Fractal Dimension Of Clusters Research Articles

Modelling nanoparticle agglomeration in the transition regime: A comparison between detailed Langevin Dynamics and population balance calculations

The prediction of nanoparticle agglomeration using population balance modelling is a difficult endeavour as continuously growing clusters undergo different collision regimes and exhibit complex, fractal-like structures. A key model parameter in this context is the collision kernel which governs the agglomeration dynamics within the particle population. In this work, we exploit a recently developed model for Brownian coagulation to solve the population balance equation for an initially monodisperse system of spherical nanoparticles in a quiescent gas. Since the model was originally developed for the limiting cases of ballistic and diffusive agglomeration, we incorporate a suitable interpolation scheme to extend the kernel expression to the more general transition regime.The results obtained from population balance calculations are validated by means of detailed Langevin Dynamics simulations where agglomerates and their individual trajectories are resolved. A comparison of agglomeration dynamics and evolving size distributions reveals strong agreement between the two approaches which corroborates the accuracy of the collision kernel model. In contrast, conventional models for the kernel are not able to reproduce the results from detailed Langevin Dynamics simulations as they noticeably underpredict the width of the agglomerate size distribution.Analysis of the agglomerates’ morphology distribution further reveals that fractal dimensions of clusters are normally distributed and can be characterized by a size-dependent mean value and standard deviation. We find, however, that the specifics of the morphology distribution do not need to be included in the kernel as omission causes only minor changes with respect to the collision dynamics. Using a size-dependent averaged fractal dimension is sufficient within PBE calculations.

Subclass Effects on Self-Association and Viscosity of Monoclonal Antibodies at High Concentrations.

The effects of a subclass of monoclonal antibodies (mAbs) on protein-protein interactions, formation of reversible oligomers (clusters), and viscosity (η) are not well understood at high concentrations. Herein, we quantify a short-range anisotropic attraction between the complementarity-determining region (CDR) and CH3 domains (KCDR-CH3) for vedolizumab IgG1, IgG2, or IgG4 subclasses by fitting small-angle X-ray scattering (SAXS) structure factor Seff(q) data with an extensive library of 12-bead coarse-grained (CG) molecular dynamics simulations. The KCDR-CH3 bead attraction strength was isolated from the strength of long-range electrostatic repulsion for the full mAb, which was determined from the theoretical net charge and a scaling parameter ψ to account for solvent accessibility and ion pairing. At low ionic strength (IS), the strongest short-range attraction (KCDR-CH3) and consequently the largest clusters and highest η were observed with IgG1, the subclass with the most positively charged CH3 domain. Furthermore, the trend in KCDR-CH3 with the subclass followed the electrostatic interaction energy between the CDR and CH3 regions calculated with the BioLuminate software using the 3D mAb structure and molecular interaction potentials. Whereas the equilibrium cluster size distributions and fractal dimensions were determined from fits of SAXS with the MD simulations, the degree of cluster rigidity under flow was estimated from the experimental η with a phenomenological model. For the systems with the largest clusters, especially IgG1, the inefficient packing of mAbs in the clusters played the largest role in increasing η, whereas for other systems, the relative contribution from stress produced by the clusters was more significant. The ability to relate η to short-range attraction from SAXS measurements at high concentrations and to theoretical characterization of electrostatic patches on the 3D surface is not only of fundamental interest but also of practical value for mAb discovery, processing, formulation, and subcutaneous delivery.

Aggregation and disaggregation processes in clusters of particles under flow: Simple numerical and theoretical insights

Aggregation and disaggregation of clusters of attractive particles under flow are studied and highlight the growth of the average steady-state cluster size as a power law of the adhesion number, a dimensionless number that quantifies the ratio of attractive forces to shear stress. Such a power-law scaling results from the competition between aggregation and disaggregation processes, as previously reported. Here we rationalize this behavior through a model based on an energy function, for which minimization yields the power-law exponent in terms of the cluster fractal dimension, in good agreement with the present simulations and with previous works.

Characterizing Experimental Monoclonal Antibody Interactions and Clustering Using a Coarse-Grained Simulation Library and a Viscosity Model.

