Explosive Fibonacci-sequence growth into unusual sector-face morphology in poly(l-lactic acid) crystallized with polymeric diluents

Abstract

Lamellar assembly in unusual sector-face pLLA spherulites from crystallization of poly(l-lactic acid) (pLLA) diluted with amorphous poly(methyl methacrylate) (pMMA). the growth and morphology of the crystalline structures is studied using polarized optical microscopy (POM), atomic-force and scanning electron microscopies (AFM, SEM). Crystals are also analyzed using differential scanning calorimetry (DSC) and small-angle X-ray scattering (SAXS). The two alternate sectored faces differ dramatically in their optical birefringence and top-surface and interior lamellar assembly. By originating from the nucleus center, an explosive fan-like sector of high-birefringence lamellae is packed by fractal growth from an initial single stalk into hundreds of branches upon reaching the periphery, with the number of stalks increasing roughly by the Fibonacci sequence along the radial distance. The exploded pattern resembles a cross-hatch grating structure, and displays a cauliflowerlike fractal-branching of optical birefringence blue/orange stripes. This finding suggests that growth with periodic branching is one of the main mechanisms to fill the ever-expanding space in the spherulitic 3D aggregates.

Unique Periodic Rings Composed of Fractal-Growth Dendritic Branching in Poly(p-dioxanone)

“…The growth cycles repeat in the same manner, with fractal branching to fill the ever-expanding space as the lamellae grow outward from the nucleus center. It can be shown later that the increase of the number of branches of the lobs of dendrites roughly follows the Fibonacci sequence, also reported earlier in many other polymer systems with dendritic growth [31][32][33][34][35]. Furthermore, interfacial crevices between the neighboring lobs of dendrites are clearly seen in the AFM height images (Figure 5a).…”

“…The number of branches was estimated and plotted as a function of distance from the nucleus center, as shown in Figure 11c. The increasing number of branches of the lobs of dendrites roughly follows the Fibonacci sequence, which is also observed in many other polymer systems with dendritic growth [31][32][33][34][35]. Figure 11 show a chopped sector of the dendritic/ringed PPDO spherulite, containing three lobs of dendrites with 6-10 branches packed with ring bands in each lob.…”

“…The number of branches was estimated and plotted as a function of distance from the nucleus center, as shown in Figure 11c. The increasing number of branches of the lobs of dendrites roughly follows the Fibonacci sequence, which is also observed in many other polymer systems with dendritic growth [31][32][33][34][35]. There is an issue concerning whether the crystallization of PPDO (from PPDO/PVPh blend) might result in the expulsion or segregation of the PVPh component from the PPDO lamellae or interphases in the aggregates.…”

Synchrotron X‐Ray Analysis and Morphology Evidence for Stereo‐Assemblies of Periodic Aggregates in Poly(3‐hydroxybutyrate) with Unusual Photonic Iridescence

“…Phthalic acid (PA), a small-molecule organic compound, can also display periodically banded patterns in solution-crystallized spherulites. [9,10] Crystals periodically branch out and form a grating-banded assembly, so the banded-patterns are displayed by polarized optical microscopy. In addition to the observation of the characteristics utilizing electron microscopy, the X-ray analysis of the electron density difference caused by the arrangement of the crystal plate can be used to infer the formation of the spherulites.…”

Review about the Application of Fractal Theory in the Research of Packaging Materials

“…For the description of non-uniform structures with rich scale features, in the mass fractal region, the box counting dimension of MF spectrum is consistent with the scattering index of the X-ray spectrometer curve, and changes from short-range mass fractal area to the long-range mass fractal region [ 36 ]. When MF evaluates and compares the heterogeneity of various porous samples, the pixel size of the sample must be the same, the sample volume must be larger than the representative volume element, the MF dimension should be normalized to the determined porosity value, and the influence of sub-resolution pores on small-scale should be ignored; the real samples can be divided into less and heterogeneous groups through the normalized fractal dimension [ 37 ]. The packaging materials of different structures and properties with MF characteristics should be treated differently.…”