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Research Papers

# Size Distribution Estimation of Three-Dimensional Particle Clusters in Metal-Matrix Nanocomposites Considering Sampling Bias

[+] Author and Article Information
Jianguo Wu

Department of Industrial, Manufacturing
and Systems Engineering,
University of Texas at El Paso,
500 W University Avenue,
Engineering Building, A-244,
El Paso, TX 79968
e-mail: jwu2@utep.edu

Yuan Yuan

IBM Research, Singapore,
10 Marina Boulevard,
Marina Bay Financial Centre Tower 2,
Singapore 18983, Singapore
e-mail: polarisyy@gmail.com

Xiaochun Li

Department of Mechanical
and Aerospace Engineering,
University of California,
Los Angeles, 48-121G Eng IV,
Los Angeles, CA 90095
e-mail: xcli@seas.ucla.edu

1Corresponding author.

Manuscript received February 20, 2017; final manuscript received April 21, 2017; published online May 25, 2017. Assoc. Editor: Donggang Yao.

J. Manuf. Sci. Eng 139(8), 081017 (May 25, 2017) (11 pages) Paper No: MANU-17-1109; doi: 10.1115/1.4036642 History: Received February 20, 2017; Revised April 21, 2017

## Abstract

Nanoparticle clustering phenomenon is a critical quality issue in metal-matrix nanocomposites (MMNCs) manufacturing. Accurate estimation of the 3D cluster size distribution based on the 2D cross section images is essential for quality assessment, quality control, and process optimization. The existing studies often draw conclusions with observable samples, which are inherently biased because large clusters are more likely to be intersected by scanning electron microscope (SEM) images compared with small ones. This paper takes into account this sampling bias and proposes two statistical approaches, namely, the maximum likelihood estimation (MLE) and the method of moments (MM), to estimate the distribution parameters accurately. Numerical studies and real case study demonstrate the effectiveness and accuracy of the proposed approaches.

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## Figures

Fig. 1

SEM images of Al2O3 reinforced MMNCs [14]: (a) nanoparticles are uniformly distributed and (b) nanoparticles form two big clusters

Fig. 2

Illustration of randomly sampled SEM images showing particles and clusters: (a) no clustering and (b) clustered

Fig. 3

Geometric relationship among R, Zc, and Rc

Fig. 4

Comparison of the proposed MLE method with Liu et al. [26] for uniform distribution (left column—(a)–(d)), truncated normal distribution (middle column—(e)–(h)), and log-normal distribution (right column—(i)–(l)). The first row ((a), (e), and (i)) shows the true and estimated pdfs; the second row ((b), (f), and (j)) shows the normalized histograms of the generated samples from the true distribution; the third row ((c), (g), and (k)) shows the normalized histograms of the samples cut by the microscopic images; the bottom row shows the normalized histograms of the circular cross sections.

Fig. 5

(a) Histogram of the MLE estimation of ru of the uniform distribution U(3,5), where the sample size n=100, and the vertical dashed line denotes the true value and (b) the RSE of the MLE estimate ru* and the estimate r̂u=max{rc1,…,rcn}

Fig. 6

Nanoparticle clusters shown in SEM images of MMNCs fabricated using ultrasonic cavitation-assisted casting process

Fig. 7

(a) Probability density functions of the simulated 2D sizes from three MLE-estimated models in comparison with the normalized histogram of real observations and (b)–(d) the empirical distribution function of the 30 observations in comparison with these of simulated ones from three candidate models

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