High Aspect Ratio Particles Research Articles

Heterogenity of graphite oxide particles obtained with wet oxidative exfoliation

Wet oxidative exfoliation of graphite is one of the most frequently applied techniques to obtain aqueous dispersions of hydrophilic graphene derivatives as required, e.g., in 3D printing, wet spinning or film casting. Due to the harsh conditions of the process, the resulting suspension is a mixture of particles with a wide distribution range both of physical dimensions and chemical properties. An aqueous graphite oxide suspension was obtained by an improved Hummers method and separated into five fractions by controlled centrifugation. The fractions were characterized and compared by various methods, revealing their diversity in size, chemical properties and application-related viscosity. The characterization methods (powder XRD, Raman spectroscopy, ATR-FTIR spectroscopy, XPS, potentiometric titration, rheology) exhibited subtle but measurable differences that exceeded the standard deviation of the techniques employed, but no systematic trend was found across the fractions in any of the properties investigated. The conditions of our centrifugal separation hardly meet the constrains of the ideal of Stokes’s law, the polydispersity of the high aspect ratio particles as well as their concentration close to the percolation limit challenge the independent sedimentation of the platelets.

Zeolite membrane separators for fire-safe Li-ion batteries – Effects of crystal shape and membrane pore structure

The state-of-the-art lithium-ion batteries (LIB) use carbonate-based liquid electrolytes and polymeric separators which pose safety issues including fire-hazard. The use of fire-retarding salt-concentrated electrolytes can address the safety issues of LIBs, but these electrolytes have high viscosity and low wettability with polymeric separators making it difficult to fabricate high performance LIBs with the electrolytes. Here we report development of zeolite membrane separators with tailored pore structure and surface chemistry for preparation of fire-safe LIBs using salt-concentrated electrolytes. The zeolite membrane separators are made of zeolite particles with a specific zeolite structure, Si/Al ratio, and different particle shapes, coated on cathode by a scalable blade coating method. Intermediate pore, pure silica MFI-type zeolite (silicalite) was selected as the fire-proof material for membrane separators which also offer high wettability for the fire-retarding salt-concentrated electrolyte of lithium bis(fluorosulfonyl)imide (LiFSI) in tri-methyl phosphate solvent (TMP) solvent. Two silicalite membrane separators of similar pore size but different porosity and tortuosity were coated on cathode (LiNi0.5Mn0.3Co0.2O2 or NMC) using plate-shaped silicalite particles of low aspect ratio (LAR) and high aspect ratio (HAR). Both zeolite separators show significantly better wettability for the LiFSI/TMP electrolyte than the commercial polypropylene (PP) separator allowing fabrication of high-performance NMC/LiFSI-TMP/graphite cells with the non-combustible silicalite separator. The new fire-safe zeolite-membrane-based LIBs with the fire-retarding electrolyte exhibit excellent charge and discharge characteristics and cycle performance (similar to the state-of-the-art, but fire-unsafe LIBs with LiPF6-carbonate electrolyte and polymer separator). The membrane separators made of HAR plate-shaped zeolite showing low interparticle porosity and high tortuosity perform better than the membrane made of LAR zeolite particles, due to a more uniform lithium-ion flux from both interparticle pores and intraparticle zeolitic pores at the separator/anode interface.

Preparation of ZrC embedded carbon rods via thermal decomposition of metal organic frameworks

Synthesis of uniform, spherical ZrC is difficult and generally not cost-effective, as is the case for most ultra-high temperature ceramics. High aspect ratio particles for reinforcement of ceramic composites are even more difficult to synthesize. Metal organic frameworks (MOF) are crystalline coordination polymers composed of multidentate organic linkers bridging metal nodes to form porous structures. Thermal decomposition of MOFs presents a new and cost-effective route to synthesis of ZrC. In this study, heat treatment at 2000 °C of MOF PCN-222 produces zirconium carbide (ZrC) within a highly anisotropic particle. The resulting rod-shaped, glass-like carbon matrix embedded with ZrC crystals is described. These rods have potential as reinforcements for high temperature applications and as a synthetic route for ultra-high temperature ceramics with unique morphologies.

