Directional solidification under stress

Directional solidification under uniaxial stress is analyzed. In the absence of stress, it is well known that the moving planar front undergoes a morphological Mullins-Sekerka (MS) instability. Under uniaxial stress, even an interface at rest develops an instability known by the names of Asaro, Tiller, and Grinfeld (ATG). This paper analyzes the coupling between these two instabilities, a situation on which we have recently given a brief account [Durand et al., Phys. Rev. Lett. 76, 3013 (1996)]. We discover that under favorable circumstances a weak uniaxial stress of the order of 1 bar leads to a dramatic change of the Mullins-Sekerka instability. The threshold, together with the microstructure scale, are shifted by amounts going up to one (or several) decade(s). This effect should open new lines of both experimental and theoretical 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 evolution of the ATG instability leads to cusps which grow unstably, leading ultimately to the fracture threshold. It is shown here that due to a subtle coupling between both instabilities, the MS bifurcation in its supercritical regime may cause the MS-ATG coupled bifurcation to be 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.