Unimodal Lognormal Distribution Research Articles

Modelling the evolution of dual-pore structure for compacted clays along hydro-mechanical paths

The evolution of dual-pore structure for compacted clays along hydro-mechanical paths exerts considerable influences on their phenomenological behavioral properties. This study aims to propose a novel approach for modelling the evolution of dual-pore structure. Here bimodal pore size density (PSD) function is considered to be obtained by superposing two unimodal lognormal distribution functions, which are inter-aggregate PSD (inter-PSD) and intra-aggregate PSD (intra-PSD). Unimodal PSD at arbitrary state is assumed to be obtained from initial one through shifting, scaling and spreading. On the basis, six evolution parameters are defined with unambiguous physical meanings. The modelling approach for evolving bimodal PSD is proposed through relating these evolution parameters to inter-aggregate and intra-aggregate void ratios respectively, where inter-aggregate and intra-aggregate void ratios are determined based on an empirical relationship between intra-aggregate void ratio and water ratio. The empirical parameter calibration in this approach is specific to stress paths. The proposed approach is applied to model the dual-pore structure evolution for compacted clays along loading path, wetting and drying paths under free condition, as well as wetting path under isochoric condition. Comparisons between the predicted and measured data show the novel approach can effectively capture the evolution of dual-pore structure along various hydro-mechanical paths.

Automated Analysis of Continuum Fields from Atomistic Simulations Using Statistical Machine Learning

Atomistic simulations of the molecular dynamics/statics kind are regularly used to study small‐scale plasticity. Contemporary simulations are performed with tens to hundreds of millions of atoms, with snapshots of these configurations written out at regular intervals for further analysis. Continuum scale constitutive models for material behavior can benefit from information on the atomic scale, in particular in terms of the deformation mechanisms, the accommodation of the total strain, and partitioning of stress and strain fields in individual grains. Herein, a methodology is developed using statistical data mining and machine learning algorithms to automate the analysis of continuum field variables in atomistic simulations. Three important field variables are focused on: total strain, elastic strain, and microrotation. The results show that the elastic strain in individual grains exhibits a unimodal lognormal distribution, while the total strain and microrotation fields evidence a multimodal distribution. The peaks in the distribution of total strain are identified with a Gaussian mixture model and methods to circumvent overfitting problems are presented. Subsequently, the identified peaks are evaluated in terms of deformation mechanisms in a grain, which, e.g., helps to quantify the strain for which individual deformation mechanisms are responsible. The overall statistics of the distributions over all grains are an important input for higher scale models, which ultimately also helps to be able to quantitatively discuss the implications for information transfer to phenomenological models.

Design and Performance Evaluation of a Laboratory-made 200 nm Precut Electrical Cascade Impactor

Increasing public concern regarding air quality has led to the development of efficient aerosol-monitoring techniques. Among the various aerosol measurement instruments based on electrical methods, in this study, an electrical cascade impactor (ECI) was designed and fabricated in our laboratory and was used to measure the real-time size distribution of submicron-sized aerosols. In the study by Park et al. (2007), it was assumed that the size distribution of incoming particles follows a unimodal lognormal distribution. However, in this study, the distribution of particles captured at each stage (including the Faraday cage) was assumed to be a unimodal lognormal distribution; hence, the incoming particles may follow any size distribution. After the particle charging characteristics were obtained for different particle sizes, experiments were performed with monodisperse test particles to determine the collection efficiency of each stage. The current measured in each stage was converted into a number based size distribution of aerosols by using the data inversion algorithm, which utilized the experimentally obtained collection efficiency. Then, a performance evaluation was performed, both in the laboratory and in the field. The results obtained by our ECI were in agreement with the scanning mobility particle sizer (SMPS) data.

Spinodal instabilities in polydisperse lyotropic nematics.

