Spontaneous parity-breaking transition in directional growth of lamellar eutectic structures

A full study of parity-broken states in the directional solidification of lamellar eutectics is performed within the boundary-integral formulation. Symmetric states cease to exist at a wavelength λ, which is approximately twice that corresponding to their minimum undercooling, whereas solutions with a broken parity, drifting transversely to the growth front, appear as a forward bifurcation. Our results suggest that if one effectively doubles the wavelength of the initially symmetric state—a situation that can be achieved via a sudden jump of the velocity V by a factor of about 4, since ${\lambda }^2$V≃const—then tilted lamellae should appear as extended states and not as ‘‘solitons.’’ We find here that parity-broken states exist for hypereutectic as well as for hypoeutectic and eutectic compositions. We have extended the derivation of the similarity equation derived previously [K. Kassner and C. Misbah, Phys. Rev. Lett. 66, 445 (1991)] to the present situation. This case involves additional subtleties, due to the loss of reflection symmetry about the growth axis. Among other results, we find that the tilt angle φ should depend on σ=${\mathit{d}}_0$l/${\lambda }^2$ and χ=l/${\mathit{l}}_{\mathit{T}}$ only, where ${\mathit{d}}_0$, l, and ${\mathit{l}}_{\mathit{T}}$ are the capillary, diffusion, and thermal lengths, respectively, and λ is the wavelength of the pattern. At large enough growth velocities V, φ≃φ(σ), while at small V the dependence on χ is strong. These predictions can be tested experimentally.