Shallow cells in directional solidification

Flow-induced changes of the morphological stability in directional solidification: localized morphologies

“…Critical pulling speed versus concentration (taken from [15]). The region inside the curve corresponds to an unstable planar interface.…”

Analysis of diffusive cellular patterns in directional solidification of bulk samples

“…The critical solidification velocity for the onset of cellular structures is estimated from the MS criterion [5,19,37] in the low velocity limit, including effects of latent heat and different thermal conductivities for the solid and liquid phases: [1] where D is the diffusion coefficient of the solute; k the distribution coefficient; m L the slope of the liquidus line in the phase diagram; [35] L v the latent heat per unit volume; and L and s the thermal conductivities of the liquid and the solid, respectively. With their numerical values, [2] a critical velocity of v c ϭ 2.1 m s Ϫ1 is predicted.…”

“…In WSA, the Landau coefficient a 1 separates supercritical (a 1 Ͼ 0) and subcritical (a 1 Ͻ 0) transitions, and is calculated here according to Merchant and Davis, including effects of latent heat, as [19] [2] with and where ␥ is the interfacial surface energy per unit area and T M the melting temperature of the pure succinonitrile. [2] We would like to point out that the results of the calculations differ from those given by Merchant and Davis [19] concerning the transition point between subcritical and supercritical transition on the neutral stability curve Figure 1 in their article. Agreement was only found when the wrong liquid diffusion coefficient of D ϭ 1.27и10 Ϫ10 m 2 s taken from Reference 20 (instead of D ϭ 1.27и10 Ϫ9 m 2 s) [36] is used.…”

“…The spectrum is compared to the experimentally determined spacing distributions after suitable normalization. Additionally, we have carried out a variation of the original formulations [5,[16][17][18][19] in the analysis by using the concentration at the cell tips instead of the concentration ahead of a planar interface and we have extended our earlier formulation for the tip concentration. [13] The experimental spacing distributions are also compared to different analytical and numerical models.…”

“…This is in contradiction to the experimental patterns approaching a steadystate configuration at least in a statistical meaning. Therefore, an extension of LSA to weak non linear stability analysis (WSA) [16][17][18][19] is necessary to describe a time-independent steady state with constant cell amplitudes. A spectrum of unstable wavelengths with different constant growth rates is predicted from the latter.…”