Convex Grains Research Articles

Sensitivity of Polarization to Grain Shape. II. Aggregates

A previous study (Paper I) investigated the polarization properties of a variety of simple convex grain shapes, some of which were found to be consistent with the observed polarization properties of interstellar dust from far-ultraviolet to far-infrared. Here, we study the optical properties of 45 nonconvex shapes, all aggregates of N equal-sized spheres. We consider N = 2, N = 3, and N = 256 random aggregates obtained from three different aggregation schemes. We also consider “trimmed” N = 256 aggregates obtained by systematically trimming initially random aggregates to increase either flattening or elongation. The “macroporosities” of the studied aggregates range from Pmacro=0.18 (for the N = 2 bisphere) to Pmacro≈0.85 (for the N = 256 “BA” aggregates). The only aggregates consistent with observations of starlight polarization and polarized thermal emission are shapes that have been trimmed to increase their asymmetry. If interstellar grains are high-porosity aggregates, there must be processes causing extreme elongation or flattening; if not, interstellar grains must be dominated by fairly compact structures, with at most moderate porosities. The ratio of polarization in the 10 μm silicate feature to starlight polarization in the optical is shown to be insensitive to porosity and shape. X-ray scattering may be the best tool to determine the porosity of interstellar grains. We propose that modest porosities of interstellar grains could be the result of “photolytic densification.” High polarization fractions observed in some Class 0 cores require processes to reduce porosities and/or increase asymmetries of aggregates in dense regions.

Sensitivity of Polarization to Grain Shape. I. Convex Shapes

Aligned interstellar grains produce polarized extinction (observed at wavelengths from the far-ultraviolet to the mid-infrared) and polarized thermal emission (observed at far-infrared and submm wavelengths). The grains must be quite nonspherical, but the actual shapes are unknown. The relative efficacy for aligned grains to produce polarization at optical versus infrared wavelengths depends on particle shape. The discrete dipole approximation is used to calculate polarization cross sections for 20 different convex shapes, for wavelengths from 0.1 to 100 μm, and grain sizes a eff from 0.05 to 0.3 μm. Spheroids, cylinders, square prisms, and triaxial ellipsoids are considered. Minimum aspect ratios required by the observed starlight polarization are determined. Some shapes can also be ruled out because they provide too little or too much polarization at far-infrared and submm wavelengths. The ratio of 10 μm polarization to integrated optical polarization is almost independent of grain shape, varying by only ±8% among the viable convex shapes; thus, at least for convex grains, uncertainties in grain shape cannot account for the discrepancy between predicted and observed 10 μm polarization toward Cyg OB2-12.

Percolation in the Boolean model with convex grains in high dimension

Percolation in the Boolean model with convex grains in high dimension

Secondary flow in ensembles of nonconvex granular particles under shear.

Studies of granular materials, both theoretical and experimental, are often restricted to convex grain shapes. We demonstrate that a nonconvex grain shape can lead to a qualitatively novel macroscopic dynamics. Spatial crosses (hexapods) are continuously sheared in a split-bottom container. Thereby, they develop a secondary flow profile that is completely opposite to that of rod-shaped or lentil-shaped convex grains in the same geometry. The crosses at the surface migrate towards the rotation center and sink there mimicking a "reverse Weissenberg effect." The observed surface flow field suggests the existence of a radial outward flow in the depth of the granular bed, thus, forming a convection cell. This flow field is connected with a dimple formed in the rotation center. The effect is strongly dependent on the particle geometry and the height of the granular bed.

