Scaling Law Exponent Research Articles

Scaling laws for nanoparticles – Online shape heterogeneity analysis by size exclusion chromatography coupled with multi-angle light scattering

Nanoparticle-based drug delivery systems are rising technologies to access challenging therapeutic targets. Following commercial success of lipid-based nanoparticles (LBNP), accruing understandings of nanoparticle structures and critical quality attributes through advanced analytics are beneficial to future clinical development and generalization of this delivery platform. The morphological attributes of nanoparticles, such as shape, can affect uptake, cell-interaction, drug release, circulation, and flow. Gaining an understanding of these structure-activity relationships in early-stage formulation development is important because mix morphologies can affect quality and potency but often exist before process control strategies are fully implemented. In this study, we used shape heterogeneous nanoparticle mixtures, containing various populations of liposomes and lipodisks, as a model system and developed an online semi-quantitative method to characterize the nanoparticle shape heterogeneity by size exclusion chromatography (SEC) coupled with multi-angle light scattering (MALS). The liposomes and lipodisks were separated in SEC when their sizes were ∼3 fold different. When the particles of different shapes were in similar sizes, size-based separation was not always feasible. Instead, light scattering data distinguished liposomes and lipodisks by the scaling law linking radius of gyration and molecular weight of the nanoparticles, enabling morphological identification. A semi-quantitative model was built based on the exponential correlation between the scaling law exponents and the ratios of liposomes and lipodisks. The model was applied to test 6 random formulations made with different compositions and manufacturing processes, and the predicted liposome percentage for 5 formulations was within 25 % absolute difference from the percentage determined by cryogenic electron microscopy (cryo-EM). We envision this method being routinely used to characterize liposome and lipodisk shape heterogeneity during formulation screening as well as on stability studies. Potentially, the method can be converted to in-process control method and extended to other categories of nanoparticles beyond liposomes.

Multiscale lineament analysis and permeability heterogeneity of fractured crystalline basement blocks

Abstract. The multiscale analysis of lineament patterns helps define the geometric scaling laws and the relationships between outcrop- and regional-scale structures in a fracture network. Here, we present a novel analytical and statistical workflow to analyze the geometrical and spatial organization properties of the Rolvsnes granodiorite lineament (fracture) network in the crystalline basement of southwestern Norway (Bømlo Island). The network shows a scale-invariant spatial distribution described by a fractal dimension D≈1.51, with lineament lengths distributed following a general scaling power law (exponent α=1.88). However, orientation-dependent analyses show that the identified sets vary their relative abundance and spatial organization and occupancy with scale, defining a hierarchical network. Lineament length, density, and intensity distributions of each set follow power-law scaling laws characterized by their own exponents. Thus, our multiscale, orientation-dependent statistical approach can aid in the identification of the hierarchical structure of the fracture network, quantifying the spatial heterogeneity of lineament sets and their related regional- vs. local-scale relevance. These results, integrated with field petrophysical analyses of fracture lineaments, can effectively improve the detail and accuracy of permeability prediction of heterogeneously fractured media. Our results also show how the geological and geometrical properties of the fracture network and analytical biases affect the results of multiscale analyses and how they must be critically assessed before extrapolating the conclusions to any other similar case study of fractured crystalline basement blocks.

Improved modeling of the solid-to-plasma transition of polystyrene ablator for laser direct-drive inertial confinement fusion hydrocodes.