Many lyotropic liquid crystals are composed of mesogens that display a considerable spread in size or shape affecting their material properties and thermodynamics via various demixing and multi-phase coexistence scenarios. Starting from a generalized Onsager theory, we formulate a generic framework that enables locating spinodal polydispersities as well as identifying the nature of incipient size fractionation for arbitrary model potentials and size distributions. We apply our theory to nematic phases of both hard rods and disks whose main particle dimension is described by a unimodal log-normal distribution. We find that both rod-based and discotic nematics become unstable at a critical polydispersity of about 20%. We also investigate the effect of doping nematic assemblies with a small fraction of large species and highlight their effect on the stability of the uniform nematic fluid. Our main finding is that while rod-based are only weakly affected by the presence of large species, doping discotic nematics with very large platelets leads to a remarkable suppression of the spinodal instabilities. This could open up routes towards controlling the mechanical properties of nematic materials by manipulating the local stability of nematic fluid and its tendency to undergo fractionation-driven microphase separation.

Plume emission statistics in turbulent Rayleigh–Bénard convection

Direct numerical simulations (DNS) of turbulent thermal convection in a $\mathit{Pr}=0.7$ fluid up to $\mathit{Ra}=10^{12}$ are used to study the statistics of thermal plumes. At various vertical locations in a cylindrical set-up with aspect ratio ${\it\Gamma}=\text{width}/\text{height}=1/3$, plumes are identified and their properties extracted. It is found that plumes are much less likely to be emitted from plate regions with large wind shear. Close to the plates, the plumes have a unimodal log–normal distribution, whereas at more central locations the distribution becomes weakly bimodal, which can be traced back to clustering of the plumes and influence of the large-scale circulation. The number of hot plumes decreases with height. The width of the plumes scales with $\mathit{Ra}$ approximately as $\mathit{Nu}^{-1}$, indicating that it is determined by the thermal boundary layer thickness.

Effects of exopolymers on particle size distributions of suspended cohesive sediments

[1] The effects of exopolymers on the particle size distributions (PSDs) of suspended cohesive sediments were investigated in laboratory using four abundant clay minerals, kaolinite, illite, Na-montmorillonite, and Ca-montmorillonite, and two exopolymers, xanthan and guar, at six different exopolymer to clay ratios (E/C; i.e., 0, 1, 2, 5, 10, and 20 wt %) to represent the compositional variability of cohesive sediments in natural waters. Results show that the clay-exopolymer suspensions possess multimodal PSD. Statistical deconvolution of the PSD curves indicates that the suspensions consist of four discrete particle groups, primary particles, flocculi, microflocs, and macroflocs, all of which exhibit a unimodal lognormal distribution. Furthermore, such deconvolution quantifies the mean size and fraction of each particle group, leading to a more quantitative understanding of PSD kinetics of these sediments. Both clay surface charges and exopolymer polarity as well as the E/C affect the PSD kinetics. While neutral guar causes flocculation for all four clay minerals, anionic xanthan only induces flocculation for kaolinite with very low surface charges, but not for the other three clay minerals with relatively high charges. The fraction of each particle group also varies with the E/C, and such complex changes depend upon the interfacial interactions between clay particles and exopolymer molecules. For each exopolymer, critical E/C exists that can lead to a maximum or minimum fraction of microflocs or macroflocs. The role of exopolymer bridging, Coulomb force, and hydrogen bond in affecting the PSD kinetics of cohesive sediments is also discussed.