The Microwave Snow Grain Size: A New Concept to Predict Satellite Observations Over Snow‐Covered Regions

AbstractSatellite observations of snow‐covered regions in the microwave range have the potential to retrieve essential climate variables such as snow height. This requires a precise understanding of how microwave scattering is linked to snow microstructural properties (density, grain size, grain shape and arrangement). This link has so far relied on empirical adjustments of the theories, precluding the development of robust retrieval algorithms. Here we solve this problem by introducing a new microstructural parameter able to consistently predict scattering. This “microwave grain size” is demonstrated to be proportional to the measurable optical grain size and to a new factor describing the chord length dispersion in the microstructure, a geometrical property known as polydispersity. By assuming that the polydispersity depends on the snow grain type only, we retrieve its value for rounded and faceted grains by optimization of microwave satellite observations in 18 Antarctic sites, and for depth hoar in 86 Canadian sites using ground‐based observations. The value for the convex grains (0.6) compares favorably to the polydispersity calculated from 3D micro‐computed tomography images for alpine grains, while values for depth hoar show wider variations (1.2–1.9) and are larger in Canada than in the Alps. Nevertheless, using one value for each grain type, the microwave observations in Antarctica and in Canada can be simulated from in‐situ measurements with good accuracy with a fully physical model. These findings improve snow scattering modeling, enabling future more accurate uses of satellite observations in snow hydrological and meteorological applications.

Influence of the grid cell geometry on 3D cellular automata behavior in intergranular corrosion

The bulk structure of stainless steels (SS) consists of mostly convex grains whose thin (one nm) planar interfaces are called grain boundaries (GB). There are industrial contexts where SS exposed to aggressive oxidative media undergo a fast degradation due to inter-granular corrosion (IGC), a preferential attack that occurs at the grain boundaries, propagating along them and leading to grains detachment. To address the IGC issue, we developed a 3D synchronous cellular automata (CA) model. Our first goal was to obtain a GB corrosion rate that could be as isotropic as possible. We performed full angular studies, with respect to the spatial orientation of a single grain boundary, in three different cases of grid (array of cells) geometry (cubic, hexagonal close-packed (HCP), face-centered cubic (FCC)). The HCP grid appears to be the best choice, as far as isotropy of the GB corrosion rate is concerned.

Reconnaissance assessment of the Stenian-Tonian granitoids of southern Tanzania for metal resources by using geological remote sensing and geochemical techniques

This study presents the use of geological remote sensing and geochemistry to determine spatial extents, geochemical characteristics and petrogenetic origin of Ndanda-Masasi metagranitoids in order to explore for important metals (Sn, W, Ta and Nb) in the metagranitoids. Geological remote sensing was done using Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) data, and major and trace elements were analysed using a combination of Inductively Coupled Plasma Mass Spectrometer (ICP-MS) and Inductively Coupled Plasma Emission Spectrometer (ICP-ES). It has been revealed that the metagranitoids and other rocks in the Ndanda-Masasi area crop out in a larger spatial coverage than how it is depicted in available geological maps. Moreover, the meta-granitoids have variable SiO2 contents, low Mg, Ni and Cr concentrations and moderate to large Fe-numbers. They classify mainly as magnesian- and ferroan-type granites and minor granodiorite and quartz monzonite. The chondrite normalized Rare Earth Elements (REE) patterns of the rocks show slight to strong enrichment of light REE over middle REE and heavy REE ((La/Yb)N = 4.2–289.1), due to their variable contents of MREE and HREE. The rocks show flat patterns of middle REE and heavy REE and have negative Eu-anomalies (mean Eu/Eu* = 0.7), with the exception few rocks in some localities yielding positive Eu-anomalies. They are characterised by variable Sr/Y (4.7–356.7), Nb/Ta (2.6–44.5) and well correlated trends with negative slopes of SiO2 vs MgO, TiO2, CaO, and Al2O3 diagrams. These geochemical features reflects a most likely magmatic formation of the rocks in the presence of plagioclase, hornblende, garnet and rutile, or/and fractionation of the phases at variable crustal depths. Numerous myrmekites forming convex grain boundaries around K-feldspar in the rocks indicate Ca- and Na-metasomatism as an important process affecting the rocks in their evolution. The omnipresent retrograde assemblage of titanite, chlorite, epidote and muscovite forming at the expense of plagioclase, hornblende, biotite and opaque (Fe-Ti-oxides?) reflects their formation during uplifting history of the rocks. Contrary to granitoids of similar age such as those of the Karagwe-Ankolean Belt, which are metalliferous, our reconnaissance survey found that the Ndanda-Masasi meta-granitoids have uneconomic concentration of most important metals.

