Fractal Self-assembly Research Articles

Fractal Branched Microwires of Organic Semiconductor with Controlled Branching and Low-Threshold Amplified Spontaneous Emission

Fractals are quite normal in nature. However, fractal self-assembly of organic semiconductors remains challenging. Herein, we develop a facile solution assembly route to access organic microwires (MWs) comprising an oligo(p-phenylenevinylene) derivative (OPV-A) with and without branching. Instead of kinetically controlled β-OPV-A microrods (MRs), thermodynamically favored α-OPV-A gives fractal branching MW patterns. As-prepared 9,10-dicyanoanthracene (DCA) alloyed assemblies function as seeds to allow for the heteroepitaxial growth of branching α-OPV-A MWs via either coassembly or two-step seeded growth. Consequently, fractal MWs with single- and multisite growth were both achieved, accompanied by tailorable branching densities and hierarchies. Thermodynamic control and a well-matched epitaxial relationship should be crucial to the formation of fractal MW patterns. Importantly, the aligned α-OPV-A MW array functions as a multichannel optical gain medium and exhibits low-threshold amplified spontaneous emission (ASE). The present work deepens the research into fractal self-assembly of functional organic semiconductors.

Interactive patches over amyloid-β oligomers mediate fractal self-assembly.

The monomeric units of intrinsically disordered proteins self-assemble into oligomers, protofilaments, and eventually fibrils which may turn into amyloid. The aggregation of these proteins is primarily studied in bulk with no restriction on their degrees of freedom. Herein we experimentally demonstrate that amyloid-β (Aβ) aggregation under diffusion-limited conditions leads to its fractal self-assembly. Confocal microscopy and scanning electron microscopy with energy dispersion x-ray analysis were used to confirm that the fractal self-assemblies were formed from Aβ rather than the salt present in the two supporting media: deionized water and phosphate buffered saline. The results from the molecular docking experiments implicated that electrostatic and hydrophobic patches on the solvent-accessible surface area of the Aβ oligomers mediate the fractal self-assembly. These implications were tested with laser light scattering experiments on the oligomers formed by breaking mature fibrils of Aβ through sonication, which were observed to self-assemble into fractals when sonicated solutions were drop casted. The electrostatic interactions modulate the fractal morphologies with pH of the solution, which leads to a morphological phase transition observed through the variation in their fractal dimension. These transitions provide experimental evidence for the existing theoretical framework in terms of different kinetic models. The higher surface-to-volume ratio of these fractal self-assemblies may have applications in drug delivery, biosensing, and other biomedical applications.

Fractal self-assembly and aggregation of human amylin.

Human amylin is an intrinsically disordered protein believed to have a central role in Type-II diabetes mellitus (T2DM). The formation of intermediate oligomers is a seminal event in the eventual self-assembled fibril structures of amylin. However, the recent experimental investigations have shown the presence of different self-assembled (oligomers, protofilaments, and fibrils) and aggregated structures (amorphous aggregates) of amylin formed during its aggregation. Here, we show that amylin under diffusion-limited conditions leads to fractal self-assembly. The pH and solvent sensitive fractal self-assemblies of amylin were observed using an optical microscope. Confocal microscopy and scanning electron microscopy (SEM) with energy dispersion X-ray analysis (EDAX) were used to confirm the fractal self-assembly of amylin in water and PBS buffer, respectively. The fractal characteristics of the self-assemblies and the aggregates formed during the aggregation of amylin under different pH conditions were investigated using laser light scattering. The hydropathy and the docking study indicated the interactions between the anisotropically distributed hydrophobic residues and polar/ionic residues on the solvent-accessible surface of the protein as the crucial interaction hot-spots for driving the self-assembly and aggregation of human amylin. The simultaneous presence of various self-assemblies of human amylin was observed through different microscopy techniques. The present study may help in designing different fractal-like nanomaterials with potential applications in drug delivery, sensing, and tissue engineering.

Anisotropy versus fluctuations in the fractal self-assembly of gold nanoparticles.