Transport and deposition in the Phébus FP circuit

In a nuclear reactor severe accident, fission products, after release from the fuel, along with related structural materials and actinides, are transported and partly deposited in the reactor coolant system, finally being injected into the containment. Both in the reactor coolant system and containment, they undergo physical and chemical processes that govern their behaviour and their airborne concentration in the containment atmosphere, which mainly determines the potential source term to the environment. The Phébus FP series provides a wide range of integral data on these processes, which are summarised in this account that covers the whole bundle geometry test series (FPT0 to FPT3). The transport in the circuit depends on the thermal hydraulic conditions (gas composition, temperature, flow rate). FPT0 and FPT1 were carried out under steam-rich conditions, while FPT2 and FPT3 were carried out with lower steam flows, leading to a period where hydrogen formed a large majority of the gas (∼90%) in the circuit during the main oxidation phase (oxidation of the Zircaloy cladding).The composition of the material transported in the circuit was dominated on average by the fission product noble gases, the volatile fission products Cs and Mo (except FPT0), the structural material Sn (from the cladding), the control rod materials Ag, In and Cd (FPT0/1/2) or B (FPT3), W and Re from the thermocouples, then other volatile fission products such as Te, Rb and I, the lower volatile Ba, and the fuel material U. For FPT2 and FPT3, whose main difference was in the control rod material, the proportions by mass are very similar if the control rod materials are excluded. Most of the materials were transported in the hot leg of the primary circuit as multi-component aerosols, with the notable exceptions of iodine (mainly in gaseous/vapour form) and Cd. In the cold leg, all materials are transmitted in aerosol form, with the notable exception of iodine in FPT3, where a large fraction reaches the containment in gaseous/vapour form. The size of aerosols is characterised by unimodal log–normal distributions; in FPT3 the results can also be interpreted in terms of bimodal distributions.Deposition is mainly concentrated where the temperatures of the wall and fluid decrease strongly, i.e. just above the bundle where the fluid cools from ∼2000°C to ∼700°C and, in the hot leg of the steam generator, from ∼700°C to ∼150°C. In the vertical line, simultaneously developing flows and changes in geometry (successive reductions in tube diameter) enhance deposition. The main processes in the circuit are: chemical transformation of vapours, vapour condensation onto structures as well as nucleation to form aerosols, or onto aerosols, aerosol agglomeration (primarily by diffusion), and thermophoretic deposition. The lower flows in FPT2/3 led to deposition being displaced upstream, e.g. onto the upper parts of the rods and also in the upstream part of the steam generator, where the presence of a large partial boron-rich blockage was deduced in FPT3. Revaporisation was observed in FPT1/2, notably for Cs, leading to delayed transport into the containment after reactor shutdown, also with some evidence for resuspension in FPT0. Post-test revaporisation experiments carried out externally on circuit samples from FPT1/3/4 aid in interpretation of the integral results. Also, the fact that temperatures fall from ∼2000°C at the exit of the bundle to ∼150°C at the exit to the containment, with fast transit times, implies that some reactions are kinetically limited so the composition at containment entry partly reflects that at higher temperatures. This applies particularly for iodine, where early presence of gaseous iodine is found in the containment (e.g. in FPT0, FPT1 and FPT3), and for carbonaceous gases resulting from B4C oxidation in FPT3, where the methane is lower than detection limits (reflecting high temperature bundle conditions, over 2000°C), whereas a non-negligible fraction would be expected under equilibrium, low temperature, containment conditions (typically about 150°C).

Steady-State Local Diffusive Fluxes in Porous Geo-Materials Obtained by Pore-Scale Simulations

Computer simulations of non-sorbing tracers diffusing in fluid-saturated porous sediment/rock were performed using pore-scale X-ray microtomographic images to reveal the following. (i) The histogram of the magnitude of the local diffusive flux vector obeys a unimodal log-normal distribution having a long positive tail. Simulations using model images were also performed to show that the flux broadening in large pores and the flux mixing at the pore network junctions are responsible for the log-normal shape. (ii) The simulation enabled us to directly visualize pore voxels with large and small fluxes, confirming the existence of transport pores and stagnant pores. Because of the unimodal nature, however, it was difficult to distinguish transport pores from stagnant pores using an objective threshold in the histogram. (iii) Another histogram of the flux vector component along the direction of the macroscopic concentration gradient was analyzed. A negative tail was found in the histogram, indicating that local counter diffusion exists in the porous geo-materials. However, the population and intensity of the counter diffusion fluxes are too small and weak to contribute to the overall diffusive transport across the porous media system. A long positive tail representing a large-flux diffusive pathway was also observed in the histogram. However, again, the population of the large-flux transport pores is small. As a result, the main conveyer of the tracer is the stagnant pores (not the transport pores), which have small positive flux values but a large population.

