Numerical calculations reveal that the target and spiral growth patterns in spherulites can be generated from the time-dependent Ginzburg-Landau equations (model C) by coupling a conserved compositional order parameter and a nonconserved crystal ordering parameter. Of particular interest is that the periodic concentric rings (target) or the spirals at the spherulitic core remain stationary in both the crystal (orientational) ordering field and the concentration field. Another intriguing observation is that the growth of target and spiral spherulites occurs in a stepwise fashion in synchronism with the rhythmic energy dissipation during crystallization. PACS numbers: 61.41. + e, 64.70.Dv, 81.10.AjSpiral and target patterns are commonly observed in excitable media and nonlinear dissipative systems involving chemical waves, liquid crystal ordering, and biological organization [1]. Similar spiral and target patterns have also been found experimentally during crystallization of polymers, organic, and inorganic materials [2-5]. One major difference is that the core of the spiral crystal is stationary as opposed to the excitable media where the core is constantly oscillating. Although a similarity in patterns does not automatically guarantee that the origin is the same, it is worth investigating the pattern forming aspects of polymer crystallization from a new perspective of phase transitions.In atomic crystals, crystallization phenomena have been generally analyzed in the context of the classical macroscopic models of phase transitions [6,7]. The governing equations treat thermodynamic variables, such as temperature and composition of the individual phases, independently with a discrete interface of zero thickness. The discontinuity in the thermodynamic variables and in the gradients often presents difficult mathematical formulations and numerical singularities [8]. Recently, a new microscopic theory has been proposed, often known as a phase-field model [8,9], by incorporating the free energy functional for a crystal phase transition into the time-dependent Ginzburg-Landau-equation model C (TDGL model C) [10][11][12]. This phase-field model has been employed in the growth of atomic crystals and recently extended to the phase transitions of binary metal alloys [10] and eutectic crystal growth [13].The present paper is a first attempt to analyze the pattern forming aspects of polymer crystallization in the context of the TDGL model C equations [11,12] pertaining to phase transitions. The purpose of this paper is to demonstrate that the target and spiral growth patterns in polymer spherulites can be generated from the TDGL model C equations. In this model, we consider the system as a whole by defining the crystal ordering as a phase field to characterize the phase (or state) of the system at each point in time ͑t͒ and space ͑r͒. We then couple the conserved concentration (or number density) and nonconserved orientational order parameters in the coupled TDGL model C equations in conjunction with the Landau-type double-wel...

A new theoretical approach which describes the formation of banded textures in liquid crystalline polymers is introduced. The approach incorporates the concept of extended curvature elastic free energy in describing the response of the director field to strain recovery. Physically, this corresponds to accounting for an additional energetic penalty for deformation of the director field over very short distances, resulting from the natural persistence of orientation in rigid polymer chains. Both analytical and numerical results indicate that periodic patterns are spontaneously generated whenever the magnitude of the second-order gradient terms exceeds a critical value. Numerical investigations of twodimensional systems reveal that elastic anisotropy has a significant effect on the aspect ratio and orientation of the patterns formed.

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