Structure and rheology of oil-continuous capillary suspensions containing water-swellable cellulose beads and fibres

Cellulose particles are used in a range of materials and applications. Here, we investigated the role of cellulose sorptivity on the rheology and microstructure of oil-continuous capillary suspensions consisting of cellulose beads and fibres at volume fractions of 0.3–0.65 and water at volume fractions up to 0.2. We hypothesised that, in spite of their water-absorbent nature, cellulose particles in oil would be structured into a percolated network by water added as an immiscible secondary phase. The interplay between cellulose bead load and water content was used to generate suspensions ranging from flowable to highly elastic with moduli near 1 MPa. While capillary bridging induced a liquid to semi-solid transition, nearer particle volume fractions of 0.65, we found a transition to a solid, wet-granular material. Liquid-solid transitions occurred even if the water was absorbed by the particles, suggesting that the increase in elasticity was not solely due to the formation of capillary bridges, but also as a result of clustering and the associated increase in effective packing volume fraction. Swapping the beads with fibres increased stiffness by 4 orders whereas swapping half of the beads at the same water fraction increased material stiffness by 3 orders of magnitude. The two conclusions of this study were that, even for water-sorptive particles, capillary and hydrophilic interactions may be used to structure oil-continuous suspensions, and that the use of high aspect ratio particles can increase the rigidity of oil-continuous capillary suspensions to a greater extent than beads at an equivalent volume fraction.

An apparatus for in-vacuum loading of nanoparticles into an optical trap.

We describe the design, construction, and operation of an apparatus that utilizes a piezoelectric transducer for in-vacuum loading of nanoparticles into an optical trap for use in levitated optomechanics experiments. In contrast to commonly used nebulizer-based trap-loading methods that generate aerosolized liquid droplets containing nanoparticles, the method produces dry aerosols of both spherical and high-aspect ratio particles ranging in size by approximately two orders of magnitude. The device has been shown to generate accelerations of order 107g, which is sufficient to overcome stiction forces between glass nanoparticles and a glass substrate for particles as small as 170nm in diameter. Particles with sizes ranging from 170nm to ∼10μm have been successfully loaded into optical traps at pressures ranging from 1bar to 0.6mbar. We report the velocity distribution of the particles launched from the substrate, and our results indicate promise for direct loading into ultra-high-vacuum with sufficient laser feedback cooling. This loading technique could be useful for the development of compact fieldable sensors based on optically levitated nanoparticles as well as matter-wave interference experiments with ultra-cold nano-objects, which rely on multiple repeated free-fall measurements and thus require rapid trap re-loading in high vacuum conditions.

Particle shape engineering for improving safety and efficacy of doxorubicin - A case study of rod-shaped carriers in resistant small cell lung cancer.

Therapeutic drug delivery is known to be influenced by interplay between various design parameters of delivery carriers which influence the drug uptake efficiency and subsequently the effectiveness of treatment. Amongst, the several design parameters such as size, shape and surface charge, particle shape is gaining attention as a crucial design parameter for development of robust and efficient delivery carriers. In this exploration, we investigated the influence of particle shape on injectability and therapeutic effectiveness of the delivery carriers using doxorubicin (DOX) conjugated polymeric microparticles. Results of injectability experiments demonstrated the influence of particle shape with anisotropic rod-shaped particles displaying increased injectability as against spherical particles. Impact of particle shape on therapeutic effectiveness was assessed against small cell lung cancer (SCLC) which was selected as a model disease. Results of cellular uptake studies revealed preferential uptake of rod-shaped particles than spherical particles in cancer cells. These results were further validated by in-vitro tumor simulation studies wherein rod-shaped particles displayed enhanced anti-tumorigenic activity along with distortion of tumor integrity against spheres. Furthermore, the impact of particle size was also assessed on cardiotoxicity, an adverse effect of DOX which limits its therapeutic use. Results illustrated that the high aspect ratio particles displayed diminished cardiotoxicity activity. These results provide valuable insights about influence of particle shape for designing efficient therapeutics.