A bimodal bulk ultra-fine-grained nickel: Experimental and micromechanical investigations

Bulk nanocrystalline and ultrafine-grained metals are materials having grain size in the submicron range and have motivate considerable attention due to their interesting physical and mechanical properties. An important issue in the field of submicron grain-sized materials is how to achieve both high strength and high ductility? It has been suggested that, one strategy for enhancing the ductility of high-strength nanocrystalline materials is to develop a bimodal grain-size distribution, in which the fine grains provide strength, and the coarser grains enable strain hardening. In this paper, we report on the micromechanical behaviour of bulk nickel samples having bimodal microstructures. The samples were processed by hot isostatic pressing of blends of nano and micrometer-sized powder particles. The resulting microstructure is a bimodal randomly distributed grains considered here as a mixture of two unimodal log-normal distributions. An efficient modelling approach (i.e. a generalized self-consistent approach) previously developed by Jiang and Weng (2004a,b) is then applied to such experimental data to investigate, among others, local plastic strains and internal stresses fields as well as local magnitudes deviations.

Retrieval of Columnar Aerosol Size Distributions from Spectral Attenuation Measurements over Central Himalayas

Extensive measurements of spectral aerosol optical depths (AODs) were made at Manora Peak, Nainital (29.4o N, 79.5o E, ~1958 m above mean sea level) in the central Himalayas, using a ten channel multi-wavelength solar radiometer during January 2002 to December 2005. Using these spectral AOD values, the columnar size distribution [CSD; nc(r)] function of aerosols have been derived. The CSD, retrieved from spectral AODs are, in general, bimodal (combination of power law and unimodal log normal distribution) with a prominent secondary (or coarse) mode occurring at a fairly large value of radius (r > 0.5 µm), while the primary (or fine) mode either does not appear explicitly or perhaps occurs below the radius  0.1 µm. The bimodal nature of CSDs indicates the presence of fine as well as coarse mode aerosols over the observational site. The effective radius, total aerosol number content and columnar mass loading computed from deduced CSD shows minimum values during winter (November to February) and maximum during summer (March to June) months. The share of sub micron and super micron aerosols to the total aerosol number concentration (Nt) indicates the dominance of sub micron aerosols to the Nt and it accounts for > 90% during the study period.

Retrieval of Columnar Aerosol Size Distributions from Spectral Attenuation Measurements over Central Himalayas

Extensive measurements of spectral aerosol optical depths (AODs) were made at Manora Peak, Nainital (29.4°N, 79.5°E,~1958 m above mean sea level) in the central Himalayas, using a ten channel multi-wavelength solar radiometer during January 2002 to December 2005. Using these spectral AOD values, the columnar size distribution [CSD; n(subscript c)(r)] function of aerosols have been derived. The CSD, retrieved from spectral AODs are, in general, bimodal (combination of power law and unimodal log normal distribution) with a prominent secondary (or coarse) mode occurring at a fairly large value of radius (r＞0.5 μm), while the primary (or fine) mode either does not appear explicitly or perhaps occurs below the radius ≌0.1 μm. The bimodal nature of CSDs indicates the presence of fine as well as coarse mode aerosols over the observational site. The effective radius, total aerosol number content and columnar mass loading computed from deduced CSD shows minimum values during winter (November to February) and maximum during summer (March to June) months. The share of sub micron and super micron aerosols to the total aerosol number concentration (N(subscript t)) indicates the dominance of sub micron aerosols to the N(subscript t) and it accounts for ＞90% during the study period.