SOME DENSE RANDOM PACKINGS GENERATED BY THE DEAD LEAVES MODEL

The intact grains of the dead leaves model enables us to generate random media with non overlapping grains. Using the time non homogeneous sequential model with convex grains, theoretically very dense packings can be generated, up to a full covering of space. For these models, the theoretical volume fraction, the size distribution of grains, and the pair correlation function of centers of grains are given.

Determination of Boolean models by densities of mixed volumes

AbstractIn Weil (2001) formulae were proved for stationary Boolean models Z in ℝd with convex or polyconvex grains, which express the densities (specific mean values) of mixed volumes of Z in terms of related mean values of the underlying Poisson particle process X. These formulae were then used to show that in dimensions 2 and 3 the densities of mixed volumes of Z determine the intensity γ of X. For d = 4, a corresponding result was also stated, but the proof given was incomplete, since in the formula for the density of the Euler characteristic V̅0(Z) of Z a term $\overline V^{(0)}_{2,2}(X,X)$ was missing. This was pointed out in Goodey and Weil (2002), where it was also explained that a new decomposition result for mixed volumes and mixed translative functionals would be needed to complete the proof. Such a general decomposition result has recently been proved by Hug, Rataj, and Weil (2013), (2018) and is based on flag measures of the convex bodies involved. Here, we show that such flag representations not only lead to a correct derivation of the four-dimensional result, but even yield a corresponding uniqueness theorem in all dimensions. In the proof of the latter we make use of Alesker’s representation theorem for translation invariant valuations. We also discuss which shape information can be obtained in this way and comment on the situation in the nonstationary case.

Support grain architecture design for additive manufacturing

Supporting the overhang section, restraining the model deformation or warping, minimizing the residual stress and controlling the cooling rate are some common functions of support structure in multiple additive manufacturing (AM) process. Since it needs to be removed at the post processing stage of fabrication, it is a considerable waste in terms of material, energy and time employed for their construction. Hence, it is advantageous to minimize the amount of support which eventually can improve the overall efficiency of the AM process. In this paper, a novel support architecture design methodology is proposed considering the amount of support volume, maximum contact interface, lower fabrication time, and ease of fabrication. First, the support needed points on the object surface are identified considering their normal direction. The points are clustered considering their uniform curvature and location. Afterward, each cluster of points is segmented iteratively into closed-convex regions i.e. grain boundary, considering the geometric factors such as aspect ratio, fill factor, and contour area to ensure the ease of fabrication and supportability. These convex grains are the model-support interface segments where a partial contact based support removal technique is implemented. Finally, self-supported slanting and pillar support structures are generated that minimizes support material consumption and consequently saving build time. The proposed research is implemented on free-form objects and the results are evaluated with the available support generator software. The result shows the improvement is build time, reduction in support volume and ease of support removal.

Grain-based Support Architecture Design for Additive Manufacturing

Supporting the overhang section, restraining the model deformation or warping, minimizing the residual stress and controlling the cooling rate are some common functions of support structure in multiple additive manufacturing (AM) process. Since it needs to be removed at the post processing stage of fabrication, it is a considerable waste in terms of material, energy and time employed for their construction. Hence, it is advantageous to minimize the amount of support which eventually can improve the overall efficiency of the AM process. In this paper, a novel support architecture design methodology is proposed considering the amount of support volume, maximum contact interface, lower fabrication time, and ease of fabrication. First, the support needed points on the object surface are identified considering their normal direction. The points are clustered with a weighted algorithm considering their uniform curvature and location. Afterward, each cluster of points is segmented iteratively into closed-convex regions i.e. grain boundary, considering the geometric factors such as aspect ratio, fill factor, and contour area to ensure the ease of fabrication and supportability. These convex grains are the model-support interface segments. Finally, self-supported slanting and pillar support structures are generated that minimizes support material consumption and consequently saving build time. The proposed research is implemented on free-form objects and the results are evaluated with the available support generator software.