In a recent report, the fractal self-assembly of gold nanoparticles (AuNPs) having a directional feature was observed in the presence of visible light. Therein, the visible light, an external parameter, was suspected to be responsible for the directional feature. Herein, we investigate the intrinsic factors, the aspect size ratio p and the size a of AuNPs, in modulating the fractal characteristics of their self-assemblies. Through light scattering experiments and microscopic imaging, we demonstrate the transition of morphologies from fractal-like to cross-shaped in gold colloidal aggregates with particles having nearly spherical and ellipsoidal shapes, respectively. The transition indicates the competitive role of anisotropy and fluctuations in deciding the morphological characteristics of the aggregates. By taking noise-reduced diffusion-limited aggregation (NRDLA) as a model system, we address the shape and size induced noise of the particles in the colloidal systems which are prone to form fractal aggregates. We qualitatively relate the noise due to the particles having a distinct aspect size ratio p and size a with the noise reduction parameter m of NRDLA. The realistic nature of the experimental systems, where the particles of different p and a are present during the growth process, is incorporated by introducing the Gaussian noise reduction in diffusion-limited aggregation (DLA). The morphological phase transition in Gaussian noise reduced DLA is characterized, and its relevance for accounting the shape and size originated noise fluctuations during the fractal growth process is discussed. The results of the present study may be used for tailored applications of AuNPs in drug delivery, biomedicine, biosensing, and cancer nanotechnology.

Self-assembly of 4-sided fractals in the Two-Handed Tile Assembly Model

We consider the self-assembly of fractals in one of the most well-studied models of tile based self-assembling systems known as the Two-Handed Tile Assembly Model (2HAM). In particular, we focus our attention on a class of fractals called discrete self-similar fractals (a class of fractals that includes the discrete Sierpinski carpet). We present a 2HAM system that finitely self-assembles the discrete Sierpinski carpet with scale factor 1. Moreover, the 2HAM system that we give lends itself to being generalized and we describe how this system can be modified to obtain a 2HAM system that finitely self-assembles one of any fractal from an infinite set of fractals which we call 4-sided fractals. The 2HAM systems we give in this paper are the first examples of systems that finitely self-assemble discrete self-similar fractals at scale factor 1 in a purely growth model of self-assembly. Finally, we show that there exists a 3-sided fractal (which is not a tree fractal) that cannot be finitely self-assembled by any 2HAM system.

Hierarchical self-assembly of fractals with signal-passing tiles

In this paper, we present high-level overviews of tile-based self-assembling systems capable of producing complex, infinite, aperiodic structures known as discrete self-similar fractals. Fractals have a variety of interesting mathematical and structural properties, and by utilizing the bottom-up growth paradigm of self-assembly to create them we not only learn important techniques for building such complex structures, we also gain insight into how similar structural complexity arises in natural self-assembling systems. Our results fundamentally leverage hierarchical assembly processes, and use as building blocks square “tile” components which are capable of activating and deactivating their binding “glues” a constant number of times each, based only on local interactions. We provide the first constructions capable of building arbitrary discrete self-similar fractals at scale factor 1, and many at temperature 1 (i.e. “non-cooperatively”), including the Sierpinski triangle.

Strict Self-Assembly of Fractals Using Multiple Hands

In this paper we consider the problem of the strict self-assembly of infinite fractals within tile self-assembly. In particular, we provide tile assembly algorithms for the assembly of a Sierpinski triangle and the discrete Sierpinski carpet within a class of models we term the h-handed assembly model (h-HAM), which generalizes the 2-HAM to allow up to h assemblies to combine in a single assembly step. Despite substantial consideration, no purely growth self-assembly model has yet been shown to strictly assemble an infinite fractal without significant modification to the fractal shape. In this paper we not only achieve this, but in the case of the Sierpinski carpet are able to achieve it within the 2-HAM, one of the most well studied tile assembly models in the literature. Our specific results are as follows: We provide a 6-HAM construction for a Sierpinski triangle that works at scale factor 1, 30 tile types, and assembles the fractal in a near perfect way in which all intermediate assemblies are finite-sized iterations of the recursive fractal. We further assemble a Sierpinski triangle within the 3-HAM at scale factor 3 and 990 tile types. For the Sierpinski carpet, we present a 2-HAM result that works at scale factor 3 and uses 1216 tile types. We further include analysis showing that the aTAM is incapable of strictly assembling the Sierpinski triangle considered in this paper, and that multiple hands are needed for the near-perfect assembly of a Sierpinski triangle and the Sierpinski carpet.