Uniwavelength lidar sensitivity to spherical aerosol microphysical properties for the interpretation of Lagrangian stratospheric observations

The determination of stratospheric particle microphysical properties from multiwavelength lidar, including Rayleigh and/or Raman detection, has been widely investigated. However, most lidar systems are uniwavelength operating at 532 nm. Although the information content of such lidar data is too limited to allow the retrieval of the full size distribution, the coupling of two or more uniwavelength lidar measurements probing the same moving air parcel may provide some meaningful size information. Within the ORACLE-O3 IPY project, the coordination of several ground-based lidars and the CALIPSO (Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation) space-borne lidar is planned during measurement campaigns called MATCH-PSC (Polar Stratospheric Clouds). While probing the same moving air masses, the evolution of the measured backscatter coefficient (BC) should reflect the variation of particles microphysical properties. A sensitivity study of 532 nm lidar particle backscatter to variations of particles size distribution parameters is carried out. For simplicity, the particles are assumed to be spherical (liquid) particles and the size distribution is represented with a unimodal log-normal distribution. Each of the four microphysical parameters (i.e. log-normal size distribution parameters, refractive index) are analysed separately, while the three others are remained set to constant reference values. Overall, the BC behaviour is not affected by the initial values taken as references. The total concentration ( N 0) is the parameter to which BC is least sensitive, whereas it is most sensitive to the refractive index ( m). A 2% variation of m induces a 15% variation of the lidar BC, while the uncertainty on the BC retrieval can also reach 15%. This result underlines the importance of having both an accurate lidar inversion method and a good knowledge of the temperature for size distribution retrieval techniques. The standard deviation ( σ) is the second parameter to which BC is most sensitive to. Yet, the impact of m and σ on BC variations is limited by the realistic range of their variations. The mean radius ( r m) of the size distribution is thus the key parameter for BC, as it can vary several-fold. BC is most sensitive to the presence of large particles. The sensitivity of BC to r m and σ variations increases when the initial size distributions are characterized by low r m and large σ. This makes lidar more suitable to detect particles growing on background aerosols than on volcanic aerosols.

Evaluation of aerosol sizing characteristic of an impactor using imaging plate technique

The activity-size distribution of radon decay products are normally determined using two approaches: direct and indirect. The present study utilises the direct approach to evaluate sizing information of a low pressure cascade impactor using imaging plate (IP) technique for radon decay products. The experiment verified the use of the collection media as suggested by the manufacturer of impactor and proposed a few improvements toward sizing characteristics of impactor. The obtained relative activity-size distribution of radon decay products presents a sharp unimodal log-normal distribution of the particle characterised by activity median aerodynamic diameter (AMAD) of 268 nm and geometric standard deviation (sigma(g)) of 1.66. The obtained data with all the suggested improvements were evaluated by the data obtained from a scanning mobility particle sizer (SMPS, Model 3934, TSI Inc), as reference data. The verification lead to a derivative area ratio of 0.803 between the reference and experimental data.

Impact of the grain size distribution on the yield stress of heterogeneous materials

When the mean grain size is larger than ∼100 nm, it is well accepted for different heterogeneous materials that the macroscopic yield stress follows the so-called Hall–Petch relation. However, in this classic formalism, only the mean grain size is considered in a semi-phenomenological way and the fact that the grains form a population of stochastic nature with different sizes and shapes is not stated. Moreover, an efficient homogenization procedure leading to the aforementioned grain size dependent behaviour from the individual properties of grains has not yet been reported in the literature mainly due to a lack of statistical description. In this paper, a recently developed self-consistent scheme making use of the “translated fields” technique for elastic–viscoplastic materials is used as micro–macro scale transition. The representative volume element is composed of grains supposed to be spherical and randomly distributed with a grain size distribution following a log-normal statistical function. The viscoplastic strain rate of the grains depends on their individual size. Intragranular plastic anisotropy and strain hardening are not considered in order to focus this work on grain size heterogeneities only. Numerical results with unimodal log-normal distributions firstly display that the overall yield stress depends not only on the mean grain size but also on the dispersion of the distribution. A decrease of the yield stress with an increase of the dispersion occurs and is more important when the mean grain size is on the order of the μm. Secondly, prediction of the evolution of the internal structure indicates an increase of the internal stresses when the dispersion is increased. Lastly, numerical results concerning bimodal grain size distributions, considered as mixtures of unimodal log-normal distributions, are discussed.