Estimation of local built-in potential of amorphous silicon thin-film solar cells by Kelvin force microscopy

The local surface potential of pin-type hydrogenated amorphous silicon (a-Si:H) thin-film solar cells has been evaluated by Kelvin force microscopy (KFM). We have also estimated the local built-in potential of the solar cells by KFM. In the surface morphology image of the solar cells, large convex grains related to the textured structure of the substrate were found. The surface potential distribution related to the surface morphology was observed in the solar cells. A similar surface potential distribution was also found in an n-type hydrogenated microcrystalline Si (µc-Si:H) film. The surface potential of the solar cells was not the same as that of the n-type film. The difference in average surface potential between the n-type hydrogenated microcrystalline Si (µc-Si:H) film and the solar cells increased with increasing built-in potential. The difference in local surface potential on large convex grains was smaller than that in the region between the large convex grains.

Local surface potential on hydrogenated microcrystalline silicon films

The local surface potential on hydrogenated microcrystalline silicon films was investigated using a Kelvin force microscope. In the surface morphology image of the films deposited on Asahi U-type substrates, large and small convex grains were observed. In i-type films, the surface potentials on the large and small convex grains were higher than those in the concave regions between large convex grains and between small convex grains, respectively. The potential difference between a large grain and the region between large grains was larger than that between a small grain and the region between small grains. The surface potential distribution on p-type films showed the same tendency as that on the i-type films. On the other hand, the surface potential distribution on n-type films showed a reverse tendency to that on the i-type films. The potential difference between a large convex grain and the region between large convex grains depended on the doping ratio and that in the p-type films was smaller than that in the n-type films.

Geometry of polycrystals and microstructure

We investigate the geometry of polycrystals, showing that for polycrystals formed of convex grains the interior grains are polyhedral, while for polycrystals with general grain geometry the set of triple points is small. Then we investigate possible martensitic morphologies resulting from intergrain contact. For cubic-totetragonal transformations we show that homogeneous zero-energy microstructures matching a pure dilatation on a grain boundary necessarily involve more than four deformation gradients. We discuss the relevance of this result for observations of microstructures involving second and third-order laminates in various materials. Finally we consider the more specialized situation of bicrystals formed from materials having two martensitic energy wells (such as for orthorhombic to monoclinic transformations), but without any restrictions on the possible microstructure, showing how a generalization of the Hadamard jump condition can be applied at the intergrain boundary to show that a pure phase in either grain is impossible at minimum energy.

Minkowski tensor density formulas for Boolean models

A stationary Boolean model is the union set of random compact particles which are attached to the points of a stationary Poisson point process. For a stationary Boolean model with convex grains we consider a recently developed collection of shape descriptors, the so called Minkowski tensors. By combining spatial and probabilistic averaging we define Minkowski tensor densities of a Boolean model. These densities are global characteristics of the union set which can be estimated from observations. In contrast local characteristics like the mean Minkowski tensor of a single random particle cannot be observed directly, since the particles overlap. We relate the global to the local properties by density formulas for the Minkowski tensors. These density formulas generalize the well known formulas for intrinsic volume densities and are obtained by applying results from translative integral geometry. Our results support the idea that the degree of anisotropy of a Boolean model may be expressed in terms of the Minkowski tensor densities. Furthermore we observe that for smooth grains the mean curvature radius function of a particle can be reconstructed from the Minkowski tensor densities. In a simulation study we numerically determine Minkowski tensor densities for non-isotropic Boolean models based on ellipses and on rectangles in two dimensions and find excellent agreement with the derived analytic density formulas. The tensor densities can be used to characterize the orientational distribution of the grains and to estimate model parameters for non-isotropic distributions. In particular, the numerically determined values for the density of the Euler characteristic allow the estimation of certain mixed functionals of the grains.

