Unbiased estimation of the Weibull scale parameter using linear least squares analysis

Unbiased estimation of the Weibull scale parameter using unweighted linear least squares (LLS) analysis was investigated in this work as a function of specimen number, N. Mean, median and mode values of the Weibull scale parameter were estimated using a Monte Carlo procedure and general probability estimator of F = (i – a)/(N + b) with a and b varying between zero and unity. Unlike the mean and mode cases, the median value of Weibull scale parameter was essentially independent of Weibull modulus, m, with the coefficient of variation decreasing with b and being a factor of five smaller compared to that of m. Optimum values of a as a function of N were obtained using the median of the Weibull scale parameter with a being found to vary between 0.4796 for N = 10 and 0.2698 for N = 200 with b = 0.     

Confidence limits for Weibull parameters estimated using linear least squares analysis

Confidence limits (at selected levels of 68.27%, 90%, 95% and 99%) for unbiased estimation of Weibull parameters obtained using linear least squares (LLS) analysis were investigated in this paper. A Monte Carlo procedure was used to obtain probability distributions for unbiased estimates of Weibull modulus, m, and Weibull scale parameter, S_o, as a function of total specimen number, N (10 ≤ N ≤ 200), and m (1 ≤ m ≤ 25). Inspection of the probability distributions indicated that confidence limits for m depended only on N whereas those for S_o depended on both N and m. Whilst the determination of confidence limits for m proved to be relatively straightforward, the respective values for S_o were obtained by fitting an empirical equation to the S_o probability distributions approximated by a Gaussian curve. Example values of m and S_o confidence limits for selected N have been presented in this work.     

Mechanical properties of porous ceramic scaffolds: Influence of internal dimensions

Highly porous ceramic scaffolds have been fabricated from a 70% SiO₂–30% CaO glass powder using stereolithography and the lost-mould process combined with gel-casting. After sintering at 1200 °C the glass crystallised to a structure of wollastonite and pseudowollastonite grains in a glassy matrix with a bulk porosity of 1.3%. All scaffolds had a simple cubic strut structure with an internal porosity of approximately 42% and internal pore dimensions in the range 300–600 μm. The mean crushing strength of the scaffolds is in the range 10–25 MPa with the largest pore sizes showing the weakest strengths. The variability of scaffold strengths has been characterised using Weibull statistics and each set of scaffolds showed a Weibull modulus of m≈3 independent of pore size. The equivalent strength of the struts within the porous ceramics was estimated to be in the range 40–80 MPa using the models of the Gibson and Ashby. These strengths were found to scale with specimen size consistent with the Weibull modulus obtained from compression tests. Using a Weibull analysis, these strengths are shown to be in accordance with the strength of 3-point bend specimens of the bulk glass material fabricated using identical methods. The strength and Weibull modulus of these scaffolds are comparable to those reported for other porous ceramic scaffold materials of similar porosity made by different fabrication routes.     

A model of growth of titanium dioxide crystals with simultaneous transformation from anatase to rutile

This study simulates the combination of crystal growth by evaporation–condensation with the potential for a phase change once the crystals have reached a critical size.Crystal growth has been simulated as a two-dimensional Monte Carlo process in which the probability of accepting or rejecting an individual evaporation–condensation step is given by the Metropolis algorithm. Crystals, taken to be of the anatase form of TiO₂, which have grown larger than a specified critical size may transform irreversibly to a second phase, taken to be rutile. The probability of conversion increases with increasing crystal size. For each critical size there exists a limited range over which the transformation kinetics follow the familiar relationship α = 1 ? exp(?ktⁿ) (where α is the fractional conversion at time, t). The dependence of the mean size of the two polymorphs and of the rutile size-spread (rms) on critical size have been investigated. The most significant aspect of the plots of the normalized size-spread, rms/mean-size, is its gradual increase as the rutile fraction increases from 0.75 to 0.98.     

Silicon carbide–titanium diboride ceramic composites

The effect of TiB₂ content on mechanical properties of silicon carbide–titanium diboride ceramic composites was studied. The hardness of the ceramics decreased from 27.8 GPa for nominally pure SiC to 24.4 GPa for nominally pure TiB₂. In contrast, fracture toughness of the ceramics increased from 2.1 MPa m^1/2 for SiC to ～6 MPa m^1/2 for SiC with TiB₂ contents of 40 vol.% or higher. Flexure strengths were measured for three composites containing 15, 20, and 40 vol.% TiB₂ and analyzed using a two parameter Weibull analysis. The Weibull modulus increased from 12 for 15 vol.% TiB₂ to 17 for 20 and 40 vol.% TiB₂. Microstructural analysis revealed microcracking in the ceramics containing 20 and 40 vol.% TiB₂. The ceramic containing 40 vol.% TiB₂ had the best combination of properties with a fracture toughness of 6.2 MPa m^1/2, hardness of 25.3 GPa, Weibull modulus of 17, and a strength of 423 MPa.     