Rigidity Percolation in Disordered 3D Rod Systems

In composite materials composed of a soft polymer matrix and rigid, high aspect-ratio particles, the composite undergoes a transition in mechanical strength when the incorporated particle phase surpasses a critical density. This phenomenon, termed rheological percolation, is well known to occur in many polymer-rod and polymer-platelet composites at a critical density that exceeds the conductivity percolation threshold (which occurs when the conducting particles form a large connected component that spans the composite). Contact percolation in rod-like composites has been routinely exploited to engineer thermal or electrical conductivity in otherwise nonconducting polymers, and the characterization of contact percolation is well established. Mechanical or rheological percolation, however, has evaded a complete theoretical explanation and predictive description. A natural hypothesis is that rheological percolation is due to a rigidity phenomenon, whereby a large rigid component of inclusions spans the composite. Here we build an algorithm to detect the rigidity percolation threshold in rod-polymer composites. We model the composites as systems of randomly distributed, soft-core (intersecting at contact) rods and study the emergence of a giant (i.e., spanning) rigid component. Building on our previous results for two-dimensional composites, we develop an approximate algorithm that identifies spanning rigid components through hierarchically identifying and compressing provably rigid motifs---equivalently, decomposing a giant rigid component into rigid assemblies of a hierarchy of successively smaller rigid components. We apply this algorithm to random monodisperse systems that are generated in Monte Carlo simulations to estimate a rigidity percolation threshold (critical density) and explore its dependence on rod aspect ratio. We show that this transition point---like its contact percolation analogue---scales inversely with the excluded volume of a rod. Moreover, this implies that the critical contact number (i.e., the number of contacts per rod at the rigidity percolation threshold) is constant for aspect ratios above some relatively low value and is lower than the prediction from Maxwell's isostatic condition.

Continuous Isolation of Particles with Varying Aspect Ratios up to Thin Needles Achieving Free-Flowing Products

The continuous vacuum screw filter (CVSF) for small-scale continuous product isolation of suspensions was operated for the first time with cuboid-shaped and needle-shaped particles. These high aspect ratio particles are very common in pharmaceutical manufacturing processes and provide challenges in filtration, washing, and drying processes. Moreover, the flowability decreases and undesired secondary processes of attrition, breakage, and agglomeration may occur intensively. Nevertheless, in this study, it is shown that even cuboid and needle-shaped particles (l-alanine) can be processed within the CVSF preserving the product quality in terms of particle size distribution (PSD) and preventing breakage or attrition effects. A dynamic image analysis-based approach combining axis length distributions (ALDs) with a kernel-density estimator was used for evaluation. This approach was extended with a quantification of the center of mass of the density-weighted ALDs, providing a measure to analyze the preservation of the inlet PSD statistically. Moreover, a targeted residual moisture below 1% could be achieved by adding a drying module (Tdry = 60 °C) to the modular setup of the CVSF.

Experimental Investigation of Polypropylene Composite Drawn Fibers with Talc, Wollastonite, Attapulgite and Single-Wall Carbon Nanotubes.

Isotactic polypropylene (PP) composite drawn fibers were prepared using melt extrusion and high-temperature solid-state drawing at a draw ratio of 7. Five different fillers were used as reinforcement agents (microtalc, ultrafine talc, wollastonite, attapulgite and single-wall carbon nanotubes). In all the prepared samples, antioxidant was added, while all samples were prepared with and without using PP grafted with maleic anhydride as compatibilizer. Material characterization was performed by tensile tests, differential scanning calorimetry, thermogravimetric analysis and Fourier transform infrared spectroscopy. Attapulgite composite fibers exhibited poor results in terms of tensile strength and thermal stability. The use of ultrafine talc particles yields better results, in terms of thermal stability and tensile strength, compared to microtalc. Better results were observed using needle-like fillers, such as wollastonite and single-wall carbon nanotubes, since, as was previously observed, high aspect ratio particles tend to align during the drawing process and, thus, contribute to a more symmetrical distribution of stresses. Competitive and synergistic effects were recognized to occur among the additives and fillers, such as the antioxidant effect being enhanced by the addition of the compatibilizer, while the antioxidant itself acts as a compatibilizing agent.

