Non-spherical Objects Research Articles

Cloaking non-spherical objects and collections of objects using the scattering cancelation method

Acoustic cloaks can be designed using transformation acoustics (TA) to guide acoustic disturbances around an object. TA cloaks, however, require the use of exotic materials such as pentamode materials [Proc. R. Soc. A. 464, pp. 2411–2434, (2008)]. Alternatively, the scattering cancelation (SC) method allows the cloaked object to interact with the acoustic wave and can be realized with isotropic materials [Phys. Rev. B., 86, 104302 (2012)]. Unfortunately, SC cloaking performance may be degraded if the shape of the cloaked object diverges from the one for which the cloak was originally designed. This study investigates the design of two-layer SC cloaks for imperfect spherical objects. The cloaking material properties are determined by minimizing the scattered field from a model of the imperfect object approximated as a series of concentric shells. Predictions from this approximate analytical model are compared with three-dimensional finite element (FE) models of the cloaked and uncloaked non-spherical shapes. Analytical and FE results are in good agreement for ka ≤ 5, indicating that the SC method is robust to object imperfections. Finally, FE models are used to explore SC cloak robustness to multiple-scattering by investigating linear arrays of cloaked objects for different incident angles. [Work supported by ONR.]

Update for BayesApp: a web site for analysis of small-angle scattering data

An update for BayesApp, a web site for analysis of small-angle scattering data, is presented. The indirect transformation of the scattering data now includes an option for a maximum-entropy constraint in addition to the conventional smoothness constraint. The maximum-entropy constraint uses an ellipsoid of revolution as a prior, and the dimensions of the ellipsoid as well as the overall noise level of the experimental data are estimated using Bayesian methods. Furthermore, a correction for slit smearing has been added. The web site also includes options for calculation of the scattering intensity from simple models as well as the estimation of structure factors for polydisperse spheres and nonspherical objects of axial ratios between 0.4 and 2.5.

Cavity method for force transmission in jammed disordered packings of hard particles.

The force distribution of jammed disordered packings has always been considered a central object in the physics of granular materials. However, many of its features are poorly understood. In particular, analytic relations to other key macroscopic properties of jammed matter, such as the contact network and its coordination number, are still lacking. Here we develop a mean-field theory for this problem, based on the consideration of the contact network as a random graph where the force transmission becomes a constraint satisfaction problem. We can thus use the cavity method developed in the past few decades within the statistical physics of spin glasses and hard computer science problems. This method allows us to compute the force distribution P(f) for random packings of hard particles of any shape, with or without friction. We find a new signature of jamming in the small force behavior P(f) ∼ f(θ), whose exponent has attracted recent active interest: we find a finite value for P(f = 0), along with θ = 0. Furthermore, we relate the force distribution to a lower bound of the average coordination number z[combining macron](μ) of jammed packings of frictional spheres with coefficient μ. This bridges the gap between the two known isostatic limits z[combining macron]c (μ = 0) = 2D (in dimension D) and z[combining macron]c(μ → ∞) = D + 1 by extending the naive Maxwell's counting argument to frictional spheres. The theoretical framework describes different types of systems, such as non-spherical objects in arbitrary dimensions, providing a common mean-field scenario to investigate force transmission, contact networks and coordination numbers of jammed disordered packings.

Use of Statistically Inhomogeneous Screens in Calibration of Lidar from the Parameters of Images of Particles for the Bottom Layer of the Atmosphere

There has been practically no discussion of the problem of remote measurement of the microstructure of scattering media consisting of polydisperse and nonspherical particles. This may be attributed to the great diversity of particles of the bottom layer of the atmosphere possessing undetermined optical and physical properties. Lidar functioning at several wavelengths with subsequent solution of an ill-posed inverse problem is usually used to determine the microstructure of polydisperse spherical particles (fog, haze, etc.). The process of solving the problem requires a priori information concerning the scattering object (for example, the assumption that the distribution function is close to a lognormal distribution). Information about the ranges of the real and imaginary components of the coefficient of diffraction is also important. A particular optimal distribution function of the particles by size, to which the measured spectral dependence of the lidar signals corresponds, is selected as the solution of the ill-posed inverse problem. The radial moments of the particle distribution function by size [1] are used to compare the results of the modeling with the microstructure of the scattering object being studied. A significant volume of adjustment parameters and a priori information about the probed object are needed to solve the ill-posed inverse problem. These features prevent us from performing remote measurements of the parameters of the microstructure of a scattering object. Methods that are used for the solution of an ill-posed inverse problem are not applicable to nonspherical objects (dust, soot, pollen, etc.), since for such particles it is important to determine the transverse dimension of a particle. Cases of thread-like inhomogeneities are also possible. Thus, for existing methods an evaluation of the reliability of the sizes of particles is critical. There is now no indicator of the size of particles available, since an optical scheme of probing cannot be adjusted to particles with given sizes. Multiwave probing is a technically complex procedure for this problem, since calibration of signals to several wavelengths is a necessary condition for its realization. A method that uses lidar operating at a single wavelength (elastic scattering lidar) is quite promising. The problem may be solved with the use of an equivalent scattering object consisting of monodisperse inhomogeneities whose basic optical parameters are the back-scatter factor and the transmission coefficient, as well as the distortion of the beam, as in the case of a scattering object [1]. The measured backscattering signal will then determine the concentration of these particles in the

