Faraday melting-freezing waves at a 4He surface


In the early part of the 19th century, Michael Faraday discovered that you can parametrically excite standing surface waves at the surface of a fluid. The easiest way to imagine how to do this, is to mount a petri dish on a loudspeaker. If you then fill the petri dish with a fluid (water or oil), and drive the loudspeaker at a frequency of 50 to 100 Hertz, say, you will at some point see that standing waves spontaneously emerge on the surface. This way of exciting surface waves is called parametric excitation, since by oscillating the fluid in the petri dish on the loudspeaker, it is basically as if gravity is periodically modulated. Such waves are often refered to as Faraday waves, and they are nowadays studied intensively in the field of nonequilibrium spontaneous pattern formation. One of their advantages of this pattern forming system is that the timescales of the patterns are very convenient experimentally.

Of course, normally crystal surfaces don't sustain capillary waves like fluid surfaces do. This is because for these to occur, the crystal would have to melt and freeze periodically while the wave is running, and the molecular processes at crystal surfaces are much too sluggish and slow to sustain such waves. Another way to think about this is to note that an enormous driving force is needed to drive the growth or melting process, and that because of the friction in the melt such waves would be highly overdamped. However, there is one exception: as Andreev predicted in 1978, crystals of Helium 4 in contact with superfluid Helium can sustain such capillary waves. In this context, they are often termed melting-freezing waves. The reason that this is possible is two-fold. First of all, the mass flow in the superfluid is frictionless. Secondly, at low temperatures a Helium surface responds very easily to a small driving force: the quantum processes make the attachment or detachment of atoms very easy, and in fact at low temperatures the ``growth resistance'' of a Helium surface results from the doppler shift of phonons that are reflected by the moving surface! This remarkable phenomenon was also predicted by Andreev, and in the eighties and nineties, several experimental groups (e.g., Andreeva et al. in the former USSR, Lipson et al. in Israel, and Balibar et al. in France) have been able to measure these waves. In fact, by measuring their dispersion and damping, one can extract both the ``growth resistance'' and the surface stiffness. Especially the experiments by Balibar and coworkers have given a wealth of data that allowed them to compare in detail with predictions from theories of the surface stiffness in terms of the properties of steps.

Since these crystallization waves resemble ordinary fluid surface waves so closely, Weeks and I wondered a few years ago whether one could not excite crystallization waves parametrically, i.e., generate Faraday crystallization waves. The calculation that we did showed that this indeed should be possible with experimentally accessible frequencies and driving amplitudes. In addition, our calculations also had a very interesting surprise. If we consider a so-called vicinal crystal surface, one which is cut under a small angle relative to the closely packed crystalline planes, the surface has steps (for ordinary crystals, these have extensively been studied with high temperature STM in the group of Frenken, formerly at AMOLF, now at the Kamerlingh Onnes Laboratory in Leiden). Such a crystal surface is highly anisotropic: height variations in the direction perpendicular to the steps correspond to a compression mode of the steps, height variations along the direction of the steps to bending modes of the steps. For this reason, Weeks and I had intuitively expected the instability threshold to be strongly anisotropic, i.e., to depend sensitively on the direction of the wavevector of the height fluctuations relative to the steps. But, at low temperatures, our analysis shows that this is not the case! The reason is that the bending mode of the steps is very stiff but weakly damped, while the compression mode is very soft, but relatively highly damped. In practice, the two effects just cancel eachother, and modes in all directions become unstable at the same time at low temperatures.

The implication of the above is that our linear theory does not predict whether above threshold, one will see square or stripe standing wave patterns (or even some other pattern). The good side of this is, that the pattern symmetry will then be determined by the nonlinearities. This opens up the possibility to infer the microscopic properties of 4He crystals by studying the nonlinear Faraday patterns!

To my knowledge, experiments on parametric excitation of crystallization waves have not been done yet. If they will be done at some point, I intend to work out the weakly nonlinear theory for the patterns above onset.

References
W. van Saarloos and J. D. Weeks, Faraday instability of crystallization waves at the 4He solid-liquid interface, Phys. Rev. Lett. 74 290 (1995).
E. Rolley, A C. Guthmann, E. Chevalier and S. Balibar, The static and dynamic properties of vicinal surfaces of 4He crystals, J. Low Temp. Phys. 99, 851 (1995).

[Pattern formation] [Wim van Saarloos] [Instituut-Lorentz]