The target performance of laser direct-drive inertial confinement fusion (ICF) can be limited by the development of hydrodynamic instabilities resulting from the nonhomegeneous laser absorption at the target surface, i.e., the laser imprint on the ablator. To understand and describe the formation of these instabilities, the early ablator evolution during the laser irradiation should be considered. In this work, an improved modeling of the solid-to-plasma transition of a polystyrene ablator for laser direct-drive ICF is proposed. This model is devoted to be implemented in hydrocodes dedicated to ICF which generally assume an initial plasma state. The present approach consists of the two-temperature model coupled to the electron, ion and neutral dynamics including the chemical fragmentation of polystyrene. The solid-to-plasma transition is shown to significantly influence the temporal evolution of both free electron density and temperatures, which can lead to different shock formation and propagation compared with an initial plasma state. The influence of the solid-to-plasma transition on the shock dynamics is evidenced by considering the scaling law of the pressure with respect to the laser intensity. The ablator transition is shown to modify the scaling law exponent compared with an initial plasma state.

Dynamical crossover in the transient quench dynamics of short-range transverse-field Ising models

Dynamical detection of quantum phases and phase transitions (QPT) in quenched systems with experimentally convenient initial states is a topic of interest from both theoretical and experimental perspectives. Quenched from polarized states, longitudinal magnetization decays exponentially to zero in time for the short-range transverse-field Ising model (TFIM) and hence, has a featureless steady state regime, which prevents it from exhibiting dynamical phase transitions of type-I. In this paper, we ask whether the transient regimes of such non-equilibrium processes probed by single-site observables, that is magnetization per site, could encode information about the underlying QPT. The decay rates of time-dependent and single-site observables exhibit a dynamical crossover that separates two dynamical regions, ordered and disordered, both of which have distinct nonequilibrium responses. We construct a dynamical order parameterlike quantity that exhibits a scaling law at the vicinity of the crossover. Our results reveal that scaling law exponent in short times at the close vicinity of the dynamical crossover is significantly different than the one predicted by analytical theory for long times. When integrability is strongly broken, the crossover boundary turns into a region that separates two other dynamical regions that act like dynamically-ordered and -disordered regimes.

Scaling laws in natural conversations among elderly people.

Language is a result of brain function; thus, impairment in cognitive function can result in language disorders. Understanding the aging of brain functions in terms of language processing is crucial for modern aging societies. Previous studies have shown that language characteristics, such as verbal fluency, are associated with cognitive functions. However, the scaling laws in language in elderly people remain poorly understood. In the current study, we recorded large-scale data of one million words from group conversations among healthy elderly people and analyzed the relationship between spoken language and cognitive functions in terms of scaling laws, namely, Zipf's law and Heaps' law. We found that word patterns followed these scaling laws irrespective of cognitive function, and that the variations in Heaps' exponents were associated with cognitive function. Moreover, variations in Heaps' exponents were associated with the ratio of new words taken from the other participants' speech. These results indicate that the exponents of scaling laws in language are related to cognitive processes.

Rational synthesis of novel biocompatible thermoresponsive block copolymer worm gels.

It is well known that reversible addition-fragmentation chain transfer (RAFT) aqueous dispersion polymerization of 2-hydroxypropyl methacrylate (HPMA) enables the rational design of diblock copolymer worm gels. Moreover, such hydrogels can undergo degelation on cooling below ambient temperature as a result of a worm-to-sphere transition. However, only a subset of such block copolymer worms exhibit thermoresponsive behavior. For example, PMPC26-PHPMA280 worm gels prepared using a poly(2-(methacryloyloxy)ethyl phosphorylcholine) (PMPC26) precursor do not undergo degelation on cooling to 6 °C (see S. Sugihara et al., J. Am. Chem. Soc., 2011, 133, 15707-15713). Informed by our recent studies (N. J. Warren et al., Macromolecules, 2018, 51, 8357-8371), we decided to reduce the mean degrees of polymerization of both the PMPC steric stabilizer block and the structure-directing PHPMA block when targeting a pure worm morphology. This rational approach reduces the hydrophobic character of the PHPMA block and hence introduces the desired thermoresponsive character, as evidenced by the worm-to-sphere transition (and concomitant degelation) that occurs on cooling a PMPC15-PHPMA150 worm gel from 40 °C to 6 °C. Moreover, worms are reconstituted on returning to 40 °C and the original gel modulus is restored. This augurs well for potential biomedical applications, which will be examined in due course. Finally, small-angle X-ray scattering studies indicated a scaling law exponent of 0.67 (≈2/3) for the relationship between the worm core cross-sectional diameter and the PHPMA DP for a series of PHPMA-based worms prepared using a range of steric stabilizer blocks, which is consistent with the strong segregation regime for such systems.

