Fractal-like Clusters Research Articles

Models for Simulation of Fractal-like Particle Clusters with Prescribed Fractal Dimension

This review article delves into the growing recognition of fractal structures in mesoscale phenomena. The article highlights the significance of realistic fractal-like aggregate models and efficient modeling codes for comparing data from diverse experimental findings and computational techniques. Specifically, the article discusses the current state of fractal aggregate modeling, with a focus on particle clusters that possess adjustable fractal dimensions (Df). The study emphasizes the suitability of different models for various Df–intervals, taking into account factors such as particle size, fractal prefactor, the polydispersity of structural units, and interaction potential. Through an analysis of existing models, this review aims to identify key similarities and differences and offer insights into future developments in colloidal science and related fields.

Control of Nanoscale In Situ Protein Unfolding Defines Network Architecture and Mechanics of Protein Hydrogels.

Hierarchical assemblies of proteins exhibit a wide-range of material properties that are exploited both in nature and by artificially by humankind. However, little is understood about the importance of protein unfolding on the network assembly, severely limiting opportunities to utilize this nanoscale transition in the development of biomimetic and bioinspired materials. Here we control the force lability of a single protein building block, bovine serum albumin (BSA), and demonstrate that protein unfolding plays a critical role in defining the architecture and mechanics of a photochemically cross-linked native protein network. The internal nanoscale structure of BSA contains “molecular reinforcement” in the form of 17 covalent disulphide “nanostaples”, preventing force-induced unfolding. Upon addition of reducing agents, these nanostaples are broken rendering the protein force labile. Employing a combination of circular dichroism (CD) spectroscopy, small-angle scattering (SAS), rheology, and modeling, we show that stapled protein forms reasonably homogeneous networks of cross-linked fractal-like clusters connected by an intercluster region of folded protein. Conversely, in situ protein unfolding results in more heterogeneous networks of denser fractal-like clusters connected by an intercluster region populated by unfolded protein. In addition, gelation-induced protein unfolding and cross-linking in the intercluster region changes the hydrogel mechanics, as measured by a 3-fold enhancement of the storage modulus, an increase in both the loss ratio and energy dissipation, and markedly different relaxation behavior. By controlling the protein’s ability to unfold through nanoscale (un)stapling, we demonstrate the importance of in situ unfolding in defining both network architecture and mechanics, providing insight into fundamental hierarchical mechanics and a route to tune biomaterials for future applications.

From monomers to agglomerates: A generalized model for characterizing the morphology of fractal-like clusters

The morphological description of fractal agglomerates is generally reduced to only two parameters, namely the mass fractal dimension and its prefactor. In the most evolved approaches, a stretching exponent is also introduced, while a packing factor is preferred to the fractal prefactor. In any case, the current analytical description of agglomerates morphology is accurate only for sufficiently large agglomerates, which is due to the limited spatial extension of the clusters that are actually quasi-fractal. In the present study, a cutoff function of the pair correlation function is considered for both larger and smaller scales. This enables a more accurate morphological description valid for any cluster size is to be given taking into account the polydispersity of the primary spheres. This new analytical morphological description relying on 5 parameters, is presented here for the first time. The physical range covered by these morphological parameters is determined based on virtually generated Diffusion Limited Cluster Agglomeration. Finally, the model is used to express the fractal prefactor and structure factors and their dependence on agglomerate size and morphological parameters is investigated.

Motility-limited aggregation of mammary epithelial cells into fractal-like clusters

Migratory cells transition between dispersed individuals and multicellular collectives during development, wound healing, and cancer. These transitions are associated with coordinated behaviors as well as arrested motility at high cell densities, but remain poorly understood at lower cell densities. Here, we show that dispersed mammary epithelial cells organize into arrested, fractal-like clusters at low density in reduced epidermal growth factor (EGF). These clusters exhibit a branched architecture with a fractal dimension of [Formula: see text], reminiscent of diffusion-limited aggregation of nonliving colloidal particles. First, cells display diminished motility in reduced EGF, which permits irreversible adhesion upon cell-cell contact. Subsequently, leader cells emerge that guide collectively migrating strands and connect clusters into space-filling networks. Thus, this living system exhibits gelation-like arrest at low cell densities, analogous to the glass-like arrest of epithelial monolayers at high cell densities. We quantitatively capture these behaviors with a jamming-like phase diagram based on local cell density and EGF. These individual to collective transitions represent an intriguing link between living and nonliving systems, with potential relevance for epithelial morphogenesis into branched architectures.

