I. Cantat et al., DIRECTIONAL SOLIDIFICATION UNDER STRESS, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 58(5), 1998, pp. 6027-6040
Directional solidification under uniaxial stress is analyzed. In the a
bsence of stress, it is well known that the moving planar front underg
oes a morphological Mullins-Sekerka (MS) instability. Under uniaxial s
tress, even an interface at rest develops an instability known by the
names of Asaro, Tiller, and Grinfeld (ATG). This paper analyzes the co
upling between these two instabilities, a situation on which we have r
ecently given a brief account [Durand et al., Phys. Rev. Lett. 76, 301
3 (1996)]. We discover that under favorable circumstances a weak uniax
ial stress of the order of 1 bar leads to a dramatic change of the Mul
lins-Sekerka instability. The threshold, together with the microstruct
ure scale, are shifted by amounts going up to one (or several) decade(
s). This effect should open new lines of both experimental and theoret
ical inquiries. A weakly nonlinear analysis is presented by means of a
Landau expansion. It is known that the MS bifurcation is subcritical
for a small enough solute partition coefficient, and is supercritical
otherwise. The ATG instability is always subcritical. The nonlinear ev
olution of the ATG instability leads to cusps which grow unstably, lea
ding ultimately to the fracture threshold. It is shown here that due t
o a subtle coupling between both instabilities, the MS bifurcation in
its supercritical regime may cause the MS-ATG coupled bifurcation to b
e supercritical. Discussions and outlooks are presented. In particular
it is appealing to speculate that the creation of giant causeways in
igneous rocks can be interpreted within the present context. [S1063-65
1X(98)07010-X].