Electromagnetic Field Released in Collision-Impact Events Generate in the Matrix Interface Fractal Scalable Invariant Geometric Triangular Chiral Hexagonal Structures

In biology, cancer is the most beautiful natural model of a chaotic system, which under uncontrolled proliferations, generates extreme disorder that finally causes intercellular collisions. The authors have described and documented fractal self-assembly of geometric triangular chiral hexagonal crystal-like complex organizations (GTCHC) and interface comet tail effect patterns in cancer processes. According to this novel observation cancer incorporates a real visualization world with a great surprising finding in biology, physics, and geology. This visualization platform literally allows us to see what would otherwise remain completely invisible. From theory to practice this irreducible geometric matrix allowed us to identify in geology, real measurable green infrared-electromagnetic stripe line in interface with hexagonal geomorphic pattern, triangular chiral pyramidal rock structures, geology well defined mirror images, template platform to bio signature characterization of ancestral primitive polar head-tail organization, embryoid and human-like shape pattern embedded as giant fossils in rocks that have never been seen before. Electromagnetic field released in collision-impact events generate in the matrix interphase fractal scalable invariant order of geometric triangular chiral hexagonal structures. The laws of biology and geology can finally be redirected to the laws of physics; specifically magnetic fields create a new kind of classification based on these fractal structural similarities of the relationship. Further interdisciplinary collaboration must be carried out to study these geometric self-assembly geological structures, ancient sediments and rocks that could provide insights into antecedents of life.

Fractal Self-Assembly of Single-Stranded DNA on Hydrophobic Self-Assembled Monolayers

The self-assembled structures possess superior stability, biocompatibility and mechanical strength, and their study can provide insight into the use of creating novel biomaterials. Atomic force microscopy (AFM) images of single-stranded DNA (ssDNA) nanostructures show that well-ordered organization, high homogeneity, and molecular dimensions fractal-shaped fibers formed on a gold substrate covered with self-assembled monolayers (SAMs) of 1-hexadecanethiol (HDT). The nanoscaled architectures of ssDNA on HDT/Au changed remarkably following the process of diffusion-limited cluster aggregation (DLA) over time. The ssDNA fibers prefer to form on hydrophobic SAMs instead of hydrophilic SAMs, and the ssDNA has to have complementary regions in their sequences. This method might not be used only for the construction of fractal patterns, but also for the design and fabrication of functional DNA-based, self-assembled materials that exhibit self-similarity at multiple length scales.

Insight into Fractal Self-Assembly of Poly(diallyldimethylammonium chloride)/Sodium Carboxymethyl Cellulose Polyelectrolyte Complex Nanoparticles

Poly(diallyldimethylammonium chloride)-sodium carboxymethyl cellulose polyelectrolyte complexes (PDDA-CMCNa PECs) solids were prepared and dispersed in NaOH aqueous solution. Self-assembly of PECs nanoparticles during the solvent evaporation was examined by field emission electron microscopy (FESEM), atomic force microscopy (AFM), and fractal dimension analysis. It was found that tree-shaped fractal patterns formed after the solvent (water) was dried at ambient temperatures, and the fractal pattern is composed of needle-shaped PEC aggregate (PECA) nanoparticles. Time-dependent FESEM observation revealed that the fractal pattern started with the formation of initial nucleon and it is growing, during which the diffusion limited aggregation (DLA) mechanism revealed and made the pattern branched. Physical insight into the DLA mechanism was discussed in detail. Effects of PEC concentrations, PEC compositions, solvent evaporation temperatures, pH of PEC dispersion, and chemical structures of PECs on the formation of self-assembled fractal pattern were studied. Generally, it was found that the morphologies, charge characters of PEC particles, and the solvent evaporation conditions play important roles during the fractal self-assembly process.