Attractive protein-protein interactions in concentrated monoclonal antibody (mAb) solutions may lead to the formation of clusters that increase viscosity. Here, we propose an analytical model that relates mAb solution viscosity to clustering by accounting for the contributions of suboptimal mAb packing within a cluster and cluster fractal dimension. The influence of short-range, anisotropic attractions and long-range Coulombic repulsion on cluster properties is investigated by analyzing the cluster-size distributions, cluster fractal dimensions, radial distribution functions, and static structure factors from a library of coarse-grained molecular dynamics simulations. The library spans a vast range of mAb charges and attractive interactions in solutions of varying ionic strength. We present a framework for combining the viscosity model and simulation library to successfully characterize the attraction, repulsion, and clustering of an experimental mAb in three different pH and cosolute conditions by fitting the measured viscosity or structure factor from small-angle X-ray scattering. At low ionic strength, the cluster-size distribution is impacted by strong charges, and both the viscosity and net charge or structure factor and net charge must be considered to deconvolute the effects of short-range attraction and long-range repulsion.

Influence of carbon binder domain on the performance of lithium‐ion batteries: Impact of size and fractal dimension

AbstractA lithium‐ion battery (LIB) cathode comprises three major components: active material, electrical conductivity additive, and binder. The combination of binder and electrical conductivity additive leads to the formation of composite clusters known as the carbon binder domain (CBD) clusters. Preparation of a LIB cathode strongly influences the dispersion of the above‐mentioned constituents leading to the formation of distinct pore and electrical conduction networks. The resulting structure thus governs the performance of LIBs. The presence of CBD is essential for the structural integrity and sufficient electrical conductivity of the LIB cathode. However, CBD abundance in LIB cathodes leads to unfavorable gravimetrical and volumetrical consequences owing to its electrochemical inertness. Increasing CBD content adds to the weight of the LIBs, thus negatively impacting the energy density. Furthermore, increased electrical conductivity is won at a cost of ionic conductivity as CBD clusters breach the pore networks in the cathode microstructure. The following study establishes a link between the various possibilities of CBD cluster size and fractal dimension that may eventualize during the mixing process of slurry preparation to the resulting microstructural properties and hence to the performance of LIBs by means of idealized cathode geometries. Since the performance determining processes occur at the microstructural scale, which are often very tedious to study via experimental research, the study makes use of spatially resolving microstructural, numerical, simulations. The results demonstrate that the CBD cluster size has a strong influence on the cathode microstructure. The CBD cluster fractal dimension on the other hand displayed a minor influence on the structural properties of the cathode, and the size of the cluster primary particles was shown to be the dominant factor. Finally, performance evaluation simulations confirmed the trends seen in structural properties with changing cluster size and fractal dimension.

Modification of the RLA model for presenting a cluster system of a composite material with a fractal filler structure

The paper proposes a modification of the diffusion-limited aggregation model to study the properties of a cluster system. A computational experiment to determine the mutual influence of the sticking probability and the volume concentration of particles on the formation of fractal clusters in a cluster system was carried out in accordance with the second-order orthogonal central compositional plan (OCCP). As a result of a computational experiment in accordance with the OCCP, an equation was obtained for the dependence of the mass fractal dimension of clusters on the volume of particle concentration and the probability of adhesion of diffusing particles and cluster particles in the adhesion zone. This dependence was obtained in a range of volume concentration of particles from 2 to 5 % and the probability of adhesion of diffusing particles and particles of clusters in the adhesion zone from 0.2 to 1.

Effect of nanoparticle aggregation on the thermal radiation properties of nanofluids: an experimental and theoretical study

Nanofluids show significant enhancement in the absorption of solar energy. However, little work has been done on the effect of nanoparticle aggregation on the thermal radiation properties of nanofluids up to now. In this work, we systematically investigate the thermal radiation properties of nanofluids with nanoparticle aggregation from experiment and theory. On the experimental side, titanium dioxide/silver (TiO2@Ag) plasmonic nanofluids are prepared in distilled water and the spectral transmittance of nanofluids with different degrees of aggregation is measured. The result shows that the size of aggregated nanoparticle cluster has a significant effect on the absorption of nanofluids. On the theoretical side, a theoretical model is developed to calculate the thermal radiation properties of nanofluids with nanoparticle aggregation. By applying the multiple sphere T-matrix (MSTM) method and the Monte Carlo (MC) method, we investigate the effects of nanoparticle aggregation, cluster fractal dimensions and cluster sizes on the thermal radiation properties of nanofluids. The results indicate that our theoretical model can predict the thermal radiation properties of nanofluids with nanoparticle aggregation more reasonably. This is of great significance to understand the mechanism that affects the thermal radiation properties of nanofluids and to actively design nanofluids.

Do String-like Cooperative Motions Predict Relaxation Times in Glass-Forming Liquids?