Movement of ablated high-<i>Z</i> plasmas

The energy leaking through a slot in the hohlraum filled with low-<i>Z</i> foams is a typical dynamic problem of the ablated high-<i>Z</i> plasmas. In this paper, we develop a simplified one-dimensional model to study the expansion-reverse process of the ablated Au plasmas, which corresponds to the closing-reopening process of a slot. Our work shows that its physical mechanism is the ablation pressure competing with radiation pressure difference and the material pressure of low-<i>Z</i> foams. The analytical formulas for the reverse time and reverse distance of the Au plasma are deduced, respectively, indicating that the cubic value for each of both peak temperatures is proportional to the density of the low-<i>Z</i> foams. The main conclusions of analytic theory are verified by numerical simulation through using the modified radiation-hydrodynamic program MULTI. It is shown that the power exponents of scaling law in high-<i>Z</i> plasma ablation keep unchanged in a wide range of density of low-<i>Z</i> foams. The range of validity of the model is discussed.

Kinetic percolation.

We demonstrate that kinetic aggregation forms superaggregates that have structures identical to static percolation aggregates, and these superaggregates appear as a separate phase in the size distribution. Diffusion limited cluster-cluster aggregation (DLCA) simulations were performed to yield fractal aggregates with a fractal dimension of 1.8 and superaggregates with a fractal dimension of D=2.5 composed of these DLCA supermonomers. When properly normalized to account for the DLCA fractal nature of their supermonomers, these superaggregates have the exact same monomer packing fraction, scaling law prefactor, and scaling law exponent (the fractal dimension) as percolation aggregates; these are necessary and sufficient conditions for same structure. The size distribution remains monomodal until these superaggregates form to alter the distribution. Thus the static percolation and the kinetic descriptions of gelation are now unified.

Intraspecific scaling laws are preserved in ventricular hypertrophy but not in heart failure.

It is scientifically and clinically important to understand the structure-function scaling of coronary arterial trees in compensatory (e.g., left and right ventricular hypertrophy, LVH and RVH) and decompensatory vascular remodeling (e.g., congestive heart failure, CHF). This study hypothesizes that intraspecific scaling power laws of vascular trees are preserved in hypertrophic hearts but not in CHF swine hearts. To test the hypothesis, we carried out the scaling analysis based on morphometry and hemodynamics of coronary arterial trees in moderate LVH, severe RVH, and CHF compared with age-matched respective control hearts. The scaling exponents of volume-diameter, length-volume, and flow-diameter power laws in control hearts were consistent with the theoretical predictions (i.e., 3, 7/9, and 7/3, respectively), which remained unchanged in LVH and RVH hearts. The scaling exponents were also preserved with an increase of body weight during normal growth of control animals. In contrast, CHF increased the exponents of volume-diameter and flow-diameter scaling laws to 4.25 ± 1.50 and 3.15 ± 1.49, respectively, in the epicardial arterial trees. This study validates the predictive utility of the scaling laws to diagnose vascular structure and function in CHF hearts to identify the borderline between compensatory and decompensatory remodeling.