SAXS: Structure Verification of an S-Layer Protein using a Fractal Mean Force Potential

Using a combination of SAXS and molecular dynamics simulations we get an integrated picture of the structure of S-layer proteins, for which no crystallized structure is currently available. S-layers are the most commonly observed cell surface structure of prokaryotic organisms and they are made up of identical protein subunits. One of the most striking properties of S-layers is that they are able to self-assemble into crystalline lattices in suspension and on various solid substrates. The resulting ordered molecular layer provides a matrix for the binding of various biomolecules and nanoparticles. Due to the complex biochemical properties of these proteins, classical techniques such as NMR or X-ray crystallography have not been able to provide an atomistic structural model for S-layer proteins. Motivated by the results obtained through the use of a fractal concept for the analysis of SAXS data in cluster physics, here we employ such fractal concept for the investigation of the structure of S-layer proteins. We fit the SAXS intensity as a function of the scattering angle using both a fractal form factor and a fractal structure factor. We compute the form factor by a Fourier transform of an average fractal delta function of characteristic units, thereby allowing us to investigate the presence of local, rather than global, electron densities in the structure of S-layer protein monomers. The structure factor is calculated by a Fourier transform of a fractal potential of mean force. Using this fitting function, we calculate the shape of the monomers, which is in good agreement with the shape obtained using molecular dynamics simulations. Our approach reveals itself as a novel means to get a detailed insight into the structure of proteins that adopt a fractal self-assembly and that lack of crystallized structure.

The fractal self-assembly of the silk protein sericin

Silk consists primarily of two proteins—a fibrous core protein, fibroin, and a glue protein, sericin, which envelops the fibroin fibers with sticky layers thus helping in the formation of cocoon, achieved by cementing the silk fibers together. Sericin, a water soluble protein, has traditionally been discarded in silk processing, despite great potential for use as a biomaterial. Here we show that this glue protein, sericin, has the ability to form self-assembled nano and microstructures with hierarchical self-similarity across length scales in the form of a diffusion-limited, fractal assembly. Sericin obtained from two silkworms, a domesticated mulberry Bombyx mori and a wild non-mulberry Antheraea mylitta, were studied, with particular insight into its structure and morphology as it dried on a surface. High-resolution atomic force microscopy was used to image the self-assembled protein, with investigations on the various factors that influenced this process. We describe the self-assembly patterns formed by the sericin protein and analyze the images based on the theory of diffusion limited aggregation to explain the fractal nature of these architectures observed. The unique physical behavior and fractal nature of this glue protein may represent a key step in understanding its biological relevance as well as the role it plays in the assembly and formation of silks.

Self-assembly of discrete self-similar fractals

In this paper, we search for theoretical limitations of the Tile Assembly Model (TAM), along with techniques to work around such limitations. Specifically, we investigate the self-assembly of fractal shapes in the TAM. We prove that no self-similar fractal weakly self-assembles at temperature 1 in a locally deterministic tile assembly system, and that certain kinds of discrete self-similar fractals do not strictly self-assemble at any temperature. Additionally, we extend the fiber construction of Lathrop et al. (2009) to show that any discrete self-similar fractal belonging to a particular class of “nice” discrete self-similar fractals has a fibered version that strictly self-assembles in the TAM.

Symmetry-Driven Spontaneous Self-Assembly of Nanoscale Ceria Building Blocks to Fractal Superoctahedra

A combination of long-term aging studies and molecular dynamics (MD) simulations has been successfully used to explain the time-dependent hierarchical assembly of ceria nanoparticles (CNPs). When the CNPs were aged in as-synthesized condition at room temperature in water, it was observed that the individual 3−5 nm CNPs result in octahedral superstructures through a fractal assembly. This hierarchical fractal self-assembly was observed despite the absence of any surfactant, at room temperature, and under atmospheric pressure. High resolution transmission electron microscopy (HRTEM) and fast Fourier transform (FFT) analysis have been used to explore the assembly of the individual nanoparticles into fractal superoctahedra. Both experimental work and theoretical analysis showed that the initial octahedral and truncated octahedral seeds symmetrically assemble and result in the superoctahedra with intermediate transformation steps.