Heating ability of magnetic nanoparticles with cubic and combined anisotropy.

The low frequency hysteresis loops and specific absorption rate (SAR) of assemblies of magnetite nanoparticles with cubic anisotropy are calculated in the diameter range of D = 20–60 nm taking into account both thermal fluctuations of the particle magnetic moments and strong magneto–dipole interaction in assemblies of fractal-like clusters of nanoparticles. Similar calculations are also performed for assemblies of slightly elongated magnetite nanoparticles having combined magnetic anisotropy. A substantial dependence of the SAR on the nanoparticle diameter is obtained for all cases investigated. Due to the influence of the magneto–dipole interaction, the SAR of fractal clusters of nanoparticles decreases considerably in comparison with that for weakly interacting nanoparticles. However, the ability of magnetic nanoparticle assemblies to generate heat can be improved if the nanoparticles are covered by nonmagnetic shells of appreciable thickness.

Polyelectrolyte character of rigid rod peptide bundlemer chains constructed via hierarchical self-assembly.

Short α-helical peptides were computationally designed to self-assemble into robust coiled coils that are antiparallel, homotetrameric bundles. These peptide bundle units, or 'bundlemers', have been utilized as anisotropic building blocks to construct bundlemer-based polymers via a hierarchical, hybrid physical-covalent assembly pathway. The bundlemer chains were constructed using short linker connections via 'click' chemistry reactions between the N-termini of bundlemer constituent peptides. The resulting bundlemer chains appear as extremely rigid, cylindrical rods in transmission electron microscopy (TEM) images. Small angle neutron scattering (SANS) shows that these bundlemer chains exist as individual rods in solution with a cross-section that is equal to that of a single coiled coil bundlemer building block of ≈20 Å. SANS further confirms that the interparticle solution structure of the rigid rod bundlemer chains is heterogeneous and responsive to solution conditions, such as ionic-strength and pH. Due to their peptidic constitution, the bundlemer assemblies behave like polyelectrolytes that carry an average charge density of approximately 3 charges per bundlemer as determined from SANS structure factor data fitting, which describes the repulsion between charged rods in solution. This repulsion manifests as a correlation hole in the scattering profile that is suppressed by dilution or addition of salt. Presence of rod cluster aggregates with a mass fractal dimension of ≈2.5 is also confirmed across all samples. The formation of such dense, fractal-like cluster aggregates in a solution of net repulsive rods is a unique example of the subtle balance between short-range attraction and long-rage repulsion interactions in proteins and other biomaterials. With computational control of constituent peptide sequences, it is further possible to deconvolute the underlying sequence driven structure-property relationships in the modular bundlemer chains.

Selective Molecular Transport in Thermoresponsive Polymer Membranes: Role of Nanoscale Hydration and Fluctuations

For a wide range of modern soft functional materials the selective transport of sub-nanometer-sized molecules (`penetrants') through a stimuli-responsive polymeric membrane is key to the desired function. In this study, we investigate the diffusion properties of penetrants ranging from non-polar to polar molecules and ions in a matrix of collapsed Poly(N-isopropylacrylamide) (PNIPAM) polymers in water by means of extensive molecular dynamics simulations. We find that the water distributes heterogeneously in fractal-like cluster structures embedded in the nanometer-sized voids of the polymer matrix. The nano-clustered water acts as an important player in the penetrant diffusion, which proceeds via a hopping mechanism through `wet' transition states: the penetrants hop from one void to another via transient water channels opened by rare but decisive polymer fluctuations. The diffusivities of the studied penetrants extend over almost five orders of magnitude and thus enable a formulation of an analytical scaling relation with a clear non-Stokesian, exponential dependence of the diffusion coefficient on the penetrant's radius for the uncharged penetrants. Charged penetrants (ions) behave differently as they get captured in large isolated water clusters. Finally, we find large energetic activation barriers for hopping, which significantly depend on the hydration state and thereby challenge available transport theories.