Application of the Weibull distribution function for modeling the kinetics of isothermal dehydration of equilibrium swollen poly (acrylic acid) hydrogel

A theoretical procedure for evaluating the density distribution function (p(E_a)) of apparent activation energies (E_a), based on modeling of the experimental isothermal conversion curves by Weibull probability function of reacting times was established. This procedure was applied on isothermal dehydration of poly (acrylic acid) hydrogel (PAAH) at four different operating temperatures (306, 324, 345 and 361 K). It was established that: (a) the experimental conversion curves at all temperatures can be described by the Weibull probability distribution function of reacting times in a wide range of the degree of conversion (0.14 ? α ? 0.95); (b) the Weibull distribution parameters (β and η) show functional dependences on the operating temperature, and (c) the apparent activation energy (E_a) varies with the degree of conversion (α). It was shown that the density distribution function of activation energies is independent on the temperature, and because of this fact, the considered function is an Eigen characteristic of the dehydration studied. The calculated p(E_a) function is in good agreement with the experimental p(E_a) function (p(E_a)_exp), obtained by the Miura procedure.     

Raman study of damage extent in graphene nanostructures carved by high energy helium ion beam

We performed spatial Raman mapping on supported monolayer graphene carved by 30 keV He⁺ beam. A tilted beam was introduced to effectively eliminate the substrate swelling. The ratio between D and G peak intensities shows that Stage 1 and Stage 2 disorder are introduced over a wider range on both sides of the 35 nm etched line. The mean defect distance L_D was estimated in these regions using the local activation model. Vacancies and amorphisations are dominant types of defects as suggested by the ratio of D and D′ peak intensities. Monte Carlo simulation on stopping range of ions was accomplished to explain the asymmetric defect formation in graphene.     

Improved Estimation of Weibull Parameters with the Linear Regression Method

Monte Carlo simulations were used to search for the optimal probability estimator for estimating Weibull parameters with the linear regression method. Compared with commonly used probability estimators, the optimal one obtained gives a more accurate estimation of the Weibull modulus and the same estimation precision of the scale parameter. It is also concluded that the maximum likelihood method results in the highest precision, however, less conservative than the linear regression method.     

Effect of temperature on structural properties of Aloe vera (Aloe barbadensis Miller) gel and Weibull distribution for modelling drying process

Aloe vera (Aloe barbadensis Miller) gel was dried at five inlet temperatures 50, 60, 70, 80 and 90 °C, in a convective dryer with a constant air flow of 2.0 ± 0.2 m/s. Rehydration ratio, water holding capacity, texture, microstructure and total polysaccharide content were evaluated. Drying kinetics was estimated using the Weibull distribution (r² > 0.97 and Chi-square < 0.0009). Values of scale and shape parameters ranged from 90.94 to 341.06 (min) and 1.43 to 1.49, respectively. Furthermore, the influence of temperature on the model parameters as well as on the quality attributes was analysed using a least significant difference test (p-value < 0.05). These effects were more evident for the long drying period (e.g. 810 min at 50 °C). However, minor alterations in the structural properties and total polysaccharide content were produced at drying temperatures of 60–70 °C, resulting in a high quality gel.     

A comparative structural study of ternary graphite intercalation compounds containing alkali metals and linear or branched amines

New donor-type ternary graphite intercalation compounds (GICs) containing alkali metals and branched amines are prepared by direct reaction of graphite with the alkali metal and amine. The new GICs include M(iC3)_0.4C_15–18 (M = Li, Na; iC3 = iso-propylamine; d_i = 0.76 nm) with a monolayer intercalate arrangement; and Na(sC4)_1.6C₁₈ (sC4 = sec-butylamine; d_i = 1.34 nm) and Na(iC4)_2.0C₂₈ (iC4 = iso-butylamine; d_i = 1.28 nm) with bilayer arrangements. Li-sC4-GIC and K-iC3-GIC are not formed using this reaction chemistry. M-iC3-GICs show galleries expanded by 0.06 nm more than that for M(nC3)_0.7–0.8C₁₆ (M = Li, Na; nC3 = n-propylamine). Unlike the case of rigid-sheet hosts where branched amines intercalate at a lower rate and with lower amine content, M-sC4-GICs and M-iC4-GICs readily form intercalate bilayers.     