Retarded hydrodynamic properties of fractal clusters

Fractal clusters are commonly encountered when working with the stability and the aggregation of colloidal suspensions. In spite of the number of studies that have focused on their stationary hydrodynamic properties, no information is currently known on their retarded hydrodynamic properties. The objective of this work is to close this gap. Clusters with a broad range of fractal dimension values, generated via Monte-Carlo simulations have been analyzed. A rigorous model based on multipole expansion of time-dependent Stokes equations has been developed, and then the full cluster resistance matrix as a function of the frequency has been computed. An attempt has been made to extend Basset, Boussinesque and Oseen equations to fractal clusters, but it was found that the corresponding hydrodynamic radius needs to be a function of frequency. In the case of translational motion, the cluster hydrodynamic radius loses any structural information at high frequencies, becoming independent of the fractal dimension, but depending only on its mass. A simplified model, based on an extension of Kirkwood–Rieseman approach has also been developed. This allows one to perform calculations for clusters with arbitrary masses and fractal dimensions, with good accuracy and very low computational time. It is the first time that the frequency dependence of hydrodynamic properties of complex non-spherical objects has been investigated.

“Ideal glassformers” vs “ideal glasses”: Studies of crystal-free routes to the glassy state by “potential tuning” molecular dynamics, and laboratory calorimetry

The ability of some liquids to vitrify during supercooling is usually seen as a consequence of the rates of crystal nucleation (and∕or crystal growth) becoming small [D. R. Uhlmann, J. Non-Cryst. Solids 7, 337 (1972)]--and thus a matter of kinetics. However, there is evidence dating back to the empirics of coal briquetting for maximum trucking efficiency [D. Frenkel, Physics 3, 37 (2010)] that some object shapes find little advantage in self-assembly to ordered structures--meaning random packings prevail. Noting that key studies of non-spherical object packing have never been followed from hard ellipsoids [A. Donev, F. H. Stillinger, P. M. Chaikin, and S. Torquato, Phys. Rev. Lett. 92, 255506 (2004); A. Donev, I. Cisse, D. Sachs, E. A. Variano, F. H. Stillinger, R. Connelly, S. Torquato, and P. M. Chaikin, Science 303, 990 (2004)] or spherocylinders [S. R. Williams and A. P. Philipse, Phys. Rev. E 67, 051301 (2003)] (diatomics excepted [S.-H. Chong, A. J. Moreno, F. Sciortino, and W. Kob, Phys. Rev. Lett. 94, 215701 (2005)] into the world of molecules with attractive forces, we have made a molecular dynamics study of crystal melting and glass formation on the Gay-Berne (G-B) model of ellipsoidal objects [J. G. Gay and B. J. Berne, J. Chem. Phys. 74, 3316 (1981)] across the aspect ratio range of the hard ellipsoid studies. Here, we report that in the aspect ratio range of maximum ellipsoid packing efficiency, various G-B crystalline states that cannot be obtained directly from the liquid, disorder spontaneously near 0 K and transform to liquids without any detectable enthalpy of fusion. Without claiming to have proved the existence of single component examples, we use the present observations, together with our knowledge of non-ideal mixing effects, to discuss the probable existence of "ideal glassformers"--single or multicomponent liquids that vitrify before ever becoming metastable with respect to crystals. We find evidence that "ideal glassformer" systems might also be highly fragile systems, approaching the "ideal glass" condition. We link this to the high "volume fragility" behavior observed in recent hard dumbbell studies at similar length∕diameter ratios [R. Zhang and K. S. Schweitzer, J. Chem. Phys. 133, 104902 (2010)]. The discussion suggests some unusual systems for laboratory study. Using differential scanning calorimetry detection of fusion points T(m), liquidus temperatures T(l), and glass transition temperatures T(g), we describe a system that would seem incapable of crystallizing before glass transition, i.e., an "ideal glassformer." The existence of crystal-free routes to the glassy state will eliminate precrystalline fluctuations as a source of the dynamic heterogeneities that are generally considered important in the discussion of the "glassy state problem [P. W. Anderson, Science 267, 1615 (1995)]."

