Rhythmic Growth of Target and Spiral Spherulites of Crystalline Polymer Blends

Abstract

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...

Diffusion-Induced Oscillations of Extended Defects

“…For v = v P the result (8) consistently reduces to the first equation in Eqs. (3). Subtracting the latter from Eq.…”

“…In all approaches the source of oscillatory defect motions is identified as an unstable regime where a reduction of the driving force leads to an increase of the defect velocity. Additional information is provided by simulations, based on phase-field models for directional solidification [10] and for nucleation [3] processes.…”

“…In all approaches the source of oscillatory defect motions is identified as an unstable regime where a reduction of the driving force leads to an increase of the defect velocity. Additional information is provided by simulations, based on phase-field models for directional solidification [10] and for nucleation [3] processes.In the present Letter we introduce an extremely simple but powerful model for the diffusion-induced oscillatory motion of a planar interface, using the language adapted to the directional solidification of a dilute binary alloy. A major advantage of our approach is that it allows a transparent and, to a large extend, analytical evaluation.…”

“…In a large number of metallic materials this leads to the formation of banded structures [1], reflecting a periodic array of layers with high and low solute concentrations where the former ones show a dendritic microstructure. The appearance of similar banding effects has recently been discussed [2] in rapid solidification of colloids.Layered structures are also generated by the oscillatory nucleation of a solid phase under the action of a diffusion field [3]. A related phenomenon is the oscillatory zoning, observed in solid solutions [4] and in natural minerals [5].…”

“…Layered structures are also generated by the oscillatory nucleation of a solid phase under the action of a diffusion field [3]. A related phenomenon is the oscillatory zoning, observed in solid solutions [4] and in natural minerals [5].…”