Giant permittivity phenomena in layered BaTiO3–Ni composites

Layered BaTiO₃–Ni cermet composites with a constant composition but diversified microstructures were produced by a rolling-and-folding processing method. These composites differ from conventional laminates in that their interface has a tendency to be wavy, with a globular or elongated second phase within a continuous matrix phase. Based on an analysis of the (di)electric properties and Monte Carlo simulations we confirmed the critical influence of the composite's microstructural characteristics on the percolation threshold. We found that the dielectric properties of the composite, when it is in the insulation regime, were controlled by the insulating BaTiO₃ phase. A giant effective permittivity of around 200 000, with modest losses of tan δ < 0.04, was measured when the percolation threshold approached the composition of the cermet. Partial decomposition and deformation of the layered structure resulted in the creation of conducting paths, whereas further homogenization again shifted the percolation threshold above the actual composition of the cermet.     

Strength and fracture analysis of silicon-based components for embedding

Single-crystalline silicon chips are widely employed in printed circuit boards (PCBs) as embedded components. Their design often requires one side patterned with metal contacts, whereas the opposite one is constituted by pure silicon. These components must possess a minimum strength to withstand the loads occurring during both production and operation of the board. In this work, the strength and fracture behaviour of miniaturised Si chips (dimensions: 2 mm × 2 mm × 0.125 mm) has been assessed under biaxial loading on both the pure silicon side and the metal-patterned side by means of a miniaturised ball-on-three-balls (B3B) fixture. Experimental results showed significant difference in the characteristic fracture load between the silicon-side (P₀ = 21.2 N, Weibull modulus m ≈ 2.6) and the metal-patterned side (P₀ = 8.6 N, m ≈ 12.3). Fracture mechanics and fractographic analyses, together with FE simulations of the loading process, helped clarifying the effect of the metal contacts on the overall fracture behaviour of the Si-chips.     

Fully reacted high strength geopolymer made with diatomite as a fumed silica alternative

Geopolymers are formed by mixing of aluminosilicate sources with alkaline meta-silicate solution at room temperature. In the current study, diatomite of Turkish origin was fully utilized as a fumed silica alternative for the preparation of geopolymer, having a typical formula of K₂O•Al₂O₃•4SiO₂•11H₂O. From XRD of this sample, a broad peak centered at 28° 2θ indicated the well-known formation of amorphous geopolymer, as well as a fully reacted microstructure of geopolymer as seen by scanning electron microscopy. Additionally, geopolymer having the same formula was made by using fumed silica, in order to compare with geopolymers prepared from diatomite. The Weibull modulus was calculated from four-point bending and compressive strength testing of both geopolymer composites. The use of diatomite as a fumed silica substitute in geopolymer production resulted in a very close flexure strength 9.2 (± 4.2 MPa) when compared to geopolymer made from fumed silica 10.2 (± 3.3 MPa). There was a significantly higher compressive strength 71 (± 13.9 MPa) and Weibull modulus (5.4), than comparable properties of geopolymer made from fumed silica, which had a compressive strength 54 (± 25.8 MPa) and Weibull modulus of 2.0. The discrepancy was attributed to some self-reinforcement of the geopolymer matrix due to unreacted diatomite.     

Heat treatment of silanized feldspathic ceramic: Effect on the bond strength to resin after thermocycling

PurposeThis study aimed to evaluate the effect of heat treatment (at 77 °C) of a silanized feldspathic ceramic on microtensile bond strength (μTBS) with a resin cement before and after being aged by thermocycling.Material and methodsTwenty-four blocks (12×10×4 mm³) of a CAD/CAM feldspathic ceramic (Vitablocks Mark II, Vita) were obtained and randomly divided into three groups, according to the surface treatment prior to the cementation: Group AS – hydrofluoric acid 10%+silane; Group S77 – silane+heating at 77 °C for 60 s; and Group AS77 – hydrofluoric acid 10%+silane+heating at 77 °C for 60 s. Ceramic blocks were cemented to composite resin blocks with a resin cement. The sets were subsequently cross-sectioned into 1 mm² beams for μTBS testing. The beams of each group were randomly divided into two subgroups: aging (thermocycling, 12,000 cycles between 5 °C and 55 °C) and non-aging (tested immediately). One-way ANOVA and Tukey's test (α=0.05) and Weibull analysis (95% CI) were used to analyze the data.ResultsGroup AS77 had the lowest pre-test failure number during the cutting among the groups. There was no significant difference (p=0.255) between the μTBS mean values of the non-aged groups. After aging, the mean value of S77 was significantly lower than those of AS77 and AS (p=0.005). There was no difference in the Weibull modulus (m) and characteristic strength (σ₀) of the aged and non-aged groups for all comparisons. Before aging, heat treatment of silanized feldspathic ceramic (non acid-etched surface) demonstrated bond strength similar to that achieved with hydrofluoric-acid-etching treatment however, it had lower bond strength after aging.ConclusionThe combination of hydrofluoric-acid-etching treatment with heat treatment silanized feldspathic ceramic did not improve the bond strength of the interface.     

