gels Article Supramolecular Fractal Growth of Self-Assembled Fibrillar Networks Pedram Nasr 1 , Hannah Leung 1 , France-Isabelle Auzanneau 2 and Michael A. Rogers 1, *   Citation: Nasr, P.; Leung, H.; Auzanneau, F.-I.; Rogers, M.A. Supramolecular Fractal Growth of Self-Assembled Fibrillar Networks. Gels 2021, 7, 46. https://doi.org/ 10.3390/gels7020046 Academic Editor: Pablo H. Di Chenna Received: 22 February 2021 Accepted: 12 April 2021 Published: 14 April 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Department of Food Science, University of Guelph, Guelph, ON N1G 2W1, Canada; pnasr@uoguelph.ca (P.N.); hleung07@uoguelph.ca (H.L.) 2 Department of Chemistry, University of Guelph, Guelph, ON N1G 2W1, Canada; fauzanne@uoguelph.ca * Correspondence: mroger09@uoguelph.ca; Tel.: +11-519-824-4120 (ext. 54327) 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. Keywords: 1,3:2,4-Dibenzylidene sorbitol; self-assembled fibrillar networks (SAFiNs); fractality; Cayley Tree; fractal dimension; solvent viscosity; supercooling; crystallization 1. Introduction 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] (D f = 1.26 (D = log 4/log 3) or Sierpinski Carpet [2] (D f = 1.89 (D = log 8/log 3)) (Figure 1), are better defined by their properties of self-similarity and non-Euclidean dimensions. Uniform objects have D f = d, while partly-filled, more open structures where density decreases radially have D f <d[3]. 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 [7–10] are routinely found in nature [11]. Understanding and ultimately controlling the fractal nature of self-assembled fibrillar networks (SAFiNs) is particularly important because the crystalline network morphology determines the macroscale properties (e.g., oil binding, elasticity, and breaking properties) and, ultimately, gel applications [12–14]. Molecular self-assembly constructs precision materials, where their supramolecular structures assemble molecule-by-molecule, by way of “bottom–up” nanofabrication, and remarkably, coding for assembly is embedded in the structural motifs of the molecule [15]. Structural motifs direct self-assembly via non- covalent interactions (i.e., hydrogen bonding [16], π–π stacking [17], and van der Waals interactions [18]). Understanding molecular coding (i.e., molecular chirality [19–25], po- sitional isomers [26–29], molecular polarity [30–32]) is an active area of inquiry for low- molecular-mass (MW < 2000 Da) organogelators (LMOGs) [21,33–39]. These highly specific interactions drive inter-molecular interactions promoting 1-dimensional (1D) growth—the precursor to gel formation. Gels 2021, 7, 46. https://doi.org/10.3390/gels7020046 https://www.mdpi.com/journal/gels