The Adam-Gibbs theory of glass formation posits that the growth in the activation barrier of fragile liquids on cooling emerges from a loss of configurational entropy and concomitant growth in "cooperatively rearranging regions" (CRRs). A body of literature over 2 decades has suggested that "string-like" cooperatively rearranging clusters observed in molecular simulations may be these CRRs-a scenario that would have profound implications for the understanding of the glass transition. The central element of this postulate is the report of an apparent zero-parameter relationship between the mass of string-like CRRs and the relaxation time. Here, we show, based on molecular dynamics simulations of multiple glass-forming liquids, that this finding is the result of an implicit adjustable parameter-a "replacement distance". This parameter is equivalent to an adjustable exponent within a generalized Adam-Gibbs relation, such that it tunes the entire functional form of the relation. Moreover, we are unable to find any objective criterion, based on the radial distribution function or the cluster fractal dimension, for selecting this replacement distance across multiple systems. We conclude that the present data do not establish that string-like cooperative rearrangements, as presently defined, are predictive of segmental relaxation via an Adam-Gibbs-like physical model.

Heterogeneity in polymer networks formed by a single copolymerization reaction: I. Gelation and pre-gel structure

Coarse-grained molecular dynamics simulations of the formation of 3- and 6-functional polymer networks explicitly shows gelation via large dendritic clusters. The heterogeneous network structures are studied in detail; clusters have pendant structures and occupy significant volume fraction. Cluster fractal dimension and the Fisher exponent for cluster size are consistent with percolation theory and kinetic gelation models. Several criteria for determining gelation yield consistent results somewhat higher than the estimate for ideal networks on regular lattices. Almost all crosslink loop formation occurs after gelation just as traditional statistical theories assume. The presence and size of these clusters provide a natural explanation for density variations seen in microscopy studies of crosslinked polymers. The second paper in this series describes how the network structure develops after gelation. Large pendant structures, and more localized defects, persist and comprise a substantial fraction of material that is not contributing to network strength.

FRACTAL ANALYSIS OF STRESS SENSITIVITY OF PERMEABILITY IN POROUS MEDIA

A permeability model for porous media considering the stress sensitivity is derived based on mechanics of materials and the fractal characteristics of solid cluster size distribution. The permeability of porous media considering the stress sensitivity is related to solid cluster fractal dimension, solid cluster fractal tortuosity dimension, solid cluster minimum diameter and solid cluster maximum diameter, Young's modulus, Poisson's ratio, as well as power index. Every parameter has clear physical meaning without the use of empirical constants. The model predictions of permeability show good agreement with those obtained by the available experimental expression. The proposed model may be conducible to a better understanding of the mechanism for flow in elastic porous media.

Growth of multiparticle aggregates in sedimenting suspensions

AbstractThe process of multiparticle aggregation in a dilute sedimenting suspension is rigorously simulated, with precise hydrodynamical interactions. The primary particles are monodisperse non-Brownian spheres at zero Reynolds number, with short-range molecular attractions. The rigid aggregates grow, as they settle downwards, by sequential particle addition – a valid assumption for dilute suspensions during the initial stages. The growth starts from doublet–particle interaction, but the indeterminate initial doublet concentration does not affect the results for cluster geometry and settling velocity. A new particle is generated far below a cluster with uniform probability density, and many trial particle–cluster relative trajectories are computed with high accuracy until a collision is found. The new cluster is then assumed to be rigid and allowed to reach a steady sedimentation regime (which is a spiral motion around the axis of steady rotation, ASR) before another particle is added, and so on. The ASR is typically far away from the cluster centre of mass. The Stokes flow solution algorithm for particle–cluster interaction works very efficiently with high-order multipoles (to order 100) and is extended to arbitrarily small particle–cluster separations by a geometry perturbation adapted from the conductivity simulations of Zinchenko (Phil. Trans. R. Soc. Lond. A, 1998, vol. 356, pp. 2953–2998). Clusters are generated to $N=100$ spheres, with extensive averaging over many growth realizations. The fractal scaling $\sim N^{0.48}$ for the cluster settling speed is quickly attained once $N\geq 25$, and the exponent 0.48 is practically independent of the strength of molecular forces. The cluster fractal dimension is predicted to be $d_f=1.91\pm 0.02$ (in contrast to the existing views that sequential addition can only produce high-$d_f$ clusters). Several average characteristics of the cluster size are also computed. The theoretical settling speed has no adjustable parameters and agrees reasonably well with prior experiments for a moderately polydisperse system in a broad range of cluster sizes.