Simultaneous 3D measurement of the translation and rotation of finite-size particles and the flow field in a fully developed turbulent water flow

We report a novel experimental technique that measures simultaneously in three dimensions the trajectories, the translation and the rotation of finite-size inertial particles together with the turbulent flow. The flow field is analyzed by tracking the temporal evolution of small fluorescent tracer particles. The inertial particles consist of a super-absorbent polymer that renders them index and density matched with water and thus invisible. The particles are marked by inserting at various locations tracer particles into the polymer. Translation and rotation, as well as the flow field around the particle are recovered dynamically from the analysis of the marker and tracer particle trajectories. We apply this technique to study the dynamics of inertial particles much larger in size (Rp/η ≈ 100) than the Kolmogorov length scale η in a von Kármán swirling water flow (Rλ ≈ 400). We show, using the mixed (particle/fluid) Eulerian second-order velocity structure function, that the interaction zone between the particle and the flow develops in a spherical shell of width 2Rp around the particle of radius Rp. This we interpret as an indication of a wake induced by the particle. This measurement technique has many additional advantages that will make it useful to address other problems such as particle collisions, dynamics of non-spherical solid objects, or even of wet granular matter.

Theory of kinetic arrest, elasticity, and yielding in dense binary mixtures of rods and spheres

We extend the quiescent and stressed versions of naïve mode coupling theory to treat the dynamical arrest, shear modulus, and absolute yielding of particle mixtures where one or more species is a nonrotating nonspherical object. The theory is applied in detail to dense isotropic "chemically matched" mixtures of variable aspect ratio rods and spheres that interact via repulsive and short range attractive site-site pair potentials. A remarkably rich ideal kinetic arrest behavior is predicted with up to eight "dynamical phases" emerging: an ergodic fluid, partially localized states where the spheres remain fluid but the rods can be a gel, repulsive glass or attractive glass, doubly localized glasses and gels, a porous rod gel plus sphere glass, and a narrow window where a type of rod glass and gel localization coexist. Dynamical complexity increases with rod length and the introduction of attractive forces between all species which both enhance gel network formation. Multiple dynamic reentrant features and triple points are predicted, and each dynamic phase has unique particle localization characteristics and mechanical properties. Orders of magnitude variation of the linear shear modulus and absolute yield stress are found as rod length, mixture composition and the detailed nature of interparticle attractions are varied. The interplay of total (high) mixture packing fraction and composition at fixed temperature is also briefly studied. The present work provides a foundation to study more complex rod-sphere mixtures of both biological and synthetic interest that include physical features such as interaction site size asymmetry, rod-sphere specific attractions, and/or Coulomb repulsion.

Determination of the Scalar Friction Factor for Nonspherical Particles and Aggregates Across the Entire Knudsen Number Range by Direct Simulation Monte Carlo (DSMC)

The friction factor of an aerosol particle depends upon the Knudsen number (Kn), as gas molecule–particle momentum transfer occurs in the transition regime. For spheres, the friction factor can be calculated using the Stokes–Millikan equation (with the slip correction factor). However, a suitable friction factor relationship remains sought-after for nonspherical particles. We use direct simulation Monte Carlo (DSMC) to evaluate an algebraic expression for the transition regime friction factor that is intended for application to arbitrarily shaped particles. The tested friction factor expression is derived from dimensional analysis and is analogous to Dahneke's adjusted sphere expression. In applying this expression to nonspherical objects, we argue for the use of two previously developed drag approximations in the continuum (Kn → 0) and free molecular (Kn → ∞) regimes: the Hubbard–Douglas approximation and the projected area (PA) approximation, respectively. These approximations lead to two calculable geometric parameters for any particle: the Smoluchowski radius, R S, and the projected area, PA. Dimensional analysis reveals that Kn should be calculated with PA/πR S as the normalizing length scale, and with Kn defined in this manner, traditional relationships for the slip correction factor should apply for arbitrarily shaped particles. Furthermore, with this expression, Kn-dependent parameters, such as the dynamic shape factor, are readily calculable for nonspherical objects. DSMC calculations of the orientationally averaged drag on spheres and test aggregates (dimers, and open and dense 20-mers) in the range Kn = 0.05–10 provide strong support for the proposed method for friction factor calculation in the transition regime. Experimental measurements of the drag on aggregates composed of 2–5 primary particles further agree well with DSMC results, with differences of less than 10% typically between theoretical predictions, numerical calculations, and experimental measurements. Copyright 2012 American Association for Aerosol Research

