasonable agreement. For epsilon > 0.5 the theoretically predicted boundaries shift toward larger values of k with increasing epsilon while experimentally determined boundaries exhibit little dependence on epsilon. The discrepancies between theory and experiment may be due to differences between the experiment parameters(M/R = 8 at Prandtl number of 100) and those of theory (M/R = 2.7 at infinite Prandtl number).

The penta-hepta defect (PHD) is a superposition of three rolls with well-defined wave numbers far from the defect. One of the rolls (with wave vector k1 below) is defect-free  while the other two each contain a single dislocation located at the PHD core.

The PHD is a wave number selection mechanism for hexagonal patterns.  For small (large) wave numbers the defect moves in a direction that adds (removes) two  rolls to (from) the system and hence increases (decreases) the overall wave number to a value closer to the center of the stable band. For   k1 = k2 = k3 the defect moves parallel to k1 to decrease the wave number . Figure (a) below shows the path of the PHD in the case where it is decreasing the  wave nmber of the pattern.. Circles represent the position of defect after every  six seconds.The direction of motion also depends on the relative magnitudes of the wave numbers. Figure (b) shows the trajectory after changing k2 by less than 5% from the value in Figure(a).

Note that the defect path , while relatively straightt on average, is a combination of climb and glide. In addition, the circles are not evenly spaced,  evidence of time dependence in the speed. Motion in the opposite direction where the wave number is being increased is qualitatively different - the path is smoother and there is less of a time dependence in the speed. The movies below illustrate the two different types of motion.

Cell Collapse

Mitosis
 

D. Semwogerere, M.F. Schatz, Evolution of hexagonal patterns from controlled initial conditions in a Benard-Marangoni experiment, submitted to Phys. Rev. Lett. (2001). (PDF format.)