Abstract
Complex morphologies, as is the case in self-assembled fibrillar networks (SAFiNs) of 1,3:2,4-Dibenzylidene sorbitol (DBS), are often characterized by their Fractal dimension and not Euclidean. Self-similarity presents for DBS-polyethylene glycol (PEG) SAFiNs in the Cayley Tree branching pattern, similar box-counting fractal dimensions across length scales, and fractals derived from the Avrami model. Irrespective of the crystallization temperature, fractal values corresponded to limited diffusion aggregation and not ballistic particle–cluster aggregation. Additionally, the fractal dimension of the SAFiN was affected more by changes in solvent viscosity (e.g., PEG200 compared to PEG600) than crystallization temperature. Most surprising was the evidence of Cayley branching not only for the radial fibers within the spherulitic but also on the fiber surfaces.
Highlights
Supramolecular Fractal Growth of Fractal or self-similar objects exhibit ‘never-ending’ identical patterns across different length scales leading to equal Hausdorff dimensions, often termed fractal dimensions (D)
Fractality is reported for numerous materials—including, but not limited to, frost [4], fat crystal networks [5], self-assembled polymers [6], and molecular gels [7,8,9,10] are routinely found in nature [11]
Dibenzylidene sorbitol (DBS) in polyethylene glycol (PEG) presents its fractality through the consistent Db, across magnifications, and through its Cayley Tree fractal branching pattern and corresponding linear Avrami determination of Df
Summary
Supramolecular Fractal Growth of Fractal or self-similar objects exhibit ‘never-ending’ identical patterns across different length scales leading to equal Hausdorff dimensions, often termed fractal dimensions (D). The Hausdorff dimensions of uniform objects—a point = 0, line = 1, square = 2, and cube = 3—are defined as topological dimensions. More complex shapes, such as the Koch Snowflake [1] (Df = 1.26 (D = log 4/log 3) or Sierpinski Carpet [2] (D = log 8/log 3)) (Figure 1), are better defined by their properties of self-similarity and non-Euclidean dimensions. Self-similarity is achieved when each part of a geometric figure has the same statistical character as the whole. Fractality is reported for numerous materials—including, but not limited to, frost [4], fat crystal networks [5], self-assembled polymers [6], and molecular gels [7,8,9,10] are routinely found in nature [11]
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