On the flow field about an electrophoretic particle

The flow field about an electrophoretic body is theoretically investigated by analytical methods. An effective boundary condition for the electric potential at particle surface is derived. This condition, which generalizes the one obtained by Levich [Physicochemical Hydrodynamics (Prentice-Hall, Englewood Cliffs, 1962), Chap. 9, p. 475], captures the effect of (convective and electromigratory) surface current in the Debye layer and is valid as far as it is legitimate to neglect ion-concentration gradient in the bulk liquid. Conditions for negligible concentration gradients are also presented and discussed. The effect of surface current determines a deviation from Morrison's “classical” theory, which predicts irrotational flow-field for any particle shape with electrophoretic velocity given by the well-known Smoluchowski formula and always directed along the applied electric field. It is shown here that in the presence of the above effect the irrotationality of the flow field is not preserved if the particle surface has non-uniform curvature. However, irrotational flow-field still subsists for a sphere and a cylinder and is analytically determined in terms of a new non-dimensional parameter, referred to as the electrophoretic number. The case of spheroidal objects is also examined in detail. In this case the flow field, though not strictly irrotational, is shown to be nearly approximated by an irrotational flow-field, which is also determined over wide ranges of electrophoretic number and spheroid aspect ratio. The quality of this approximation is expressed as a relative error on the Helmholtz-Smoluchowski condition and numerically evaluated both in longitudinal and transverse configuration. The limiting cases of spheroid degenerating into a needle and a disk are also addressed. In all above cases the respective mobilities deviate from Smoluchowski's formula and depend on the electrophoretic number. An important effect of surface ion-transport in the double layer is anisotropy of electrophoretic mobility for non-spherical objects. That always bears a bias of the electrophoretic velocity with respect to the applied electric field when the latter is not collinear with a symmetry axis of the body. For a cylinder, bias is always toward the axis. For a spheroid, it is generally toward the polar axis; however, bias toward the equatorial axis is predicted for moderately oblate spheroids. In general, the bias angle is remarkable, which is of potential consequence in technical applications of electrophoresis. Comparison of results of the present theory with experimental work from the literature is presented and discussed.

Stabilization of Miniemulsion Droplets by Cerium Oxide Nanoparticles: A Step toward the Elaboration of Armored Composite Latexes

Stable methyl methacrylate (MMA) miniemulsions were successfully prepared using for the first time cerium oxide (CeO(2)) nanoparticles as solid stabilizers in the absence of any molecular surfactant. The interaction between MMA droplets and CeO(2) nanoparticles was induced by the use of methacrylic acid (MAA) as a comonomer. Both MAA and CeO(2) contents played a key role on the diameter and the stability of the droplets formed during the emulsification step. Cryo-transmission electron microscopy (TEM) images of the suspensions formed with 35 wt % of CeO(2) showed the presence of polydisperse 50-150 nm spherical droplets. More surprisingly, some nonspherical (likely discoidal) objects that could be the result of the sonication step were also observed. The subsequent polymerization of these Pickering miniemulsion droplets led to the formation of composite PMMA latex particles armored with CeO(2). In all cases, the conversion was limited to ca. 85%, concomitant with a loss of stability of the latex for CeO(2) contents lower than 35 wt %. This stability issues were likely related to the screening of the cationic charges present on CeO(2) nanoparticles upon polymerization. TEM images showed mostly spherical particles with a diameter ranging from 100 to 400 nm and homogeneously covered with CeO(2). Besides, for particles typically larger than 200 nm, a buckled morphology was observed supporting the presence of residual monomer at the end of the polymerization and consistent with the limited conversion. The versatility of these systems was further demonstrated using 35 wt % of CeO(2) and replacing MMA by n-butyl acrylate (BA) either alone or in combination with MMA. Stable monomer emulsions were always obtained, with the droplet size increasing with the hydrophobicity of the oil phase, pointing out the key influence of the wettability of the solid stabilizer. The polymerization of Pickering miniemulsion stabilized by CeO(2) nanoparticles proved to be an efficient strategy to form armored composite latex particles which may find applications in coating technology.