A Fast Algorithm for Invasion Percolation

We present a computationally fast Invasion Percolation (IP) algorithm. IP is a numerical approach for generating realistic fluid distributions for quasi-static (i.e., slow) immiscible fluid invasion in porous media. The algorithm proposed here uses a binary-tree data structure to identify the site (pore) connected to the invasion cluster that is the next to be invaded. Gravity is included. Trapping is not explicitly treated in the numerical examples but can be added, for example, using a Hoshen–Kopelman algorithm. Computation time to percolation for a 3D system having \(N\) total sites and \(M\) invaded sites at percolation goes as \(O(M \log M)\) for the proposed binary-tree algorithm and as \(O(M N)\) for a standard implementation of IP that searches through all of the uninvaded sites at each step. The relation between \(M\) and \(N\) is \(M = N^{D/E}\), where \(D\) is the fractal dimension of an infinite cluster and \(E\) is Euclidean space dimension. In numerical practice, on finite-sized cubic lattices with invasion structures influenced by the injection boundary and boundary conditions lateral to the flow direction, we observe the scaling \(M = N^{0.852}\) in 3D (valid through the second decimal place) instead of \(M= N^{0.843}\) based on the infinite cluster fractal dimension \(D=2.53\).

Population Balance Modeling of Aggregation and Coalescence in Colloidal Systems

A complex interplay between aggregation and coalescence occurs in many colloidal polymeric systems and determines the morphology of the final clusters of primary particles. To describe this process, a 2D population balance equation (PBE) based on cluster mass and fractal dimension is solved, employing a discretization method based on Gaussian basis functions. To prove the general reliability of the model and to show its potential, parametric simulations are performed employing both diffusion‐limited‐cluster aggregation (DLCA) and reaction‐limited‐cluster‐aggregation (RLCA) kernels and different coalescence rates. It turns out that in both DLCA and RLCA regimes, a faster coalescence leads to smaller sized and more compact clusters, whereas a slow coalescence promotes the formation of highly reactive fractals, resulting in larger aggregates.

Colloidal gelation with variable attraction energy

We present an approximation scheme to the master kinetic equations for aggregation and gelation with thermal breakup in colloidal systems with variable attraction energy. With the cluster fractal dimension df as the only phenomenological parameter, rich physical behavior is predicted. The viscosity, the gelation time, and the cluster size are predicted in closed form analytically as a function of time, initial volume fraction, and attraction energy by combining the reversible clustering kinetics with an approximate hydrodynamic model. The fractal dimension df modulates the time evolution of cluster size, lag time and gelation time, and of the viscosity. The gelation transition is strongly nonequilibrium and time-dependent in the unstable region of the state diagram of colloids where the association rate is larger than the dissociation rate. Only upon approaching conditions where the initial association and the dissociation rates are comparable for all species (which is a condition for the detailed balance to be satisfied) aggregation can occur with df = 3. In this limit, homogeneous nucleation followed by Lifshitz-Slyozov coarsening is recovered. In this limited region of the state diagram the macroscopic gelation process is likely to be driven by large spontaneous fluctuations associated with spinodal decomposition.

Fractals and spatial statistics of point patterns

The relationship between fractal point pattern modeling and statistical methods of parameter estimation in point-process modeling is reviewed. Statistical estimation of the cluster fractal dimension by using Ripley’s K-function has advantages in comparison with the more commonly used methods of box-counting and cluster fractal dimension estimation because it corrects for edge effects, not only for rectangular study areas but also for study areas with curved boundaries determined by regional geology. Application of box-counting to estimate the fractal dimension of point patterns has the disadvantage that, in general, it is subject to relatively strong “roll-off” effects for smaller boxes. Point patterns used for example in this paper are mainly for gold deposits in the Abitibi volcanic belt on the Canadian Shield. Additionally, it is proposed that, worldwide, the local point patterns of podiform Cr, volcanogenic massive sulphide and porphyry copper deposits, which are spatially distributed within irregularly shaped favorable tracts, satisfy the fractal clustering model with similar fractal dimensions. The problem of deposit size (metal tonnage) is also considered. Several examples are provided of cases in which the Pareto distribution provides good results for the largest deposits in metal size-frequency distribution modeling.

Testing and Evaluation on Fractal Characteristic of Cementing Materials Powder

The cement powder materials are evaluated by fractal theory. Fractal characteristic parameters are analysed and evaluated by the method of laser particle size analyzer, microscope, transmission electron microscope. In this paper, cement powder materials has good self-similarity. Fractal dimension of grading is tested and evaluated by laser particle size analyzer, fractal dimension of cluster is tested by microscope, fractal characteristic of particle distribution is tested by transmission electron microscope. Compared with traditional weight of screen residue and specific surface area, those methods are more careful. Fineness fractal parameters and activity of ultra fine fly ash and pulverized slag have good linear dependence relation that is evaluated by fractal dimensions. Fractal characteristics provide an important basis to further explore the inherent relation of cement powder materials and concrete material density effect.