Air-jet atomization for preparation of lipid nanoparticles: Dependence of size on solute concentration

Engineering nanoparticles for applications such as drug delivery, needs understanding of dependence of particle size on process and precursor parameters. The size of nanoparticles prepared through the aerosol route using solution droplet drying technique is traditionally governed by a 1/3rd scaling law between dry particle diameter and precursor solute concentration. Here, we experimentally investigate this relationship, for precursor solutions of lipids in organic solvents. Nanoparticles were prepared using air-jet atomization of precursor solutions of stearic acid in organic solvents (acetone, chloroform, methanol and ethanol), in a pulse heat aerosol reactor, at a process temperature of 50°C. The relationship between median mobility diameter of prepared nanoparticles (40–155nm) and solute concentration was governed by a scaling law exponent (SLE) in the range 0.104–0.182. The observed exponents showed significant deviations from the classical 1/3rd law. The experimental investigation indicated an unexpected increase in particle number concentration with increasing solute concentration for all solvents studied. Simulations with a critical super saturation based solute diffusion model and theoretical calculations of size dependent external factors such as coagulation and wall losses during transit, could not explain the deviation in SLE, nor the increased number concentrations. These considerations point to possible effects of solute concentration on the mechanism of droplet formation during the atomization process. A preliminary model based on an area conservation law has been proposed to explain the twin effects of deviation in SLE and increased droplet number concentration with increasing solute concentration. The present study provides empirical data on the relationship between dry particle diameter and solute concentration in the selected lipid-organic solvent systems. Also, it opens avenues for further understanding nanoparticle formation by air-jet atomization.

Influence of segmented vessel size due to limited imaging resolution on coronary hyperemic flow prediction from arterial crown volume.

Computational predictions of the functional stenosis severity from coronary imaging data use an allometric scaling law to derive hyperemic blood flow (Q) from coronary arterial volume (V), Q = αV(β) Reliable estimates of α and β are essential for meaningful flow estimations. We hypothesize that the relation between Q and V depends on imaging resolution. In five canine hearts, fluorescent microspheres were injected into the left anterior descending coronary artery during maximal hyperemia. The coronary arteries of the excised heart were filled with fluorescent cast material, frozen, and processed with an imaging cryomicrotome to yield a three-dimensional representation of the coronary arterial network. The effect of limited image resolution was simulated by assessing scaling law parameters from the virtual arterial network at 11 truncation levels ranging from 50 to 1,000 μm segment radius. Mapped microsphere locations were used to derive the corresponding relative Q using a reference truncation level of 200 μm. The scaling law factor α did not change with truncation level, despite considerable intersubject variability. In contrast, the scaling law exponent β decreased from 0.79 to 0.55 with increasing truncation radius and was significantly lower for truncation radii above 500 μm vs. 50 μm (P< 0.05). Hyperemic Q was underestimated for vessel truncation above the reference level. In conclusion, flow-crown volume relations confirmed overall power law behavior; however, this relation depends on the terminal vessel radius that can be visualized. The scaling law exponent β should therefore be adapted to the resolution of the imaging modality.

Is there enough fertile soil to feed a planet of growing cities?

We analyze a scaling law for the consumption of agricultural soil by cities. The nonlinear dependence of the size of the city on the number of inhabitants gives rise to an equation for population dynamics. We found the asymptotic limit of the solution for this equation, given by the carrying capacity in terms of number of inhabitants that can be fed. The carrying capacity as a function of the scaling law exponent is computed numerically, showing that the exponent must be very small to ensure a food sustainability. We suggest a bound for the value of this exponent and analyze the reliability of the scaling law for major cities.

Memory effect in M ≥ 7 earthquakes of Taiwan

The M ≥ 7 earthquakes that occurred in the Taiwan region during 1906–2006 are taken to study the possibility of memory effect existing in the sequence of those large earthquakes. Those events are all mainshocks. The fluctuation analysis technique is applied to analyze two sequences in terms of earthquake magnitude and inter-event time represented in the natural time domain. For both magnitude and inter-event time, the calculations are made for three data sets, i.e., the original order data, the reverse-order data, and that of the mean values. Calculated results show that the exponents of scaling law of fluctuation versus window length are less than 0.5 for the sequences of both magnitude and inter-event time data. In addition, the phase portraits of two sequent magnitudes and two sequent inter-event times are also applied to explore if large (or small) earthquakes are followed by large (or small) events. Results lead to a negative answer. Together with all types of information in study, we make a conclusion that the earthquake sequence in study is short-term corrected and thus the short-term memory effect would be operative.

