![Fig. 9. (a) Al6061-CNT powders mechanically alloyed for 30 min, (b-d) three types of the approximation geometries and (e) optical image of Al6061-CNT composite wit powders mechanically alloyed for 3 min. The SEM images of the particles mechanically alloyed for 30 min are shown in Fig. 9(a). Due to the complexity of the particle shapes, The SEM images of the fabricated Al6061-CNT samples are shown in Fig. 7. The CNT agglomerations were shown as black area in the SEM. The fraction of CNT agglomeration area over the total image area was measured by software Image] and is listed in Table 2 as Aq. In the first 10 min, the CNT de-clustering was very slow due to rupturing of CNT spherical structure. After the CNT-metal powder was mechanically alloyed for 40 min, CNT agglomerated area was rarely observed. Since the agglomerated CNT weight fraction can be obtained from the images (Wg/W; =Aa/Aqo), the dispersion can be written as d=1—Ag/Ago, where Ago is the CNT agglomeration](https://figures.academia-assets.com/7119884/figure_008.jpg)
![Fig. 5. As received CNT and Al6061-CNT powders mechanically alloyed for 10 min. total number of particles involved in welding or fracturing can be obtained by Eq. (20a) and t can be determined according to [17]: In Fig. 3(a)-(c), the particle evolution process is manually divided into two steps: deforming (Fig. 3(b)) and welding (Fig. 3(c)). In Fig. 3(a), a surface area, AA,“), will be embedded at time t. Because of the deformation from ball milling, this surface area changes to AA.) in Fig. 3(b), which is embedded during the welding process at time t+ At (In Fig. 3(c)).](https://figures.academia-assets.com/7119884/figure_005.jpg)
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Key research themes
1. How can the distribution of shortest path lengths in random networks be analytically characterized and approximated?
This research theme probes the analytical characterization, closed-form expressions, and approximations for the distribution of shortest path lengths (DSPL) in various random network models, such as random regular graphs (RRGs) and Erdős-Rényi (ER) networks. Understanding the DSPL is critical as it underpins large-scale network structure, affects dynamical processes like signal propagation, and reflects the small-world property of networks. The theme involves deriving moment generating functions, tail distributions, and recursive equations to obtain mean, variance, and full distributions, and examines scaling laws with network size and degree distribution.
Key finding: This paper provides a closed-form expression for the moment generating function of the distribution of shortest path lengths in random regular graphs (RRGs), leading to exact formulas for the mean and variance. It refines... Read more
Key finding: The authors develop two analytical approaches, Recursive Shells Approach (RSA) and Recursive Paths Approach (RPA), to calculate the full distribution of shortest path lengths in Erdős-Rényi (ER) networks across regimes of... Read more
Key finding: Although primarily focused on length-biased distributions, this paper introduces a novel length-biased weighted Lindley distribution and derives its moment generating function and moments. The methodological framework for... Read more
2. What are the theoretical developments and applications of length-biased distributions and their generalizations?
This theme focuses on the definition, properties, and generalizations of length-biased distributions, which arise when sampling is proportional to size or length, affecting the observed distributions of random variables. Research involves constructing new families (e.g., length-biased inverse Gaussian, weighted Lindley, Marshall-Olkin length-biased exponential), studying their statistical properties (moments, moment-generating functions, entropy), proposing parameter estimation methods, and demonstrating applications in reliability, lifetime data, hydrology, and other fields. These works contribute to expanding the class of models available for analyzing biased or weighted observational data.
Key finding: The paper introduces the two-sided length-biased inverse Gaussian (TS-LBIG) distribution and its generalization, addressing the difficulty that the original TS-LBIG's PDF and moment generating function lack closed-form... Read more
Key finding: This study proposes the length-biased weighted Lindley distribution, deriving explicit mathematical properties including moment generating function, moments, and reliability functions. It develops parameter estimation methods... Read more
Key finding: By applying the Marshall-Olkin methodology to the length-biased exponential distribution, this paper defines a new flexible distribution family (MOLBE) with an additional shape parameter. It derives analytical expressions for... Read more
3. How can mechanical alloying processes be modeled to predict length distribution and breakage of carbon nanotubes in metal matrix composites?
This theme addresses the challenges in preserving and characterizing the length and distribution of carbon nanotubes (CNTs) during mechanical alloying processes used to disperse CNTs in metallic powders for composite fabrication. Key problems include CNT clustering, breakage due to high-energy milling, and difficulties in measuring embedded CNTs. Research develops mathematical and experimental models to predict CNT length evolution by incorporating effects of particle welding, fracturing, and embedding, enabling better control of CNT preservation and enhancement of composite properties.
Key finding: This study quantifies CNT breakage during high-energy ball milling of Al6061 powder and CNT mixtures by collecting CNTs from particle surfaces and measuring length distribution. It introduces a mathematical model predicting... Read more
Key finding: Through mechanical alloying experiments and modeling, this work demonstrates that CNT clustering leads to early-stage breakage, which is mitigated once CNTs are embedded inside metal particles, reducing further fragmentation.... Read more