High-Resolution R2R-Compatible Printing of Carbon Nanotube Conductive Patterns Enabled by Cellulose Nanocrystals

Carbon nanotubes (CNTs) with enhanced properties compared to conventional materials are a leading material of choice for fabricating next-generation electronic devices. Nanocellulose-based conductive component-enabled electronics also offer great potential for commercial scalability of environmentally friendly, sustainable, flexible, wearable electronics. Printing these functional materials through R2R printing will enable the economic and high-throughput production of next-generation electronic devices. However, the lateral resolution during R2R printing of these materials is currently limited due to the enhanced aggregation behavior of these high-aspect ratio particles, and the lower lateral resolution limits the performance of the fabricated devices. This article demonstrates high-resolution, R2R-compatible printing of conductive patterns of CNTs using cellulose nanocrystals (CNCs) through the topographical discontinuous dewetting and liquid-bridge transfer patterning technique. The CNC dispersion obtained through acid hydrolysis of spinifex grass biomass was used as a sustainable functional ink and deposited as a structural wetting layer, which necessarily allowed the subsequent deposition of a conductive CNT layer to form high-resolution conductive patterns. Conductive patterns with lateral feature sizes down to ∼4.5 μm were reliably printed and those with feature sizes down to ∼925 nm were also possible. The high-resolution conductive CNC/CNT patterns could be printed on different hydrophilic substrates, including flexible, transparent CNC films, for use in devices. This study represents a proof-of-concept for the realization of the economic and environmentally friendly printing of high-resolution nanocellulose/carbon-based electronics.

Modelling the particle trajectory and melting behaviour of non-spherical ice crystal particles

Existing ice crystal icing codes commonly neglect non-spherical particle rotation behaviour. This leads to uncertainties in modelling ice crystal icing as ice crystals are typically non-spherical. This paper develops a two-dimensional framework to model the particle trajectory and melting behaviour of rotating non-spherical ice crystal particles. Non-spherical particle translational and rotational motions are resolved using a framework of unit complex numbers. The effect of rotation on particle heat and mass transfer is implemented using a sub-model dependent on both rotational and translational particle Reynolds numbers. A test case of flow through a swan neck duct is presented. The model is validated through comparison to the discrete phase model in ANSYS Fluent and experimental water droplet impingement test data. Comparison of the developed framework with other simplified particle tracking methods (without modelling the non-spherical particle rotation) is first conducted. These results show large differences in the trajectory, velocity and melt ratio of individual particles and in overall particle impingement behaviours, especially for high aspect ratio particles. Particle rotation can indirectly affect single particle’s melting behaviour through its effect on particle trajectory and velocity. Both aspect ratio and porosity are seen to enhance particle melting behaviour and affect particle impingement behaviour. The numerical uncertainties in the DNS-based correlations of the particle forces and torques employed are also discussed, as well as the performance of the developed framework for the case of a flow past a test article.

Flow-Particle Coupling in a Channel Flow Laden with Elongated Particles: The Role of Aspect Ratio

A turbulent channel flow laden with elongated, fiber-like particles is investigated experimentally by optical techniques. The flow-particle inter-coupling is analyzed in the case of particles with an aspect ratio of 40 and 80, at two volume fractions, 10−5 and 10−4. An image processing technique is presented, which is employed to simultaneously obtain carrier flow velocimetry data and distribution and orientation data of dispersed particles. Turbulence enhancement is reported in the near-wall region, with a higher level of increase associated with higher aspect ratio particles. Comparison to fiber data suggests that this mechanism of turbulence modulation stems from a particles orientational behavior. The preferential particle distribution is reported to be dependent on the aspect ratio in the region close to the wall. The probability density function of the fibers’ orientation angle appears to be independent of the particle aspect ratio once it is conditioned to the fibers’ characteristic size.

The effects of powder reuse on the mechanical response of electron beam additively manufactured Ti6Al4V parts

High cost of metal powders has increased the demand for recycling of unmelted powder in electron beam powder bed fusion additive manufacturing process. However, powder characteristics are likely to change during manufacturing, recovery and reuse. It is important to track the evolution of powder characteristics at different stages of recycling to produce components with consistent properties. The present work evaluates the changes in Ti6Al4V powder properties during manufacturing by characterising powder particles at different locations in the powder bed; recovery and reuse, through evaluating the effects of the powder recovery system and sieving for 10 build cycles. Heterogeneous powder degradation occurred during manufacturing with the particles closer to the melt zone showing higher oxygen content and thicker α laths with β phase boundaries. Most of them had a hard-sintered and agglomerated powder morphology in contrast to particles at the edges of the powder bed. Recovery and reuse resulted in a refined particle size distribution, but only marginal change in powder morphology. The increased oxygen caused a slight increase in the yield and tensile strengths of the build. The effect of powder reuse on material elongation, hardness and Charpy impact energy was negligible. The high cycle fatigue performance deteriorated with reuse due to the increased lack-of-fusion defects. This might be attributed to the voids formed in the powder bed due to decrease in the number of fine particles coupled with an increase in the number of high-aspect ratio particles.