Memory effect in M ≥ 6 earthquakes of South-North Seismic Belt, Mainland China

The M ≥ 6 earthquakes occurred in the South–North Seismic Belt, Mainland China, during 1901–2008 are taken to study the possible existence of memory effect in large earthquakes. The fluctuation analysis technique is applied to analyze the sequences of earthquake magnitude and inter-event time represented in the natural time domain. Calculated results show that the exponents of scaling law of fluctuation versus window length are less than 0.5 for the sequences of earthquake magnitude and inter-event time. The migration of earthquakes in study is taken to discuss the possible correlation between events. The phase portraits of two sequent magnitudes and two sequent inter-event times are also applied to explore if large (or small) earthquakes are followed by large (or small) events. Together with all kinds of given information, we conclude that the earthquakes in study is short-term correlated and thus the short-term memory effect would be operative.

Scaling in Electrical Conductivity Measurements and Rheological Measurements of Monarch 700 Dispersions Stabilized by Polymers

For making stable dispersions of graphitic carbon black (Monarch 700), the effectiveness of three dispersants/polymers (hypermer LP1, hypermer B246, and OLOA 11000) in xylene is investigated. Hypermer LP1 (polyhydroxystearic acid) is a homopolymer and hypermer B246 (PEG 30-dipolyhydroxystearate) is a polyhydroxystearic acid/polyethylene oxide/polyhydroxystearic acid ABA block copolymer, while OLOA 11000 (polyisobutylene succinimide) has a polar head group (polyamine) attached to a hydrocarbon chain (polyisobutylene). Well-dispersed graphitic carbon black dispersions were prepared using dispersants at optimum concentrations. Percolation threshold and rheological threshold were determined by analyzing the variations in electrical conductivity and elastic modulus with concentration of carbon black. Above threshold concentration, scaling law was applied to experimental data of rheology (dynamic measurements) and electrical conductivity measurements to evaluate quality of the system. Effectiveness of polymers was investigated on the basis of value of critical exponent (t and t′, respectively) in scaling power law. Hypermer LP1 was proved to be a poor dispersant for Monarch 700 dispersions while other two polymers were found to be effective stabilizers.

Conformational transition of polyelectrolyte chains extending over the de Gennes regime in slitlike nanochannels

The confinement-induced conformational transitions of the polyelectrolyte chain are characterized with the coarse-grained Brownian dynamics simulations and the blob theory. Submicron-sized biopolymer xanthan is chosen as a model polyelectrolyte taking into account both flexible and semiflexible chains for comparison. Confined flexible and semiflexible chains exhibit a nonmonotonic variation in size in weak confinements, where the relative radius of gyration shows a dip and then increases when decreasing the channel width. The rigid chain, realized at low screening, exhibits a sigmoidal transition without minima in size. Major attention should be on the dependence of scaling law exponents on the screening effect of the solution in the moderate confinement of the de Gennes regime. Our findings are expected to provide useful information and new insight into the confined polyelectrolytes when relevant micro/nanochannels are designed and fabricated.