THE GRAVITY FIELD OF A CUBE

We calculate the Newtonian gravitational potential and field of a cubic, homogeneous asteroid and we a pply it to the orbit of possible satellites. Large astronom ical objects such as stars or planets, naturally te nd to form spherical shapes due to the dominance of the gravit ational forces, but as a thought experiment, we con sider the properties of a planet in the form of a perfect cub e. We investigate the formation of stable orbits ar ound such cubic objects, for the case of a static as well as a rotating cube employing the method of Poincare sections. The calculation of the gravitational field around non-s pherical objects has a significant role in space mi ssions to investigate asteroid belt objects that require calc ulating orbits around a large non-spherical mass. T he calculation of such non-spherical fields also has r elevance in identifying deposits or beds of ores in side the Earth, by measuring gravitational anomalies.

Sphericities of non-spherical objects

Sphericity, as one of the most important shape parameter for non-spherical objects, is extensively applied in evaluating the porosity or packing density of particles. In this paper, the sphericities of common non-spherical objects are deduced and investigated. Maximum sphericities and optimum shapes of these objects are presented as well. A decreasing order of sphericity from sphere (1.0) to regular tetrahedron (0.671) for objects with constant sphericity is given. Similar trends are found in most sphericity–aspect ratio relationships, which exhibit single peak and the sphericity increases with the growth of aspect ratio before the peak point and decreases afterward. The peak loci of aspect ratio are all around 1.0 which makes the shape approaching to a sphere. The information in the paper could be useful as literature for general application.

Effect of the solvent composition on the morphology of nano-objects synthesized via RAFT polymerization of benzyl methacrylate in dispersed systems

A hydrophilic macromolecular RAFT (reversible addition–fragmentation chain transfer) agent (macroRAFT agent) composed of 50 mol% methacrylic acid and 50 mol% poly(ethylene oxide) monomethyl ether methacrylate end-capped by a reactive trithiocarbonate group (P(MAA-co-PEOMA)) was used in the polymerization of benzyl methacrylate (BzMA) in different media, ethanol–water and 1,4-dioxane–water mixtures. Depending on the solvent composition, the polymerization showed features of either a dispersion polymerization (monomer soluble in the initial medium) or an emulsion polymerization (monomer insoluble in the initial medium). In all cases, the RAFT mechanism led to the in situ formation of well-defined amphiphilic P(MAA-co-PEOMA)-b-PBzMA block copolymers that self-assembled during the growth step into self-stabilized nano-objects, according to a polymerization-induced micellization process. For a given composition of the block copolymer, the final morphology depended strongly on the solvent composition. The presence of the organic co-solvent was favorable to the formation of fibers while an increased amount of water favored the formation of spherical particles. Compared to the ethanol–water system, in which the non-spherical objects existed only above 77–80 vol% of ethanol, in 1,4-dioxane–water mixtures the morphological transition was observed at a lower proportion of organic co-solvent (close to 20 vol%). For a given molar mass of the macroRAFT agent and an increased molar mass of the PBzMA block in a given solvent composition (ethanol–water, 95/5, v/v), the morphology changed from spheres to fibers and then to large spheres or vesicles. The molar mass window in which fibers were obtained was wider than that observed in pure water at pH 5 using the same macroRAFT agent [X. Zhang et al., Macromolecules, 2011, 44, 4149].

Scattering from highly extended acoustical objects using multiple precision computation

The extended boundary condition (or “T-matrix”) technique of Waterman [J. Acoust. Soc. Am., 45, 1417-1429 (1969)] has recently been used to study scattering from highly non-spherical axisymmetric air-filled acoustical objects in water. Evaluation of the T-matrix for extended objects is frequently problematical, typically requiring the inversion of very large and highly ill-conditioned matrices. A second issue is computation of the matrix elements themselves, requiring accurate numerical quadrature of the surface integrals around the scattering object. In this work, a computational scheme was implemented using MATLAB®, and a freely downloadable software package “mptoolbox,” which enables numerical computations to be performed to arbitrary degrees of precision. Here, it was implemented with 250 bits of precision specified for the mantissa part of the number, leading to computations with about 75 decimal places of accuracy. Using a multiple precision method significantly improved the stability and accuracy of both types of operations and indicates a powerful tool for the successful implementation of many scattering and other acoustical problems demanding high precision computational schemes. [Work supported by ONR.]