Fabrication of diamond p–i–p–i–p structures and their electrical and electroluminescence properties under high electric fields

Diamond p–i–p–i–p device structures with highly insulating (i) and p-type (p) regions were laterally fabricated using a microscopic etching process of high-pressure/high-temperature-synthesized (HPHT) Ib-type (N-included) diamond with focused ion beams and a homoepitaxial chemical-vapor-deposition (CVD) process. The i regions (at room temperature (RT)) were composed of thinned HPHT diamond with a thickness of 0.5 or 1 μm while the p regions were composed of boron-doped CVD diamond films with a thickness of 0.2 or 0.5 μm. Thus, high electric fields up to ≈ 1 × 10⁷ V/cm were successfully applied to the thinned i regions. When the electrical property of the p–i–p–i–p structures was measured at RT either in a vacuum (5 × 10^− 5 Pa) or in 1-atm ambient N₂ and Ar gases, dramatic current increases were observed for all the specimens examined at the voltages, V, slightly above the threshold voltages, V_th, depending on the i region thickness. For V > V_th, we observed electroluminescence whose spectra well corresponded to cathodoluminescence spectra taken from the devices. Furthermore, in the measurements in the gases, additional current increases appeared at V substantially higher than V_th due to gas discharges under high electric fields existing outside the diamond devices. These observed phenomena are discussed in relation to an avalanche current amplification process and the band structure of the p–i–p–i–p devices.     

Nanoindentation of WC–Co hardmetals

WC–Co cemented carbide has been investigated using instrumented indentation with maximum applied loads from 0.1 to 10 mN. The hardness and indentation modulus of individual phases and the influence of crystallographic orientation of WC on the hardness and indentation modulus have been studied. The hardness of the Co binder was approximately 10 GPa and that of WC grains up to 50 GPa with relatively large scatter under the indentation load of 1 mN. Investigation of the role of crystallographic orientation of WC grains on hardness at 10 mN load revealed average values of H_ITbasal = 40.4 GPa (E_ITbasal = 674 GPa) and H_ITprismatic = 32.8 GPa (E_Itprismatic = 542 GPa), respectively. The scatter in the measured values at low indentation loads is caused by the effects of surface and sub-surface characteristics (residual stress, damaged region) and at higher loads by “mix-phase” volume below the indenter.     

Advanced multifunctional graphene aerogel – Poly (methyl methacrylate) composites: Experiments and modeling

In this work, graphene aerogel (GA)–poly (methyl methacrylate) (PMMA) composites are first developed by backfilling PMMA into the pores of the GAs, providing uniform distribution of multi-layer reduced graphene oxide (m-rGO) sheets in the PMMA matrix. Electrical, mechanical and thermal properties of the as-prepared GA–PMMA composites are investigated by two-probe, microindentation and comparative infrared techniques respectively. As graphene loadings increase from 0.67 to 2.50 vol.%, the composites exhibit significant increases in electrical conductivity (0.160–0.859 S/m), microhardness (303.6–462.5 MPa) and thermal conductivity (0.35–0.70 W/m K) from that of pure PMMA as well as graphene–PMMA composites prepared by traditional dispersion methods. Thermal boundary resistance between graphene and PMMA is estimated to be 1.906 × 10⁻⁸ m² K/W by an off-lattice Monte Carlo algorithm that takes into account the complex morphology, size distribution and dispersion of m-rGO sheets.     

Study of kinetics in the biosorption of lead onto native and chemically treated olive stone

In this research, olive stone was used as precursor for the development of new biosorbents for lead ions. Chemical treatments were analyzed in terms of their effects on physical–chemical properties and kinetics of lead removal. A kinetic study of the biosorption of lead ions by olive stone was analyzed according to six different kinetic models (pseudo first, pseudo second, pseudo n-order, Elovich, solid diffusion and double exponential models). The biosorption kinetic data were successfully described with pseudo-nth order and double exponential models for all biosorbents. The double exponential model allowed estimating the values of external and internal mass transfer coefficients. The values of external mass transfer coefficient (k_e) ranged from 42.62 × 10⁻⁶ to 508.3 × 10⁻⁶ m min⁻¹ and the internal mass transfer coefficient (k_i) from 3.76 × 10⁻⁶ to 73.4 × 10⁻⁶ m min⁻¹. On the other hand, the analysis of experimental data showed that chemical treatments of the biomass led to increase biosorption capacity of the native biomass.