TRANSITION FROM CURVED TO ANGULAR TEXTURES
Convection
in a binary mixture is characterized by traveling-wave convection, but if the
Rayleigh number is increased above a certain threshhold, the propagation
velocity goes to zero and stationary convection is observed. We have found
that the stationary convection patterns observed in this regime are different
from those observed in a pure liquid. Near the traveling-wave transition the
patterns are angular, consisting of geometric domains of straight rolls,
whereas at higher Rayleigh number, the texture of the pattern is curved.
For
an account of the theoretical basis of convection textures, see
"Ingredients of a Theory of Convective Textures Close to Onset",
M. C. Cross, Physical Review A25, 1065 (1982).
The
basic idea is that a stable convection pattern is one which maintains a
wavenumber close to it's optimum value (roughly twice the height of the
convection cell), and has a minimum of curvature. Straight rolls are
obviously the best solution, but in a geometry in which straight rolls are
not possible, (such as our cylindrical cell) a compromise is necessary. We
beleive the angular and curved textures results from competition between
wavenumber and curvature optimization.
OBSERVATIONS IN AN 8% ETHANOL MIXTURE
We
first observed the transition in a mixture of 8% ethanol (by weight) in water.
This mixture supports a strongly nonlinear, large amplitude traveling-wave
convection state.
THE CURVED TEXTURE
This
pattern was observed at reduced Rayleigh number 3.7.
THE ANGULAR TEXTURE
This
pattern was observed at reduced Rayleigh number 1.7, just above the
transition to traveling-wave convection.
LOCAL CURVATURE IN THE CURVEDTEXTURE
In
this image, the curvature is evenly distributed along the roll
boundaries. This is a favorable curvature distribution.
LOCAL CURVATURE IN THE ANGULARTEXTURE
In
this image, the curvature is zero in most parts of the pattern, but very
intense along the boudaries of the straight domains. This is an unfavorable
curvature distribution.
LOCAL WAVENUMBER IN THE CURVEDTEXTURE
In
this image, there is intense wavenumber frustration in the region near
the defects where the domain boundaries meet. This is an unfavorable
wavenumber distribution.
LOCAL WAVENUMBER IN THE ANGULARTEXTURE
In
this image, the wavenumber is uniform and equal to the system's prefered
wavenumber. This is a favorable wavenumber distribution.
ILLUSTRATION OF THETWO TEXTURES
Our
understanding of this transition is that the pattern wants to optimize
both its wavenumber distribution (by having it uniform) and its curvature
distribution (by having is uniform). The defects in the pattern make it
impossible for both conditions to be fulfilled at once. Notice that in the
sketch above, which illustrates the arrangement of rolls near a defect, the
angular texture has sharp corners (bad for wavenumber), put perfect
wavenumber, and the curved texture has better curvature (more smooth) but
wavenumber distortion. We believe that the transition from a curved to
angular texture is a crossover from optimization of wavenumber at the
expense of curvature (angular texture), to optimization of curvature at the
expense of wavenumber.
OBSERVATIONS IN AN 1% ETHANOL MIXTURE, AND IN PURE WATER
In
a 1% ethanol mixture, the traveling-wave state is still observed, but it is
of smaller amplitude, and occurs at lower Rayleigh number. In pure water,
there is no traveling-wave state. We found that the angular texture could be
observed in the 1% mixture, but that it did not occur in pure water,
indicating that it is associated to the transition to traveling-waves.
CONVECTIONPATTTERN IN THE 1% MIXTURES
The
two images show the convection pattern at Rayleigh number 1.7 and 1.3,
respectively. Although the topology of the pattern is different than that
observed with the 8% mixture, the transition to an angular state is clear.
CURVATURE IN THE1% MIXTURES
Here
the curvature of the two patterns shown above is calculated. The lines of
intense curvature in the left-hand picture indicate the angular texture.
CONVECTIONPATTERN IN THE PURE WATER
Here
we try to repeat the experiment in pure water. In the mixture, the angular
state forms as we lower the Rayleigh number and approach the traveling-wave
state. In the pure fluid, there is not traveling-wave state, and the angular
texture does not form as we lower the Rayleigh number and approach the onset
value.
CURVATURE IN THE PURE WATER
The
curvature maps confirm that the angular texture does not form in the pure
fluid.
In
this summary, we have tried to give an intuitive argument as to why the
curved and angular textures are related to the competition between
wavenumber and curvature frustration. In our paper (cited above) we
introduce an effective free energy based on the Swift-Hohenberg model which
seems to capture the essence of the transition.
We
believe that the transition is of general interest, because it illustrates
the difference between curved patterns, such are typically observed in
Rayleigh-Benard convection, and angular patterns, such as are found in
ferromagnetic garnets, for instance. (See Seul, et. al., Science 254, 1616 (1991)).