Electroformation of Giant Vesicles on Indium Tin Oxide (ITO)-Coated Poly(ethylene terephthalate) (PET) Electrodes.

Electroformation of cell-sized lipid membrane vesicles (giant vesicles, GVs) from egg yolk phosphatidylcholine was examined using a poly(ethylene terephthalate) sheet coated with indium tin oxide (ITO-PET) as the electrode material. With sinusoidal ac voltage, GV formation occurred in a similar manner to that on an ITO-glass electrode widely used in electroformation. Difference in the specific electrical resistance of ITO-PET did not significantly affect electroformation. The present results indicate that ITO-PET may be used as more flexible and less expensive electrode material in electroformation. In order to obtain insights into electroformation, other electric voltage forms, static dc and dc pulses, were also tested in place of commonly used sinusoidal ac. Under the present conditions, the best GV formation was observed with dc pulses of negative polarity. The result with static dc demonstrated that the mechanical vibration of swelling lipid seen with sinusoidal ac voltage was not essential for GV formation. On the positive electrode, the electroswelling of lipid mainly yielded non-spherical membranous objects. Pre-application of positive dc voltage on lipid hindered GV formation in electroswelling of the lipid with ac.

Full 3D translational and rotational optical control of multiple rod‐shaped bacteria

The class of rod-shaped bacteria is an important example of non-spherical objects where defined alignment is desired for the observation of intracellular processes or studies of the flagella. However, all available methods for orientational control of rod-shaped bacteria are either limited with respect to the accessible rotational axes or feasible angles or restricted to one single bacterium. In this paper we demonstrate a scheme to orientate rod-shaped bacteria with holographic optical tweezers (HOT) in any direction. While these bacteria have a strong preference to align along the direction of the incident laser beam, our scheme provides for the first time full rotational control of multiple bacteria with respect to any arbitrary axis. In combination with the translational control HOT inherently provide, this enables full control of all three translational and the two important rotational degrees of freedom of multiple rod-shaped bacteria and allows one to arrange them in any desired configuration.

486 The sphingosine kinase-1 survival pathway is a molecular target for the tumour-suppressive tea and wine polyphenols in prostate cancer

We present a new model for a variety of electric polarization effects on oblate and prolate homogeneous and single-shell spheroids. For homogeneous spheroids the model is identical to the Laplace model. For single-shell spheres of cell-like geometry the calculated difference of the induced dipole moments is in the thousandths range. To solve Laplace's equation for nonspherical single-shell objects it is necessary to assume a confocal shell, which results in different cell membrane properties in the pole and equator regions, respectively. Our alternative model addresses this drawback. It assumes that the disturbance of the external field due to polarization may project into the medium to a characteristic distance, the influential radius. This parameter is related to the axis ratio of the spheroid over the depolarizing factors and allows us to determine the geometry for a finite resistor-capacitor model. From this model the potential at the spheroid's surface is obtained and, consequently, the local field inside a homogeneous spheroid is determined. In the single-shell case, this is the effective local field of an equivalent homogeneous spheroid. Finally, integration over the volume yields the frequency-dependent induced dipole moment. The resistor-capacitor approach allowed us to find simple equations for the critical and characteristic frequencies, force plateaus and peak heights of deformation, dielectrophoresis and electrorotation for homogeneous and single-shell spheroids, and a more generalized equation for the induced transmembrane potential of spheroidal cells.

A new Guinier–Porod model

Small-angle scattering (SAS) curves are characterized by two main features: the Guinier region and the Porod region. Standard linear plots are available to fit SAS data and obtain a radius of gyration and a Porod exponent. A new Guinier–Porod empirical model is introduced to fit SAS data from spherical as well as nonspherical objects such as rods or platelets. It also applies to shapes intermediate between spheres and rods or between rods and platelets. The new model is used to fit SAS data from a Pluronic solution that sequentially forms unimers, then spherical micelles, then cylindrical micelles, then lamellar micelles upon heating. This single model can fit structures associated with all four phases as well as the intermediate structures.

Perceiving the Time to Contact of Rotating Non-Spherical Objects

Perceiving the Time to Contact of Rotating Non-Spherical Objects