On the local approximation of mean densities of random closed sets

Mean density of lower dimensional random closed sets, as well as the mean boundary density of full dimensional random sets, and their estimation are of great interest in many real applications. Only partial results are available so far in current literature, under the assumption that the random set is either stationary, or it is a Boolean model, or it has convex grains. We consider here non-stationary random closed sets (not necessarily Boolean models), whose grains have to satisfy some general regularity conditions, extending previous results. We address the open problem posed in (Bernoulli 15 (2009) 1222-1242) about the approximation of the mean density of lower dimensional random sets by a pointwise limit, and to the open problem posed by Matheron in (Random Sets and Integral Geometry (1975) Wiley) about the existence (and its value) of the so-called specific area of full dimensional random closed sets. The relationship with the spherical contact distribution function, as well as some examples and applications are also discussed.

Local photovoltaic characterization for silicon thin film solar cells using a scanning probe microscope

The photovoltaic characterization of silicon thin film solar cells with nano-scale spatial resolution is demonstrated by a new characterization method developed using a scanning microscope. In the surface topography of p-i-n type amorphous silicon thin film solar cells, large and small convex grains corresponding to the textured surface of an Asahi U-type substrate and the crystalline grains in the n-type microcrystalline silicon layer are found. In the local photo-current image, distribution in local photo-current correlated with the structure of the n-type microcrystalline silicon layer is observed in the p-i-n type amorphous silicon (a-Si:H) thin film solar cells. The local surface potential in amorphous silicon thin film solar cells with light irradiation is also evaluated. In the local surface potential image of the p-i-n a-Si:H thin film solar cells without light irradiation, the local surface potential on the large convex grains found in the surface topography is smaller than that in the concave region between the large convex grains. Similar distribution of the local surface potential is observed in the local surface potential images with light irradiation. The local surface potential difference between the large convex grains and the concave region decreases with increasing power of the irradiation light.

Strong interlocking of nonconvex particles in random packings

We present a numerical study of random packings made of nonconvex grains. These particles are built by the agglomeration of overlapping spheres in order to control their sphericity φ. The contact number C is found to be much larger than the coordination number Z, providing a significant difference with convex grains. The packing properties are found to be highly dependent on the morphological parameters of the grains : packing fractions as low as 0.3 have been reached. More importantly, the way nonconvex grains develop multiple contacts, i.e., interlocking, is found to be a relevant effect in such packings. Interlocking provides more stability to loose packings.

Study on 3D von Neumann equation with anisotropy for convex grains

Understanding the laws of grain growth in three dimensions is one of the classic problems of materials science. By considering the anisotropy in real polycrystalline structure and the relationship between the integral of surface mean curvature and the mean caliper diameter of a convex individual grain, three-dimensional von Neumann equation for accurate grain growth rate is studied. The result shows that accurate grain growth rate of a convex grain is related to the grain mean caliper diameter, the sum of the length of grain edges and the corresponding dihedral exterior angles. This result is verified by Kelvin tetrakaidecahedron and the only five convex regular polyhedra.

Scattering properties and structure functions of Boolean models

The Boolean model is a successful short-range order model for ensembles of overlapping grains possessing many applications. Convex grains, randomly distributed in space with a certain density, are characterized by their SAS correlation function γ 0( r). Characteristics, such as volume fraction c (porosity 1 − c) and sample correlation function γ( r) result. The function γ 0( r) of unknown grains can be re-constructed from γ( r). Information on size and shape of the grains results. The chord length distributions of grain phase and matrix phase are analyzed. The latter distribution is an exponential distribution. The distribution law of the grain phase is determined explicitly, incorporating formulas for the moments.