Taylor dispersion of elongated rods

Particles transported in fluid flows, such as cells, polymers, or nanorods, are rarely spherical. In this study, we numerically and theoretically investigate the dispersion of an initially localized patch of passive elongated Brownian particles constrained to one degree of rotational freedom in a two-dimensional Poiseuille flow, demonstrating that elongated particles exhibit an enhanced longitudinal dispersion. In a shear flow, the rods translate due to advection and diffusion and rotate due to rotational diffusion and their classical Jeffery's orbit. The magnitude of the enhanced dispersion depends on the particle's aspect ratio and the relative importance of its shear-induced rotational advection and rotational diffusivity. When rotational diffusion dominates, we recover the classical Taylor dispersion result for the longitudinal spreading rate using an orientationally averaged translational diffusivity for the rods. However, in the high-shear limit, the rods tend to align with the flow and ultimately disperse more due to their anisotropic diffusivities. Results from our Monte Carlo simulations of the particle dispersion are captured remarkably well by a simple theory inspired by Taylor's original work. For long times and large Peclet numbers, an effective one-dimensional transport equation is derived with integral expressions for the particles' longitudinal transport speed and dispersion coefficient. The enhanced dispersion coefficient can be collapsed along a single curve for particles of high aspect ratio, representing a simple correction factor that extends Taylor's original prediction to elongated particles.

Combined FIB/SEM tomography and TEM analysis to characterize high aspect ratio Mg-silicate particles inside silica-based optical fibres

The evolution of particles, during the fabrication step of particle-containing glass fibres, was investigated. The aim of this study was to characterize quantitatively the composition, size and shape distribution of Mg-silicate particles within the initial silica-based preform and the final fibre, these information being crucial to better understand the mechanisms involved during the drawing step. A combined experimental approach using Scanning electron microscopy, Transmission electron microscopy and FIB/SEM tomography was implemented. The particle composition was found to stay unchanged during the drawing step. On the other hand, strong modifications of their shape within the fibre is observed and discussed. • The microstructure of high aspect ratio particles within optical fibres is analysed by electron microscopy techniques • The measured composition of particles is the one of the enstatic phase • 3D imaging of the particles by FIB/SEM tomography reveals their morphology changes induced by the drawing step

Bioinspired particle engineering for non-invasive inhaled drug delivery to the lungs

Pulmonary drug delivery is governed by several biophysical parameters of delivery carriers, such as particle size, shape, density, charge, and surface modifications. Although much attention has been given to other parameters, particle shape effects have rarely been explored. In this work, we assess the influence of particle shape of inhaled delivery carriers on their aerodynamic properties and macrophage uptake by using polymeric microparticles of different geometries ranging in various sizes. Doxorubicin was conjugated to the polymer particles and the bioconjugates were characterized. Interestingly, the results of in-vitro lung deposition, performed using a next generation impactor, demonstrated a significant improvement in the aerodynamic properties of the rod-shaped particles with a high aspect ratio as compared to spherical particles with the same equivalent volume. The results of a macrophage uptake experiment demonstrate that the high aspect ratio particles were phagocytosed less than spherical particles. Furthermore, the cytotoxicity of these doxorubicin-conjugated particles was determined against murine macrophages, resulting in reduced toxicity when treated with high aspect ratio particles as compared to spherical particles. This project provides valuable insights into the influence of particle shape on aerodynamic properties and primary defense mechanisms in the peripheral lungs, while using polymeric microparticles of various sizes and geometries. Further systematic development can help translate these findings to preclinical and clinical studies for designing efficient inhalable delivery carriers.