Swarming Migration of Co-attracting Mesenchymal Cells into Fractal-Like Epithelial Clusters

Cell-secreted factors can function as biochemical “signal relays” to coordinate collective migration over long distances. For instance, cells “co-attract” by releasing chemokines that they also chemotax towards, driving aggregation and cohesion during neural crest development, mesenchymal condensation, and neutrophil homing. Here, we show that mammary epithelial cells cultured in reduced serum conditions display a mutual attraction that results in self-organizing migration patterns. Ordinarily, the migration of spatially dispersed cells in EGF-supplemented media can be understood as a persistent random walk. Instead, cells cultured in EGF-depleted media display directed migration towards each other through co-attraction, resulting in multicellular aggregation. Subsequently, aggregated cells establish cell-cell junctions with E-cadherin, reminiscent of a mesenchymal-to-epithelial transition (MET). Furthermore, cells exhibit substantial morphological changes, extending outward protrusions towards other cells in the proximity. As a consequence, these aggregates display a highly branched, fractal-like architecture. This biological behavior has physical analogies with diffusion-limited aggregation of colloidal particles, which results in a gelation or jamming transition even at very subconfluent cell densities. We implement a computational model based on dissipative particle dynamics that recapitulates clustering and aggregation. These results may reveal new quantitative insights into swarming cell migration during neural crest development, mesenchymal condensation, and tumor metastasis.

Flowing suspensions of carbon black with high electronic conductivity for flow applications: Comparison between carbons black and exhibition of specific aggregation of carbon particles

Flow batteries and flow capacitors are promising technologies to store and generate electrical power. However, to increase their energy performances, low viscosity, electronic conductive suspensions loaded with active material are required. Comparing the behavior of three types of carbon black particles in water suspensions, we show that compressed acetylene carbon black particles suspensions display a slow variation of viscosity and conductivity as a function of concentration. It allows reaching intermediate viscosity (1 Pa s for a shear rate of 10 s−1) with high electronic conductivity between 0.1 and 5 mS/cm. This behavior is very promising for flow applications. At small range, attractive van der Walls interactions between carbon aggregates dominate. However, at longer range, compressed acetylene carbon black particles are highly attractive in water. After shearing with emulsifier, fractal-like shape clusters are obtained through a diffusion limited aggregation process. These fractal clusters constitute the building blocks of a flexible connected network. By contrast, for the two other investigated carbons, an energy barrier has to be overcome to enable aggregation. The clusters are compact and result from a reaction limited aggregation process. For these two carbons, the conductivity and the viscosity vary abruptly at percolation which is not suitable for flow devices.

Discrete dipole approximation for black carbon-containing aerosols in arbitrary mixing state: A hybrid discretization scheme

An accurate and efficient simulation of light scattering by an atmospheric black carbon (BC)-containing aerosol—a fractal-like cluster of hundreds of carbon monomers that is internally mixed with other aerosol compounds such as sulfates, organics, and water—remains challenging owing to the enormous diversities of such aerosols׳ size, shape, and mixing state. Although the discrete dipole approximation (DDA) is theoretically an exact numerical method that is applicable to arbitrary non-spherical inhomogeneous targets, in practice, it suffers from severe granularity-induced error and degradation of computational efficiency for such extremely complex targets. To solve this drawback, we propose herein a hybrid DDA method designed for arbitrary BC-containing aerosols: the monomer-dipole assumption is applied to a cluster of carbon monomers, whereas the efficient cubic-lattice discretization is applied to the remaining particle volume consisting of other materials. The hybrid DDA is free from the error induced by the surface granularity of carbon monomers that occurs in conventional cubic-lattice DDA. In the hybrid DDA, we successfully mitigate the artifact of neglecting the higher-order multipoles in the monomer-dipole assumption by incorporating the magnetic dipole in addition to the electric dipole into our DDA formulations. Our numerical experiments show that the hybrid DDA method is an efficient light-scattering solver for BC-containing aerosols in arbitrary mixing states. The hybrid DDA could be also useful for a cluster of metallic nanospheres associated with other dielectric materials.

Shear-Induced Reactive Gelation.

