The solidification phenomenon of a liquid has been one of the most fascinating firstorder phase transitions, involving a complex interplay of many physical effects. The simplest solidification front is the planar interface. Usually, the solid-liquid interface is an active free boundary from which latent heat is liberated during solid-liquid phase transformation. Preferentially, this latent heat is conducted away from the convex solid-liquid tips, but it may be trapped in the case of concave interface, thereby leading to the directional growth of interfacial morphologies. Consequently, these interfaces are generally rough with convex and concave curvatures. Among them, the basic patterns are dendrites and seaweeds. A structure with pronounced orientational order is termed as dendrite, and without apparent orientational order it is called seaweed. The dendritic shape is a symmetric needle crystal with a parabolic tip, the sides of which are influenced by a secondary branching. The seaweed morphology was originally introduced on the basis of experimental observations under the name of dense-branching morphology, which is characterized by repeating tip splitting at the interface front [1]. The basic element of the seaweed structure is a doublon, which consists of two fingers with a liquid channel running along the axis of the symmetry between them. Experimentally, Akamatsu[2] showed that the seaweed growth involves a continual creation and elimination of doublons. In directional solidification, the solid-liquid interface is subjected to morphological instabilities in an externally imposed directional temperature gradient. Various interface morphologies have been observed [2-4] depending on the physical properties of the materials. Especially, the anisotropic properties of the solid-liquid interface play a particularly important role in deter mining the stability of the dendrites as well as the transformation between seaweed and dendrite patterns. Thus, an in-depth understanding of the interface morphology during solidification is essential to shed light on the pattern formation aspects of crystal solidification.Theoretical investigations of the solid-liquid phase transition have long been the subject of considerable interest. Among these, there are two main streams: one is the Edited by Avraam I. Isayev