An In Situ and Real-Time Plasmonic Approach of Seed/Adhesion Layers: Chromium Buffer Effect at the Zinc/Alumina Interface

The effect of additives on metal/oxide interfaces is explored in situ and in real time on evaporated films by a combination of surface science techniques, among which a very flexible optical method shows a unique ability to scrutinize the growth and wetting properties of supported clusters that involve several elements. The study focuses on Cr at the Zn/α-Al2O3(0001) interface at 300 K. A particular interest of the present interface is that Zn does not stick at all on bare alumina. The sticking and morphology of both Cr and Zn films during their growth are analyzed from sub-monolayer to multilayer thicknesses. After an initial oxidation reaction with residual OH groups, shown to be detrimental to Zn adhesion, Cr growth proceeds through the formation of high aspect ratio particles that percolate around an average thickness of 10 Å. With regard to Zn growth on a Cr deposit, two very distinct stages can be distinguished. In the sub-monolayer thickness range, Cr forms a seed layer that drastically increases the Zn sticking coefficient from 0 to nearly 1 due to a diffusion length of physisorbed Zn adatoms before desorption larger than Cr island separation; Zn clusters are anchored on the Cr seeds that they encapsulate, but their wetting behavior is dictated by the interaction with alumina. In a second stage, as soon as the Cr film percolates, it forms an adhesion layer on which Zn grows in a nearly two-dimensional mode. In all cases, Cr films are stable upon annealing. On Cr-covered alumina, the Zn desorption energy is enhanced as compared to bare surfaces, which, in line with atomistic simulations, is assigned to the formation of more favorable Cr–Al2O3 and Cr–Zn than Zn–Al2O3 bonds. Generally speaking, the ability demonstrated herein of small amounts of additives to dramatically increase the adhesion of films is of great practical interest. It shows that noncontinuous and partially oxidized films of additives, closer to realistic cases of application, can strongly enhance the sticking of films. Also, anchoring a functional film by discrete predeposited seeds can keep its properties intact.

Bioinspired Polymeric High Aspect Ratio Particles with Asymmetric Janus Functionalities.

Polymeric particles with intricate morphologies and properties have been developed based on bioinspired designs for applications in regenerative medicine, tissue engineering, and drug delivery. However, the fabrication of particles with asymmetric functionalities remains a challenge. Janus polymeric particles are an emerging class of material with asymmetric functionalities; however, they are predominantly spherical in morphology, made from non-biocompatible materials, and made using specialized fabrication techniques. We therefore set out to fabricate nonspherical Janus particles inspired by high aspect ratio filamentous bacteriophage using polycaprolactone polymers and standard methods. Janus high aspect ratio particles (J-HARPs) were fabricated with a nanotemplating technique to create branching morphologies selectively at one edge of the particle. J-HARPs were fabricated with maleimide handles and modified with biomolecules such as proteins and biotin. Regioselective modification was observed at the tips of J-HARPs, likely owing to the increased surface area of the branching regions. Biotinylated J-HARPs demonstrated cancer cell biotin receptor targeting, as well as directional crosslinking with spherical particles via biotin-streptavidin interactions. Lastly, maleimide J-HARPs were functionalized during templating to contain amines exclusively at the branching regions and were dual-labeled orthogonally, demonstrating spatially separated bioconjugation. Thus, J-HARPs represent a new class of bioinspired Janus material with excellent regional control over biofunctionalization.

Quadrature by Parity Asymptotic eXpansions (QPAX) for Scattering by High Aspect Ratio Particles

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 4 May 2021Accepted: 07 September 2021Published online: 06 December 2021Keywordsboundary integral methods, asymptotic analysis, numerical quadrature, scatteringAMS Subject Headings41A60, 65D30, 65R20Publication DataISSN (print): 1540-3459ISSN (online): 1540-3467Publisher: Society for Industrial and Applied MathematicsCODEN: mmsubt

Crystal Comets: A Geometric Model for Sculpting Anisotropic Particles from Emulsions.

Microscopic high aspect ratio particles have many applications including enhanced delivery of active ingredients and food stability. Here, we develop a simple, scalable process that produces particles with a continuously controllable aspect ratio. Oil-in-water emulsion droplets are quenched and crystallize in the presence of surfactants that facilitate the ejection of the solid oil phase from its liquid precursor. Tuning the ejection and crystallization rates to be comparable, by adjusting the surfactant concentration and quench depth, promotes anisotropic particle growth by continuously ejecting solidified oil from the precursor droplet as the crystallization proceeds. We predict the accessible morphologies using an analytical geometric model that indicates a nonconstant contact angle during the crystallization process. We see that the crystal aspect ratio is dependent on the surfactant concentration, which can be explained as a variation of the maximum growth angle achieved during crystallization.