In this work, we describe a method for the production of porous polymer materials in the form of particles characterized by narrow pore size distribution using the principle of shear-induced reactive gelation. Poly(styrene-co-divinylbenzene) primary particles with diameter ranging from 80 to 200 nm are used as building blocks, which are assembled into fractal-like clusters when exposed to high shear rates generated in a microchannel. It was found that independent of the primary particle size, it is possible to modulate the internal structure of formed fractal-like aggregates having fractal dimension ranging from 2.4 to 2.7 by varying the residence time in the microchannel. Thermally induced postpolymerization was used to increase the mechanical resilience of such formed clusters. Primary particle interpenetration was observed by SEM and confirmed by light scattering resulting in an increase of fractal dimension. Nitrogen sorption measurements and mercury porosimetry confirmed formation of a porous material with surface area ranging from 20 to 40 m(2)/g characterized by porosity of 70% and narrow pore size distribution with an average diameter around 700 nm without the presence of any micropores. The strong perfusive character of the synthesized material was confirmed by the existence of a plateau of the height equivalent to a theoretical plate measured at high reduced velocities using a chromatographic column packed with the synthesized microclusters.

Cooperative self-assembly of peptide gelators and proteins.

Molecular self-assembly provides a versatile route for the production of nanoscale materials for medical and technological applications. Herein, we demonstrate that the cooperative self-assembly of amphiphilic small molecules and proteins can have drastic effects on supramolecular nanostructuring of resulting materials. We report that mesoscale, fractal-like clusters of proteins form at concentrations that are orders of magnitude lower compared to those usually associated with molecular crowding at room temperature. These protein clusters have pronounced effects on the molecular self-assembly of aromatic peptide amphiphiles (fluorenylmethoxycarbonyl- dipeptides), resulting in a reversal of chiral organization and enhanced order through templating and binding. Moreover, the morphological and mechanical properties of the resultant nanostructured gels can be controlled by the cooperative self-assembly of peptides and protein fractal clusters, having implications for biomedical applications where proteins and peptides are both present. In addition, fundamental insights into cooperative interplay of molecular interactions and confinement by clusters of chiral macromolecules is relevant to gaining understanding of the molecular mechanisms of relevance to the origin of life and development of synthetic mimics of living systems.

Coherent backscattering by discrete random media composed of clusters of spherical particles

The approximate method of accounting for the weak localization effect in discrete random media makes it possible to calculate the intensity and the degree of linear polarization of radiation reflected by a semi-infinite layer containing randomly oriented clusters of identical nonabsorbing spherical particles. The paper is focused on the analysis of the influence of the medium properties – the concentration of the clusters, their structure, the sizes and the number of constituent monomers – on the phase dependence of the characteristics of the interference (coherent) component of radiation. Three types of clusters were chosen for the analysis: a fractal-like cluster and two random clusters with different packing densities of monomers. Three values of the size parameters of monomers were considered: 0.5, 1.0, and 1.5. The chosen values of the refractive index of monomers correspond to that of magnesium oxide in the visible spectral range. This allowed us to compare the theoretical phase curves of the degree of linear polarization of light reflected by the medium with the data of measurements by Lyot. For all of the considered cluster types, starting from some minimal number of constituent monomers, which is typical of the specified cluster's type and the monomer's size, the characteristics of the coherent component were found to be practically independent of the number of constituents. In particular, in the phase-angle range from 0° to 15°, the branch of negative polarization measured for a layer of magnesium oxide by Lyot can be well described with several models of the media containing either the fractal-like clusters or the random clusters with the packing density larger than 30%. In these models, the number and sizes of constituent monomers vary. To select from the whole set of “suitable” clusters those the optical characteristics of which actually correspond to characteristics of an elementary volume of the specified medium, additional information is required. For example, this can be the results of polarization measurements in a wider phase interval that includes a zone of the dominating influence of incoherent (diffuse) scattering.

Improving Device Efficiency of Polymer/Fullerene Bulk Heterojunction Solar Cells Through Enhanced Crystallinity and Reduced Grain Boundaries Induced by Solvent Additives

The power conversion efficiency of a device incorporating a crystalline polymer/fullerene thin film improves from 5% to 7.3% – a relative increase of 45% – when an additive, diiodohexane (DIH), is present during processing. The DIH-processed active layer exhibits substantially enhanced polymer crystallinity and smaller fractal-like fullerene clusters. Detailed facts of importance to specialist readers are published as ”Supporting Information”. Such documents are peer-reviewed, but not copy-edited or typeset. They are made available as submitted by the authors. Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.