Effect of synthesis conditions on the microstructure of TEOS derived silica hydrogels synthesized by the alcohol-free sol–gel route

Silica matrices synthesized from a pre-hydrolysis step in ethanol followed by alcohol removal at low pressure distillation, and condensation in water, are suitable for encapsulation of biomolecules and microorganisms and building bioactive materials with optimized optical properties. Here we analyze the microstructure of these hydrogels from the dependence of I(q) data acquired from SAXS experiments over a wide range of silica concentration and pH employed in the condensation step. From the resulting data it is shown that there is a clear correlation between the microscopic parameters—cluster fractal dimension (D), elementary particle radius (a) and cluster gyration radius (R)—with the attenuation of visible light when the condensation step proceeds at pH < 6. At higher pHs, there is a steep dependence of the cluster density (~R D−3) with the condensation pH, and non-monotonous changes of attenuance are less than 20%, revealing the complexity of the system. These results, which were obtained for a wide pH and silica concentration range, reinforce the idea that the behavior of gels determined in a restricted interval of synthesis variables cannot be extrapolated, and comparison of gelation times is not enough for predicting their properties.

Rheological characterization of edible films made from collagen colloidal particle suspensions

A rheological characterization of collagen edible films is proposed through the experimental evaluation of shear elastic modulus, toughness, hydration capability and stress and stretch ratio at fracture. These properties are obtained from hydration, simple extension and compression tests and analyzed through the BST (Blatz, Sharda, & Tschoegl, 1974) hyperelastic model. Also precursor collagen particle suspensions, with solid concentrations in the range 0.5–4% w/w, are studied to evaluate their thixotropic responses in sudden imposed shear rates and shear loops. These responses are a consequence of particle aggregations leading to cluster formations. The particle size distribution function of suspensions is determined via scanning electron microscopy. Films are formed by casting and then fixed through either a coagulation process with a salt solution or a chemical process involving covalent cross-links with glutaraldehyde. Results indicate that these collagen films behave as colloidal particulate networks composed of particles and clusters. Also it is found that the fractal dimension of clusters is an intrinsic property of collagen particle suspensions, independently from the maturation time. Clusters in these suspensions are formed with a rather open architecture (low fractal value) due to rather large particle attractive forces. The evaluation of the BST rheological parameters allows one the estimation of the energy amount required to get film fracture for different collagen and cross-linker concentrations. Since the final films have to satisfy several quality requirements before using, the interplay between their stiffness and toughness is presented and discussed.

Linking Trehalose Self-Association with Binary Aqueous Solution Equation of State

In aqueous solutions, trehalose possesses a high propensity to form hydrogen bonds with water as well as other trehalose molecules. This hydrogen bonding not only affects water structure but also promotes extensive concentration dependent aggregation of trehalose molecules, which may impact trehalose's role as a protective cosolute to biomacromolecules. To study the association of trehalose in aqueous solutions over a wide concentration range, we used molecular dynamics simulations based on two different force fields, as well as vapor pressure osmometry. By analyzing trehalose cluster size and fractal dimension in simulations, we estimate the cluster percolation threshold at 1.5-2.2 m. Experimentally, trehalose solutions showed positive deviations from ideal van't Hoff's law that grew with concentration. These variations in osmotic pressure can be explained using a simple equation of state that accounts for the repulsive excluded volume interactions between trehalose molecules as well as attractions reflected in sugar clustering. We find that simulations with both applied force fields result in reasonable representations of the solution equation of state. However, in contrast to experiments, the balance between the repulsive and attractive trehalose-trehalose interactions in simulations results in a slightly negative deviation from ideality, probably due to the moderately overaggregative nature of the force fields used.

Fractal dimension of cell clusters in effusion cytology

In this study, the fractal dimension (FD) of the ball-like tight clusters of known metastatic adenocarcinoma cells is compared with the tight cell clusters of the benign cells in the effusion fluid to find out the role of FD in differentiating malignant from benign cluster in effusion. A total of 68 and 72 images of cluster of cells from 12 cases of benign and 13 cases of metastatic pleural and peritoneal effusion were studied for FD. The mean FD of the cluster of cells of benign and malignant effusion was 1.4801 ± 0.23260 and 1.7175 ± 0.09006, respectively. Mann-Whitney U-test showed significant difference of FD between the cell clusters of malignant and benign effusion cases (P < 0.0001). Cytological features along with the measurement of FD of cell clusters in effusion cytology may be helpful in differentiating benign from malignant cases.