Superparamagnetic resonance of ferric ions in devitrified borate glass

Drastic changes are observed in magnetic resonance spectra of borate glasses doped with low concentrations of iron oxide, after heat treatment above the glass transition temperature. A sharp gef = 4.3 line, characteristic of diluted Fe3+ ions in the vitreous matrix, progressively disappears in the course of treatment. A new two-line pattern grows in the vicinity of gef ≈ 2.0, consisting of two superimposed features: a narrow one and a broad one. The narrow feature is dominant at lower treatment temperatures, while the broader one grows rapidly at higher treatment temperatures. Computer simulations succeed in fitting the spectra as being due to superparamagnetic resonance of static monodomain magnetic particles, randomly distributed in the vitreous matrix and characterised by a magnetocrystalline anisotropy constant, a demagnetisation field and a line shape function depending on the particle volume. The two-line pattern observed is shown to result from a unimodal log-normal distribution of particle diameters, with the most probable diameter value increasing with treatment temperature. Such a distribution results from a very inhomogeneous iron distribution in the glass matrix.

Patterns and Limitations in Limnoplankton Size Spectra

The hypothesis that biomass is uniformly distributed over logarithmic size classes was evaluated with plankton samples from 15 temperate lake sites in southern Quebec. Over the size range from 0.2 to 1600 μm equivalent spherical diameter (ESD), biomass tended to increase between logarithmic size classes at a median rate of 7%, and numerical abundance declined strongly with increasing size. The slope of the normalized biomass spectrum (reflecting overall trends in the distribution) becomes significantly steeper as total phosphorus decreases and total biomass declines. Hence, more oligotrophic systems have a more uniform biomass distribution and proportionately more small organisms. Over the observed size range, most samples were dominated by a phytoplankton mode between 20 and 50 μm ESD. Because of this dominance and because of the variability of the samples, most sample distributions could not be shown to differ significantly from unimodal lognormal distributions. Efforts to fit bimodal and trimodal distributions met with limited success, likely because only 39 size classes could be used. Simple mathematical descriptors of size spectra are insensitive to substantial departures from both linear and lognormal models, and therefore, variaion in the parameters of such descriptions are ineffective correlates of departures from average patterns.

Relationship between body weight and the incidence of positive axillary nodes at mastectomy for breast cancer.

The question, whether obesity is associated with an increased incidence of positive axillary nodes at mastectomy for breast cancer, was studied in two quite different hospital populations; one from a large urban teaching hospital (Montefiore) and one from a medium-sized Air Force medical center (Malcolm Grow). In the Montefiore population, the answer was "yes"; 67% of the node-positive patients, but only 31% of the node-negative patients were obese (20% or more above ideal weight) (p less than 0.05). In the Malcolm Grow population, the answer was "no"; 20% of the node-positive and 20% of the node-negative patients were obese. The different answers, we believe, are due to the biological differences between the populations; the Montefiore population was shorter (by an average of 1.7 inches), heavier (by an average of 20 lbs), and more obese. The incidence of obesity was about three times as high in the Montefiore population (52% versus 20%; p less than 0.02) and it contained a statistically distinct subpopulation of obese patients, while the few obese patients in the Malcolm Grow population constituted merely the upper tail of a unimodal log-normal distribution of weight in that population. We propose that it is possible to demonstrate a relationship of obesity to node-positivity in the Montefiore population but not in the Malcolm Grow population because obesity was highly prevalent in the former and almost nonexistent in the latter. It seems self-evident that it is not possible to demonstrate an effect of obesity in a population if that population manifests no significant obesity, statistically speaking, but disregarding this principle, we believe, may account for the controversy in the literature about whether obesity is a risk factor in breast cancer.