Numerical calculation of Mueller matrices of randomly distributed soot cluster agglomerates

Cluster-cluster aggregation model based on Monte Carlo method is used to simulate the structure of the fractal-like soot cluster agglomerates consisting of spherical primary particles，and Mueller-matrix elements of fractal-like soot cluster agglomerates are calculated based on the discrete dipole approximation method. The numerical relations between the Mueller-matrix elements and the total number and the diameter of the primary sphere in agglomerate are discussed for different incident wavelengths，which provides an effective theoretical method for studing the formation mechanism，structural characteristics and scattering characteristics of randomly distributed cluster agglomerates.

Controlled Aggregation of Functionalized Gold Nanoparticles with a Novel Conjugated Oligomer

A new route is developed to control the self-assembly of gold nanoparticles (AuNPs) functionalized with a novel pyridyl-ended porphyrin-oligo(p-phenylene vinylene) conjugated oligomer (P-OPV-Py) into branched-rods, chain-networks, uniform fractal-like Au clusters, and larger nanoparticles. The techniques of optical spectroscopy, transmission electron microscopy (TEM) and dynamic light scattering (DLS) are used to characterize the self-assembly in various solvent systems (pure toluene, CHCl(3)/toluene, THF/toluene and n-butanol/toluene). A combination of the ligand/AuNPs molar ratio and relative concentration serves as the driving force to control the size and shape of P-OPV-Py capped AuNPs.

Fractal aggregates of the Pt nanoparticles synthesized by the polyol process and poly(N-vinyl-2-pyrrolidone) reduction

Small-angle X-ray scattering was used to characterize the size and aggregation behavior of the Pt nanoparticles synthesized by the polyol process and the unusual poly(N-vinyl-2-pyrrolidone) (PVP) reduction. With formaldehyde (HCHO) as the reduction agent, the Pt nanoparticles synthesized in aqueous solutions with a high PVP/PtCl4 weight ratio were characterized by short rods with a 70% polydispersity in rod length. The size and size distribution of the rod-like Pt nanoparticles (3 nm in rod length and 2 nm in rod diameter) are consistent with the corresponding transmission electron microscopy image. With a comparable PVP/PtCl4 weight ratio in the aqueous solution containing HCHO, the high number density of reduced Pt nanoparticles led to a fractal-like aggregation with a fractal dimension of 2.1 and a correlation length of ~30 nm. We also demonstrated that Pt nanoparticles can be synthesized by PVP reduction at 323 K without HCHO. The particle size and the clustering behavior of the Pt nanoparticles reduced by PVP are closely related to the PVP concentration in the solution. Both the Pt nanoparticles synthesized in the commonly used polyol process and the unusual PVP reduction form fractal-like clusters via the PVP–metal nanoparticle association when the number density of the Pt nanoparticles in the solutions is high.

Coherent backscattering effects for discrete random media: Numerical and theoretical results

Manifestation of the backscattering enhancement phenomenon in the reflection matrix elements of the coherent component of scattered radiation is considered. The dependence of the coherent backscattering effects on the microphysical properties of the medium scatterers are investigated. It is shown that random media of fractal-like clusters exhibit brightness and polarization opposition effects, which are like those observed for some atmosphereless Solar system bodies. Conditions for a bimodal angle dependence in the degree of linear polarization are discussed and the manifestation of the enhanced backscattering phenomenon in the intensity of scattered radiation is studied.

Homogeneous aggregation in magnetic fluids. Theoretical model of fractal-like cluster formation

This paper is devoted to the theoretical study of diffusion-limited aggregation in magnetic fluids taking into account the decrease of free particle concentration outside the aggregates. The internal structure of these aggregates is investigated.

Primordial structure of massive black hole clusters

We describe a mechanism of the primordial black holes formation that can explain the existence of a population of supermassive black holes in galactic bulges. The mechanism is based on the formation of black holes from closed domain walls. The origin of such domain walls could be a result of the evolution of an effectively massless scalar field during inflation. The initial non-equilibrium distribution of the scalar field imposed by background de Sitter fluctuations gives rise to the spectrum of black holes, which covers a wide range of masses—from superheavy ones down to deeply subsolar. The primordial black holes of smaller masses are concentrated around the most massive ones within a fractal-like cluster.