Passive scalar decay in chaotic flows with boundaries

This paper considers the long-time decay rate of a passive scalar in two-dimensional flow. The focus is on the effects of boundary conditions for kinematically prescribed velocity fields with random or periodic time dependence. Scalar evolution is followed numerically in a periodic geometry for families of flows that have either a slip or a no-slip boundary condition on a square or plane layer subdomain . The boundary conditions on the passive scalar are imposed on the boundary of by restricting to a subclass invariant under certain symmetry transformations. The scalar field obeys constant (Dirichlet) or no-flux (Neumann) conditions exactly for a flow with the slip boundary condition and approximately in the no-slip case. At late times the decay of a passive scalar is exponential in time with a decay rate γ(κ), where κ is the molecular diffusivity. Scaling laws of the form γ(κ) ≃ Cκα for small κ are obtained numerically for a variety of boundary conditions on flow and scalar, and supporting theoretical arguments are presented. In particular when the scalar field satisfies a Neumann condition on all boundaries, α ≃ 0 for a slip flow condition; for a no-slip condition we confirm results in the literature that α ≃ 1/2 for a plane layer, but find α ≃ 2/3 in a square subdomain where the decay is controlled by stagnant flow in the corners. For cases where there is a Dirichlet boundary condition on one or more sides of the subdomain , the exponent measuring the decay of the scalar field is α ≃ 1/2 for a slip flow condition and α ≃ 3/4 for a no-slip condition. The scaling law exponents α for chaotic time-periodic flows are compared with those for similarly constructed random flows.

A General Principle of Neural Arbor Branch Density

The tree-like structures of a neuron that are responsible for distributing (axons) or collecting (dendrites) information over a region of the brain are called arbors. The size of the territory occupied by an arbor and the density of the arbor branches within that territory are important for computation because these factors determine what fraction of a neural map is sampled by a single cell and at what resolution [1]. Arbor territory size and branch density can vary by many orders of magnitude; however, we have identified a universal relationship between these two physical properties revealing a general neural architectural design principle. All of the arbors (axons and dendrites) we have studied (including fish retinal ganglion cells, rodent Purkinje cells, and the cortical arbors of various neural classes from rat, cat, monkey, and human) are found to be systematically less dense when they cover larger territories. This relationship can be described as a power law. Of several simple biological explanations explored, we find that this relationship is most consistent with a design principle that conserves the average number of connections between pairs of arbors of different sizes.

Electrophoresis of a Charged Droplet in a Dielectric Liquid for Droplet Actuation

Electrophoretic motion of a charged droplet in a dielectric fluid under an electric field has been investigated experimentally for use as a microdroplet actuation method. The effects of the droplet size, electric field strength, and electrolyte concentration and ion species on the charging of an aqueous droplet have been examined. The amount of electrical charging has been measured by two different methods: indirect measurement using the image analysis of droplet motion and direct measurement using the electrometer. Quantitative comparison of the droplet charge measured experimentally and the theoretical value of a perfectly conductive sphere shows that an aqueous droplet is less charged than the corresponding perfectly conductive sphere. The limiting effect on electrical charging is more significant for an electrolyte droplet, and the effect is positively correlated to the electrolyte concentration rather than the ion species. This implies that the low electrical conductivity of water is not a major cause of the limiting effect. The scaling law of the charging amount for a deionized water droplet nearly follows that of the perfect conductor, whereas for an electrolyte droplet, the scaling law exponent is slightly higher. Some advantages and potentials of the current droplet actuation method are also discussed in comparison with the conventional ones.

Scaling of the viscosity of the Lennard-Jones chain fluid model, argon, and some normal alkanes

In this work, we have tested the efficiency of two scaling approaches aiming at relating shear viscosity to a single thermodynamic quantity in dense fluids, namely the excess entropy and the thermodynamic scaling methods. Using accurate databases, we have applied these approaches first to a model fluid, the flexible Lennard-Jones chain fluid (from the monomer to the hexadecamer), then to real fluids, such as argon and normal alkanes. To enlarge noticeably the range of thermodynamics conditions for which these scaling methods are applicable, we have shown that the use of the residual viscosity instead of the total viscosity is preferable in the scaling procedures. It has been found that both approaches, using the adequate scaling, are suitable for the Lennard-Jones chain fluid model for a wide range of thermodynamic conditions whatever the chain length when scaling law exponents and prefactors are adjusted for each chain length. Furthermore, these results were found to be well respected by the corresponding real fluids.